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Abstract： In this paper， half-sweep iteration concept applied on quadraturedifference schemes with Gauss-Seidel （GS） iterative method in solving linear Fredholm integro-differential equations. The combinations of discretization schemes of repeated trapezoidal and Simpson’s1 3with central difference schemes are analyzed. The formulation and the implementation of the proposed methods are explained in detail. In addition， several numerical experiments and computational complexity analysis were also carried out to validate the presentation of the schemes and methods. Thefindings show that， the HSGS iteration method is superior to the standard GS method. As well the high order quadrature scheme produced more accurate approximation solution compared to combination of repeated trapezoidal-central difference schemes.
Key words： Linear Fredholm integro-differential equations； Simpson’s scheme； Central difference； Half-Sweep Gauss-Seidel
with the Dirichlet boundary conditions y（a） = A1and y（b） = B1， where K（x，t）， g（x）， q（x） and p（x） are defined variables，λis a real parameter whereas y（x） is the unknown function to be determined. In this paper， we focus on numerical solutions forfirst and second order linear integro-differential equations of Fredholm types. In many application areas， it is necessary to use the numerical approach to discretize problem （1） to generate system of linear equation then solved by numerical methods such as Lagrange interpolation [1] and Taylor polynomial [2] and rationalized Haar functions [3]， Tau [4]， Conjugate Gradient [5]， GMRES [6] and collocation methods [7]. However in this paper we emphasize quadrature-difference schemes [8] to derive the approximation equation to generate system of linear equations. In addition to that， in this paper， we proposed a new half-sweep quadrature-difference discretization scheme which is combination of half-sweep reduction technique [9] on standard quadrature-difference schemes.
In this paper， two combinations of half-sweep discretization schemes namely half sweep repeated trapezoidal-central difference （HSRT-HSCD） and repeated Simpson-central difference （HSRS-HSCD） schemes will be implemented to discretize problem （1） to generate system of linear equations. Then the generated linear system will be solved iteratively by using half-sweep Gauss-Seidel （HSGS） method. In point of fact， the HSGS represents combination of half-sweep iteration concept on standard Gauss-Seidel （GS） which is also known as Full-Sweep Gauss Seidel （FSGS） method. The concept of the half-sweep iteration has been introduced by Abdullah [9] via Explicit Decoupled Group （EDG） iterative method to solve two-dimensional Poisson equation.Then， the idea of half-sweep iteration concept also identified as the complexity reduction approach [9] extensively studied by many researchers [10–13].
Key words： Linear Fredholm integro-differential equations； Simpson’s scheme； Central difference； Half-Sweep Gauss-Seidel
with the Dirichlet boundary conditions y（a） = A1and y（b） = B1， where K（x，t）， g（x）， q（x） and p（x） are defined variables，λis a real parameter whereas y（x） is the unknown function to be determined. In this paper， we focus on numerical solutions forfirst and second order linear integro-differential equations of Fredholm types. In many application areas， it is necessary to use the numerical approach to discretize problem （1） to generate system of linear equation then solved by numerical methods such as Lagrange interpolation [1] and Taylor polynomial [2] and rationalized Haar functions [3]， Tau [4]， Conjugate Gradient [5]， GMRES [6] and collocation methods [7]. However in this paper we emphasize quadrature-difference schemes [8] to derive the approximation equation to generate system of linear equations. In addition to that， in this paper， we proposed a new half-sweep quadrature-difference discretization scheme which is combination of half-sweep reduction technique [9] on standard quadrature-difference schemes.
In this paper， two combinations of half-sweep discretization schemes namely half sweep repeated trapezoidal-central difference （HSRT-HSCD） and repeated Simpson-central difference （HSRS-HSCD） schemes will be implemented to discretize problem （1） to generate system of linear equations. Then the generated linear system will be solved iteratively by using half-sweep Gauss-Seidel （HSGS） method. In point of fact， the HSGS represents combination of half-sweep iteration concept on standard Gauss-Seidel （GS） which is also known as Full-Sweep Gauss Seidel （FSGS） method. The concept of the half-sweep iteration has been introduced by Abdullah [9] via Explicit Decoupled Group （EDG） iterative method to solve two-dimensional Poisson equation.Then， the idea of half-sweep iteration concept also identified as the complexity reduction approach [9] extensively studied by many researchers [10–13].