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Abstract
In this MSc thesis we will handle important problems appeared in many phenomena in different disciplines, Engineering, Viscoelasticity, Thermoelasticity, physics, chemistry, economics and other fields. Namely, Space fractional advection-dispersion equation. Space-time fractional advection-dispersion equations. Multi-term fractional-differential equations. The thesis is organized as follows: Chapter one: In this chapter, we give basic definitions in fractional calculus (Riemann-liouville and caputo’s) and brief introduction to the spectral methods and orthogonal polynomials. Chapter two: In this chapter, we give two numerical algorithms for solving two kinds of space fractional linear advection-dispersion problems. The proposed numerical solutions are spectral and they are buile on assuming the approximate solutions as certain double shifted tchebyshev basis. We study the convergence and error analysis of the double shifted tchebyshev basic are carefully investigated aiming to illustrate the correctness and feasibility of the proposed double expansion