Splitting method for an inverse source problem in parabolic differential equations: Error analysis and applications

Abstract

In this work, we present a numerical method based on a splitting algorithm to find the solution of an inverse source problem with the integral condition. The source term is reconstructed by using the specified data and by employing the Lie splitting method, we decompose the equation into linear and nonlinear parts. Each subproblem is solved by the Fourier transform and then by combining the solutions of subproblems, the solution of the original problem is computed. Moreover, the framework of strongly continuous semigroup (or C0‐semigroup) is employed in error analysis of operator splitting method for the inverse problem. The convergence of the proposed method is also investigated and proved. Finally, some numerical examples in one, two, and three‐dimensional spaces are provided to confirm the efficiency and capability of our work compared with some other well‐known methods.

Splitting Physics-Informed Neural Networks for Inferring the Dynamics of Integer- and Fractional-Order Neuron Models

“…In this work we introduce a new approach for solving neuron models that combines operator splitting methods with physics-informed neural networks (PINNs). Operator splitting methods have been successfully applied in various fields of physics and engineering [7,12,16,19,25,36,54,55], while PINNs provide a powerful tool for approximating the solution of differential equations. A general introduction to the splitting method can be found in [27,46].…”