# Fitted Reproducing Kernel Method for Solving a Class of Third-Order Periodic Boundary Value Problems

## TL;DR

This paper introduces a fitted reproducing kernel method within a Hilbert space framework to efficiently solve third-order periodic boundary value problems, providing accurate series solutions with proven convergence.

## Contribution

The paper develops a novel fitted reproducing kernel algorithm for third-order periodic boundary value problems, offering a new analytical-numerical solution approach with demonstrated effectiveness.

## Key findings

- The method produces uniformly convergent series solutions.
- Numerical examples confirm high accuracy and efficiency.
- The approach is applicable to various differential equations in physics and engineering.

## Abstract

In this article, the reproducing kernel Hilbert space [0, 1] is employed for solving a class of third-order periodic boundary value problem by using fitted reproducing kernel algorithm. The reproducing kernel function is built to get fast accurately and efficiently series solutions with easily computable coefficients throughout evolution the algorithm under constraint periodic conditions within required grid points. The analytic solution is formulated in a finite series form whilst the truncated series solution is given to converge uniformly to analytic solution. The reproducing kernel procedure is based upon generating orthonormal basis system over a compact dense interval in Sobolev space to construct a suitable analytical-numerical solution. Furthermore, experiments result of some numerical examples are presented to illustrate the good performance of the presented algorithm. The results indicate that the reproducing kernel procedure is powerful tool for solving other problems of ordinary and partial differential equations arising in physics, computer and engineering fields.

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Source: https://tomesphere.com/paper/1704.04837