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|Numerical and Analytical Solutions of Fractional-Order Differential Equations and System of Fractional-Order Differential Equations
|Abdul Wali Khan University, Mardan
|The fractional partial differential equations (FPDEs) are considered to be the modern tool in calculus for modeling various phenomena in applied sciences and engineering. In ordinary calculus it was difficult to model nonlinear real world phenomena in nature and therefore the subject of fractional calculus gained importance among the researchers. In this connection, the approximate and analytical solutions of FPDEs are of much importance to describe the accu rate dynamics of the important physical processes. In connection with the above statement, the mathematicians have developed and used many approximate and analytical techniques to deter mine the solutions of some important mathematical models having some real world problems representation. Although, it is very difficult to calculate the analytical and even the approximate solutions of certain nonlinear FPDEs and system of FPDEs but the mathematicians are still suc cessful to do their best in this regard. In the current research work, we extended the idea of how to apply the analytical techniques to get the solutions of certain fractional problems by using modified analytical techniques. The fractional derivatives are described in term of Caputo op erator. The generalized analytical schemes for various fractional-order problems are derived in very simple and straight forward procedures. For verification of the derived results, the analyti cal solutions of few illustrative examples are presented. The results of the suggested schemes are presented, using tables and graphs. It is observed that the results obtained are matching well with the actual solutions of the problems. The solution of each problem at fractional-order derivative is achieved and shown to be very closed to the solutions at integer-order of that problem. Many physical phenomena such as fractional-order convection-diffusion equations, acoustic waves equations, fractional order gas dynamics equations, telegraph model and fractional-order Burg er’s equation have a significant role in different area of physical engineering and science. It is a tedious task for researchers to calculate the approximate and analytical solutions of the above mentioned 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, the generalized mathematical models of various fractional-order differential equations are solved by using different analytical techniques with Mohand decomposition method, Variational iteration transform method, Variational Homotopy perturbation method, Elzaki decomposition method. The generalized schemes provide a sophis ticated procedure to find the solution of any particular FPDEs. In chapter 3, an analytical investigation of Convection-diffusion and Gas dynamics equation s by using Variational iteration transform method is studied. In chapter 4, certain analytical iv solution of important acoustic wave equations with the help of the Variational Homotopy per turbation method. In Chapter 5, fractional-order telegraph equations using Mohand decompo sition method are discussed. In Chapter 6, consist of fractional-view investigation of system of Burger’s equations by Homotopy perturbation transform method (HPTM). Chapter 7, consist of fractional-view investigation of Noyes-Fields models by Elzaki decomposition method (EDM). In all of the above techniques, we have used the Adomian decomposition method, Variational iteration method, Variational Homotopy perturbation method, Laplace transformation, Elzaki transformation and Mohand transformation to developed some new efficient hybrid techniques. The convergent series form solutions are obtained by using the proposed techniques. It is ob served 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 solutions are analyzed. Moreover, due to simple and straightforward implementation the proposed techniques can be extended for the solution of other higher non-linear fractional partial different equations.
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