Local Polynomial Regression Solution for Differential Equations with Initial and Boundary Values

Liyun Su,Fenglan Li,Tianshun Yan,Yanyong Zhao

doi:10.1155/2013/530932

Liyun Su, Fenglan Li + Show 2 more

Open Access

https://doi.org/10.1155/2013/530932

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Abstract

Numerical solutions of the linear differential boundary issues are obtained by using a local polynomial estimator method with kernel smoothing. To achieve this, a combination of a local polynomial-based method and its differential form has been used. The computed results with the use of this technique have been compared with the exact solution and other existing methods to show the required accuracy of it. The effectiveness of this method is verified by three illustrative examples. The presented method is seen to be a very reliable alternative method to some existing techniques for such realistic problems. Numerical results show that the solution of this method is more accurate than that of other methods.

Highlights

Linear ordinary differential equation boundary value problems are very common theoretical and practical issues
A local polynomial-based fitting method has been proposed for numerical solutions of differential equations with boundary values
Comparisons of the computed results with exact solutions and existing methods showed that the method has the capability of solving the differential equations with boundary values

Summary

Introduction

Linear ordinary differential equation boundary value problems are very common theoretical and practical issues. The shooting method is the most widely used Both single and multishooting methods: (1) select the initial value of needing to be solved; (2) use the differential equation initial value algorithm to compute step by step and get to the end of boundary value; (3) compute the error function between boundary value to compute and real boundary value readjust the initial value of needing to be solved until the error is up to a certain level of accuracy. The current method is of a general nature and can be used for solving some partial differential equations arising in various areas

Local Polynomial Estimator for Differential Equations

Local Polynomial Estimator

Example Simulation

Initial Value Problem of Homogeneous Linear Differential Equation

Conclusions