Approximation of mild solutions of the linear and nonlinear elliptic equations

TL;DR

This paper develops a modified series-based method to approximate solutions of ill-posed linear and semi-linear elliptic equations, providing error estimates and demonstrating effectiveness through numerical examples.

Contribution

It introduces a new modified approach for approximating solutions of ill-posed elliptic equations with error analysis and numerical validation.

Findings

The method effectively approximates solutions of elliptic equations.

Error estimates are established for various cases.

Numerical examples confirm the method's feasibility and efficiency.

Abstract

In this paper, we investigate the Cauchy problem for both linear and semi-linear elliptic equations. In general, the equations have the form \[ \frac{\partial^{2}}{\partial t^{2}}u\left(t\right)=\mathcal{A}u\left(t\right)+f\left(t,u\left(t\right)\right),\quad t\in\left[0,T\right], \] where $\mathcal{A}$ is a positive-definite, self-adjoint operator with compact inverse. As we know, these problems are well-known to be ill-posed. On account of the orthonormal eigenbasis and the corresponding eigenvalues related to the operator, the method of separation of variables is used to show the solution in series representation. Thereby, we propose a modified method and show error estimations in many accepted cases. For illustration, two numerical examples, a modified Helmholtz equation and an elliptic sine-Gordon equation, are constructed to demonstrate the feasibility and efficiency of the proposed method.