Abstract
In this article, we report the derivation of high accuracy finite difference method based on arithmetic average discretization for the solution of Un=F(x,u,u´)+∫K(x,s)ds , 0 x s < 1 subject to natural boundary conditions on a non-uniform mesh. The proposed variable mesh approximation is directly applicable to the integro-differential equation with singular coefficients. We need not require any special discretization to obtain the solution near the singular point. The convergence analysis of a difference scheme for the diffusion convection equation is briefly discussed. The presented variable mesh strategy is applicable when the internal grid points of the solution space are both even and odd in number as compared to the method discussed by authors in their previous work in which the internal grid points are strictly odd in number. The advantage of using this new variable mesh strategy is highlighted computationally.
Highlights
We consider the non-linear differential equation with a source function in integral form: The two point boundary conditions associated with (1) are given by: (2)where 0, 1 are finite constants
The presented variable mesh strategy is applicable when the internal grid points of the solution space are both even and odd in number as compared to the method discussed by authors in their previous work in which the internal grid points are strictly odd in number
Mohanty and Dhall [13] have proposed a three-point third order variable mesh method for the solution of non-linear integro-differential Equation (1), which is applicable only when the internal grid points of the solution region are odd in number
Summary
We consider the non-linear differential equation with a source function in integral form:. Mohanty et al ([912]) have discussed a family of third order variable mesh methods for the solution of two point non-linear boundary value problems and obtained convergent solution for singular problems. Mohanty and Dhall [13] have proposed a three-point third order variable mesh method for the solution of non-linear integro-differential Equation (1), which is applicable only when the internal grid points of the solution region are odd in number. We propose an efficient third order variable mesh method based on arithmetic average discretization for the solution of non-linear integro-differential Equation (1), which is applicable when the internal grid points of the solution region are both odd and even in number.
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