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Title of Thesis

Mesh Free Collocation Method For Numerical Solution Of Initial-boundary-value Problems Using Radial Basis Functions

Author(s)

Arshed Ali

Institute/University/Department Details
Faculty of Engineering Sciences / GIK Institute of Engineering Sciences and Technology, Topi
Session
2009
Subject
Engineering Sciences
Number of Pages
169
Keywords (Extracted from title, table of contents and abstract of thesis)
Modelnonlinear, Equations, Mesh, Collocation, Initial, Problems, Regularized, Radial, Basis, Functions, Finite

Abstract
Nonlinear partial differential equations are often used to understand and modelnonlinear processes arising in many branches of science and engineering. For most of partial differential equations a general closed-form analytical solution is not available and therefore use of numerical methods always remains an important alternative for the
solution of partial differential equations. Several numerical methods are developed for the solution of partial differential equations including finite difference methods, finite
element methods, spectral methods and spline methods. However numerical methods posses some limitations such as mesh generation, slow rate of convergence, spatial
dependence, stability, low accuracy and difficult to implement in complex geometries.One of domain type methods is known as radial basis functions method, which is a truly meshless method, infinitely differentiable, numerically accurate, stable, very high rate of convergence, spatial independence and flexible with respect to complex
geometry. The main difference between the mesh free radial basis functions method and classical mesh-based methods is that the radial basis functions can be extended to
the entire domain of influence without diving into elements.
In this thesis, we present mesh free radial basis functions method based on collocation principle for numerical solution of various time dependent nonlinear partial differential
equations namely, Regularized Long Wave (RLW) equation, Modified Regularized Long Wave (MRLW) equation, Modified Equal Width Wave (MEW) equation, Klein-
Gordon Schrödinger (KGS) equations, Klein-Gordon Zakharov (KGZ) equations, Two dimensional Coupled Burgers’ equations and Two dimensional Reaction-Diffusion Brusselator equations. Different radial basis functions are used for this purpose. First order forward and second order central difference approximation is employed to the
time derivative. The elementary stability and convergence of the proposed method are discussed. Accuracy of the method is assessed in terms of various error norms, number of nodal points and time step size. Performance of the proposed method is validated through examples from literature. Apart from ease of implementation, better accuracy is obtained. Comparison with existing methods such as finite difference methods, finite element methods, boundary element methods and spline methods is made to show the superiority and simple applicability of the mesh free method.

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1,706 KB
S. No. Chapter Title of the Chapters Page Size (KB)
1 0 CONTENTS

 

vii
108 KB
2

1

INTRODUCTION

1.1 Literature survey and motivation
1.2 Radial Basis Functions
1.3 Matrix method for stability analysis
1.4 Unsymmetric collocation method
1.5 Limitations of the RBF method
1.6 Solitary wave and Soliton

1
234 KB
3 2 MESHFREE COLLOCATION METHOD FOR THE NUMERICAL SOLUTION OF THE REGULARIZED LONG WAVE EQUATION

2.1 Introduction
2.2 Construction of the RBF Methods
2.3 Stability Analysis
2.4 Numerical tests and results
2.5 Conclusion

15
660 KB
4 3 MODIFIED REGULARIZED LONG WAVE EQUATION

3.1 Introduction
3.2 The governing equation and construction of the RBF Method
3.3 Stability Analysis
3.4 Numerical tests and results
3.5 Conclusion

40
356 KB
5 4 MODIFIED EQUAL WIDTH WAVE EQUATION

4.1 Introduction
4.2 Construction of the proposed method
4.3 Stability analysis
4.4 Test problems and discussion
4.5 Conclusion

62
328 KB
6 5 COUPLED KLEIN-GORDON EQUATIONS

5.1 Introduction
5.2 Construction of the RBF method
5.3 Numerical Tests and results
5.4 Conclusion

86
339 KB
7 6 TWO DIMENSIONAL COUPLED BURGERS’ EQUATIONS

6.1 Introduction
6.2 Construction of the RBF method
6.3 Stability Analysis
6.4 Numerical Tests and discussion
6.4 Conclusion

109
545 KB
8 7 TWO DIMENSIONAL REACTION-DIFFUSION BRUSSELATOR SYSTEM

7.1 Introduction
7.2 Construction of the RBF method
7.3 Stability Analysis
7.4 Numerical Tests and discussion
7.5 Conclusion

133
382 KB
9 8 REFERENCES

152


142 KB