Space-time multi-domain spectral collocation method for the Fisher’s equation
DOI: https://doi.org/10.3846/mma.2026.25075Abstract
A space-time multi-domain Legendre-Gauss-Lobatto collocation method is developed to solve Fisher’s equation with nonhomogeneous boundary conditions, addressing aliasing errors caused by nonlinear terms. For large-scale problems in the spatial domain, a multi-domain collocation method is proposed, where appropriate boundary conditions are imposed at the common interfaces of adjacent subdomains to ensure the global smoothness of the numerical solution. A quasi-orthogonal approximation is constructed to fit both the function values at the interval endpoints and those at interior points. The convergence and stability of the scheme are analyzed using the Petrov-Galerkin spectral method with numerical integration. Numerical experiments demonstrate the effectiveness and accuracy of the proposed method.
Keywords:
space-time spectral collocation methods for Fisher’s equations, Legendre-Gauss-Lobatto nodes, composite multi-domain quasi-orthogonal approximation and the Petrov-Galerkin spectral methodHow to Cite
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Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University.
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References
W.M. Abd-Elhameed, M.A. Abdelkawy, N.M.A. Alsafri and A.G. Atta. Numerical treatment of the time-fractional Kuramoto–Sivashinsky equation using a combined Chebyshev-collocation approach. Fractal and Fractional, 9(11):727, 2025. https://doi.org/10.3390/fractalfract9110727
W.M. Abd-Elhameed, A.H. Al-Mehmadi, N.M.A. Alsafri, O.M. Alqubori, M. Adel and A.G. Atta. Spectral collocation approach for solving the time-fractional Kuramoto-sivashinsky equation using the Fibonacci coefficient polynomials. AIMS Mathematics, 10(8):18070–18093, 2025. https://doi.org/10.3934/math.2025805
W.M. Abd-Elhameed, O.M. Alqubori, N.M.A. Alsafri, A.K. Amin and A.G. Atta. A matrix approach by convolved Fermat polynomials for solving the fractional Burger’s equation. Mathematics, 13(7):1135, 2025. https://doi.org/10.3390/math13071135
D. Ghoshal. Fisher’s equation for a decaying brane. J. High Energ. Phys., 2011(15):1–9, 2011. https://doi.org/10.1007/JHEP12(2011)015
B.Y. Guo. Spectral Methods and Their Applications. World Scientific, 1998. https://doi.org/10.1142/3662
B.Y. Guo. Error estimation of Hermite spectral method for nonlinear partial differential equations. Math. Comp., 68(227):1069–1078, 1999. https://doi.org/10.1090/s0025-5718-99-01059-5
B.Y. Guo and T.J. Wang. Composite Laguerre-Legendre spectral method for exterior problems. Adv. in Comp. Math., 32:393–429, 2010. https://doi.org/10.1007/s10444-008-9112-5
P.K. Gupta and S. Verma. A numerical study of the nonlinear reaction-diffusion equation with different type of absorbent term by homotopy analysis method. Z. Naturforsch A, 67(10–11):621–627, 2012. https://doi.org/10.5560/zna.2012-0066
Y. Gurefe and E. Misirli. Exp-function method for solving nonlinear evolution equations with higher order nonlinearity. Comput. Math. Appl., 61(8):2025–2030, 2011. https://doi.org/10.1016/j.camwa.2010.08.060
T. Kadri and K. Omrani. A second-order accurate difference scheme for an extended Fisher’s-Kolmogorov equation. Comput. Math. Appl., 61(2):451–459, 2011. https://doi.org/10.1016/j.camwa.2010.11.022
D.A. Kopriva. Implementing spectral methods for partial differential equations. Springer, 2009. https://doi.org/10.1007/978-90-481-2261-5
D. Li and C. Zhang. Split Newton iterative algorithm and its application. Appl. Math. Comput., 217(5):2260–2265, 2010. https://doi.org/10.1016/j.amc.2010.07.026
L. Liu and H.P. Ma. Space-time spectral method for parabolic inverse problem with unknown control parameter. J. Numer. Meth. Comput. Appl., 41(1):19–26, 2020. https://doi.org/10.12288/szjs.2020.1.19
Y.N. Ma, T.J. Wang and B.B. Li. Space-time spectral collocation method for Korteweg-de Vries equation. J. Numer. Meth. Comput. Appl., 42(4):351–360, 2021. https://doi.org/10.12288/szjs.s2020-0692
