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Title: On the Solution of Certain Fractional Order Partial Differential and Integro-Differential Equations
Authors: Asma Ali Abolgasem Elbeleze
Keywords: variational iteration
Issue Date: Apr-2015
Publisher: Universiti Sains Islam Malaysia
Abstract: Fractional partial differential equation (FPDEs) and integro-differential equations of integer and fractional order (IDEs&FIDEs) appear more and more frequently in various areas and engineering applications.There is a growing need to find the solution of these equations. However,most these equation are difficult or impossible to solve analytically. As a consequence,an effective and easy-to-use numerical and approximate methods are needed.Over the last decades several analytical/ approximate methods have been developed to solve FPDEs and FIDEs. The main objective of this thesis is to solve FPDEs,IDEs and FIDEs, where the homotopy perturbation method (HPM) and the variational iteration method (VIM) were used to solve them.In case,when we applied HPM to FPDEs, the existence of unique solution was proved, convergence analysis of HPM applied to these type of equations was discussed.Maximum absolute truncated error of HPM series solution was estimated.Modified homotopy perturbation method (MHPM) is an efficient method for calculating analytical approximate solutions for fractional linear and nonlinear partial differential equatiions where numerical results illustrate that the method was capable of reducing the volume of computational work as compared to standard homotopy perturbation method. A Comparative study between MHPM solutions with exact solution and solutions obtained by using other methods such as Laplace homotopy perturbation method was conducted. The homotopy perturbation method coupled with Sumudu transform (HPSTM) was successfully applied to get the approximate analytical solution of fractional partial differential equations. Further,the fractional variational iteration method (FVIM) used to solve fractional partial differential equations.On the other hand, the variable-Coefficient Singular partial differential equations of fractional order (SFPDEs) were solved by using homotopy perturbation method and variational iteration method, the methods provided the solutions in terms of convergent series with easily computable component even with the presence of singularities.The existence of unique solution was proved, convergence analysis of HPM applied to these types of equations was discussed,maximum absolute truncated error of HPM series solution was estimated.Some applications of these equations which will be solved by HPM and VIM were Black-Sholes European option pricing equations, fractional Klein-Gordon equations,fractional Burgers equation,linear inhomogeneous fractional wave equation, singular linear vibration equation of fractional order.On the other hand,new modified homotopy perturbation method (NMHPM) which is based on HPM and an improved version it used to solve IDEs and the exact solution was obtained just in two steps.Further, homotopy perturbation method and variational iteration method were applied for solving FIDEs. The fractional derivatives were used in this thesis Caputo and Riemann-Loiuville senses.Numerical results showed that the two approaches are easy to implement and accurate when applied to FPDEs,IDEs and FIDEs. Comparison achieved a very good approximation with the actual solution of all tested model problems using the both methods.Also, we emphasise that these methods are applicable for many other types of FPDEs,IDEs and FIDEs.
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