F1000Research F1000Research 2046-1402 F1000 Research Limited London, UK 10.12688/f1000research.173252.1 Research Article Articles Simultaneous Numerical Determination of Two Time-dependent Coefficients in Second Order Parabolic Equation With Nonlocal Initial and Boundary Conditions

[version 1; peer review: 3 approved, 1 approved with reservations]

 A.J.Al-Shatrah Mohammed Data Curation Formal Analysis Investigation Methodology Software Visualization Writing – Original Draft Preparation 1 Sabah Hussein Mohammed Conceptualization Formal Analysis Resources Software Supervision Validation Writing – Review & Editing https://orcid.org/0000-0002-9456-4303 a 2 Ministry of Education, Directorate of Education Thi Qar, Thi Qar, 64001, Iraq University of Baghdad Al-Jaderyia Campus College of Science, Baghdad, Baghdad Governorate, Iraq mmmsh@sc.uobaghdad.edu.iq

No competing interests were disclosed.

 10 2 2026 2026 15 228 29 1 2026 Copyright: © 2026 A.J.Al-Shatrah M and Sabah Hussein M 2026 This is an open access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Background

This study establishes a mathematically consistent and computational framework for the simultaneous identification of two time-dependent coefficients in a one-dimensional second-order parabolic partial differential equation. The considered problem is governed by nonlocal initial, boundary, and integral overdetermination conditions.

 Methods

The direct problem is solved using the Crank-Nicolson finite difference method (FDM), which ensures unconditional stability and second-order accuracy in both spatial and temporal discretizations. The corresponding inverse problem is reformulated as a nonlinear regularized least-squares optimization problem and efficiently solved used the MATLAB subroutine lsqnonlin from the optimization Toolbox. Due to the intrinsic, ill-posedness of the inverse formulation, small input data errors lead to big output errors. Then, Tikhonov regularization, is employed to enhance numerical stability and robustness.

 Results

Extensive numerical experiments are carried out under exact and noisy data to evaluate the numerical accuracy and convergence behavior of the method. The results confirm that the regularization technique effectively damps numerical oscillations, minimizes reconstruction error, and ensures reliable recovery of the unknown coefficients. Sensitivity analysis further reveals the essential role of the regularization parameter in controlling the trade-off between stability and accuracy.

 Conclusions

The proposed approach provides an accurate and computationally efficient tool for IP in heat transfer, diffusion processes, and related applied sciences.

 Finite difference method Crank-Nicolson Tikhonov regularization Inverse problem Coefficient problem Ill-posed problem Nonlocal conditions Parabolic equation. The author(s) declared that no grants were involved in supporting this work. 1. Introduction

Inverse problems (IPs), arise naturally in many scientific and engineering disciplines as they are typically modeled by differential equations. In the context of ordinary differential equations, the direct problem refers to obtaining solutions for a given system, whereas IP focuses on reconstructing the governing system from observed characteristics. Historically, the role of IPs was recognized in celestial mechanics. 1 In practical applications, IPs are commonly encountered when one seeks to determine unknown causes from their observed outcomes, in contrast to direct problems where effects are predicted from known causes. Compared with direct problems, IPs, are often more challenging to solve due to their ill-posedness, solutions may fail to exist, may not be unique, or may not change continuously with the input data. 2 Such problems appear in almost every scientific and technological domain, particularly in models derived from social and physical systems. Since most of these models are expressed through differential and integral equations, their analysis requires not only solving the equations but also interpreting the system’s behavior under varying conditions, provided that sufficient information is available to fully describe the model. IPs, associated with such equations arise in a wide range of scientific and engineering applications, as well as in the modeling of social processes. Many physical phenomena are described by mathematical models in the form of initial and boundary value problems for partial differential equations (PDEs). Frequently, these models involve differential and integral equations. 3 In boundary type IPs, the goal is to recover unknown boundary conditions, which usually leads to classical ill-posedness because the absence of continuous dependence of the solution on the input data. Numerical solution (NS) of direct mathematical physics problems is presently a well-studied matter. In solving multi-dimensional boundary value problems, difference methods and the finite element method are widely used. Partial differential equations give a basis for applied mathematical models, and their solution can be found in mathematical physics equations and some relations, including additional relations, boundary, and initial conditions, among other elements. 4 The following reconstruction problems serve as examples of IP applications in daily life, tomography, initial condition estimation of transient problems in conductive heat transfer, detection of non-metallic materials beneath the surface using reflected radiation methods, intensity, position estimation of illuminated radiation from a biological source using experimental radiation measures, and thermal source intensity estimation with functional dependence in space. 5– 9 On the other side, there are three different kinds of differential equations, Second-order equations are the most crucial for applications, Elliptic, Parabolic, and hyperbolic equations are examples of these equations. Yuldashev and other authors studied elliptic type integrate differential equations in 10, 11 while hyperbolic and parabolic were studied in 12– 14 and 15– 17 respectively. The alternating direction explicit method used for reconstructing solutions is also efficient and unconditionally stable, in 18 is restructured into a nonlinear regularized least-square optimization problem, and is effectively resolved using the MATLAB subroutine lsqnonlin from the optimization toolbox. The Crank-Nicolson (C-N) FDM together with the TR was used in 19– 21 the effectiveness of the computational method was shown together with the proof of the solution’s existence and uniqueness. To find a stable and accurate approximate solution of finite differences. In 22– 26 the study shows that for the IP with the time-dependent source coefficient in heat equations, there’s a smooth solution pair as extended as a single data point at an observation point and by providing an explicit formula for the time-dependent source coefficient, additional information at the observation point improves our comprehension of the behavior of the problem. This paper’s IP classical solution for a second order parabolic equation has been shown to be exist and unique by Elvin Azizbayov. 27 As a result, the main objective of the current work is a numerical realization of such a problem. In this paper, the article is divided into five sections, Section 2, includes a mathematical formulation of the IP under study. Section 3, a numerical method for solving the forward problem using C-N FDM. In Section 4, the IPs mathematical solution is shown with an initial guess, while, in Section 5, discuss the numerical problem and explain the obtained results.

 2. Mathematical formulation

As a mathematical model, we consider D T be a rectangular region that is defined by D T : { ( v , τ ) : 0 ≤ v ≤ 1 , 0 ≤ τ ≤ T } and ( T > 0 ) , be a fixed number, the one-dimensional IP of determining of unknown functions { w ( v , τ ) , a ( τ ) , b ( τ ) } for the upcoming parabolic equation, f ( v , τ ) = c ( τ ) w τ ( v , τ ) − w vv ( v , τ ) − a ( τ ) w ( v , τ ) − b ( τ ) g ( v , τ ) , ( v , τ ) ∈ D T , with, the nonlocal initial conditions (ICs) w ( v , 0 ) + δw ( v , T ) + ∫ 0 T Ƥ ̂ ( τ ) w ( v , τ ) dτ = ζ ( v ) , v ∈ [ 0 , 1 ] , with, periodic boundary condition, w ( 0 , τ ) = γw ( 1 , τ ) , τ ∈ [ 0 , T ] , and, nonlocal integral condition ∫ 0 1 w ( v , τ ) dv = 0 , τ ∈ [ 0 , T ] ,

and the overdetermination conditions w ( v i , τ ) = z i ( τ ) ( i = 1 , 2 ) , τ ∈ [ 0 , T ] , where, γ , δ ≥ 0 , v i ∈ ( 0 , 1 ) , ( i = 1 , 2 ) ; v 1 ≠ v 2 , are fixed numbers, c ( τ ) > 0 , g ( v , τ ) , f ( v , τ ) , Ƥ ̂ ( τ ) ≥ 0 , ζ ( v ) , z i ( τ ) ( i = 1 , 2 ) are given functions, w ( v , τ ) , a ( τ ) , b ( τ ) are the sought functions ( v is space component and τ is time component).

2.1 Definition 1

Consider the triplet { w ( v , τ ) , a ( τ ) , b ( τ ) } represents a classical solution, to the IP ( 1 ) − ( 5 ) , if the functions w ( v , τ ) , a ( τ ) , and b ( τ ) satisfies the following conditions:

The function w ( v , τ ) and its derivatives w τ ( v , τ ) , w v ( v , τ ) , w vv ( v , τ ) are continuous in the domain D T .

The functions a ( τ ) , and b ( τ ) are continuous on [ 0 , T ] .

Equation (1) and conditions ( 2 ) − ( 5 ) are satisfied in the classical (usual) sense.