Z.P. Mao and X.H. Meng. Physics-informed neural networks with residual/gradient-based adaptive sampling methods for solving partial differential equations with sharp solutions. Appl. Math. Mech., 44(7):1069–1084, 2023. https://doi.org/10.1007/s10483-023-2994-7
T.J. Newman, E.B. Kolomeisky and J. Antonovics. Population dynamics with global regulation: the conserved Fisher’s equation. Phys. Rev. Lett., 92(22):228103, 2004. https://doi.org/10.1103/PhysRevLett.92.228103
L. Roman-Miller and P. Broadbridge. Exact integration of reduced Fisher’s equation reduced Blasius equation and the Lorenz model. J. Math. Anal. Appl., 251(1):65–83, 2000. https://doi.org/10.1006/jmaa.2000.7020
A. Secer and M. Cinar. A Jacobi wavelet collocation method for fractional Fisher’s equation in time. Therm. Sci., 24(1):119–129, 2020. https://doi.org/10.2298/TSCI20S1119S
F. Shayanfard and H.L. Dastjerdi. A multistep collocation method for approximate solution of Volterra integro-differential equations of the third kind. Comput. Appl. Math., 43(176):176, 2024. https://doi.org/10.1007/s40314-024-02635-4
J. Shen, T. Tang and L.L. Wang. Spectral Methods: Algorithms, Analysis and Applications. Springer, 2011. https://doi.org/10.1007/978-3-540-71041-7
John C. Strikwerda. Finite difference schemes and partial differential equations. Society for Industrial and Applied Mathematics, 1989.
M. Tamsir and M.J. Huntul. A numerical approach for solving Fisher’s reactiondiffusion equation via a new kind of spline functions. Ain Shams Eng. J.,
12(3):3157–3165, 2021. https://doi.org/10.1016/j.asej.2020.11.024
J. Tan and T.J. Wang. A multi-domain spectral collocation method for the Fokker -Planck equation in an infinite channel. Math. Comput. Simul., 220:533– 551, 2024. https://doi.org/10.1016/j.matcom.2024.02.014
J. Tan and T.J. Wang. Spectral collocation method coupled with domain decomposition for exterior problems of the Fisher’s equation. Comput. Math. Appl., 175:289–312, 2024. https://doi.org/10.1016/j.camwa.2024.10.003
J.G. Tang and H.P. Ma. Single and multi-interval Legendre spectral methods in time for parabolic equations. Numer. Meth. Part. D. E., 22(5):1007–1034, 2006. https://doi.org/10.1002/num.20135
H. Tian, J. Zhang and L.L. Ju. A spectral collocation method for nonlocal diffusion equations with volume constrained boundary conditions. Appl. Math. Comput., 370:124930, 2020. https://doi.org/10.1016/j.amc.2019.124930
L.N. Wang, Q. Tong, L.J. Yi and M.Z. Zhang. Legendre-Gauss-Radau spectral collocation method for nonlinear second-order initial value problems with applications to wave equations. J. Comput. Math., 42(1):217–247, 2024. https://doi.org/10.4208/jcm.2203-m2021-0244
T.J. Wang. Generalized Laguerre spectral method for Fisher’s equation on a semi-infinite interval. Inter. J. Comput. Math., 92(5):1039–1052, 2015. https://doi.org/10.1080/00207160.2014.920833
T.J. Wang and B.Y. Guo. Composite generalized Laguerre-Legendre pseudospectral method for exterior problems. Comm. Comp. Phys., 5(2–4):350–375, 2009. https://doi.org/10.4208/cicp.2009.v5.p350
T.J. Wang and Y.J. Jiao. A fully discrete pseudospectral method for Fisher’s equation on the whole line. Appl. Numer. Math., 120:243–256, 2017. https://doi.org/10.1016/j.apnum.2017.06.002
A.M. Wazwaz. The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations. Appl. Math. Comput., 188(2):1467–1475, 2007. https://doi.org/10.1016/j.amc.2006.11.013
Y. Yang and Z.Y. Tang. Mapped spectral collocation methods for Volterra integral equations with noncompact kernels. Appl. Numer. Math., 160:166–177, 2021. https://doi.org/10.1016/j.apnum.2020.10.001
G.Q. Yao, Z.Q. Wang and C. Zhang. A multi-domain hybrid spectral collocation method for nonlinear Volterra integral equations with weakly singular kernel. J. Comput. Appl. Math., 444:115785, 2024. https://doi.org/10.1016/j.cam.2024.115785
U. Zrahia and P. Bar-Yoseph. Space-time spectral element method for solution of second-order hyperbolic equations. Comput. Methods Appl. Mech. Engrg., 116:135–146, 1994. https://doi.org/10.1016/S0045-7825(94)80017-0
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Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University.
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