Numerical solutions of linear fredholm integro-differential equations of the second kind using quadrature-difference methods - Jalius, Chriscella (2016)
Fredholm integro-differential equation (FIDE) is an equation which is the unknown functions appears under the sign of derivative and also integral sign. Therefore, the formulation of numerical quadrature rules and finite difference method are applied for solving first-order and second-order linear FIDE of the second kind. The finite difference method is used for ordinary differential equations part, while composite quadrature rules are applied for the integral part of FIDE. Numerical solutions of linear FIDE by using quadrature-difference methods are proposed in this thesis. There are four types of formulation proposed in this thesis which are composite Simpsons 3/8 rule with first derivative of 5-point finite difference, composite Simpsons 3/8 rule with second derivative of 5-point finite difference, composite Booles rule with first derivative of 7-point finite difference and composite Booles rule with second derivative of 7-point finite difference. These formulations will be used to produce an approximation equations in order to discretize the FIDE into a system of linear algebraic equation. The system of linear algebraic equation will be solved by using Gauss elimination method. An algorithm and a coding of the proposed methods are developed in this thesis. The source of the coding for solving linear FIDE is developed by using C programming with constant step size. The four types of formulation which based on quadrature rules and finite difference method are implemented for solving Type 1 and Type 2 of first-order and secondorder linear FIDE. In this thesis, the boundary condition will be considered in solving the second-order linear FIDE. Moreover, the order of accuracy of the proposed method are studied in this thesis. Finally, the numerical experiments were carried out in order to examine the accuracy of the proposed method. The results indicated that the proposed methods are suitable for solving first-order and second-order linear FIDE of the second kind.