Thesis (h.D.) - Cairo University - Faculty of Science - Department of Mathematics

Stochastic partial differential equations become very powerful extensions of deter- ministic partial differential equations. Under appropriate assumptions, the existence and uniqueness of solutions to such equations are ensured. However, in many cases, these solutions are not given explicitly, therefore the numerical approximations are used to study the properties of these models. The theme of this thesis is numerical studies for solving certain partial differential equations involving uncertainties.These uncertainties arise in the form of forcing terms or initial data. We convert the underlying stochastic partial differential equations into stochastic ordinary differential equations by discretizing the spatial in spatial direction. We then solving the new resultant stochastic ordinary differential equations using dif- ferent scenarios.That scenarios are Euler-Maryuma procedure, method of lines and high-order ordinary differential equations solver. To evaluate uncertainty in the calcu- lations, we estimate the expectation by using repeated generators of random numbers. We then obtaining different sample paths, and considering the expectation. When the number of samples grows, the predicted estimated value converges to the true estimate.The idea behind this study is: to enhanced the performance of the deterministic numerical methods when it is used on stochastic models, analyze similarities between the exact solution and numerical approximations for different sample paths and accel- erate the convergence rate of the solutions.The main contributions in this thesis, we obtained more accurate numerical solutions for more general models, captured real physical phenomena not easily obtained in deterministic cases and we accelerated the convergence rate. The thesis handles the space fractional version of the Stochastic Advection-Diffusion Equation using the stochastic compact {uFB01}nite difference scheme

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