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Title of Thesis

Numerical Solution Of Boundary Value And Initial-Boundary-value Problems Using Spline Functions

Author(s)

Fazal-i-haq

Institute/University/Department Details
Faculty of Engineering Sciences / GIK Institute of Engineering Sciences and Technology, Topi
Session
2009
Subject
Engineering Sciences
Number of Pages
159
Keywords (Extracted from title, table of contents and abstract of thesis)
Quartic, Quintic, Error, Solution, Numerical, Propagation, Nonlinear, approximations, Boundary, Initial, Spline, Benchmark, Functions, Problems

Abstract
The following two types of problems in differential equations are investigated:
(i) Second and sixth-order linear and nonlinear boundary-value problems inordinary differential equations using non-polynomial spline functions.
(ii) One dimensional nonlinear Initial-boundary-value problems in partialdifferential equations using B-spline collocation method.
Polynomial splines, non-polynomial splines and B-splines are introduced. Somewell known results and preliminary discussion about convergence analysis of boundary-value problems and stability theory are described.Quartic non-polynomial spline functions are used to develop numerical methods for computing approximations to the solution of linear, nonlinear and system of secondorderboundary-value problems and singularly perturbed boundary-value problems.Convergence analysis of the method is discussed.Numerical methods for computing approximations to the solution of linear andnonlinear sixth-order boundary-value problems with two-point boundary conditions are developed using septic non-polynomial splines. Second-, Fourth- and Sixth-orderconvergence is obtained.Numerical method based on collocation method using quartic B-spline functions forthe numerical solution of one-dimensional modified equal width (MEW) wave equationis developed. The scheme is shown to be unconditionally stable using Von-Neumannapproach. Propagation of a single wave, interaction of two waves and Maxwellian initial condition are discussed.
Algorithms based on quartic and Quintic B-spline collocation methods are designed or the numerical solution of the modified regularized long wave (MRLW) equation.Stability analysis is performed. Propagation of a solitary wave, interaction of multiplesolitary waves, and generation of train of solitary waves are also investigated.                                                                                                                      Quartic and quintic B-spline functions have been used to develop collocationmethods for the numerical solution of Kuramoto-Sivashinsky (KS) equation. Also,using splitting technique, the equation is reduced to a problem of second order in space.
Using error norms L2 and L∞ and conservative properties of mass, momentum andenergy, accuracy and efficiency of the suggested methods is established through comparison with the existing numerical techniques. Performance of the algorithms is tested through application of the methods on benchmark problems.

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

 

vii
103 KB
2

1

INTRODUCTION

1.1 Historical note and literature survey
1.2 Existence and uniqueness of two-point boundary-value problems
1.3 Polynomial and non-polynomial spline functions
1.4 B-Splines
1.5 Von Neumann stability analysis
1.6 Convergence of two-point boundary-value problems
1.7 Newton’s method
1.8 Summary

1
168 KB
3 2 NON-POLYNOMIAL SPLINES METHODS FOR THE SOLUTION OF SECOND-ORDER BOUNDARY-VALUE PROBLEMS

2.1 Introduction
2.2 Numerical Methods
2.2 Fuzzy Time series
2.3 Convergence Analysis
2.4 Numerical results and discussion
2.5 Conclusion

17
344 KB
4 3 NON-POLYNOMIAL SPLINES APPROACH TO THE SOLUTION OF SIXTH-ORDER BOUNDARY-VALUE PROBLEMS

3.1 Introduction
3.2 Numerical Methods
3.3 Numerical methods of different orders
3.4 Properties of the coefficient matrix A0
3.5 Convergence
3.6 Numerical results and discussion
3.7 Conclusion

52
214 KB
5 4 COLLOCATION METHOD USING QUARTIC B-SPLINE FOR NUMERICAL SOLUTION OF THE MODIFIED EQUAL WIDTH EQUATION

4.1 Introduction
4.2 Quartic B-spline solution
4.3 Stability analysis
4.4 Test problems and discussion
4.5 FFBPNN
4.5 Conclusion

77
171 KB
6 5 SOLITARY WAVE SOLUTIONS OF THE MODIFIED REGULARIZED LONG WAVE EQUATION

5.1 Introduction
5.2 The B-spline collocation methods
5.3 Stability of the proposed scheme
5.4 Numerical Tests and Results
5.5 Conclusion

96
184 KB
7 6 COLLOCATION METHOD USING B-SPLINES FOR NUMERICAL SOLUTION OF KURAMOTO-SIVASHINSKY EQUATION

6.1 Introduction
6.2 The B-spline collocation methods
6.3 Numerical validation
6.4 Conclusion

123
369 KB
8 7 REFERENCES

140


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