High Accuracy Arithmetic Average Discretization for Non-Linear Two Point Boundary Value Problems with a Source Function in Integral Form

ABSTRACT

In this article, we report the derivation of high accuracy finite difference method based on arithmetic average discretization for the solution of*U*^{n}=*F*(*x,u,u*´)+∫*K*(*x,s*)d*s* , 0 <*x* < 1, 0 < *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.

In this article, we report the derivation of high accuracy finite difference method based on arithmetic average discretization for the solution of

KEYWORDS

Variable Mesh, Arithmetic Average Discretization, Non-Linear Integro-Differential Equation, Diffusion Equation, Simpson’s 1/3 Rd Rule, Singular Coefficients, Burgers’ Equation, Maximum Absolute Errors

Variable Mesh, Arithmetic Average Discretization, Non-Linear Integro-Differential Equation, Diffusion Equation, Simpson’s 1/3 Rd Rule, Singular Coefficients, Burgers’ Equation, Maximum Absolute Errors

Cite this paper

nullR. Mohanty and D. Dhall, "High Accuracy Arithmetic Average Discretization for Non-Linear Two Point Boundary Value Problems with a Source Function in Integral Form,"*Applied Mathematics*, Vol. 2 No. 10, 2011, pp. 1243-1251. doi: 10.4236/am.2011.210173.

nullR. Mohanty and D. Dhall, "High Accuracy Arithmetic Average Discretization for Non-Linear Two Point Boundary Value Problems with a Source Function in Integral Form,"

References

[1] H. B. Keller, “Numerical Methods for Two Point Boun-dary Value Problems”, Blaisdell, London, 1968.

[2] P. J. Davis and P. Rabinowitz, “Method of Numerical Integra-tion,” 2nd Edition, Academic Press, New York, 1970.

[3] G. M. Phillips, “Analysis of Numerical Itera-tive Methods for Solving Integral and Integro-Differential Equations,” Computer Journal, Vol. 13, No. 3, 1970, pp. 297-300. doi:10.1093/comjnl/13.3.297

[4] P. Linz, “A Method for Approximate Solution of Linear Integro-Differential Equations,” SIAM Journal on Numerical Analysis, Vol. 11, No. 1, 1974, pp. 137-144. doi:10.1137/0711014

[5] V. Lakshmikantham and M. R. M. Rao, “Theory of Integro-Differential Equations,” Gordon and Breach, London, 1995.

[6] K. E. Atkinson, “The Numerical Solution of Integral Equations of the Second Kind,” Cambridge University Press, Cambridge, 1997. doi:10.1017/CBO9780511626340

[7] R. P. Agarwal and D. O’Regan, “Integral and Integro-Differential Equa-tions: Theory, Method and Applications,” Gordon and Breach, London, 2000.

[8] M. K. Jain, S. R. K. Iyenger and G. S. Subramanyam, “Variable Mesh Methods for the Numerical Solution of Two Point Singular Perturbation Problems,” Computer Methods in Applied Mechanics and Engineering, Vol. 42, 1984, pp. 273-286. doi:10.1016/0045-7825(84)90009-4

[9] R. K. Mohanty, “A Family of Variable Mesh Methods for the Estimates of (du/dr) and the Solution of Nonlinear Two Point Boundary Value Problems with Singularity,” Journal of Computational and Applied Mathematics, Vol. 182, No. 1, 2005, pp. 173-187. doi:10.1016/j.cam.2004.11.045

[10] R. K. Mohanty and N. Khosla, “A Third Order Accurate Variable Mesh TAGE Iterative Method for the Numerical Solution of Two Point Nonlinear Singular Boundary Value Prob-lems,” International Journal of Computer Mathematics, Vol. 82, No. 10, 2005, pp. 1261-1273. doi:10.1080/00207160500113504

[11] R. K. Mohanty and N. Khosla, “Application of TAGE Iterative Algo-rithms to an Efficient Third Order Arithmetic Average Variable Mesh Discretization for Two Point Non-Linear Boundary Value Problems,” Applied Mathematics and Computations, Vol. 172, No. 1, 2006, pp. 148-162. doi:10.1016/j.amc.2005.01.134

[12] R. K. Mohanty, “A Class of Non-Uniform Mesh Three Point Arithmetic Av-erage Discretization for and the estimates of ,” Applied Ma-thematics and Computations , Vol. 183, No. 1, 2006, pp. 477-485. doi:10.1016/j.amc.2006.05.071

[13] R. K. Mohanty and D. Dhall, “Third Order Accurate Variable Mesh Discretization and Application of TAGE Iterative Method for the Non-Linear Two-Point Boundary Value Problems with Homogeneous Functions in Integral Form,” Applied Mathematics and Computations, Vol. 215, 2009, pp. 2024-2034. doi:10.1016/j.amc.2009.07.046

[14] G. Evans, “Practical Numerical Integration,” John Wiley & Sons, New York, 1993.

