Mixed Jacobi-Fourier spectral method for Fisher equation
Abstract
In this paper, we propose a mixed Jacobi-Fourier spectral method for solving the Fisher equation in a disc. Some mixed Jacobi-Fourier approximation results are established, which play important roles in numerical simulation of various problems defined in a disc. We use the generalized Jacobi approximation to simulate the singularity of solution at the regional center. This also simplifies the theoretical analysis and provides a sparse system. The stability and convergence of the proposed scheme are proved. Numerical results demonstrate the efficiency of this new algorithm and coincide well with the theoretical analysis.
Keyword : Fisher equation in a disc, mixed Jacobi-Fourier approximation, spectral method, problem with end-point weak singularity, nonlinear problem
How to Cite
Jiao, Y., Wang, T., Shi, X., & Liu, W. (2018). Mixed Jacobi-Fourier spectral method for Fisher equation. Mathematical Modelling and Analysis, 23(2), 240-261. https://doi.org/10.3846/mma.2018.016
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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[21] D. Olmos and B.D. Shizgal. A pseudospectral method of solution of Fisher’s equation. Journal of Computational and Applied Mathematics, 193(1):219–242, 2006. https://doi.org/10.1016/j.cam.2005.06.028
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[26] T.-J. Wang. Generalized Laguerre spectral method for Fisher’s equation on a semi-infinite interval. International Journal of Computer Mathematics, 92(5):1039–1052, 2015. https://doi.org/10.1080/00207160.2014.920833
[27] T.-J. Wang and Y.-J. Jiao. A fully discrete pseudospectral method for Fisher’s equation on the whole line. Applied Numerical Mathematics, 120(Supplement C):243–256, 2017. https://doi.org/10.1016/j.apnum.2017.06.002
[28] A.-M. Wazwaz and A. Gorguis. An analytic study of Fisher’s equation by using Adomian decomposition method. Applied Mathematics and Computation, 154(3):609–620, 2004. https://doi.org/10.1016/S0096-3003(03)00738-0
[29] X.-H. Yu and Z.-Q. Wang. Mixed Fourier-Jacobi spectral method for two-dimensional Neumann boundary value problems. East Asian Journal on Applied Mathematics, 1(3):284–296, 2011. https://doi.org/10.4208/eajam.281010.200411a
[30] X.-H. Yu and Z.-Q. Wang. Jacobi spectral method with essential imposition of Neumann boundary condition. Applied Numerical Mathematics, 62(8):956–974, 2012. https://doi.org/10.1016/j.apnum.2012.03.004
[2] Ch. Bernardi and Y. Maday. Polynomial interpolation results in Sobolev spaces. Journal of Computational and Applied Mathematics, 43(1):53–80, 1992. https://doi.org/10.1016/0377-0427(92)90259-Z
[3] J.P. Boyd. Polynomial series versus sinc expansions for functions with corner or endpoint singularities. Journal of Computational Physics, 64(1):266–270, 1986. https://doi.org/10.1016/0021-9991(86)90031-8
[4] J.P. Boyd. The asymptotic Chebyshev coefficients for functions with logarithmic endpoint singularities: mappings and singular basis functions. Applied Mathematics and Computation, 29(1):49–67, 1989. https://doi.org/10.1016/0096-3003(89)90039-8
[5] J.P. Boyd and F. Yu. Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike polynomials, Logan-Shepp ridge polynomials, Chebyshev-Fourier series, cylindrical Robert functions, Bessel-Fourier expansions, square-to-disk conformal mapping and radial basis functions. Journal of Computational Physics, 230(4):1408–1438, 2011. https://doi.org/10.1016/j.jcp.2010.11.011
[6] N.F. Britton. Reaction-Diffusion Equations and Their Applications to Biology. Academic Press, Inc., London, 1986.
[7] R.A. Fisher. The wave of advance of advantageous genes. Annals of Eugenics, 7(4):355–369, 1937. https://doi.org/10.1111/j.1469-1809.1937.tb02153.x
[8] B.-Y. Guo. Gegenbauer approximation and its applications to differential equations on the whole line. Journal of Mathematical Analysis and Applications, 226(1):180–206, 1998. https://doi.org/10.1006/jmaa.1998.6025
[9] B.-Y. Guo. Spectral Methods and Their Applications. World Scientific, Singapore, 1998.
