Application of the Maximum Principle for Differential Equations Together with Fifth-degree B-spline in Deciding Error Boundaries of Fourth OrderObstacle Boundary Value Problems

殘差修正法求fourth order obstacle boundary value problems解的誤差分析

王啟昌(Chi-Chang Wang) ； 李宗乙(Zong-Yi Lee)

《興國學報》 11期 (2010/01) Pp. 1-21

https://doi.org/10.29950/JHKUM.201001.0001

摘要

本文利用微分方程最大值原理的觀念配合漸進疊代的技巧建立4階obstacle型邊界值問題的單調性關係以求得解的較大、較小近似解與誤差分析。數值求解以fifth-degree B-spline為基本函數來離散微分方程式後配合本文所提出的方法「殘差修正法」在離散的格點上加入殘差修正量，使得原本複雜的不等式拘束數學規劃（mathematical programming）問題得以轉換成簡單的等式疊代問題。數值結果發現不論在線性或非線性的問題皆可以得到近似解的誤差邊界，此外本方法在修正過程中具有將較大與較小近似解修正成大致對稱於正確解兩側的特性，因此發現即使在格點數相當少的情況下所得的平均近似解依然相當接近正確解。

關鍵字

最大值原理 ； B樣線 ； 殘值 ； 較大與較小近似解 ； 誤差邊界 ； obstacle問題

並列摘要

This paper deals with application of the maximum principle in combination with the asymptotic iteration method to seek upper and lower approximate solutions of fourth order obstacle boundary value problems. To obtain numerical solutions, the fifth-degree B-spline is utilized as a basic function to discretize differential equations, followed by adoption of the ”Residual Correction Method” proposed in this paper to add residual correction quantity in discretized grid points to convert constraint mathematical programming problems of inequalities that were complex originally into simpler equational iteration problems. Numerical results show that boundaries of errors between approximate solutions and exact solutions can be decided for both linear and non-linear problems. In addition, this approach is characterized by the advantage in correcting the acquired upper and lower approximate solutions to enable them to be distributed on two sides of the exact solutions roughly in a symmetrical way, thus making the mean approximate solutions remain considerably close to the exact solutions, even if the number of calculation grid points is rather small.

並列關鍵字

maximum principle ； B spline ； residual ； upper and lower solutions ； Error Boundaries ； obstacle problems

參考文獻

Al-Sadi, E. A.,Noor, M. A.(2002).Quartic spline method for solving fourth order obstacle boundary value problems.J. Comput. Appl. Math.143,107-116.

Al-Said, E. A., Noor, M. A.,Kaya, D.,Al-khalid, K.(2004).Finite difference method for solving fourth-order obstacle problems.Int. J. Comput. Math..81,741-748.

Al-Said, E. A.,Noor, M. A.(1995).Computational methods for fourth order obstacle boundary value problems.Comm. Appl. Nonlinear Anal..2(3),73-83.

Al-Said, E. A.,Noor, M. A.(1998).Numerical solutions of a system of fourth order boundary value problems.Internat. J. Comput. Math..71,347-355.

Al-Said,,Noor, M. A.,Rassias, T. M.(2006).Cubic splines method for solving fourth-order obstacle problems.Applied Mathematics and Computation.174,180-187.