Please use this identifier to cite or link to this item: http://prr.hec.gov.pk/jspui/handle/123456789/12946
Title: Existence Theory and Numerical Solutions of Boundary Value Problems for Fractional Differential Equations
Authors: Ali, Amjad
Keywords: Mathematics
Issue Date: 2017
Publisher: University of Malakand, Malakand
Abstract: In this dissertation, we investigate different types of boundary value problems of nonlinear fractional order differential equations. The concerned research is associated to the existence and uniqueness of solutions, Hyers–Ulam type stability and numerical analysis for fractional order differential equations. We develop sufficient conditions for existence and uniqueness of solutions for fractional differential equations, with the help of classical fixed point theory of Laray Schauder type, Banach contraction type and Topological degree method. Further, we investigate the conditions for stability analysis of fractional differential equations. One of the important area of fraction differential equation is known as hybrid fractional differential equation. Hybrid fractional differential equations has an efficient techniques used for modeling various dynamical phenomenon. Therefore, for investigation sufficient conditions for existence and uniqueness of solutions hybrid fractional differential equations, we have used Hybrid fixed point theory established by Dhage and develop sufficient conditions for the existence and uniqueness of solutions of hybrid fractional differential equations. On the other hand, in most cases the nonlinear fractional differential equations are very complicated to obtained an exact analytic solution. Although, if an exact solution is possible, that needed very complicated calculations. Therefore, we paid a strong attention to the numerical solution of fractional differential equations and fractional partial differential equations. We have developed some powerful and an efficient numerical techniques for the approximate solutions of both linear and nonlinear fractional order differential equations. The established technique are based on Laplace transform coupled with Adomain polynomials to obtain the aforesaid solutions in the form of convergent series. Further, we also develop another interesting and useful method based on operational matrices obtained via using Lagendre polynomials. With the help of these mentioned techniques, we solve both linear and nonlinear ordinary as well as partial fractional order differential equations. We consider some fractional order differential equations for illustrative purposes and numerical approximations of their solutions are obtained using MAPLE and MATLAB. The numerical results obtained via aforesaid techniques, are compared with other standard techniques. Which shows, that how the Laplace transform coupled with Adomain polynomials and operational matrices obtained by Legendre polynomials are more effective and reliable, than the standard ordinary differential equations solvers.
Gov't Doc #: 16932
URI: http://prr.hec.gov.pk/jspui/handle/123456789/12946
Appears in Collections:PhD Thesis of All Public / Private Sector Universities / DAIs.

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