A Novel Numerical Approach for Solving Initial Value Problems in Heat Equations Using Variational Regularization and Intelligent Particle Swarm Optimization

Abstract

In this article, we use the variational approach as a regularization tool to solve the initial value problem that appears in a heat partial differential equation. Although the temperature obtained at time t=T>0 is known, the initial temperature distribution remains unknown. By using the separation of variables method, the partial differential equation is transformed into a Fredholm integral equation of the first kind. We then apply a discretization algorithm to reduce the integral equation to a system of linear algebraic equations, commonly referred to as an inverse linear operator problem. The variational regularization method is employed to obtain a regularized solution. We also present a fundamental analysis of this method for solving inverse problems. Furthermore, we describe the application of the Intelligent Particle Swarm Optimization (IPSO) technique to determine the optimal regularization parameter. Our results demonstrate that integrating particle swarm optimization with variational optimization is both effective and computationally feasible.