Abstract
Problem statement: This research purposely brought up to solve complicated equations such as partial differential equations, integral equations, Integro-Differential Equations (IDE), stochastic equations and others. Many physical phenomena contain mathematical formulations such integro-differential equations which are arise in fluid dynamics, biological models and chemical kinetics. In fact, several formulations and numerical solutions of the linear Fredholm integro-differential equation of second order currently have been proposed. This study presented the numerical solution of the linear Fredholm integro-differential equation of second order discretized by using finite difference and trapezoidal methods. Approach: The linear Fredholm integro-differential equation of second order will be discretized by using finite difference and trapezoidal methods in order to derive an approximation equation. Later this approximation equation will be used to generate a dense linear system and solved by using the Generalized Minimal Residual (GMRES) method. Results: Several numerical experiments were conducted to examine the efficiency of GMRES method for solving linear system generated from the discretization of linear Fredholm integro-differential equation. For the comparison purpose, there are three parameters such as number of iterations, computational time and absolute error will be considered. Based on observation of numerical results, it can be seen that the number of iterations and computational time of GMRES have declined much faster than Gauss-Seidel (GS) method. Conclusion: The efficiency of GMRES based on the proposed discretization is superior as compared to GS iterative method.
Highlights
Integro-Differential Equation (IDE) is an important branch of modern mathematics and arises frequently in many applied areas which include engineering, mechanics, physics, chemistry, astronomy, biology, economics, potential theory and electrostatics (Kurt and Sezer, 2008)
We focus on second order linear Fredholm integrodifferential equation
Generalized Minimal Residual (GMRES) iterative method will be used for solving linear algebraic equations produced by the discretization of the second-order linear Fredholm integro-differential equations by using quadrature and finite difference methods
Summary
Integro-Differential Equation (IDE) is an important branch of modern mathematics and arises frequently in many applied areas which include engineering, mechanics, physics, chemistry, astronomy, biology, economics, potential theory and electrostatics (Kurt and Sezer, 2008). IDE is an equation that the unknown function appears under the sign of integration and it contains the derivatives of the unknown function It can be classified into Fredholm equations and Volterra equations. Earlier numerical treatment has been done for first order integro-differential equation (Aruchunan and Sulaiman, 2009). In this conjunction, there are many iterative methods under the category of Krylov subspaces have been proposed widely to be one of the feasible and successful classes of numerical algorithms for solving linear systems. GMRES iterative method will be used for solving linear algebraic equations produced by the discretization of the second-order linear Fredholm integro-differential equations by using quadrature and finite difference methods. In order to compare the efficiency of the GMRES method, Gauss-Seidel (GS) method was used for numerical comparison
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