2.2 Existence of the inverse problem classical solution

2.2.1 Theorem 1[28]: Let the assumptions ( μ 1 ) − ( μ 4 ) and the condition ( ( E ( T ) + 2 ) H ( T ) + G ( T ) + Q ( T ) ) ( E ( T ) + 2 ) < 1 , where, μ 1 ) ζ ( v ) ∈ C 2 [ 0 , 1 ] , ζ ( 3 ) ( v ) ∈ L 2 ( 0 , 1 ) , ζ ( 0 ) = γζ ( 1 ) , ζ ′ ( 0 ) = ζ ′ ( 1 ) , ζ ′ ′ ( 0 ) = γζ ′ ′ ( 1 ) ; where, a ̂ = 1 − γ 1 + γ and b ̂ = γ 1 + γ ; γ ≠ ± 1 . μ 2 ) f ( v , τ ) ∈ C v , τ 2 , 0 ( D T ) , f vvv ( v , τ ) ∈ L 2 ( D T ) , f ( 0 , τ ) = γf ( 1 , τ ) , f v ( 0 , τ ) = f v ( 1 , τ ) , f vv ( 0 , τ ) = γ f vv ( 1 , τ ) , ( γ ≠ ± 1 ) , 0 ≤ τ ≤ T . μ 3 ) g ( v , τ ) ∈ C v , τ 2 , 0 ( D T ) , g vvv ( v , τ ) ∈ L 2 ( D T ) , g ( 0 , τ ) = γg ( 1 , τ ) , g v ( 0 , τ ) = g v ( 1 , τ ) , g vv ( 0 , τ ) = γ g vv ( 1 , τ ) , ( γ ≠ ± 1 ) , 0 ≤ τ ≤ T . μ 4 ) δ ≥ 0 , c ( τ ) ∈ C [ 0 , T ] , c ( τ ) > 0 , 0 ≤ Ƥ ̂ ( τ ) ∈ C [ 0 , T ] , z i ( τ ) ∈ C 1 [ 0 , T ] , z ( τ ) = z 1 ( τ ) g ( v 2 , τ ) − z 2 ( τ ) g ( v 1 , τ ) ≠ 0 , ( i = 1 , 2 ) , 0 ≤ τ ≤ T . where, E 1 ( T ) = 2 ( 1 + δ ) − 1 ‖ ζ ( v ) ‖ L 2 ( 0 , 1 ) + 2 T ( 1 + δ ( 1 + δ ) − 1 ) ‖ 1 c ( τ ) ‖ C [ 0 , T ] ‖ f ( v , τ ) ‖ L 2 ( D T ) ) , E 2 ( T ) = 2 10 ‖ ζ ( 3 ) ( v ) ‖ L 2 ( 0 , 1 ) + 2 10 T ( 1 + δ ) ‖ 1 c ( τ ) ‖ C [ 0 , T ] ‖ f vvv ( v , τ ) ‖ L 2 ( D T ) , E 3 ( T ) = 4 3 ‖ ζ ( 3 ) ( v ) ( 1 − b ̂ − a ̂ v ) − 3 a ̂ ζ ( 2 ) ( v ) ‖ L 2 ( 0 , 1 ) + 4 3 T ( 1 + δ ) ‖ 1 c ( τ ) ‖ C [ 0 , T ] ‖ f vvv ( v , τ ) ( 1 − b ̂ − a ̂ v ) − 3 a ̂ f vv ( v , τ ) ‖ L 2 ( D T ) + 12 2 | a ̂ | ( 1 + δ ) ‖ 1 c ( τ ) ‖ C [ 0 , T ] ‖ ζ ( 3 ) ( v ) ‖ L 2 ( 0 , 1 ) + 12 3 | a ̂ | ‖ 1 c ( τ ) ‖ C [ 0 , T ] 2 ‖ c ( τ ) ‖ C [ 0 , T ] T T ( 1 + δ ) 2 ‖ f vvv ( v , τ ) ‖ L 2 ( D T ) , H 1 ( T ) = ( 1 + δ ( 1 + δ ) − 1 ) T ‖ 1 c ( τ ) ‖ C [ 0 , T ] , H 2 ( T ) = 6 T ( 1 + δ ) ‖ 1 c ( τ ) ‖ C [ 0 , T ] , H 3 ( T ) = 6 T ( 1 + δ ) ‖ 1 c ( τ ) ‖ C [ 0 , T ] + 6 2 | a ̂ | ‖ 1 c ( τ ) ‖ C [ 0 , T ] 2 ‖ c ( τ ) ‖ C [ 0 , T ] T 2 ( 1 + δ ) 2 , G 1 ( T ) = ( 1 + δ ) − 1 T ‖ Ƥ ̂ ( τ ) ‖ C [ 0 , T ] , G 2 ( T ) = 6 T ‖ Ƥ ̂ ( τ ) ‖ C [ 0 , T ] , G 3 ( T ) = 3 T ‖ Ƥ ̂ ( τ ) ‖ C [ 0 , T ] [ 2 + 4 | a ̂ | ‖ 1 c ( τ ) ‖ C [ 0 , T ] ( 1 + δ ) ] , Q 1 ( T ) = ( 1 + δ ( 1 + δ ) − 1 ) ‖ 1 c ( τ ) ‖ C [ 0 , T ] T ‖ b ( τ ) ‖ C [ 0 , T ] ‖ g ( v , τ ) ‖ L 2 ( D T ) , Q 2 ( T ) = 4 3 T ( 1 + δ ) ‖ 1 c ( τ ) ‖ C [ 0 , T ] ‖ g vvv ( v , τ ) ‖ L 2 ( D T ) , Q 3 ( T ) = 6 2 T ( 1 + δ ) ‖ 1 c ( τ ) ‖ C [ 0 , T ] ( ‖ g vvv ( v , τ ) ( 1 − b ̂ − a ̂ v ) − 3 a ̂ g vv ( v , τ ) ‖ L 2 ( D T ) + | a ̂ | ‖ 1 c ( τ ) ‖ C [ 0 , T ] ‖ c ( τ ) ‖ C [ 0 , T ] T ( 1 + δ ) ‖ g vvv ( v , τ ) ‖ L 2 ( D T ) ) , where, E 4 ( T ) = E 1 ( T ) + E 2 ( T ) + E 3 ( T ) , H 4 ( T ) = H 1 ( T ) + H 2 ( T ) + H 3 ( T ) , G 4 ( T ) = G 1 ( T ) + G 2 ( T ) + G 3 ( T ) , and , Q 4 ( T ) = Q 1 ( T ) + Q 2 ( T ) + Q 3 ( T ) , E 5 ( T ) = ‖ [ z ( τ ) ] − 1 ‖ C [ 0 , T ] { ‖ z 1 ( τ ) ( c ( τ ) z 2 ′ ( τ ) − f ( v 2 , τ ) ) − z 2 ( τ ) ( c ( τ ) z 1 ′ ( τ ) − f ( v 1 , τ ) ) ‖ C [ 0 , T ] + ( ∑ k = 1 ∞ λ k − 2 ) 1 2 [ 2 2 [ ( 1 + 4 3 | a ̂ | ‖ 1 c ( τ ) ‖ C [ 0 , T ] ( 1 + δ ) ) ‖ ζ ( 3 ) ( v ) ‖ L 2 ( 0 , 1 ) + ‖ ζ ( 3 ) ( v ) ( 1 − b ̂ − a ̂ v ) − 3 a ̂ ζ ( 2 ) ( v ) ‖ L 2 ( 0 , 1 ) ] + 2 2 T ( 1 + δ ) ‖ 1 c ( τ ) ‖ C [ 0 , T ] [ ( 1 + 6 2 | a ̂ | ‖ 1 c ( τ ) ‖ C [ 0 , T ] ‖ c ( τ ) ‖ C [ 0 , T ] T ( 1 + δ ) ) ‖ f vvv ( v , τ ) ‖ L 2 ( D T ) ) + ‖ f vvv ( v , τ ) ( 1 − b ̂ − a ̂ v ) − 3 a ̂ f vv ( v , τ ) ‖ L 2 ( D T ) ] ] ‖ | g ( v 2 , τ ) | + | g ( v 1 , τ ) | ‖ C [ 0 , T ] ‖ a ̂ v + b ̂ ‖ C [ 0 , 1 ] } , E 6 ( T ) = ‖ [ z ( τ ) ] − 1 ‖ C [ 0 , T ] { ‖ g ( v 2 , τ ) ( c ( τ ) z 1 ′ ( τ ) − f ( v 1 , τ ) ) − g ( v 1 , τ ) ( c ( τ ) z 2 ′ ( τ ) − f ( v 2 , τ ) ) ‖ C [ 0 , T ] + ( ∑ k = 1 ∞ λ k − 2 ) 1 2 [ 2 2 [ ( 1 + 4 3 | a ̂ | ‖ 1 c ( τ ) ‖ C [ 0 , T ] ( 1 + δ ) ) ‖ ζ ( 3 ) ( v ) ‖ L 2 ( 0 , 1 ) + ‖ ζ ( 3 ) ( v ) ( 1 − b ̂ − a ̂ v ) − 3 a ̂ ζ ( 2 ) ( v ) ‖ L 2 ( 0 , 1 ) ] + 2 2 T ( 1 + δ ) ‖ 1 c ( τ ) ‖ C [ 0 , T ] [ ( 1 + 6 2 | a ̂ | ‖ 1 c ( τ ) ‖ C [ 0 , T ] ‖ c ( τ ) ‖ C [ 0 , T ] T ( 1 + δ ) ) ‖ f vvv ( v , τ ) ‖ L 2 ( D T ) ) + ‖ f vvv ( v , τ ) ( 1 − b ̂ − a ̂ v ) − 3 a ̂ f vv ( v , τ ) ‖ L 2 ( D T ) ] ] х ‖ | z 1 ( τ ) | + | z 2 ( τ ) | ‖ C [ 0 , T ] ‖ a ̂ v + b ̂ ‖ C [ 0 , 1 ] } , H 5 ( T ) = 2 T ‖ [ z ( τ ) ] − 1 ‖ C [ 0 , T ] ( ∑ k = 1 ∞ λ k − 2 ) 1 2 ( 1 + δ ) ‖ 1 c ( τ ) ‖ C [ 0 , T ] ( 1 + 3 2 | a ̂ | ‖ 1 c ( τ ) ‖ C [ 0 , T ] ‖ c ( τ ) ‖ C [ 0 , T ] T ( 1 + δ ) ) ‖ | g ( v 2 , τ ) | + | g ( v 1 , τ ) | ‖ C [ 0 , T ] ‖ a ̂ v + b ̂ ‖ C [ 0 , 1 ] , H 6 ( T ) = 2 T ‖ [ z ( τ ) ] − 1 ‖ C [ 0 , T ] ( ∑ k = 1 ∞ λ k − 2 ) 1 2 ( 1 + δ ) ‖ 1 c ( τ ) ‖ C [ 0 , T ] ( 1 + 3 2 | a ̂ | ‖ 1 c ( τ ) ‖ C [ 0 , T ] ‖ c ( τ ) ‖ C [ 0 , T ] T ( 1 + δ ) ) ‖ | z 1 ( τ ) | + | z 2 ( τ ) | ‖ C [ 0 , T ] ‖ a ̂ v + b ̂ ‖ C [ 0 , 1 ] , G 5 ( T ) = 2 T ‖ [ z ( τ ) ] − 1 ‖ C [ 0 , T ] ( ∑ k = 1 ∞ λ k − 2 ) 1 2 ‖ Ƥ ̂ ( τ ) ‖ C [ 0 , T ] ( 1 + 2 3 ( 1 + δ ) ‖ 1 c ( τ ) ‖ C [ 0 , T ] ) ‖ | g ( v 2 , τ ) | + | g ( v 1 , τ ) | ‖ C [ 0 , T ] ‖ a ̂ v + b ̂ ‖ C [ 0 , 1 ] , G 6 ( T ) = 2 T ‖ [ z ( τ ) ] − 1 ‖ C [ 0 , T ] ( ∑ k = 1 ∞ λ k − 2 ) 1 2 ‖ Ƥ ̂ ( τ ) ‖ C [ 0 , T ] ( 1 + 2 3 ( 1 + δ ) ‖ 1 c ( τ ) ‖ C [ 0 , T ] ) ‖ | z 1 ( τ ) | + | z 2 ( τ ) | ‖ C [ 0 , T ] ‖ a ̂ v + b ̂ ‖ C [ 0 , 1 ] , Q 5 ( T ) = 2 2 T ( 1 + δ ) ‖ 1 c ( τ ) ‖ C [ 0 , T ] ‖ [ z ( τ ) ] − 1 ‖ C [ 0 , T ] ( ∑ k = 1 ∞ λ k − 2 ) 1 2 [ ( 1 + 6 2 | a ̂ | ‖ 1 c ( τ ) ‖ C [ 0 , T ] ‖ c ( τ ) ‖ C [ 0 , T ] T ( 1 + δ ) ) ‖ g vvv ( v , τ ) ‖ L 2 ( D T ) + ‖ g vvv ( v , τ ) ( 1 − b ̂ − a ̂ x ) − 3 a ̂ g vv ( v , τ ) ‖ L 2 ( D T ) ‖ | g ( v 2 , τ ) | + | g ( v 1 , τ ) | ‖ C [ 0 , T ] ‖ a ̂ v + b ̂ ‖ C [ 0 , 1 ] , Q 6 ( T ) = 2 2 T ( 1 + δ ) ‖ 1 c ( τ ) ‖ C [ 0 , T ] ‖ [ z ( τ ) ] − 1 ‖ C [ 0 , T ] ( ∑ k = 1 ∞ λ k − 2 ) 1 2 [ ( 1 + 6 2 | a ̂ | ‖ 1 c ( τ ) ‖ C [ 0 , T ] ‖ c ( τ ) ‖ C [ 0 , T ] T ( 1 + δ ) ) ‖ g vvv ( v , τ ) ‖ L 2 ( D T ) + ‖ g vvv ( v , τ ) ( 1 − b ̂ − a ̂ x ) − 3 a ̂ g vv ( v , τ ) ‖ L 2 ( D T ) ‖ | z 1 ( τ ) | + | z 2 ( τ ) | ‖ C [ 0 , T ] ‖ a ̂ v + b ̂ ‖ C [ 0 , 1 ] , where, E ( T ) = E 4 ( T ) + E 5 ( T ) + E 6 ( T ) , H ( T ) = H 4 ( T ) + H 5 ( T ) + H 6 ( T ) , G ( T ) = G 4 ( T ) + G 5 ( T ) + G 6 ( T ) , and Q ( T ) = Q 4 ( T ) + Q 5 ( T ) + Q 6 ( T ) , holds, then problem ( 1 ) − ( 3 ) , has a unique solution in the ball K = K R , of the space X k 3 , where, λ k = 2 kπ ; ( k = 0 , 1 , … ) .

