الفهرس 
Riemann-Liouville fractional integralOnly 14 pages are availabe for public view
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Abstract
Problems involving the physical phenomena of time-fractional advection dispersion equation are formulated in the form of partial differential and integral equation of fractional order.
The present thesis is mainly concerned with developing the analytical methods used and applied to solve such type of equations. Also, The first Stokes` problem and Rayleigh-Stokes equations and energy equation have been treated and solved .
A time fractional partial differential equation is considered, where the fractional derivative is defined in the Caputo sense. Laplace transform along with an intermediate step of Mellin transform have been applied to achieve an exact solution in terms of H-function and the complement error function. A number of special cases are also considered.
Exact solution of the time-fractional advection-dispersion equation with reaction term has been obtained. The solution is achieved by using Fourier and Laplace transforms to get the formulas of the fundamental solution, which are expressed explicitly in terms of Fox’s H-function by making use of the relationship between Fourier and Mellin transforms.
Fourier sine transform and Laplace integral transform are used for solving the Stokes` first problem and the Rayleigh-Stokes problem for a generalized second grade fluid with fractional derivative. Exact solutions for both the velocity and temperature have been achieved. The solutions of the classical problem for both stokes` first problem and Rayleigh- Stokes problem have been obtained as limiting cases.
The fundamental solution of the fractional diffusion equation of distributed order in time is obtained based on its Mellin-Barnes integral representation. Such solution is proved to be related via a Laplace transform to the Fox-Wright functions. The results of the time fractional diffusion equation of a single order are also recalled and then re-obtained from the general theory. Also some results concerning the solution of fractional advection-dispersion equation of distributed order are obtained.
Key Words: Fractional derivatives, Laplace transform, Fourier transform, Mellin transform, Fox’s H-function , advection dispersion equation, Rayeil Stokes equations.