A Robust Septic Hermite Collocation technique for Dirichlet Boundary Condition Heat Conduction Equation

TL;DR

This paper introduces a new Septic Hermite Collocation Method for solving 1D heat conduction equations with Dirichlet boundary conditions, demonstrating its accuracy and effectiveness through numerical comparisons.

Contribution

The paper presents a novel Septic Hermite Collocation Method using Chebyschev and Legendre polynomial roots for the first time in heat conduction problems.

Findings

The scheme provides highly accurate numerical solutions.

It is effective for both linear and nonlinear heat conduction problems.

The method outperforms some existing techniques in accuracy.

Abstract

In the present manuscript, approximate solution for 1D heat conduction equation will be sought with the Septic Hermite Collocation Method (SHCM). To achieve this goal, by means of the roots of both Chebyschev and Legendre polinomials used at the inner collocation points, the pseudo code of this method is found out and applied using Matlab, one of the widely used symbolic programming platforms. Furthermore, to illustrate the accuracy and effectiveness of this newly presented scheme, a comparison among analytical and numerical values is investigated. It has been illustrated that this scheme is both accurate and effective one and at the same time can be utilized in a successful way for finding out numerical solutions of several problems both linear and nonlinear.