[10] B.-Y. Guo. Error estimate of Hermite spectral method for nonlinear partial differential equation. Mathematics of Computation, 68(227):1067–1078, 1999. https://doi.org/10.1090/S0025-5718-99-01059-5
[11] B.-Y. Guo. Gegenbauer approximation in certain Hilbert spaces and its applications to singular differential equations. SIAM Journal on Numerical Analysis, 37(2):621–645, 1999. https://doi.org/10.1137/S0036142998342161
[12] B.-Y. Guo and Z.-X. Chen. Analytic solutions of the Fisher equation. Journal of Physics A: Mathematical and General, 24(3):645–650, 1991. https://doi.org/10.1088/0305-4470/24/3/023
[13] B.-Y. Guo and L.-L. Wang. Jacobi interpolation approximations and their applications to singular differential equations. Advances in Computational Mathematics, 14(3):227–276, 2001. https://doi.org/10.1023/A:1016681018268
[14] B.-Y. Guo and L.-L. Wang. Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces. Journal of Approximation Theory, 128(1):1–41, 2004. https://doi.org/10.1016/j.jat.2004.03.008
[15] R. Jiwari. A Haar wavelet quasilinearization approach for numerical simulation of Burgers’ equation. Computer Physics Communications, 183(11):2413–2423, 2012. https://doi.org/10.1016/j.cpc.2012.06.009
[16] D.S. Jones and B.D. Sleeman. Differential Equations and Mathematical Biology. Allen Unwin, London, 1983.
[17] J.D. Logan. An Introduction to Nonlinear Differential Equations, 2nd edition. John Wilcy & Sons, Inc., Hoboken, New Jersey, New York, 2008.
[18] T. Matsushima and P.S. Marcus. A spectral method for polar coordinates. Journal of Computational Physics, 120(2):365–374, 1995. https://doi.org/10.1006/jcph.1995.1171
[19] R.C. Mittal and R. Jiwari. Numerical study of Fisher’s equation by using differential quadrature method. International Journal of Information and Systems Sciiences, 5(1):143–160, 2009.
[20] J.D. Murray. Mathematical Biology ll: Spatial Models and Biomedical Applications. Springer-Verlag, Berlin Heidelberg, 1993.
[21] D. Olmos and B.D. Shizgal. A pseudospectral method of solution of Fisher’s equation. Journal of Computational and Applied Mathematics, 193(1):219–242, 2006. https://doi.org/10.1016/j.cam.2005.06.028
[22] J. Shen, T. Tang and L.-L. Wang. Spectral Methods: Algorithms, Analysis and Applications. Springer-Verlag, Berlin Heidelberg, 2011.
[23] J. Smoller. Shock Waves and Reaction-Diffusion Equations, volume 258 of Grundlehren der mathematischen Wissenschaften. Springer, New-York, 1983.
[24] F. Stenger. Numerical methods based on Whittaker cardinal, or sinc functions. SIAM Review, 23(2):165–224, 1981. https://doi.org/10.1137/1023037
[25] A. Verma, R. Jiwari and M.E. Koksal. Analytic and numerical solutions of nonlinear diffusion equations via symmetry reductions. Advances in Difference Equations, 2014(1):229, 2014. https://doi.org/10.1186/1687-1847-2014-229
[26] T.-J. Wang. Generalized Laguerre spectral method for Fisher’s equation on a semi-infinite interval. International Journal of Computer Mathematics, 92(5):1039–1052, 2015. https://doi.org/10.1080/00207160.2014.920833
[27] T.-J. Wang and Y.-J. Jiao. A fully discrete pseudospectral method for Fisher’s equation on the whole line. Applied Numerical Mathematics, 120(Supplement C):243–256, 2017. https://doi.org/10.1016/j.apnum.2017.06.002
[28] A.-M. Wazwaz and A. Gorguis. An analytic study of Fisher’s equation by using Adomian decomposition method. Applied Mathematics and Computation, 154(3):609–620, 2004. https://doi.org/10.1016/S0096-3003(03)00738-0
[29] X.-H. Yu and Z.-Q. Wang. Mixed Fourier-Jacobi spectral method for two-dimensional Neumann boundary value problems. East Asian Journal on Applied Mathematics, 1(3):284–296, 2011. https://doi.org/10.4208/eajam.281010.200411a
[30] X.-H. Yu and Z.-Q. Wang. Jacobi spectral method with essential imposition of Neumann boundary condition. Applied Numerical Mathematics, 62(8):956–974, 2012. https://doi.org/10.1016/j.apnum.2012.03.004