Reconstruction of Two Time-Dependent Coefficients in 1D Parabolic Equations Using Tikhonov Regularization method

Abstract

In this paper we solved a parabolic partial differential equation with nonlocal time condition, Neumann boundary condition and nonlocal integral condition. To solving direct problem, we applied Crank-Nicolson finite difference which insure stability and convergent. In order to solve the inverse problem and recovering two coefficients time-depend and the solution by using the two-point measurements, which it is ill-posed nonlinear problem, applied Tikhonov regularization by using Nonlinear Least Squares method in MATLAB's Optimization Toolbox "lsqnonlin" with Trust-Region-Reflective algorithm, the regularization parameter selected from the range { to } by using trial and error strategy. To insure the evaluated of numerical solution using root mean square error {RMSE} proving that our methodology recovers the composite coefficients with acceptable accuracy.