3. FDM scheme for direct (Forward) problem

The direct problem will be resolved in this section, i.e., when unknown, a ( τ ) , c ( τ ) and b ( τ ) are assumed to be given, in order, to solve this problem, Crank-Nicolson (C-N) FDM scheme had been used for finding the solutions of the nonlocal problem given by Equation (1)–(4) . Where the domain D T had been divided into M × N mesh with spatial step size ∆ v = 1 M and the time step size ∆ τ = T N where M and N are given positive integers. The grid points have been given by v i = i ∆ v , i = 0 , M , ¯ τ j = j ∆ τ , j = 0 , N , ¯

These quantities’ discretized form is given as follows, w ( v i , τ j ) ≕ w i , j , a ( τ j ) ≕ a j , b ( τ j ) ≕ b j , c ( τ j ) ≕ c j , f ( v i , τ j ) ≕ f i , j , g ( v i , τ j ) ≕ g i , j , and ζ ( v i ) ≕ ζ i for i = 0 , M , ¯ j = 0 , N . ¯

Applying the C-N FDM, for the discretizes Equation (1) , which approximated as, w i , j + 1 − w i , j ∆ τ = 1 2 ( d ~ j w i + 1 , j − 2 w i , j + w i − 1 , j ( ∆ v ) 2 + a ~ j w i , j + b ~ j g i , j + d ~ j f i , j + d ~ j + 1 w i + 1 , j + 1 − 2 w i , j + 1 + w i − 1 , j + 1 ( ∆ v ) 2 + a ~ j + 1 w i , j + 1 + b ~ j + 1 g i , j + 1 + d ~ j + 1 f i , j + 1 ) , i = 1 , M , ¯ j = 0 , N , ¯ where, d ~ j = 1 c j , a ~ j = a j c j , b ~ j = b j c j , d ~ j + 1 = 1 c j + 1 , a ~ j + 1 = a j + 1 c j + 1 , and, b ~ j + 1 = b j + 1 c j + 1 simplifying the above Equation (28), get the following difference equation, − A ̂ j + 1 w i − 1 , j + 1 + ( 1 + E ̂ j + 1 ) w i , j + 1 − A ̂ j + 1 w i + 1 , j + 1 = A ̂ j w i − 1 , j + ( 1 − E ̂ j ) w i , j + A ̂ j w i + 1 , j + ∆ τ 2 H ̂ + ∆ τ 2 D ̂ , where, A ̂ j = ∆ τ d ~ j 2 ( ∆ v ) 2 , A ̂ j + 1 = ∆ τ d ~ j + 1 2 ( ∆ v ) 2 , E ̂ j + 1 = ( 2 A ̂ j + 1 − ∆ τ 2 a ~ j + 1 ) , E ̂ j = ( 2 A ̂ j − ∆ τ 2 a ~ j ) , H ̂ = b ~ j g i , j + b ~ j + 1 g i , j + 1 , and, D ̂ = d ~ j f i , j + d ~ j + 1 f i , j + 1 .

