Abstract
The inverse problem of reconstructing the time-dependent coefficients, along with the temperature in a one-dimensional parabolic equation with initial and Neumann boundary conditions supplemented by non-local integral and boundary specification conditions is, for the first time, numerically solved. This is challenging and interesting nonlinear inverse problem which has many significant applications in various fields of physics and mechanics. From the literature, we already know that this inverse problem has a unique solution. However, the problem is still ill-posed (very slight errors in the additional input may cause relatively significant errors in the output coefficients) by being unstable to noise in the input data. For the numerical realization, we apply the Crank–Nicolson finite difference method (FDM) together with the Tikhonov regularization to find a stable and accurate numerical solution of finite differences. The resulting nonlinear system of parabolic equation is solved computationally using the MATLAB subroutine lsqnonlin. Both exact and numerically simulated noisy input data are inverted. Numerical results presented for two examples show the efficiency of the computational method and the accuracy and stability of the numerical solution even in the presence of noise in the input data. This study will be significant to researchers working on computational and mathematical methods for solving inverse coefficients reconstruction problems with applications in heat transfer and porous media.