A higher order frozen Jacobian iterative method for solving Hamilton-Jacobi equations
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Authors
Ebraheem O. Alzahrani
- Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia.
Eman S. Al-Aidarous
- Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia.
Arshad M. M. Younas
- Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia.
Fayyaz Ahmad
- Dipartimento di Scienza e Alta Tecnologia, Universita dell'Insubria, Via Valleggio 11, Como 22100, Italy.
- ment de Fisica i Enginyeria Nuclear, Universitat Politecnica de Catalunya, Comte d'Urgell 187, 08036 Barcelona, Spain.
- UCERD Islamabad, Pakistan.
Shamshad Ahmad
- Heat and Mass Transfer Technological Center, Technical University of Catalonia, Colom 11, 08222 Terrassa, Spain.
Shahid Ahmad
- Department of Mathematics, Government College University Lahore, Lahore, Pakistan.
Abstract
It is well-known that the solution of Hamilton-Jacobi equation may have singularity i.e., the solution is
non-smooth or nearly non-smooth. We construct a frozen Jacobian multi-step iterative method for solving
Hamilton-Jacobi equation under the assumption that the solution is nearly singular. The frozen Jacobian
iterative methods are computationally very efficient because a single instance of the iterative method uses a
single inversion (in the scene of LU factorization) of the frozen Jacobian. The multi-step part enhances the
convergence order by solving lower and upper triangular systems. The convergence order of our proposed
iterative method is \(3(m - 1)\) for \(m \geq 3\). For attaining good numerical accuracy in the solution, we use
Chebyshev pseudo-spectral collocation method. Some Hamilton-Jacobi equations are solved, and numerically
obtained results show high accuracy.
Share and Cite
ISRP Style
Ebraheem O. Alzahrani, Eman S. Al-Aidarous, Arshad M. M. Younas, Fayyaz Ahmad, Shamshad Ahmad, Shahid Ahmad, A higher order frozen Jacobian iterative method for solving Hamilton-Jacobi equations, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 12, 6210--6227
AMA Style
Alzahrani Ebraheem O., Al-Aidarous Eman S., Younas Arshad M. M., Ahmad Fayyaz, Ahmad Shamshad, Ahmad Shahid, A higher order frozen Jacobian iterative method for solving Hamilton-Jacobi equations. J. Nonlinear Sci. Appl. (2016); 9(12):6210--6227
Chicago/Turabian Style
Alzahrani, Ebraheem O., Al-Aidarous, Eman S., Younas, Arshad M. M., Ahmad, Fayyaz, Ahmad, Shamshad, Ahmad, Shahid. "A higher order frozen Jacobian iterative method for solving Hamilton-Jacobi equations." Journal of Nonlinear Sciences and Applications, 9, no. 12 (2016): 6210--6227
Keywords
- Hamilton-Jacobi equations
- frozen Jacobian iterative methods
- systems of nonlinear equations
- Chebyshev pseudo-spectral collocation method.
MSC
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