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Abstract
In this thesis, we use the Trotter-Kato theorem for approximating the solutions of some abstract differential equations. The finite right Caputo derivative will be introduced and some of its properties will be given. The finite left and right Caputo derivatives lead to introduce a finite Riesz fractional derivative in the Caputo sense, which could be considered as the fractional power of the Laplacian operator in finite domains. By the finite Riesz derivative, we can modell the dynamics of many anomalous phenomena in super-diffusive processes and also we can present a new diffusion-wave equation. A new version of the Adomian decomposition method will be introduced to solve a certain partial differential equations of fractal orders in finite domains, the generalized Adomian decomposition method has been successively used to find the explicit and numerical solutions of the time/space fractional partial differential equations. A different examples of special interest with fractional time and space derivatives of order α > 0 are considered and solved by means of the generalized Adomian decomposition method. The behavior of Adomian solutions and the effects of different values of α are shown graphically for some examples.