An efficient iterative technique for solving initial value problem for multidimensional partial differential equations

TL;DR

This paper introduces a new iterative method for efficiently solving initial value problems in multidimensional PDEs, avoiding discretization and linearization, with proven convergence and demonstrated effectiveness through numerical examples.

Contribution

The paper presents a novel iterative scheme that reduces computational complexity for multidimensional PDEs without discretization or linearization, supported by convergence analysis.

Findings

Method successfully solves high-dimensional heat and wave PDEs.

Convergence and error estimates are rigorously established.

Numerical examples confirm reliability and efficiency.

Abstract

A new iterative technique is presented for solving of initial value problem for certain classes of multidimensional linear and nonlinear partial differential equations. Proposed iterative scheme does not require any discretization, linearization or small perturbations and therefore significantly reduces numerical computations. Rigorous convergence analysis of presented technique and an error estimate are included as well. Several numerical examples for high dimensional initial value problem for heat and wave type partial differential equations are presented to demonstrate reliability and performance of proposed iterative scheme.