الفهرس 
IntroductionOnly 14 pages are availabe for public view
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Abstract
Many phenomena in science and engineering are represented by fractional differential equations like optical fibers, nuclear physics, ecology, fluid dynamics, biology, solid state of physics and quantum field theory. The fractional partial differential equations (FPDEs) does not depend only on the instant time, but also on the history of the previous time. In other words, previous values of the solution and the derivatives in fractional order differential equations are required to obtain a solution at a particular instance. The memory effect of the convolution in the fractional integral gives the equation increased expressive power. In this thesis, we will present a historical overview of the fractional calculus, followed by basic definitions of fractional calculus such as the definition of Riemann and the definition of Caputo. In light of the q-homotopy analysis transform method (q-HATM), a powerful algorithm has been developed for the solution of linear and nonlinear partial differential equations (PDE) of fractional order with applications in engineering, finance, economics, fundamental science, and applied mathematics, that is a combination of homotopy analysis method and Laplace transform method. The q-HATM has been applied to solve some of fractional evolution equations which are widely used in many applied fields such as Kundu–Eckhaus equation, massive Thirring model, fourth order cubic nonlinear Schrodinger equation, and coupled nonlinear evolution equations. Many temporal and spatial patterns were produced in biochemistry which turned into mathematical problems through these phenomena through various mathematical models, especially reaction-diffusion systems in which some of the equations to which they belong were resolved such as Gray-Scott model, Glycolysis model, Chaffee–Infante equation. And applied to solve some fractional physical differential equations such as coupled Ramani equation and completely integrable single Drinfeld-Sokolov-Satsuma-Hirota equation of sixth order. The proposed algorithm offers approximate solutions that are very close to the exact solution while avoiding the complexity that exists in many other methods. The natural frequency of the fractional of the solution varies with the change of the fractional derivative. Because of the presence of parameter ~, which controls the area of convergence and improves the accuracy of the method, this method is one of the best all other approximation methods. The obtained series solution is comparative with the exact solution by the correlation coefficient.