Commun. Comput. Phys., 12 (2012), pp. 1417-1433.


A Novel Numerical Method of O(h^4) for Three-Dimensional Non-Linear Triharmonic Equations

R. K. Mohanty 1*, M. K. Jain 2, B. N. Mishra 3

1 Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi-110 007, India.
2 Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi-110 016, India.
3 Department of Mathematics, Utkal University, Vani Vihar, Bhubaneswar-751 004, India.

Received 8 September 2010; Accepted (in revised version) 6 January 2012
Available online 8 May 2012
doi:10.4208/cicp.080910.060112a

Abstract

In this article, we present two new novel finite difference approximations of order two and four, respectively, for the three dimensional non-linear triharmonic partial differential equations on a compact stencil where the values of u, $\partial^2u/\partial n^2$ and $\partial^4u/\partial n^4$ are prescribed on the boundary. We introduce new ideas to handle the boundary conditions and there is no need to discretize the derivative boundary conditions. We require only 7- and 19-grid points on the compact cell for the second and fourth order approximation, respectively. The Laplacian and the biharmonic of the solution are obtained as by-product of the methods. We require only system of three equations to obtain the solution. Numerical results are provided to illustrate the usefulness of the proposed methods.

AMS subject classifications: 65N06
PACS: 02.60.Lj, 02.70.Bf
Key words: Finite differences, three dimensional non-linear triharmonic equations, fourth order compact discretization, Laplacian, biharmonic, maximum absolute errors.

*Corresponding author.
Email: rmohanty@maths.du.ac.in (R. K. Mohanty), bn_misramath@hotmail.com (B. N. Mishra)
 

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