Some Methods for Solving Initial and Boundary Value Problems with Nonlocal Conditions

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:
(1) Discuss the existence and uniqueness of the solution with some nonlocal initial value problems for the non-linear ordinary differential equations via some types of fixed point theorems. Also some numerical methods are used to solve special types of nonlocal initial value problems for the non-linear ordinary differential equations.
(2) Give solutions for some types of the nonlocal initial and boundary value problems for linear eigenvalue problems of the ordinary differential equations.
(3) Use some numerical methods to solve the initial-boundary value problem that consists of the one-dimensional hyperbolic and parabolic equations with two nonlocal non-linear integral boundary conditions. These methods depend on Douglas’s equation and Crank-Niklson finite difference scheme, Taylor’s expansion and some quadrature rules say Simpson’s 1/3 rule.