The C-N FDM discretizes Equations (2)–(4) which approximated as, w i , 0 + δ w i , j + I ( v i ) = ζ i , i = 1 , M , ¯ where I ( v i ) is integral in Equation (2), and we have, I ( v i ) = ∆ τ 2 ( Ƥ ̂ ( τ 0 ) w ( v i , τ 0 ) + Ƥ ̂ ( τ N ) w ( v i , τ N ) + 2 ∑ j = 1 N − 1 Ƥ ̂ ( τ j ) w ( v i , τ j ) ) ,

Finally, the trapezoidal rule and C-N discretizes integral condition ( 4 ) as; w ( v 0 , τ j ) + w ( v M , τ j ) + 2 ∑ i = 1 M − 1 w ( v i , τ j ) = 0 , i = 0 , M ¯ w i , 0 = ζ i , i = 1 , M , ¯ j = 0 , w 0 , j = γ w 1 , j , j = 0 , N − 1 , ¯ i = 0 ,

From Equation (29) , with the nonlocal initial conditions w ( v , 0 ) = ζ ( v i ) , (assuming Ƥ ( τ ) = δ = 0 ), and using Equations (30)–(33), with each time step τ j , we can be described in matrix form as L W j + 1 = L 1 W j + ∆ τ 2 V ̂ , i = 1 , M ¯ , j = 0 , N − 1 , ¯ L = [ 1 0 0 0 0 … 0 0 0 − 2 − A ̂ j + 1 1 + E ̂ j + 1 − A ̂ j + 1 0 0 … 0 0 0 0 0 − A ̂ j + 1 1 + E ̂ j + 1 − A ̂ j + 1 0 … 0 0 0 0 … … … ⋱ ⋱ ⋱ ⋮ … ⋮ ⋮ 0 0 0 0 0 … 0 − A ̂ j + 1 1 + E ̂ j + 1 − A ̂ j + 1 1 2 2 2 2 … 2 2 2 1 ] ( M ) х ( M ) , with, L 1 = [ 0 A ̂ j w 1 , j + ( 1 − E ̂ j ) w 2 , j + A ̂ j w 3 , j A ̂ j w 2 , j + ( 1 − E ̂ j ) w 3 , j + A ̂ j w 4 , j A ̂ j w 3 , j + ( 1 − E ̂ j ) w 4 , j + A ̂ j w 5 , j ⋮ A ̂ j w M − 2 , j + ( 1 − E ̂ j ) w M − 1 , j + A ̂ j w M , j A ̂ j w M − 1 , j + ( 1 − E ̂ j ) w M , j + A ̂ j w M + 1 , j 0 ] , and, V ̂ = ( H ̂ + D ̂ ) , where, V ̂ = [ 0 ( b j g 2 , j + b j + 1 g 2 , j + 1 ) + ( d ~ j f 2 , j + d ~ j + 1 f 2 , j + 1 ) ( b j g 3 , j + b j + 1 g 3 , j + 1 ) + ( d ~ j f 3 , j + d ~ j + 1 f 3 , j + 1 ) ( b j g 4 , j + b j + 1 g 4 , j + 1 ) + ( d ~ j f 4 , j + d ~ j + 1 f 4 , j + 1 ) ⋮ ( b j g M − 1 , j + b j + 1 g M − 1 , j + 1 ) + ( d ~ j f M − 1 , j + d ~ j + 1 f M − 1 , j + 1 ) ( b j g M , j + b j + 1 g M , j + 1 ) + ( d ~ j f M , j + d ~ j + 1 f M , j + 1 ) 0 ] W j + 1 = [ w 1 , j + 1 , w 2 , j + 1 , … , w M + 1 , j + 1 , w M , j + 1 ] T , W j = [ w 1 , j , w 2 , j , … , w M − 1 , j , w M , j ] T , j = 0 , N − 1 , ¯ the Equation (4) desired output data will be approximated using trapezoidal rule as 0 = ∫ 0 1 H ( v ) w ( v , τ j ) dv = ∆ τ 2 ( w ( v 0 , τ j ) + w ( v M , τ j ) + 2 ∑ k = 1 M − 1 w ( v k , τ j ) ) .

3.1 Stability and convergence analysis

The adopted C-N finite difference scheme used to solve the direct problem is unconditionally stable and second order accurate in both spatial and temporal discretization. A brief von Neumann stability analysis shows that the amplification factor satisfies | G | = 1 , for all mesh ratios, which guarantees the unconditional stability of the scheme. The local truncation error is of order ϑ ( ( ∆ v ) 2 , ( ∆ τ ) 2 ) confirming that the method provides second order accuracy and reliable convergence toward the analytical solution as the mesh is refined. Hence, the C-N formulation ensures both numerical stability and accuracy for the present problem.

 3.2 Examples for direct problem

3.2.1 Example 1: Assume an illustration for the direct problem given by ( 1 ) − ( 4 ) with D T : { 0 ≤ v ≤ 1 , 0 ≤ τ ≤ 1 } , and the input data are, f ( v , τ ) = − 1 6 e τ ( 1 + τ ) v 3 ( 2 v − 1 ) 3 ( 3 v − 1 ) + 1 44 ( 133 + 1188 e 3 τ + 176 π 2 ( 3 τ − 1 ) − τ ( 6 τ + 1 ) ) Sin ( 2 πv ) , z ( τ ) = 17 256 e τ ( 3 τ − 1 ) , c ( τ ) = 1 + 9 e 3 τ , τ ∈ [ 0 , 1 ] , v ∈ [ 0 , 1 ] , and, the analytical solution is, w ( v , τ ) = ( 3 τ − 1 ) Sin ( 2 πv ) , g ( v , τ ) = e τ ( 2 v 2 − v ) 3 ( 3 v − 1 ) , a ( τ ) = 1 44 ( 1 + 2 τ ) , with , b ( τ ) = 1 6 ( 1 + τ ) , ( v 1 , v 2 ) = ( 1 4 , 3 4 ) , τ ∈ [ 0 , T ] , ( v , τ ) ∈ D T

We investigate the accuracy of the solution using diverse mesh grid size, M = N ∈ { 20,40,80,160 } . In Figure 1 (a)–(d), an excellent agreement between the exact and numerical solutions can be clearly observed, indicating high accuracy. It can be noted that as the number of mesh points increases, the accuracy of the obtained solution improves, which reflects the convergence and stability of the proposed numerical scheme. Moreover, the absolute error graph at each mesh shows that the maximum magnitude of error didn’t exceed ( 10 − 5 ) , as depicted in Figure 1 (a)–(d).

The graphs showing analytical and computational distribution with absolute error for the direct problem <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> <mml:mo>−</mml:mo> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>4</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula>, for example1 <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>a</mml:mi> <mml:mn>1</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> <mml:mo>−</mml:mo> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi mathvariant="normal">d</mml:mi> <mml:mn>1</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> </mml:math> </inline-formula> represent the size of mesh <inline-formula> <mml:math display="inline"> <mml:mi mathvariant="normal">N</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">M</mml:mi> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo stretchy="true">{</mml:mo> <mml:mn>20</mml:mn> <mml:mo>,</mml:mo> <mml:mn>40</mml:mn> <mml:mo>,</mml:mo> <mml:mn>80</mml:mn> <mml:mo>,</mml:mo> <mml:mn>160</mml:mn> <mml:mo stretchy="true">}</mml:mo> </mml:mrow> </mml:math> </inline-formula>, respectively.

The obtained results clearly indicate that the mesh size has a noticeable influence on the numerical accuracy. As the spatial and temporal mesh are refined (i.e., as M and N increase), the numerical solution becomes smoother and shows a closer agreement with the analytical one. This behavior confirms the expected second-order convergence of the C-N finite difference scheme in both time and space. Beyond a certain refinement level (e.g., M = N ≥ 80 ), further mesh subdivision produces only negligible changes, implying that the proposed schemeis numerically stable and nearly mesh-independent. Whilst, Figure 2 and Figure 3, show the numerical outcomes for the required information w ( v i , τ ) , ( i = 1 , 2 ) , evaluated at various mesh sizes such that (s.t.), observed that an excellent agreement is obtained.

