Mixed Jacobi-Fourier spectral method for Fisher equation
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.
This work is licensed under a Creative Commons Attribution 4.0 International License.
 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
 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
 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
 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
 N.F. Britton. Reaction-Diﬀusion Equations and Their Applications to Biology. Academic Press, Inc., London, 1986.
 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
 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
 B.-Y. Guo. Spectral Methods and Their Applications. World Scientiﬁc, Singapore, 1998.
 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
 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
 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
 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
 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
 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
 D.S. Jones and B.D. Sleeman. Diﬀerential Equations and Mathematical Biology. Allen Unwin, London, 1983.
 J.D. Logan. An Introduction to Nonlinear Diﬀerential Equations, 2nd edition. John Wilcy & Sons, Inc., Hoboken, New Jersey, New York, 2008.
 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
 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.
 J.D. Murray. Mathematical Biology ll: Spatial Models and Biomedical Applications. Springer-Verlag, Berlin Heidelberg, 1993.
 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
 J. Shen, T. Tang and L.-L. Wang. Spectral Methods: Algorithms, Analysis and Applications. Springer-Verlag, Berlin Heidelberg, 2011.
 J. Smoller. Shock Waves and Reaction-Diﬀusion Equations, volume 258 of Grundlehren der mathematischen Wissenschaften. Springer, New-York, 1983.
 F. Stenger. Numerical methods based on Whittaker cardinal, or sinc functions. SIAM Review, 23(2):165–224, 1981. https://doi.org/10.1137/1023037
 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
 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
 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
 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
 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
 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