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Title of Thesis

Boundary Value Problems For Fractional Differential Equations Existence Theory And Numerical Solutions

Author(s)

Mujeeb ur Rehman

Institute/University/Department Details
Centre For Advanced Mathematics And Physics / National University Of Sciences And Technology, Islamabad
Session
2011
Subject
Mathematics
Number of Pages
166
Keywords (Extracted from title, table of contents and abstract of thesis)
Solutions, Boundary, Theory, Value, Numerical, Problems, Existence, Fractional, Equations, Differential

Abstract
Fractional calculus can be considered as supper set of conventional calculus in the sense that it extends the concepts of integer order differentiation and integration to an arbitrary (real or complex) order. This thesis aims at existence theory and numerical solutions to fractional differential equations.Particular focus of interest are the boundary value problems for fractional order differential equations.This thesis begins with the introduction to some basic concepts, notations and definitions from fractional calculus, functional analysis and the theory of wavelets. Existence and uniqueness results are established for boundary value problems that include, two–point, three–point and multi–point problems.Sufficient conditions for the existence of positive solutions and multiple positive solutions to scalar and systems of fractional differential equations are established using the Guo–Krasnoselskii cone expansion and compression theorems.Owning to the increasing use of fractional differential equations in basic sciences and engineering, there exists strong motivation to develop efficient, reliable numerical methods.In this work wavelets are used to develop a numerical scheme for solution of the boundary value problems for fractional ordinary and partial differential equations. Some new operational matrices are developed and used to reduce the boundary value problems to system of algebraic equations.Matlab programmes are developed to compute the operational matrices.The simplicity and efficiency of the wavelet method is demonstrated by aid of several examples and comparisons are made between exact and numerical solutions.

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5,826 KB
S. No. ChapterTitle of the ChaptersPageSize (KB)
10CONTENTSiv
35 KB
21INTRODUCTION1
115 KB
32PRELIMINARIES

2.1 Some special functions
2.2 Fractional calculus
2.3 Fixed Point Theorems
2.4 Wavelets

7
863 KB
43EXISTENCE AND UNIQUENESS OF SOLUTIONS

3.1 Two point boundary value problems (I)
3.2 Two point boundary value problems (II)
3.3 Three point boundary value problems (I)
3.4 Three point boundary value problems (II)
3.5 Multi point boundary value problems
3.6 Boundary value problems with integral boundary conditions

31
282 KB
54EXISTENCE AND MULTIPLICITY OF POSITIVE SOLUTIONS

4.1 Positive solutions for three point boundary value problems (I)
4.2 Positive solutions to three point boundary value problems (II)

59
198 KB
65EXISTENCE AND MULTIPLICITY OF POSITIVE SOLUTIONS FOR SYSTEMS OF FRACTIONAL DIFFERENTIAL EQUATIONS

5.1 Positive solutions for a coupled system
5.2 Positive solutions to a system of fractional differential equations with three point boundary conditions

74
190 KB
76NUMERICAL SOLUTIONS TO FRACTIONAL DIFFERENTIAL EQUATIONS BY THE HAAR WAVELETS

6.1 Numerical solutions to fractional ordinary differential equations
6.2 Numerical solutions to fractional partial differential equations

93
1,830 KB
87NUMERICAL SOLUTIONS TO FRACTIONAL DIFFERENTIAL EQUATIONS BY THE LEGENDRE WAVELETS

7.1 An operational matrices of fractional order integration
7.2 Numerical solutions of fractional differential equations

125
2,238 KB
98CONCLUSIONS135
81 KB
109REFERENCES AND APPENDIX146
250 KB