The graphs showing the exact and numerical values for, required output <inline-formula> <mml:math display="inline"> <mml:mi>w</mml:mi> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:msub> <mml:mi>v</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>τ</mml:mi> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula> equation <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>5</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula>, for example 1 <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>a</mml:mi> <mml:mn>2</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> <mml:mo>−</mml:mo> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi mathvariant="normal">d</mml:mi> <mml:mn>2</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> </mml:math> </inline-formula> with the size of mesh <inline-formula> <mml:math display="inline"> <mml:mi mathvariant="normal">N</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">M</mml:mi> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo stretchy="true">{</mml:mo> <mml:mn>20</mml:mn> <mml:mo>,</mml:mo> <mml:mn>40</mml:mn> <mml:mo>,</mml:mo> <mml:mn>80</mml:mn> <mml:mo>,</mml:mo> <mml:mn>160</mml:mn> <mml:mo stretchy="true">}</mml:mo> </mml:mrow> </mml:math> </inline-formula>, respectively. The graphs showing the exact and numerical values for, required output <inline-formula> <mml:math display="inline"> <mml:mi>w</mml:mi> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:msub> <mml:mi>v</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>τ</mml:mi> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula> equation <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>5</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula>, for example 1 <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>a</mml:mi> <mml:mn>3</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> <mml:mo>−</mml:mo> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi mathvariant="normal">d</mml:mi> <mml:mn>3</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> </mml:math> </inline-formula> with the size of mesh <inline-formula> <mml:math display="inline"> <mml:mi mathvariant="normal">N</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">M</mml:mi> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo stretchy="true">{</mml:mo> <mml:mn>20</mml:mn> <mml:mo>,</mml:mo> <mml:mn>40</mml:mn> <mml:mo>,</mml:mo> <mml:mn>80</mml:mn> <mml:mo>,</mml:mo> <mml:mn>160</mml:mn> <mml:mo stretchy="true">}</mml:mo> </mml:mrow> </mml:math> </inline-formula>, respectively.

3.2.2. Example 2: Assume an illustration for the direct problem given by ( 1 ) − ( 4 ) with D T : { 0 ≤ v ≤ 1 , 0 ≤ τ ≤ 1 } , and the input data are, f ( v , τ ) = 1 2 e τ ( ( 2 v − 1 ) ( 6 + ( 5 + τ ) ( v − 1 ) v ) − 2 Sin ( πv ) 1 + τ ) , z ( τ ) = − 3 e 2 τ 32 2 , c ( τ ) = 1 , and, the analytical solution is, w ( v , τ ) = e τ ( 1 − v ) ( v − 1 2 ) v , g ( v , τ ) = e τ Sin ( πv ) , a ( τ ) = 6 + τ , with , b ( τ ) = 1 ( 1 + τ ) , ( v 1 , v 2 ) = ( 1 4 , 3 4 ) , τ ∈ [ 0 , T ] , ( v , τ ) ∈ D T

The accuracy of the numerical solution is examined using several mesh grid size, M = N ∈ { 20,40,80,160 } . Figure 4 ( a ) − ( d ) , clearly demonstrates an excellent agreement between the analytical and numerical solutions, confirming the reliability of the proposed approach. It can be noted that as the number mesh becomes finer, the numerical accuracy improves, indicating the expected convergence and stability of the C-N finite difference scheme. Furthermore, the absolute error plots at different mesh sizes reveal that the maximum error magnitude remains below ( 10 − 8 ) , as illustrated in Figure 4 ( a ) − ( d ) .

The graphs showing analytical and computational distribution with absolute error for the direct problem <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> <mml:mo>−</mml:mo> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>4</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula>, for example2 <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>a</mml:mi> <mml:mn>4</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> <mml:mo>−</mml:mo> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi mathvariant="normal">d</mml:mi> <mml:mn>4</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> </mml:math> </inline-formula> represent the size of mesh <inline-formula> <mml:math display="inline"> <mml:mi mathvariant="normal">N</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">M</mml:mi> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo stretchy="true">{</mml:mo> <mml:mn>20</mml:mn> <mml:mo>,</mml:mo> <mml:mn>40</mml:mn> <mml:mo>,</mml:mo> <mml:mn>80</mml:mn> <mml:mo>,</mml:mo> <mml:mn>160</mml:mn> <mml:mo stretchy="true">}</mml:mo> </mml:mrow> </mml:math> </inline-formula>, respectively.

Similarly, in Example 2, the refinement of the spatial and temporal mesh has a direct impact on the numerical accuracy. As the grid is successively refined, the computed solution aligns more closely with the analytical one, and the absolute error distribution becomes smoother across the domain. This confirms that the proposed C-N finite difference formulation maintains its second-order convergence and numerical stability. It can therefore be concluded that the results are consistent and nearly insensitive to further mesh refinement beyond the tested resolutions, indicating satisfactory mesh-independence of the method. Whilst, Figure 5 and Figure 6, show the numerical outcomes for the required information w ( v i , τ ) , ( i = 1 , 2 ) , evaluated at various mesh sizes s.t. observed that an excellent agreement is obtained.

The graphs showing the exact and numerical values for, required output <inline-formula> <mml:math display="inline"> <mml:mi>w</mml:mi> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:msub> <mml:mi>v</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>τ</mml:mi> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula> equation <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>5</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula>, for example 2 <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>a</mml:mi> <mml:mn>5</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> <mml:mo>−</mml:mo> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi mathvariant="normal">d</mml:mi> <mml:mn>5</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> </mml:math> </inline-formula> with the size of mesh <inline-formula> <mml:math display="inline"> <mml:mi mathvariant="normal">N</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">M</mml:mi> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo stretchy="true">{</mml:mo> <mml:mn>20</mml:mn> <mml:mo>,</mml:mo> <mml:mn>40</mml:mn> <mml:mo>,</mml:mo> <mml:mn>80</mml:mn> <mml:mo>,</mml:mo> <mml:mn>160</mml:mn> <mml:mo stretchy="true">}</mml:mo> </mml:mrow> </mml:math> </inline-formula>, respectively. The graphs showing the exact and numerical values for, required output <inline-formula> <mml:math display="inline"> <mml:mi>w</mml:mi> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:msub> <mml:mi>v</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>τ</mml:mi> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula> equation <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>5</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula>, for example 2 <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>a</mml:mi> <mml:mn>6</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> <mml:mo>−</mml:mo> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi mathvariant="normal">d</mml:mi> <mml:mn>6</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> </mml:math> </inline-formula> with the size of mesh <inline-formula> <mml:math display="inline"> <mml:mi mathvariant="normal">N</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">M</mml:mi> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo stretchy="true">{</mml:mo> <mml:mn>20</mml:mn> <mml:mo>,</mml:mo> <mml:mn>40</mml:mn> <mml:mo>,</mml:mo> <mml:mn>80</mml:mn> <mml:mo>,</mml:mo> <mml:mn>160</mml:mn> <mml:mo stretchy="true">}</mml:mo> </mml:mrow> </mml:math> </inline-formula>, respectively. 4. Inverse problem solution for <xref ref-type="disp-formula" rid="e1 e2 e3 e4 e5">Equations (1)–(5)</xref>

For the nonlinear IP ( 1 ) − ( 5 ) , we seek precise and stable identification of w ( v , τ ) , a ( τ ) and b ( τ ) , that mean potential term a ( τ ) , and b ( τ ) is unknown. The one-dimensional second-order parabolic equation together with w ( v , τ ) satisfies the problem given by Equation (1)–(5), the problem is reformulated as a nonlinear least-squares minimization task. The resulting minimization problem is solved numerically using the lsqnonlin routine available in MATLAB’s optimization Toolbox, which implements a trust-region reflective algorithm suitable for nonlinear least squares problems with bound constraints. Due to the ill-posedness of such problems, especially in the presence of noise data, Tikhonov Regularization (TR) is employed to stabilize the solution. The regularization parameter β ≥ 0 , is selected carefully to balance the trade-off between measured and the numerically computed solution. For this purpose, MATLAB routine lsqnonlin is employed, to minimize the Tikhonov Regularized objective functional, the procedure begins with a suitably selected initial guess. The TR functional is derived based on condition ( 5 ) , and the functional error is incorporated as follows: F ( a ) = ∑ i = 1 2 ‖ w ( v i , τ ) − z i ( τ ) ‖ 2 L 2 [ 0 , τ ] + β ( ‖ a ( τ ) ‖ 2 L 2 [ 0 , τ ] + ‖ b ( τ ) ‖ 2 L 2 [ 0 , τ ] ) ,

where β > 0 , is a regularization parameter. The discretization form of ( 29 ) is F ( a _ ) = ∑ i = 1 2 ∑ j = 1 N [ w ( v i , τ j ) − z i ( τ j ) ] 2 + β ( ∑ j = 1 N a j 2 + ∑ j = 1 N b j 2 ) ,

The unregularized case, i.e., β = 0 , produces the regular a nonlinear least-squares functional, which is inherently unstable when dealing with noisy data. The MATLAB routine lsqnonlins is used to minimize F under some physical constraint. The following parameters had been used for the subroutine:

(Maxlter) maximum number of iterations = 10 6 х ( N ) .

Solution and Objective function tolerance = 10 − 10

The lower and upper bounds on the component of the vector a _ are − 10 − 2 and 10 2 , respectively.