[15] R. S. Varga, “Matrix Iterative Analysis,” Springer-Verlag, Berlin, 2000. doi:10.1007/978-3-642-05156-2

[16] D. M. Young, “Iterative Solution of Large Linear Systems,” Dover Pub-lication, New York, 2003.

[17] C. T. Kelly, “Iterative Methods for Linear and Non-linear equations,” SIAM Publication, Philadelphia, 1995.

[18] D. J. Evans, “Itera-tive Methods for Solving Nonlinear Two Point Boundary Value Problems,” International Journal of Computer Mathematics, Vol. 72, No. 3, 1999, pp. 395-401. doi:10.1080/00207169908804862

[1] H. B. Keller, “Numerical Methods for Two Point Boun-dary Value Problems”, Blaisdell, London, 1968.

[2] P. J. Davis and P. Rabinowitz, “Method of Numerical Integra-tion,” 2nd Edition, Academic Press, New York, 1970.

[3] G. M. Phillips, “Analysis of Numerical Itera-tive Methods for Solving Integral and Integro-Differential Equations,” Computer Journal, Vol. 13, No. 3, 1970, pp. 297-300. doi:10.1093/comjnl/13.3.297

[4] P. Linz, “A Method for Approximate Solution of Linear Integro-Differential Equations,” SIAM Journal on Numerical Analysis, Vol. 11, No. 1, 1974, pp. 137-144. doi:10.1137/0711014

[5] V. Lakshmikantham and M. R. M. Rao, “Theory of Integro-Differential Equations,” Gordon and Breach, London, 1995.

[6] K. E. Atkinson, “The Numerical Solution of Integral Equations of the Second Kind,” Cambridge University Press, Cambridge, 1997. doi:10.1017/CBO9780511626340

[7] R. P. Agarwal and D. O’Regan, “Integral and Integro-Differential Equa-tions: Theory, Method and Applications,” Gordon and Breach, London, 2000.

[8] M. K. Jain, S. R. K. Iyenger and G. S. Subramanyam, “Variable Mesh Methods for the Numerical Solution of Two Point Singular Perturbation Problems,” Computer Methods in Applied Mechanics and Engineering, Vol. 42, 1984, pp. 273-286. doi:10.1016/0045-7825(84)90009-4

[9] R. K. Mohanty, “A Family of Variable Mesh Methods for the Estimates of (du/dr) and the Solution of Nonlinear Two Point Boundary Value Problems with Singularity,” Journal of Computational and Applied Mathematics, Vol. 182, No. 1, 2005, pp. 173-187. doi:10.1016/j.cam.2004.11.045

[10] R. K. Mohanty and N. Khosla, “A Third Order Accurate Variable Mesh TAGE Iterative Method for the Numerical Solution of Two Point Nonlinear Singular Boundary Value Prob-lems,” International Journal of Computer Mathematics, Vol. 82, No. 10, 2005, pp. 1261-1273. doi:10.1080/00207160500113504

[11] R. K. Mohanty and N. Khosla, “Application of TAGE Iterative Algo-rithms to an Efficient Third Order Arithmetic Average Variable Mesh Discretization for Two Point Non-Linear Boundary Value Problems,” Applied Mathematics and Computations, Vol. 172, No. 1, 2006, pp. 148-162. doi:10.1016/j.amc.2005.01.134

[12] R. K. Mohanty, “A Class of Non-Uniform Mesh Three Point Arithmetic Av-erage Discretization for and the estimates of ,” Applied Ma-thematics and Computations , Vol. 183, No. 1, 2006, pp. 477-485. doi:10.1016/j.amc.2006.05.071

[13] R. K. Mohanty and D. Dhall, “Third Order Accurate Variable Mesh Discretization and Application of TAGE Iterative Method for the Non-Linear Two-Point Boundary Value Problems with Homogeneous Functions in Integral Form,” Applied Mathematics and Computations, Vol. 215, 2009, pp. 2024-2034. doi:10.1016/j.amc.2009.07.046

[14] G. Evans, “Practical Numerical Integration,” John Wiley & Sons, New York, 1993.

[15] R. S. Varga, “Matrix Iterative Analysis,” Springer-Verlag, Berlin, 2000. doi:10.1007/978-3-642-05156-2

[16] D. M. Young, “Iterative Solution of Large Linear Systems,” Dover Pub-lication, New York, 2003.

[17] C. T. Kelly, “Iterative Methods for Linear and Non-linear equations,” SIAM Publication, Philadelphia, 1995.

[18] D. J. Evans, “Itera-tive Methods for Solving Nonlinear Two Point Boundary Value Problems,” International Journal of Computer Mathematics, Vol. 72, No. 3, 1999, pp. 395-401. doi:10.1080/00207169908804862