This paper examines the finding of spacewise dependent heat source function in pseudoparabolic equation with initial and homogeneous Dirichlet boundary conditions, as well as the final time value / integral specification as additional conditions that ensure the uniqueness solvability of the inverse problem. However, the problem remains ill-posed because tiny perturbations in input data cause huge errors in outputs. Thus, we employ Tikhonov’s regularization method to restore this instability. In order to choose the best regularization parameter, we employ L-curve method. On the other hand, the direct (forward) problem is solved by a finite difference scheme while the inverse one is reformulated as an optimization problem. The

     The problem of reconstruction of a timewise dependent coefficient and free boundary at once in a nonlocal diffusion equation under Stefan and heat Flux as nonlocal overdetermination conditions have been considered. A Crank–Nicolson finite difference method (FDM) combined with the trapezoidal rule quadrature is used for the direct problem. While the inverse problem is reformulated as a nonlinear regularized least-square optimization problem with simple bound and solved efficiently by MATLAB subroutine lsqnonlin from the optimization toolbox. Since the problem under investigation is generally ill-posed, a small error in the input data leads to a huge error in the output, then Tikhonov’s regularization technique is app

     This paper investigates the simultaneous recovery for two time-dependent coefficients for heat equation under Neumann boundary condition. This problem is considered under extra conditions of nonlocal type. The main issue with this problem is the solution unstable to small contamination of noise in the input data. The Crank-Nicolson finite difference method is utilized to solve the direct problem whilst the inverse problem is viewed as nonlinear optimization problem. The later problem is solved numerically using optimization toolbox from MATLAB. We found that the numerical results are accurate and stable.

This paper presents a numerical solution to the inverse problem consisting of recovering time-dependent thermal conductivity and  heat source coefficients  in the one-dimensional  parabolic heat equation.   This  mathematical  formulation  ensures that the inverse problem  has a unique  solution.   However, the problem  is still  ill-posed since small errors  in the input data lead to a drastic  amount  of errors in the output coefficients.  The  finite  difference method  with  the Crank-Nicolson  scheme is adopted  as a direct  solver of the problem in a fixed domain.   The inverse problem is solved sub

In this article, an inverse problem of finding timewise-dependent thermal conductivity has been investigated numerically. Numerical solution of forward (direct) problem has been solved by finite-difference method (FDM). Whilst, the inverse (indirect) problem solved iteratively using Lsqnonlin   routine  from MATLAB. Initial guess for unknown coefficient expressed by explicit relation   based on nonlocal overdetermination conditions and intial input data .The obtained numrical results are presented and discussed in several figures and tables. These results are accurate and stable even in the presense of noisy data.

This paper investigates the recovery for time-dependent coefficient and free boundary for heat equation. They are considered under mass/energy specification and Stefan conditions. The main issue with this problem is that the solution is unstable and sensitive to small contamination of noise in the input data. The Crank-Nicolson finite difference method (FDM) is utilized to solve the direct problem, whilst the inverse problem is viewed as a nonlinear optimization problem. The latter problem is solved numerically using the routine optimization toolbox lsqnonlin from MATLAB. Consequently, the Tikhonov regularization method is used in order to gain stable solutions. The results were compared with their exact solution and tested via

     In this paper, we consider a two-phase Stefan problem in one-dimensional space for parabolic heat equation with non-homogenous Dirichlet boundary condition. This problem contains a free boundary depending on time. Therefore, the shape of the problem is changing with time. To overcome this issue, we use a simple transformation to convert the free-boundary problem to a fixed-boundary problem. However, this transformation yields a complex and nonlinear parabolic equation. The resulting equation is solved by the finite difference method with Crank-Nicolson scheme which is unconditionally stable and second-order of accuracy in space and time. The numerical results show an excellent accuracy and stable solutions for tw

     In this research, an unknown space-dependent force function in the wave equation is studied. This is a natural continuation of [1] and chapter 2 of [2] and [3], where the finite difference method (FDM)/boundary element method (BEM), with the separation of variables method, were considered. Additional data are given by the one end displacement measurement. Moreover, it is a continuation of [3], with exchanging the boundary condition, where  are extra data, by the initial condition. This is an ill-posed inverse force problem for linear hyperbolic equation. Therefore, in order to stabilize the solution, a zeroth-order Tikhonov regularization method is provided. To assess the accuracy, the minimum error between

     This article aims to determine the time-dependent heat coefficient together with the temperature solution for a type of semi-linear time-fractional inverse source problem by applying a method based on the finite difference scheme and Tikhonov regularization. An unconditionally stable implicit finite difference scheme is used as a direct (forward) solver. While by the MATLAB routine lsqnonlin from the optimization toolbox, the inverse problem is reformulated as nonlinear least square minimization and solved efficiently. Since the problem is generally incorrect or ill-posed that means any error inclusion in the input data will produce a large error in the output data. Therefore, the Tikhonov regularization technique is applie