The Modified Alternative Direction Iteration Method for Solving Partial Differential Equations with Application to Chronic wounds of Diabetes Patients

  Alternating Direction Implicit method (ADI) was first suggested by Peaceman and Rachford in the mid-50s of the last century for solving systemsb of algebraic equations in two dimension of  aces[peaceman and rachford,1955], which results from the finite difference discretization method for solving PDEs; [Peaceman and Rachford,1955]. From iterative method’s perspective, the ADI method may be considered as a special relaxation method, where a big system is simplified into a number of smaller subsystems, such that each of them may be solved efficiently and the solution of the whole system is then obtained
from the solutions of the sub-systems in an iterative method approach, [Al-Saif and Al-Kanani,2011]. The main theme of this thesis may be directed toward three objective:
 The first objective is to explain and clarify, in details, the alternative direction iteration method in a simple way for each type of differential equations, which is produced by rearranging the Crank-Nicholson formulation and discuss the stability, convergent and consistency of the solution  using normal time steps. we preposed a new alternative direction iteration method depending on another time step, a new formula derived to give alternative direction iteration method  more accurate. The second objective is to derive and study  the system associated with the infected equations of patients with diabetes then the effect of oxygen in the treatment of  infected wounds,  the obtained system of related equations was  of the first dimension, also  the alternative direction iteration method could  be genarlized to solve the equations of  the second dimension. Then we went  to re-model the part of this system of equations of the second dimension as a prelude to solve. Finally, the third objective  is devoted to solve  a system
of equations that  re-modeled  by using the discussed alternative direction iteration method the results that are obtaind to  find out how the stability of the system output and the accuracy of the results affect on the stages of treatment.