Numerical solution of some class of nonlinear partial differential equations using wavelet-based full approximation scheme. (English) Zbl 07205470
Summary: In the last decades, wavelets have become a dominant tool for having applications in almost all the areas of engineering and science such as numerical simulation of partial differential equations (PDEs). The performance of the conventional numerical methods has been found to involve some difficulty to observe fast convergence in low computational time. To overcome this difficulty, we presented wavelet-based full approximation scheme (WFAS) for the numerical solution of some class of nonlinear PDEs using Daubechies wavelet intergrid operators. The numerical results obtained by this scheme are compared with the exact solution to reveal the accuracy and also speed up convergence in lesser computational time as compared with the existing schemes. Some test problems are presented to show the applicability and attractiveness of WFAS.
Keywords:
multiresolution analysis; Daubechies wavelet intergrid operators; partial differential equations; full approximation schemeSoftware:
WesselingReferences:
| [1] | Allahviranl, T., Armand, A. and Pirmuhammadi, S. [2014] “ Variational homotopy perturbation method: An efficient scheme for solving partial differential equations in fluid mechanics,” J. Math. Comput. Sci.9, 362-369. |
| [2] | Arora, G. and Singh, B. K. [2013] “ Numerical solution of Burgers’ equation with modified cubic B-spline differential quadrature method,” Appl. Math. Comput.224, 166-177. · Zbl 1334.65031 |
| [3] | Avudainayagam, A. and Vani, C. [2004] “ Wavelet based multigrid methods for linear and nonlinear elliptic partial differential equations,” Appl. Math. Comput.148, 307-320. · Zbl 1044.65092 |
| [4] | Bastian, A., Burmeistier, J. and Horton, G. [1990] “ Implementation of a parallel multigrid method for parabolic partial differential equations in parallel algorithms for PDEs,” Proc. 6th GAMM Seminar Kiel, pp. 19-21. |
| [5] | Bateman, H. [1915] “ Some recent researches on the motion of fluids,” Mon. Weather Rev.43, 163-170. |
| [6] | Biazar, J. and Aminikhah, H. [2009] “ Exact and numerical solutions for non-linear Burger’s by VIM,” Math. Comput. Model.49, 1394-1400. · Zbl 1165.65395 |
| [7] | Boggess, A. and Narcowich, F. J. [2009] A First Course in Wavelets with Fourier Analysis (John Wiley and Sons, New Jersey). · Zbl 1185.42001 |
| [8] | Brandt, A. [1977] “ Multi-level adaptive solutions to boundary-value problems,” Math. Comput.31, 333-390. · Zbl 0373.65054 |
| [9] | Briggs, W. L., Henson, V. E. and McCormick, S. F. [2000] A Multigrid Tutorial (SIAM, Philadelphia). · Zbl 0958.65128 |
| [10] | Bujurke, N. M., Salimath, C. S., Kudenatti, R. B. and Shiralashetti, S. C. [2007] “ A fast wavelet-multigrid method to solve elliptic partial differential equations,” Appl. Math. Comput.185(1), 667-680. · Zbl 1107.65347 |
| [11] | Dahmen, W., Kurdila, A. and Oswald, P. [1997] Multi-scale Wavelet Methods for Partial Differential Equations (Academic Press, San Diego). · Zbl 1528.65002 |
| [12] | Daubechies, I. [1988] “ Orthonormal bases of compactly supported wavelets,” Commun. Pure Appl. Math.41, 909-996. · Zbl 0644.42026 |
| [13] | Daubechies, I. [1992] “ Ten lectures on wavelets,” CBMS-NSF Regional Conf. Series in Applied Mathematics, Vol. 61, pp. 1-342, SIAM, Philadelphia. · Zbl 0776.42018 |
| [14] | Debnath, L. [2002] Wavelet Transforms and Their Applications (Springer Science/Business Media, New York). · Zbl 1019.94003 |
| [15] | Hariharan, G. and Kannan, K. [2010] “ Haar wavelet method for solving some nonlinear Parabolic equations,” J. Math. Chem.48, 1044-1061. · Zbl 1207.35183 |
| [16] | Hariharan, G. and Kannan, K. [2013] “ The wavelet methods to linear and nonlinear reaction — diffusion model arising in mathematical chemistry,” J. Math. Chem.51, 2361-2385. · Zbl 1314.65133 |
| [17] | Hariharan, G., Kannan, K. and Sharma, K. R. [2009] “ Haar wavelet method for solving Fisher’s equation,” Appl. Math. Comput.211, 284-292. · Zbl 1162.65394 |
| [18] | Jiwari, R. [2015] “ A hybrid numerical scheme for the numerical solution of the Burgers’ equation,” Comput. Phys. Commun.188, 59-67. · Zbl 1344.65082 |
| [19] | Karasozen, B. [2018] “ Energy stable discontinuous Galerkin finite element method for the Allen-Cahn equation,” Int. J. Comput. Methods15(1), 1850013. · Zbl 1404.65173 |
| [20] | Mohammed, G. [2010] “ Adomain decomposition method for nonlinear wave-like equations with variable coefficient,” Appl. Math. Sci.4, 2431-2444. · Zbl 1219.35010 |
| [21] | Shiralashetti, S. C., Kantli, M. H. and Deshi, A. B. [2016] “ New wavelet-based full-approximation scheme for the numerical solution of nonlinear elliptic partial differential equations,” Alexandria Eng. J.55, 2797-2804. |
| [22] | Shiralashetti, S. C., Kantli, M. H. and Deshi, A. B. [2018] “ A new wavelet multigrid method for the numerical solution of elliptic-type differential equations,” Alexandria Eng. J.57, 203-209. |
| [23] | Trottenberg, U., Oosterlee, C. and Schuller, A. [2001] Multigrid (Academic Press, London, San Diego). · Zbl 0976.65106 |
| [24] | Wesseling, P. [1992] An Introduction to Multigrid Methods (John Wiley, Chichester). · Zbl 0760.65092 |
| [25] | Wu, Y. and Xiao, J. [2017] “ Implementation of the multiscale stochastic finite element method on elliptic PDE problems,” Int. J. Comput. Methods14(1), 1750003. · Zbl 1404.65282 |
| [26] | Yuzbasi, S. [2016] “ A numerical method for solving second-order linear partial differential equations under Dirichlet, Neumann and Robin boundary conditions,” Int. J. Comput. Methods14(2), 1750015. · Zbl 1404.65203 |
| [27] | Zhi, S., Yongyan, C. and Chen, Q. [2012] “ Solving 2D and 3D Poisson equations and biharmonic equations by the Haar wavelet method,” Appl. Math. Model.36, 5143-5161. · Zbl 1254.65138 |
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