### Abstract:

This research is a simplified concept of the technique of solving ordinary differential equations as integral equations. No one can deny the effective and distinguished role performed by the integral equations in the fields of mechanics and physics where many of the threshold values in mathematical physics are returned to integral equations. The objective of this research is to device a unified method for solving ordinary linear and linear differential equations from the higher ranks. Moreover, integral equations are considered an important tool in many branches of analysis like functional analysis and random evloutionaries. This subject is considered to be a principal discipline in old and modern mathematics and also considered as an indispensible strong support for the specialists in the fields in physics, chemistry, mechanics and mathematics. In the following, we will confine all the study to the second type of the integral equations. Fredholm integral equation of the second type will have the greatest share of this study. The reason for this is that at most a person encounters this kind of integral equations when finding out solutions for threshold values problems in integral equations and mathematical physics. Then the theory of integral equations of the second type are to a larger easier than theory of integral equations of the first type. The method adopted in this research is the method of Coloumb team for solving linear and non-linear equations and some repetitive methods. Given the importance characterizing the subject of solving the integral equations, the concept of an integral equation was defined. Also some types of special equations and the methods of their solution were explored in the first and second chapters. How this equation will be solved by the core status and deteriorating core was explained. The results arrived by the study is the application of this research where in it the problems of threshold and primary values are solved for ordinary differential equations after transforming them to integral equations. Also the concept of Dirac delta and Green's function with some examples were shown. the study recommends the device of a technique or a transform to solve the partial differential equations as integral equations.

### Description:

A Dissertation Submitted to the University of Gezira in Partial Fulfillment of theRequirements for the Award of the Degree of Master of Science in Mathematics Department of Mathematics ,Faculty of Mathematical and Computer Sciences January, 2014