Study of Some Analytical Techniques, Based on Transformations with Decompostion Method to Solve Fractional Differential Equations

Abstract

Many physical phenomena that have been modeled by using fractional partial differential e quations (FPDEs) have a significant role to model different physical phenomena in various areas of mathematics and applied sciences, are needed to solve numerically as well as ana lytically. Many physical phenomena such as fractional-order Zakharov Kuznetsov equation, Helmholtz equation, the system of KdV equations, Swift-Hohenberg equations, Diffusion e quation, Heat equation, Wave equation, parabolic equation, Kuramoto-Sivashinsky equations, third order Kortewege-De Vries equations, the system of FPDEs, Whitham-Kaup equations and time-fractional multi-dimensional Navier-Stokes equations have a significant role in different area of physical engineering and science. It is a tedious task for researchers to determine the ap proximate and analytical solutions of the above mentioned there fractional-order mathematical models. However, many of them are successful to solve certain FPDEs. In the current thesis, certain analytical techniques are extended to solve some important mathematical models such as In chapter 2, the generalized mathematical models of various fractional-order differential equa tions are solved by using different analytical techniques such as the Laplace Adomian decom position method, Natural decomposition method, q-Homotopy analysis method, Laplace varia tional iteration method and Mohand decomposition method. The generalized schemes provide a sophisticated procedure to find the solution of any particular FPDEs. In chapter 3, an analytical investigation of Zakharov-Kuznetsov equation of fractional-order, Helmholtz equation, Swift-Hohenberg equations and system of KdV equations are investigated by using Laplace Adomian decomposition method (LADM). In chapter 4, the certain analytical solution of important physical models such as Diffusion, Heat, Wave and parabolic equations of fractional-order are investigated with the help of the natural decomposition method (NDM). The present technique is the mixture of natural transform (NT) with Adomian decomposition method (ADM). Chapter 5, is related to the analytical solution of Kuramoto-Sivashinsky equations us ing Laplace variational iteration method. Chapter 6, is consist of fractional-view investigation of Kortewege-De Vries equations and scheme of PDEs by Mohand decomposition method (MD M). Similarly, in chapter 7 and 8 the solutions of Whitham-Kaup equations and Navier-Stokes equa tions of fractional-orders are investigated by using natural decomposition method, q-homotopy analysis transform method and Laplace variational iterative method. In all of the above techniques, we have used the ADM, variational iteration method (VIM) and iv q-Homotopy analysis method (q-HAM) along with different transformations such as Laplace transformation, Natural transformation and Mohand transformation to developed some new ef ficient hybrid techniques. The convergent series form solutions are obtained by using the pro posed techniques. It is observed that these series form solutions have convergence rate towards the actual solution of the problems. The solutions graphs are plotted to show the closed contact of the obtained results and actual results of the targeted models. The various fractional-order solutions (FOS) are calculated for each problem. It is observed that FOS have the higher rate of convergence to integer-order results. The strong contact between the obtained and actual solu tions are analyzed. Moreover, due to simple and straightforward implementation the proposed techniques can be extended for the solution of other high non-linear fractional partial different equations