The IP ( 1 ) − ( 5 ) are resolved via both precise and noisy measurement ( 5 ) . By including a random error, the noisy data is numerically simulated: z ε ( τ j ) = z ( τ j ) + ε j j = 0 , N . ¯ where ε , is random Gaussian normal distribution vectors with mean zero and standard deviations σ , given by σ = ρ х max τϵ [ 0 , T ] | z ( τ ) | where, ρ is the percentage of noise. We use the MATLAB function “ normrnd”, as: ε _ = normrnd ( 0 , σ , N ) , to generate the random variables ε _ = ( ε j ) and, j = 0 , N . ¯

5. Results and discussion

To evaluate the precision and stability of the numerical methods, a set of numerical experiments is conducted. These experiments are designed to evaluate the accuracy of the computed results by simulating realistic measurement conditions, where noise is introduced into the input data. To quantitatively evaluate the discrepancy between the exact and the numerically computed solutions, the root mean squares errors (RMSE) are utilized by the following expression RMSE ( a ) = 1 N ∑ i = 1 N ( a i exact − a i numerical ) 2 , we take, T = 1 for simplicity.

5.1. Example 3: IP

Consider the IP ( 1 ) − ( 5 ) with the following input data, f ( v , τ ) = ( − 2 + 4 π 2 ( 3 + τ ) − τ ( 7 + 2 τ ) ) Sin ( 2 πv ) − τ 2 ( 7 + 2 τ ) Sin ( 4 πv ) , τ ∈ [ 0 , T ] , z ( τ ) = − 1 4 15 + 5 ( 33 − 8 2 ) ( 3 + τ ) ( 7 + 2 τ ) , c ( τ ) = 1 , τ ∈ [ 0 , T ] , where γ = 2 , and, the analytical solution for this IP is provided as follows, w ( v , τ ) = ( 3 + τ ) Sin ( 2 πv ) , g ( v , τ ) = ( 7 + 2 τ ) Sin ( 4 πv ) , a ( τ ) = 2 τ + 1 , with , b ( τ ) = τ 2 , ( v 1 , v 2 ) = ( 1 8 , 2 5 ) , τ ∈ [ 0 , T ] , ( v , τ ) ∈ D T

The time-dependent coefficient a ( τ ) , and b ( τ ) , are then reconstructed (in case, M = N = 80 , is taken), and consider the case of noise-free in the measurements, see Table 1, this mean p = 0 , in ( 37 ) and, no regularization applied, Figure 7 shows the objective function ( 37 ) without regularization i.e. β = 0 , while, in Figure 9, noise without regularization.

Numerical information for no regularization <inline-formula> <mml:math display="inline"> <mml:mi>β</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> </inline-formula> and noise <inline-formula> <mml:math display="inline"> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> </inline-formula>.
M = N 80
RMSE a( τ ) 0.0241
RMSE b( τ ) 2.1592 × 10 − 4
Computational time 3594 Sec
No. of Iterations 16
Obj. Function 1.2221× 10 − 10
The graph showing the objective function for the IP <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> <mml:mo>−</mml:mo> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>5</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula>, in example 3 when the size of mesh <inline-formula> <mml:math display="inline"> <mml:mi mathvariant="normal">N</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">M</mml:mi> <mml:mo>=</mml:mo> <mml:mn>80</mml:mn> </mml:math> </inline-formula>. The graphs <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>a</mml:mi> <mml:mn>8</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula> and <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>b</mml:mi> <mml:mn>8</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula> showing the reconstructed coefficient <inline-formula> <mml:math display="inline"> <mml:mi>a</mml:mi> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula> and <inline-formula> <mml:math display="inline"> <mml:mi>b</mml:mi> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula> in comparison with exact value for the IP <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> <mml:mo>−</mml:mo> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>5</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula>, in above example when the size of mesh <inline-formula> <mml:math display="inline"> <mml:mi mathvariant="normal">N</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">M</mml:mi> <mml:mo>=</mml:mo> <mml:mn>80</mml:mn> </mml:math> </inline-formula>. The graph showing the objective function when <inline-formula> <mml:math display="inline"> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.01</mml:mn> <mml:mo>%</mml:mo> <mml:mo>,</mml:mo> </mml:math> </inline-formula> no regularization for the IP <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> <mml:mo>−</mml:mo> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>5</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula>, in example 2, when the size of mesh <inline-formula> <mml:math display="inline"> <mml:mi mathvariant="normal">N</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">M</mml:mi> <mml:mo>=</mml:mo> <mml:mn>80</mml:mn> </mml:math> </inline-formula>.

We investigate the time-dependent coefficient a ( τ ) and b ( τ ) under both exact and perturbed measurements, as demonstrated in Figures 7 − 12. The associated results are unstable, as illustrated in Figure 9- Figure 10. This reveals that the problem is not properly posed, a noticeable instability appears in the results, which is anticipated due to the ill-posed nature of the investigated IP. The unregularized solutions exhibit noticeable instability even in the noise-free case, demonstrating the sensitivity of the problem to small perturbations in the input data. The TR approach is used by incorporating the penalty term β ( ‖ a ( τ ) ‖ 2 L 2 [ 0 , τ ] + ‖ b ( τ ) ‖ 2 L 2 [ 0 , τ ] ) into the classical least-squares formulation, as presented in Equation (37). In Figure 11, p = 0.01 % noise strategy is employed to enhance the solution’s robustness with regularization parameters, β ∈ { 10 − 3 , … , 10 − 10 } , yielding numerically stable regularized reconstructions of a ( τ ) and b ( τ ) , as shown in Table 2. These results indicate that smaller values of β are effective when the data is not contaminated.

The graphs <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>a</mml:mi> <mml:mn>10</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula> and <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>b</mml:mi> <mml:mn>10</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula> showing the exact and (unstable) numerical solution for <inline-formula> <mml:math display="inline"> <mml:mi>a</mml:mi> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula> and <inline-formula> <mml:math display="inline"> <mml:mi>b</mml:mi> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula> when <inline-formula> <mml:math display="inline"> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.01</mml:mn> <mml:mo>%</mml:mo> <mml:mo>,</mml:mo> </mml:math> </inline-formula> no regularization for the IP <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> <mml:mo>−</mml:mo> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>5</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula>, in example 3, when the size of mesh <inline-formula> <mml:math display="inline"> <mml:mi mathvariant="normal">N</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">M</mml:mi> <mml:mo>=</mml:mo> <mml:mn>80</mml:mn> </mml:math> </inline-formula>. The graph showing the objective function when <inline-formula> <mml:math display="inline"> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.01</mml:mn> <mml:mo>%</mml:mo> </mml:math> </inline-formula> noise with regularization, <inline-formula> <mml:math display="inline"> <mml:mi>β</mml:mi> <mml:mspace width="0.25em"/> <mml:mo>∈</mml:mo> <mml:mspace width="0.25em"/> <mml:mrow> <mml:mo stretchy="true">{</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>7</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="true">}</mml:mo> </mml:mrow> </mml:math> </inline-formula>, for the IP <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> <mml:mo>−</mml:mo> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>5</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula>, in example 3, when the size of mesh <inline-formula> <mml:math display="inline"> <mml:mi mathvariant="normal">N</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">M</mml:mi> <mml:mo>=</mml:mo> <mml:mn>80</mml:mn> </mml:math> </inline-formula>. The graphs <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>a</mml:mi> <mml:mn>12</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula> and <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>b</mml:mi> <mml:mn>12</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula> showing the exact and (stable) numerical solution for <inline-formula> <mml:math display="inline"> <mml:mi>a</mml:mi> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula> and <inline-formula> <mml:math display="inline"> <mml:mi>b</mml:mi> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula> when <inline-formula> <mml:math display="inline"> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.01</mml:mn> <mml:mo>%</mml:mo> <mml:mo>,</mml:mo> </mml:math> </inline-formula> and regularization, <inline-formula> <mml:math display="inline"> <mml:mi>β</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mo stretchy="true">{</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>7</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="true">}</mml:mo> </mml:mrow> </mml:math> </inline-formula>, for the IP <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> <mml:mo>−</mml:mo> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>5</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula>, in example 3, when the size of mesh <inline-formula> <mml:math display="inline"> <mml:mi mathvariant="normal">N</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">M</mml:mi> <mml:mo>=</mml:mo> <mml:mn>80</mml:mn> </mml:math> </inline-formula>. Numerical information for various regularization parameters and noise <inline-formula> <mml:math display="inline"> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.01</mml:mn> <mml:mo>%</mml:mo> <mml:mo>,</mml:mo> <mml:mo stretchy="true">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>80</mml:mn> </mml:math> </inline-formula>).
β 10 − 3 10 − 4 10 − 5 10 − 6 10 − 7 10 − 8 10 − 9 10 − 10
RMSE a( τ ) 0.3302 0.1220 0.0714 0.0617 0.1189 0.2408 0.2835 0.2865
RMSE b( τ ) 0.0552 0.0225 0.0259 0.0344 0.0365 0.0373 0.0377 0.0377
Computational time 84320 Sec 9211 Sec 1453 Sec 1451 Sec 3139 Sec 6277 Sec 5814 Sec 5506 Sec
No. of Iterations 401 43 6 6 14 22 22 22

Next, we will study the proportion of noise contamination with a ratio of p = 0.01 % noise, in ( 38 ) via ( 39 ) , the associated results are unstable, as illustrated. This reveals that the problem is not properly, posed and small error in the data input cause large errors in the output solution, thus, the problem needs to be regularized. As a consequence, for the situation of p = 0.01 % noise, the best selection for the regularization parameters is 10 − 6 for a ( τ ) and 10 − 4 for b ( τ ) as in Table 2, which results in the lowest RMSE values.

