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Computational and Applied Mathematics Journal

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Second Order Iterative Techniques for Boundary Value Problems and Fredholm Integral Equations

Computational and Applied Mathematics Journal
Vol.3 , No. 3, Publication Date: Aug. 25, 2017, Page: 13-21

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Authors

[1]	I. K. Youssef, Department of Mathematics, Ain Shams University, Cairo, Egypt.
[2]	R. A. Ibrahim, Department of Mathematics and Engineering Physics, Faculty of Engineering_Shoubra, Benha University, Cairo, Egypt.

Abstract

Comparison between the performance of the second and first degree stationary iterative techniques is performed on two representations to two point boundary value problems of the second and fourth orders. The numerical treatment of the Fredholm integral equation representation for the second and fourth order boundary value problems (BVP) has illustrated the effective use of their integral representation. The finite difference method with the same accuracy is employed to construct linear systems from the differential or the equivalent Fredholm form. Second degree linear stationary iterative methods are extensively used. The second degree Gauss-siedle (S. G. S) method is presented, by measuring the asymptotic rates of convergence of the sequence; we are able to determine when the Gauss-Seidle second-degree iterative method is superior to its corresponding first-degree one. Two numerical examples are considered, one of them is of the second degree BVP and the other is fourth order BVP. All calculations are done with the help of computer algebra system (MATHEMATICA 10.2).

Keywords

BVP, Fredholm Integral Equations, Second Degree Iterative Methods, Rate of Convergence, Finite Difference Method, Gauss-Siedle Method

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