Thesis

The nonlocal conditions for the boundary or initial value problems appear when values of the function on the boundary or on the initial are connected to values inside the domain. Such problems are known as
nonlocal problem.The aim of this work is to study some types of nonlocal problems. This study includes the following aspects:

 The main objective of this thesis is divided in to three directions which are:
The first one is to study and overview the main and basic concepts of stochastic calculus, as well as, studying stochastic ordinary differential equations. The second objective is to study explicit stochastic Runge-Kutta methods, then generalize this scheme for semi-explicit, implicit and mixed schemes and study theirs numerical stability. The third objective is to introduce variable step size method for solving

 This work concerns with the nonlocal problems and the main theme of it can be divide into three categories, which can be listed as follows:
First: Some analytical methods namely, the separation of  variables and the eigenfunction expansion method to solve some types of linear  partial differential equations with nonlocal conditions, are presented.   

         The use of fractional orders differential and integral operators in mathematical models has been increasingly widespread in recent years. The non-linear multi-term fractional (arbitrary) order differential equation has been considered. Its solution existences and uniqueness are proved by transform it into a linear system of equations.