Use residual correction method to calculate the upper and lower solutions of initial value problem

doi:10.1016/j.amc.2006.01.002

Applied Mathematics and Computation

Volume 181, Issue 1, 1 October 2006, Pages 29-39

https://doi.org/10.1016/j.amc.2006.01.002 Get rights and content

Abstract

This article attempts to establish solutions’ monotonic relation with residuals of initial value problem through the concept of maximum principle for differential equations, then discretizes the equation by the simple spline approximation for initial value problem, converts the once complex inequation constraint mathematical programming problem into a simple problem of equation iteration by applying the new “residual correction method for initial value problem” put forth by this article, and obtains the upper and lower solutions of initial value problem. Results from numerical validation indicate this method is accurate, simple and fast, the approximate solutions obtained can correctly analyze error range and the new method put forth hereby helps increase the accuracy of mean approximate solutions.

Introduction

This article aims to find a faster way to calculate the upper and lower solutions of initial value problem. Although the theory of maximum principle for differential equation problems [1], [2] has long been used to estimate the error range between approximate solution and exact solution of a differential equation, as the relation of residuals with approximate solutions will have to be converted into a mathematical programming problem under constraint conditions of an inequation and the calculation and procedures are rather complex, such method has failed to be applied to complex problems. As far as I know, recently, only Chang and Lee [3] used genetic algorithms method to study single pendulum problems.

In order to obtain the upper and lower solutions of initial value problem, the article tries a new “residual correction method for initial value problem”. Its concrete steps are: firstly use the obtained initial value spline approximation expression to discretize differential equation, and then use iteration technique for residual correction described in this article to covert the once complex inequation constraint problem into a simple problem of equation iteration. Therefore, the method can quickly solve mathematical programming problems under inequation constraint conditions.

Section snippets

The maximum principle

In general, in order to obtain the upper and lower solutions of a differential equation, a monotonic relation (if exist) of the residual function (also known as residual) of a differential equation with solution must be established by using the concept of differential equation maximum principle. Therefore, before introducing residual correction method, it is necessary to say something about maximum principle. Firstly, the maximum value for the first order initial value problem can be simply

Result and discussion

In this section, examples are given to explain how to apply residual correction method to the first and second order initial value problems. While in order to research how big the error of approximate solutions is, we first define mean approximate solution $\bar{u}$ , maximum possible error E_max and mean approximate error E_a are $\bar{u} (t) = \frac{\overset{⌢}{u} + \overset{⌣}{u}}{2},$ $E_{\max} (t) = \frac{\overset{⌢}{u} - \overset{⌣}{u}}{Min (| \overset{⌢}{u} |, | \overset{⌣}{u} |)},$ $E_{a} (t) = \frac{\bar{u} - u}{u},$ where u(t), $\overset{⌣}{u} (t)$ and $\overset{⌢}{u} (t)$ are the exact solution, lower and upper approximate solutions respectively. $\bar{u} (t)$ is the average

Conclusions

Through validation of two examples of this article, we can find the residual correction method put forth by this article can quickly correct the residuals of approximate solutions on initial value problem of a differential equation and satisfy the condition that corrected residuals are all greater than or smaller than zero. Under the support of monotonicity of a differential equation, the upper and lower approximation solutions of an initial value problem can always been obtained correctly. As

Acknowledgements

Thanks for the subsidy of the Outlay 93-2212-E-432-001 given by National Science Council, the Republic of China, to help us finish this special research successfully.

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A numerical solution of the Burgers’ equation using cubic B-splines
Applied Mathematics and Computation
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There are more references available in the full text version of this article.

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Use residual correction method to calculate the upper and lower solutions of initial value problem

Abstract

Introduction

Section snippets

The maximum principle

Result and discussion

Conclusions

Acknowledgements

Applied Mathematics and Computation

International Journal of Heat and Mass Transfer

Applied Mathematics and Computation