 6. Conclusions

In this study, the identification of the unknown time-dependent coefficient a ( τ ) and b ( τ ) in a one-dimensional parabolic PDE are investigated. The problem is formulated from a numerically evaluated nonlocal integral subjected to initial and boundary conditions. The direct problem is solved using the Crank-Nicolson (C-N) scheme, while the IP is transformed into a nonlinear least squares optimization problem using the lsqnonlin routine in MATLAB. An example test cases are employed in numerical experiments to evaluate the accuracy and stability of the proposed method. According to the results, using TR significantly enhances the solution’s stability. When the root-mean square error (RMSE) numbers are their lowest, the most effective approach is identified. Furthermore, numerical results have been provided to illustrate the accuracy and stability of the numerical method. It found that applying TR method stabilizes the solution. Cases of numerical results exist with noise (with and without) regularization. In the first case, consider when p = 0.01 % noise and without regularization as in Figure 9- Figure 10 in example 3, the results are unstable and highly oscillated, on the contrary. Other case, when p = 0.01 % noise introduced with β ∈ { 10 − 3 , … , 10 − 10 } , regularization, obtain are stable and accurate results for the reconstructed coefficient a ( τ ) and b ( τ ) , which are (stable) numerical solutions. As for the remaining cases, we observe that the results are stable and steady when regularization values are 10 − 6 for a ( τ ) and, 10 − 4 for b ( τ ) in example 3, as shown in Table 2.

 Data availability

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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Competing interests: No competing interests were disclosed.

 9 3 2026 Copyright: © 2026 Hazanee A 2026 This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. recommendation approve

The manuscript presents a numerical method for solving an inverse problem to determine two coefficients. The Crank-Nicolson finite difference method, combined with Tikhonov regularization, is employed. The MATLAB toolbox ‘lsqnonlin’ is used to solve the nonlinear regularized least-squares optimization problem. Two numerical examples are provided for the direct problem, and one numerical example is presented for the inverse problem.

This paper is interesting as it addresses the simultaneous estimation of two unknown time-dependent coefficients in an inverse problem. The manuscript includes numerical examples for both the direct and inverse problems, which help demonstrate the effectiveness of the proposed method. Therefore, the manuscript is recommended for indexing.

Please find the attachment for the recommendation for revision.

Review

Section 2, Page 4 : The theorem on the existence and uniqueness of the inverse problem is presented with reference [28]. However, reference [28] is missing from the reference list.

Section 2, Page 6 : This section discusses the existence and uniqueness of the inverse problem

(1)–(5). However, the theorem statement refers only to problems (1)–(3), mentioned at the end of the theorem statement. Please verify this with the cited reference.

Section 3, Page 6 : In this inverse problem, the unknown functions are a(τ) and b(τ), while c(τ) is assumed to be known in both the direct and inverse problems. However, in the first line of

Section 3, c(τ) is described as an unknown function that is taken to be known for the direct problem. This statement may confuse the reader and should be clarified as ‘i.e., when unknown a(τ) and b(τ) are assumed to be given’.

Section 3, Page 7 : Please recheck Equation (34) together with the definitions of L, L1, V and W.

Section 3.2 : Although the main focus of the paper is the inverse problem, two numerical examples are presented for the direct problem, while only one example is given for the inverse problem. It would be beneficial to include at least one example addressing both the direct and inverse problems in order to demonstrate how the regularization method retrieves the unknown coefficients.

Section 3.2, Page 15, Figure 8 : The figure does not appear on the same page where it is referenced. Please indicate the example number for Figure 8.

Section 3.2, Page 15, Figure 9 : The figure was mentioned in Example 3, not Example 2. Please correct it.

Section 3.2, Page 16, Figure 12 : There are too many curves corresponding to different parameter values, which makes the figure difficult to read. It is recommended to present only the most representative parameter case and compare it with the no-noise case in order to better illustrate how the regularization method retrieves the unstable solution.

Section 3.2, Page 16, Table 2 : It would be helpful if Table 2 also included results without noise for comparison, in order to better demonstrate the effect of the regularization method.

Is the work clearly and accurately presented and does it cite the current literature?

Yes

If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable

Are all the source data underlying the results available to ensure full reproducibility?

Yes

Is the study design appropriate and is the work technically sound?

Yes

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Yes

Reviewer Expertise:

Inverse problem

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

Sabah Hussein Mohammed

Competing interests: No competing interests were disclosed.

 14 3 2026

Dear Dr. Areena Hazanee 

We sincerely thank the reviewer for their careful reading and valuable comments.

All of the suggested corrections have been incorporated into the revised manuscript. Below are the detailed responses to each of the reviewers' comments.

The citation [28] has been corrected to [27] in the revised manuscript to ensure consistency with the references list.

The statement of the theorem has been updated to refer to problem (1)-(5)

The first sentence of Section 3 has been revised to clarify that the coefficients a(τ) , c(τ) and b(τ) are assumed to be given.

The definitions and equation have been rechecked and revised for accuracy. The corrected expressions are now consistent and correctly defined before equation (34).

The wording for Example 3 has been revised to better highlight that it demonstrates both the direct problem and the corresponding inverse problem. The revised sentence is: ''Example 3 illustrates the solution of the IP together with the corresponding direct problem in order to demonstrate the effectiveness of the proposed regularization approach.''

The caption of Figure 8 has been revised as follows: ''Figure 8. The graphs (a8) and (b8) showing the reconstructed coefficients a(τ) and b(τ) in comparison with the exact value for the IP (1)-(5) in Example 3 when the mesh size N=M=80.

Thank you for pointing this out. The reference to Figure 9 has been has been corrected to Example 3, as it was indeed intended for this example.

The caption for Figure 12 has been updated to clarify the influence of different values of the regularization parameter β on the reconstructed coefficients a(τ) and b(τ) in the presence of noisy data. The revised caption is as follows: ''Figure 12. The graphs (a12) and (b12) showing the exact solution and regularized numerical reconstructions of a(τ) and b(τ) when p=0.01% and several curves corresponding to different values of the regularization parameter β ∈ {10−3,…,10−7} , are presented to illustrate the influence of the regularization parameter on the stability of the reconstructed solution for the IP (1)-(5) in Example 3 when the mesh size N=M=80.

The results for the case with noise level p=0% have been presented in Table 1 . Therefore, Table 2, focuses on the case with p=0.01% to illustrate the impact of the regularization technique on the solution.

Best Regards

 10.5256/f1000research.191048.r458506 Reviewer response for version 1 Hameed Raad Awad 1 Referee https://orcid.org/0000-0001-9500-8563 Tikrit University, Tikrit, Iraq

Competing interests: No competing interests were disclosed.

 6 3 2026 Copyright: © 2026 Hameed RA 2026 This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. recommendation approve

The paper investigates an interesting and highly relevant topic in the field of inverse problems, with practical applications in heat transfer and diffusion processes. The combination of the Crank-Nicolson method for the forward (Direct) problem and Tikhonov regularization for the inverse (indirect) problem is a mathematically sound approach. The numerical examples provided (Examples 1, 2 (for direct problem), and 3 (for inverse problem)) effectively demonstrate the proposed method's capability to reconstruct the unknown coefficients even when the input data is perturbed by noise.

Below some points of the evaluation of the manuscript

 

In Abstract section: In the Methods subsection, the phrase "efficiently solved used the MATLAB subroutine" contains a grammatical error and should be corrected to "efficiently solved using the MATLAB subroutine

In Introduction section: There are several run-on sentences that impede clarity. For instance, "Compared with direct problems, IPs, are often more challenging to solve due to their ill-posedness, solutions may fail to exist, may not be unique, or may not change continuously with the input data". This should be split into multiple sentences.

Punctuation and Spacing: The manuscript suffers from excessive and incorrect comma usage (e.g., "In this paper, the article is divided into five sections, section 2, includes a mathematical formulation..."). A thorough language copy-edit is needed.

 

In Tables: The text refers to "Table 1" and "Table 2" summarizing the numerical information, RMSE, and computational time. Ensure that the captions in the final manuscript clearly define the units for computational time (e.g., seconds) and that all abbreviations (like "Obj. Function") are introduced properly.

before the equation 3   with periodic boundary conditions in the spatial variable v on the interval    ,

 Overall, I recommend the acceptance of manuscript after addressing the minor changes.

Sincerely,

 Dr. Raad A. Hameed

Reviewer

Is the work clearly and accurately presented and does it cite the current literature?

