The theme of the thesis can be divided to two major aspects. The first is to classify and to describe methods of solving integral equations, including approximate and numerical methods. Also, some integral transforms are discussed and applied to solve integral and differential equations. Some examples are given, and solved numerically, approximately or exactly, to illustrate the procedures of the considered methods. The second aspect of the thesis deals with finding the speed of the seismic waves in the layers of the ground. This problem is formulated as an inverse mathematical problem involving a non - 1i near integral equation of a certain type. In fact, two problems are formulated, and solved according to whether the seismic sensors are located on a vertical or horizontal line . For each considered case a computer program is written and implemented the results are given in tabular or graphical forms with comparison with the exact solution or by computing thee error resulting from substituting the approximate solution in t he integral equation .