الفهرس 
Survey of the Main Mathematical Concepts
Solving Fokker-Planck Equation by New Iterative MethodsOnly 14 pages are availabe for public view
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Abstract
Preface-i-PrefaceThe subject of this thesis relates to a branch of mathematics callednumerical analysis. In this section we discuss how to use some numericalmethods to find analytical and approximate solutions to the problems oflinear and nonlinear differential equations with integer and fractionaldifferentiation.The non-linear ordinary and partial differential equations representedmany practical problems in fluid mechanics. The study of the boundary valueproblems flow saturated porous media has received considerable interest.In general, the nonlinear partial differential equations (NPDEs) havemodeled Nonlinear complex phenomena in various scientific fields. Theinvestigation of analytical, approximate and exact solutions of NPDEs willhelp to be better understanding the complex phenomena.Also, the differential equations of fractional order, are the generalizedtype of the classical differential equations of integer order. Recently, thefractional differential equations have been the focus of many researchersbecause of their frequent appearance in many applications in viscoelasticity,physics, biology, engineering and fluid mechanics. Therefore, considerableattention has been given to the solutions of differential equations of fractionalorder, of fluid mechanics and physics interest. Most nonlinear differentialequations of fractional order do not have exact analytic solutions, sonumerical and approximation techniques, such as Jacobi collocation method[28], Galerkin method [29], He’s frequency-amplitude formulation andenergy balance methods [30], max-min and Hamiltonian methods [31, 32],variational iteration method [3, 33, 35, 36] and Adomian decompositionmethod [37-41] must be used. The Picard method [62] and local fractional-iinewiteration method [63] are relatively new techniques to provide analyticalapproximation to nonlinear problems and they are particularly valuable astools for researchers, because they provide immediate and visible symbolicterms of analytic solutions, as well as numerical approximate solutions tononlinear differential equations without linearization or discretization. In lastyears, the application of the proposed methods in extended for fractionaldifferential equations.This thesis consists of five chapters and a list of references.In chapter one of thesis, we review briefly the basic concepts anddefinitions for partial differential equations, fractional calculus, localfractional calculus Mittage Leffler function, sine function, cosine function,hyperbolic sine function and hyperbolic cosine function.In chapter two of thesis, we propose the new iterative method andintroduce the integral iterative method to solve linear and nonlinear Fokker-Planck equations and some similar equations. Fokker-Planck equation (FPE),first applied to investigate the brownian motion of particles, is now largelyemployed in various generalized forms, in physics, engineering biology, andchemistry. The results obtained by the two methods are compared with thoseobtained by both Adomian decomposition and variational iteration methods.Comparison shows that the two methods are more effective and convenient touse and overcome the difficulties arising in calculating Adomian polynomialsand Lagrange multipliers, which means that the considered methods cansimply and successfully be applied to a large class of problems.It should be noted that the results of this chapter has been published in” Mathematical Problems in Engineering Journal ”,[78].-iii-In chapter three of thesis, we use the Variational Iteration Methods(VIM) and Adomian Decomposition Method (ADM) methods to findanalytical solutions of the fractional Form of Unsteady AxisymmetricSqueezing Fluid Flow with Slip and No-Slip Boundaries.The importance of fluid flow through channels began since long timeago in various applications in the life specially in agriculture. Recently,squeezing flow has large attention of the researchers and scientists due to itswide range of applications in various fields like chemical, food industries andin bio-mechanics. Practical applications of squeezing flows in the mentionedfields are modeling of lubrication systems, compression and injectionmodeling and polymer processing. These flows are induced by applyingvertical velocities or normal stresses by means of a moving boundary, thatcan be frequently noted in many hydro-dynamical tools and machines.The first work in squeezing flows was laid down by Stefan [17] inwhich he developed an ad-hoc asymptotic solution of Newtonian fluids. Anexplicit solution of the squeeze flow, considering inertial terms, has done byThorp [18]. However, Gupta and Gupta showed that the solution in [18] failsto satisfy boundary conditions [19]. In [20] Ran et. al. used the homotopyanalysis method to obtain an explicit series solution for the squeezing flowbetween two infinite plates. Also, by this method in [21] Rashidi et. al.established an analytical solution to the unsteady squeezing flow of a secondgradefluid between two circular plates. Moreover, the processes of polymerextrusion are modeled by squeezing flow of viscous fluids in [22]. For moredetails see [23-27,53,54]The goal of this chapter, is to prepare and use these methods forsolving the fractional order form of an unsteady axisymmetric flow of-ivnonconducting,Newtonian fluid squeezed between two circular plates withslip and no-slip boundaries because of the importance of this problem and itswide applications in many subjects in fluid mechanic. In addition, thecomparisons between the results obtained by the proposed methods withthose obtained by some other methods are considered to confirm the powerand efficiency of these methods in solving this kind of problems.