Generalization of Separation of Variables n-Harmonic Equation m Dimension and Unbounded Boundary Value Problem

TL;DR

This paper extends the separation of variables method to n-harmonic equations in multiple dimensions, addressing complexities in higher-order PDEs and unbounded boundary conditions, with applications of convolution.

Contribution

It generalizes the separation of variables technique using multinomial theorem for n-harmonic equations in m dimensions and unbounded boundary value problems.

Findings

Successfully generalized the separation of variables for higher-order PDEs

Solved n-harmonic equations in multiple dimensions with unbounded boundaries

Applied convolution to enhance solution methods

Abstract

The method of separation of variables is significant, it has been applied to physics, engineering , chemistry and other fields. It allows to reduce the diffculity of problems by separating the variables from partial differential equation system into ordinary differential equations system. However, this method has complexity in higher order partial differential equations. In this reserach, we generalize this method by using multinomial theorem of n-harmonic equation to solve n-harmonic equation with $m$ dimension and then solving an important class of partial differential equations with unbounded boundary conditions. Additionaly, application of convolution.