arrow
Volume 3, Issue 3
A New High Accuracy Off-Step Discretisation for the Solution of 2D Nonlinear Triharmonic Equations

Swarn Singh, Suruchi Singh & R. K. Mohanty

East Asian J. Appl. Math., 3 (2013), pp. 228-246.

Published online: 2018-02

Export citation
  • Abstract

In this article, we derive a new fourth-order finite difference formula based on off-step discretisation for the solution of two-dimensional nonlinear triharmonic partial differential equations on a 9-point compact stencil, where the values of $u$, $(∂^{2}u/∂n^2)$ and $(∂^{4}u/∂n^{4})$ are prescribed on the boundary. We introduce new ways to handle the boundary conditions, so there is no need to discretise the boundary conditions involving the partial derivatives. The Laplacian and biharmonic of the solution are obtained as a by-product of our approach, and we only need to solve a system of three equations. The new method is directly applicable to singular problems, and we do not require any fictitious points for computation. We compare its advantages and implementation with existing basic iterative methods, and numerical examples are considered to verify its fourth-order convergence rate.

  • AMS Subject Headings

65N06

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{EAJAM-3-228, author = {}, title = {A New High Accuracy Off-Step Discretisation for the Solution of 2D Nonlinear Triharmonic Equations}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {3}, number = {3}, pages = {228--246}, abstract = {

In this article, we derive a new fourth-order finite difference formula based on off-step discretisation for the solution of two-dimensional nonlinear triharmonic partial differential equations on a 9-point compact stencil, where the values of $u$, $(∂^{2}u/∂n^2)$ and $(∂^{4}u/∂n^{4})$ are prescribed on the boundary. We introduce new ways to handle the boundary conditions, so there is no need to discretise the boundary conditions involving the partial derivatives. The Laplacian and biharmonic of the solution are obtained as a by-product of our approach, and we only need to solve a system of three equations. The new method is directly applicable to singular problems, and we do not require any fictitious points for computation. We compare its advantages and implementation with existing basic iterative methods, and numerical examples are considered to verify its fourth-order convergence rate.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.140713.130813a}, url = {http://global-sci.org/intro/article_detail/eajam/10856.html} }
TY - JOUR T1 - A New High Accuracy Off-Step Discretisation for the Solution of 2D Nonlinear Triharmonic Equations JO - East Asian Journal on Applied Mathematics VL - 3 SP - 228 EP - 246 PY - 2018 DA - 2018/02 SN - 3 DO - http://doi.org/10.4208/eajam.140713.130813a UR - https://global-sci.org/intro/article_detail/eajam/10856.html KW - High accuracy finite differences, off-step discretisation, two-dimensional nonlinear triharmonic equations, Laplacian, biharmonic, triharmonic, maximum absolute errors. AB -

In this article, we derive a new fourth-order finite difference formula based on off-step discretisation for the solution of two-dimensional nonlinear triharmonic partial differential equations on a 9-point compact stencil, where the values of $u$, $(∂^{2}u/∂n^2)$ and $(∂^{4}u/∂n^{4})$ are prescribed on the boundary. We introduce new ways to handle the boundary conditions, so there is no need to discretise the boundary conditions involving the partial derivatives. The Laplacian and biharmonic of the solution are obtained as a by-product of our approach, and we only need to solve a system of three equations. The new method is directly applicable to singular problems, and we do not require any fictitious points for computation. We compare its advantages and implementation with existing basic iterative methods, and numerical examples are considered to verify its fourth-order convergence rate.

Swarn Singh, Suruchi Singh & R. K. Mohanty. (1970). A New High Accuracy Off-Step Discretisation for the Solution of 2D Nonlinear Triharmonic Equations. East Asian Journal on Applied Mathematics. 3 (3). 228-246. doi:10.4208/eajam.140713.130813a
Copy to clipboard
The citation has been copied to your clipboard