Yes

If applicable, is the statistical analysis and its interpretation appropriate?

Yes

Are all the source data underlying the results available to ensure full reproducibility?

Yes

Is the study design appropriate and is the work technically sound?

Yes

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Yes

Reviewer Expertise:

partial differential equations

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

Sabah Hussein Mohammed

Competing interests: No competing interests were disclosed.

 14 3 2026

Dear Prof. Dr. Raad A. Hameed 

We sincerely thank the reviewers for their careful reading and valuable comments.

All of the suggested corrections have been incorporated into the revised manuscript. Below are the detailed responses to each of the reviewers' comments.

The sentence has been corrected in the revised manuscript to: ''efficiently solved using the MATLAB subroutine lsqnonlin from the MATLAB optimization Toolbox''.

The introduction section has been carefully revised and long run-on sentences have been split into shorter sentences to improve clarity and readability.

The manuscript has been carefully edited to correct punctuation and spacing issues. Unnecessary commas have been removed and several sentences have been rewritten to improve grammatical accuracy and readability.

The captions of Table 1 and Table2 have been revised to clearly indicate the unit of computational time (second). In addition, all abbreviations used in the Tables, including RMSE and objective function, have been properly defined in the revised manuscript.

The description preceding Equation (3) has been revised to clearly state the periodic boundary conditions in the spatial variable v over the specified interval.

Best Regards 

10.5256/f1000research.191048.r458507 Reviewer response for version 1 Tekin Ibrahim 1 Referee https://orcid.org/0000-0001-6725-5663 Alanya Alaaddin Keykubat University, Antalya, Turkey

Competing interests: No competing interests were disclosed.

 6 3 2026 Copyright: © 2026 Tekin I 2026 This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. recommendation approve-with-reservations

The manuscript studies the simultaneous numerical identification of two time-dependent coefficients in a second-order parabolic equation subject to nonlocal initial and boundary conditions. Inverse coefficient problems for parabolic equations are well known to be ill-posed and mathematically challenging, and the simultaneous reconstruction of multiple time-dependent parameters represents a nontrivial extension of standard inverse formulations. Therefore, the topic is relevant and potentially valuable for the inverse problems and applied mathematics community.

The manuscript presents a numerical reconstruction framework and provides computational experiments intended to demonstrate the effectiveness of the proposed approach. While the general structure of the paper is acceptable, several issues should be addressed in order to improve the scientific clarity, contextualization within the existing literature, and transparency of the computational results.

The following points should be considered by the authors:

Language and presentation:

The manuscript would benefit from careful language editing. Several sentences are grammatically unclear or stylistically weak, which occasionally makes the mathematical arguments difficult to follow.

Reference correction:

On page 4, reference [28] appears to be incorrectly cited and should be corrected to [27].

Literature context:

The Introduction currently provides a limited overview of the recent literature on inverse coefficient problems for parabolic equations. The authors should expand the literature review to better position their contribution relative to existing numerical and analytical studies, particularly those addressing nonlocal boundary conditions and overdetermination data.

Computational efficiency:

In Table 1, the reported computational time is approximately 3594 seconds for only 16 iterations. For a one-dimensional parabolic inverse problem, this computational cost appears relatively high. The authors should therefore:

1.Specify the hardware and software environment used for the computations.

2.Provide a brief discussion of the computational complexity of the proposed algorithm.

3.Clarify whether the reported time corresponds to a single reconstruction or multiple runs.

Addressing these points would substantially improve the clarity and scientific positioning of the manuscript.

Suggested Additional References (for the Introduction/Literature Review)

Huzyk, et al., 2023 - Ref 1.

Azizbayov, et al., 2025 - Ref 2.

These studies may help broaden the discussion of related inverse coefficient problems and numerical approaches.

The manuscript addresses a relevant inverse problem and proposes a numerical approach that may be of interest to researchers working on inverse parabolic problems. However, several clarifications and moderate revisions are required before the article can be considered fully scientifically sound. The requested revisions mainly concern improvements to the literature review, language clarity, and the discussion of computational aspects.

Is the work clearly and accurately presented and does it cite the current literature?

Partly

If applicable, is the statistical analysis and its interpretation appropriate?

I cannot comment. A qualified statistician is required.

Are all the source data underlying the results available to ensure full reproducibility?

No source data required

Is the study design appropriate and is the work technically sound?

Yes

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Yes

Reviewer Expertise:

Invese problems, Applied Mathematics

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

References : Coefficient inverse problem for the strongly degenerate parabolic equation. Carpathian Mathematical Publications .2023;15(1) : 10.15330/cmp.15.1.52-65 52-65 10.15330/cmp.15.1.52-65 : An Inverse Problem for a Parabolic Equation with Nonlocal Boundary Conditions and Two-point Overdetermination. European Journal of Pure and Applied Mathematics .2025;18(4) : 10.29020/nybg.ejpam.v18i4.6796 10.29020/nybg.ejpam.v18i4.6796 Sabah Hussein Mohammed

Competing interests: No competing interests were disclosed.

 14 3 2026

Dear Dr. Ibrahim Tekin 

We sincerely thank the reviewer for their careful reading and valuable comments.

All of the suggested corrections have been incorporated into the revised manuscript. Below are the detailed responses to each of the reviewers' comments.

The manuscript has been carefully revised to improve language clarity and readability. Several sentences throughout the introduction and numerical methodology sections have been edited.

The incorrect citation on page 4 has been corrected from [28] to [27]

The introduction has been expanded to include additional recent studies on inverse coefficient problems for parabolic equations with nonlocal boundary conditions. In particular, the works of Huzyk et al. (2023) and Azizbayov and Safarova (2025) have been incorporated.

The hardware and software environment used for the computations has been added in the numerical results section. A brief discussion of the computational complexity has also been included, and it has been clarified that the reported time corresponds to a single reconstruction run.

Best Regards 

10.5256/f1000research.191048.r458503 Reviewer response for version 1 Balasim Alla Tareq 1 Referee mathematics, college of basic education, Baghdad, Iraq

Competing interests: No competing interests were disclosed.

 28 2 2026 Copyright: © 2026 Balasim AT 2026 This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. recommendation approve

I have reviewed the manuscript entitled "Simultaneous Numerical Determination of Two Time-dependent Coefficients in Second Order Parabolic Equation With Nonlocal Initial and Boundary Conditions."

The paper addresses an important problem in inverse partial differential equations and proposes a stable numerical framework using the Crank-Nicolson finite difference method combined with Tikhonov regularization. 

Strengths: -

The study presents a clear scientific objective and demonstrates the effectiveness of inverse problems (IPs) in reconstructing unknown coefficients.

The numerical methodology is appropriate and shows reliable convergence and stability. 

Results confirm the robustness of the approach under both exact and noisy data.

Minor Corrections Required:

1. The introduction contains several long and complex sentences that should be revised for clarity and readability.

2. References (2–5) are not consistent in formatting and should be unified according to the journal’s style. 3. There is no reference 28.

4. The stability and convergence are not rigorously proven. The authors are encouraged to provide a theoretical proof or cite appropriate references supporting these results.

5. Equation 37 should be placed directly below (s a regularization parameter. The discretization form of (29) is).

6. Figures 3-6 should be before section 4.

Recommendation: The manuscript is scientifically sound and of clear value. I recommend acceptance after minor corrections.

Best regards,

Is the work clearly and accurately presented and does it cite the current literature?

Yes

If applicable, is the statistical analysis and its interpretation appropriate?

I cannot comment. A qualified statistician is required.

Are all the source data underlying the results available to ensure full reproducibility?

No source data required

Is the study design appropriate and is the work technically sound?

Yes

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Yes

Reviewer Expertise:

Numerical analysis, Finite difference method, Group iterative method, Dynamical system.

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

Sabah Hussein Mohammed

Competing interests: No competing interests were disclosed.

 14 3 2026

Dear Dr. Alaa Tariq 

We sincerely thank the reviewer for their careful reading and valuable comments.

All of the suggested corrections have been incorporated into the revised manuscript. Below are the detailed responses to each of the reviewers' comments.

 

The introduction section has been carefully revised to improve clarity and readability. Several long and complex sentences have been rewritten and simplified while preserving the scientific meaning.

Thank you for pointing this out. References (2-5) have been revised and reformatted to ensure consistency with the journal's citation style.

The incorrect citation [28] has been corrected to [27] in the revised manuscript to ensure consistency with the references list.

A stability and convergence discussion has been included in section 3.1. In addition, appropriate references supporting the unconditional stability and second-order convergence of the Crank-Nicolson finite difference scheme have been added.

Equation (37) has been moved and placed directly below the sentence '' as a regularization parameter. The discretization from of (29) is'' to improve the logical flow of the presentation.

Figures 3-6 have been reposition al in the revised manuscript and are now placed before section 4 in order to maintain the correct sequence between the numerical illustration and the corresponding discussion.

Best Regards