Variable Mesh Size Exponential Finite Difference Method for the Numerical Solutions of Two Point Boundary Value Problems
Abstract
In this article, we presented a non-uniform mesh size high order exponential finite difference scheme for the numerical solutions of two point boundary value problems with Dirichlet's boundary conditions. Under appropriate conditions, we have discussed the local truncation error and the convergence of the proposed method. Numerical experiments have been carried out to demonstrate the use and high order computational efficiency of the present method in several model problems. Numerical results showed that the proposed method is accurate and convergent. The order of accuracy is at least cubic which is in good agreement with the theoretically established order of the method.
Downloads
References
Ascher, U., Mattheij, R. M. M. AND Russell, R. D., Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Prentice-Hall, Englewood Cliffs, NJ (1988).
Collatz, L ., Numerical Treatment of Differential Equations (3/ed.). Springer Verlag, Berlin,(1966).
Keller, H. B., Numerical Methods for Two Point Boundary Value Problems. Blaisdell Publ. Co.,New York, Waltham, Ma(1968).
Stoer, J. and Bulirsch , R., Introduction to Numerical Analysis (2/e). Springer-Verlag, Berlin Heidelberg (1991).
Baxley, J. V., Nonlinear Two Point Boundary Value Problems in Ordinary and Partial Differential Equations (Everrit, W.N. and Sleeman, B.D. Eds.), Springer-Verlag, New York (1981), 46-54.
Roos, H.G., Stynes, M. and Tobiska, L., Numerical Methods for Singularly Perturbed Differential Equations. Springer, New York (1996).
Hairer, E., Norsett, S. P. and Wanner, G., Solving Ordinary Differential Equations I Nonstiff Problems (Second Revised Edition). Springer-Verlag New York, Inc. New York, NY, USA (1993).
M.K. Jain, Numerical Solution of Difierential Equations, Wiley Eastern Limited, New Delhi, (1984).
Hemker, P.W. and Miller, J.J.H., Numerical Analysis of Singular Perturbation Problems. Academic Press, New York, (1979).
Van Niekerk, F.D., A nonlinear one step methods for initial value problems. Comp. Math. Appl. 13(1987), 367-371.
Pandey, P. K., Nonlinear Explicit Method for first Order Initial Value Problems. Acta Technica Jaurinensis, Vol. 6, No. 2 (2013), 118-125.
Pandey, P. K., Solving Two Point Boundary Value Problems for Ordinary Differential Equations Using Exponential Finite Difference Method. Bol. Soc. Paran. Mat. (2014), **-**.
Varga, R. S., Matrix Iterative Analysis, Second Revised and Expanded Edition. Springer-Verlag, Heidelberg, (2000).
Golub, G. H. and Van Loan, C.F., Matrix Computations. John Hopkins University Press , Baltimore, USA (1989).
Henrici, P., Discrete Variable Methods in Ordinary Differential Equations. John Wiley and Sons, New York, (1962).
Volkov, Y. S. and Miroshnichenko, V. L. , Norm estimates for the inverses of matrices of monotone type and totally positive matrics. Siberian Mathematical Journal Vol. 50, No. 6,(2009), 982-987.
Ahlberg, J. H. and Nilson, E. N. , Convergence properties of the spline fit. J. SIAM, 11, No. 1, (1963), 95-104.
Scott, M. R. and Watts, H. A., Computational solution of linear two point boundary value problems via orthonormalization. SIAM J. Numer. Anal. 14 (1977), 40-70.
O’Riordan, E. and Stynes, M., A uniformely accurate finite element method for a singularly perturbed one-dimensional reaction-diffusion problem. Math. Comput.,47 (1986),555-570.
Vukoslavcevic, V. and Surla, K., Finite element method for solving self adjoint singularly perturbed boundary value problems. Math. Montisnigri,8 (1996), 89-96.
Schatz, A. H. and Wahlbin, L. B., On the finite element method for singularly perturbed reaction diffusion problems in two and one dimension. Math. Comput., 40 (1983), 47-89.
Kadalbajoo, M. K. and Panwar, D. K., Variable mesh finite difference method for self-adjoint singularly perturbed two-point boundary value problems. Journal of Computational Mathematics, Vol.28, No. 5 (2010), 711-724.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
