Numerical Processing of Nonlinear Fredholm Integro-Differential Equations by Applying Projection Techniques

The main focus of this thesis is to present a numerical study of Fredholm integral equations of the nonlinear integro-differential type. This includes examining both the regular and weakly singular cases, as well as the fractional case. To obtain numerical solutions for these equations, we use popular projection methods like the Galerkin method and the collocation method, along with classical orthogonal polynomials. The primary benefit of this approach is that it allows us to transform the main equations for each case into a nonlinear algebraic system. We can then use iterative methods to solve these systems efficiently. To show the accuracy and effectiveness of our approach, we present several numerical examples throughout the thesis. These examples demonstrate how our numerical process accurately solve the given equations, which further confirms the effectiveness of our proposed method. Keywords: Orthogonal polynomials. Fredholm integro-differential equations. Galerkim method. Collocation technique. Nonlinear equation