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This thesis is concerned primarily with spline functions and their application to solving a Fredholm integral equation of the second kind with non-degenerate kernel. Some mathematical preliminares are introduced at the begining. The problem of piecewise linear and cubic interpolation is discussed in one and two dimensions. Many theorens concerning the rates of convergence of these interpolation formulae are given. Bicubic spline interpolation is introduced and the relevant convergence theorems are proved. A method of solution of the Fredholm integral equation of the second kind with degenerate kernel is described. By approximating a non-degenerate kernel by means of a bicubic approximation (which, if rewritten in a basis function form, is degenerate) leads to an approximate method of obtaining a solution to the Fredholm second kind equation. Convergence results are proved. Numerical results are presented verifying the convergence theorems. Finally there is an appendix with graphs of the basis functions of the cubic splines in one dimension