Error analysis of Legendre-Galerkin spectral method for a parabolic equation with Dirichlet-Type non-local boundary conditions
Abstract
An efficient Legendre-Galerkin spectral method and its error analysis for a one-dimensional parabolic equation with Dirichlet-type non-local boundary conditions are presented in this paper. The spatial discretization is based on Galerkin formulation and the Legendre orthogonal polynomials, while the time derivative is discretized by using the symmetric Euler finite difference schema. The stability and convergence of the semi-discrete spectral approximation are rigorously set up by following a novel approach to overcome difficulties caused by the non-locality of the boundary condition. Several numerical tests are included to confirm the efficacy of the proposed method and to support the theoretical results.
Keyword : spectral methods, Galerkin method, parabolic equation, non-local boundary conditions, error estimate
How to Cite
Chattouh, A., & Saoudi, K. (2021). Error analysis of Legendre-Galerkin spectral method for a parabolic equation with Dirichlet-Type non-local boundary conditions. Mathematical Modelling and Analysis, 26(2), 287-303. https://doi.org/10.3846/mma.2021.12865
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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M. Tatari and M. Dehghan. On the solution of the non-local parabolic partial differential equations via radial basis functions. Appl. Math. Model., 33(3):1729– 1738, 2009. https://doi.org/10.1016/j.apm.2008.03.006
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S. Zhou and M. Cui. Approximate solution for a variable-coefficient semilinear heat equation with nonlocal boundary conditions. Int. J. Comput. Math., 86(12):2248–2258, 2009. https://doi.org/10.1080/00207160903229881
Z. Avazzadeh and H. Hassani. Transcendental bernstein series for solving reaction–diffusion equations with nonlocal boundary conditions through the optimization technique. Numer. Methods Partial Differ. Equ., 35(6):2258–2274, 2019. https://doi.org/10.1002/num.22411
C. Bernardi and Y. Maday. Spectral methods. Handbook of Numerical Analysis, 5:209–485, 1997. https://doi.org/10.1016/S1570-8659(97)80003-8
A. Borhanifar, S. Shahmorad and E. Feizi. A matrix formulated algorithm for solving parabolic equations with nonlocal boundary conditions. Numer. Algor., 74(4):1203–1221, 2017. https://doi.org/10.1007/s11075-016-0192-x
C. Canuto, M. Hussaini, A. Quarteroni and T. Zang. Spectral Methods: Fundamentals in Single Domains, volume 23. Springer Science & Business Media, 01 2006. https://doi.org/10.1007/978-3-540-30726-6
V. Capasso and K. Kunisch. A reaction-diffusion system arising in modeling man-environment diseases. Quart. Appl. Math., 46(3):431–450, 1988. https://doi.org/10.1090/qam/963580
R. Čiegis, G. Jankevičiūtė, T. Leonavičienė and A. Mirinavičius. On stability analysis of finite-difference schemes for some parabolic problems with nonlocal boundary conditions. Numer. Func. Anal. Opt., 35(10):1308–1327, 2014. https://doi.org/10.1080/01630563.2014.908208
R. Čiegis and N. Tumanova. Numerical solution of parabolic problems with nonlocal boundary conditions. Numer. Func. Anal. Opt., 31(12):1318–1329, 2010. https://doi.org/10.1080/01630563.2010.526734
R. Čiegis and N. Tumanova. On construction and analysis of finite difference schemes for pseudoparabolic problems with nonlocal boundary conditions. Math. Model. Anal., 19(2):281–297, 2014. https://doi.org/10.3846/13926292.2014.910562
R. Čiupaila, M. Sapagovas and K. Pupalaigė. M-matrices and convergence of finite difference scheme for parabolic equation with integral boundary condition. Math. Model. Anal., 25(2):167–183, 2020. https://doi.org/10.3846/mma.2020.8023
M. Cui. Compact finite difference schemes for the time fractional diffusion equation with nonlocal boundary conditions. Compu. Appl. Math., 37(3):3906–3926, 2018. https://doi.org/10.1007/s40314-017-0553-7
J.H. Cushman, H. Xu and F. Deng. Nonlocal reactive transport with physical and chemical heterogeneity: Localization error. Water Resour. Res., 31:2219– 2237, 1995. https://doi.org/10.1029/95WR01396
D. Glotov, W.E. Hames, A.J. Meir and S. Ngoma. An integral constrained parabolic problem with applications in thermochronology. Comput. Math. App., 71(11):2301–2312, 2016. https://doi.org/10.1016/j.camwa.2016.01.017
L. Hu, L. Ma and J. Shen. Efficient spectral-Galerkin method and analysis for elliptic PDEs with non-local boundary conditions. J. Sci. Comput., 68(2):417– 437, 2016. https://doi.org/10.1007/s10915-015-0145-x
F. Ivanauskas, V. Laurinavičius, M. Sapagovas and A. Nečiporenko. Reactiondiffusion equation with nonlocal boundary condition subject to PIDcontrolled bioreactor. Nonlinear Anal. Model. Control., 22(2):261–272, 2017. https://doi.org/10.15388/NA.2017.2.8
K. Jakubėlienė, R. Čiupaila and M. Sapagovas. Semi-implicit difference scheme for a two-dimensional parabolic equation with an integral boundary condition. Math. Model. Anal., 22(5):617–633, 2017. https://doi.org/10.3846/13926292.2017.1342709
M. Khaksarfard, Y. Ordokhani and E. Babolian. An approximate method for solution of nonlocal boundary value problems via Gaussian radial basis functions. SeMA J., 76(1):123–142, 2019. https://doi.org/10.1007/s40324-018-0165-1
Y. Liu. Numerical solution of the heat equation with nonlocal boundary conditions. J. Comput. Appl. Math., 110(1), 1999. https://doi.org/10.1016/S0377-0427(99)00200-9
R. Michael, J.H. William and A.N. John. Mathematical problems in viscoelasticity, volume 35 of Pitman Monographs and Surveys in Pure and Applied Mathematics. Longman Scientific & Technical, 1987.
L. Mu and H. Du. The solution of a parabolic differential equation with nonlocal boundary conditions in the reproducing kernel space. Appl. Math. Comput., 202(2):708–714, 2008. https://doi.org/10.1016/j.amc.2008.03.008
M. Sapagovas, R. Čiupaila, Ž. Jokšienė and T. Meškauskas. Computational experiment for stability analysis of difference schemes with nonlocal conditions. Informatica, 24(2):275–290, 2013. https://doi.org/10.15388/Informatica.2013.396
M. Sapagovas, T. Meškauskas and F. Ivanauskas. Influence of complex coefficients on the stability of difference scheme for parabolic equations with non-local conditions. Appl. Math. Comput., 332:228–240, 2018. https://doi.org/10.1016/j.amc.2018.03.072
M. Sapagovas, O. Štikonienė,ˇ K. Jakubėlienė and R. Čiupaila. Finite difference method for boundary value problem for nonlinear elliptic equation with nonlocal conditions. Bound. Value Probl., 2019(1):94, 2019. https://doi.org/10.1186/s13661-019-1202-4
S. Shahmorad, A. Borhanifar and B. Soltanalizadeh. An approximation method based on matrix formulated algorithm for the heat equation with nonlocal boundary conditions. Comput. Methods Appl. Math., 12(1):92–107, 2012. https://doi.org/10.2478/cmam-2012-0006
V.V. Shelukhin. A non-local in time model for radionuclides propagation in Stokes fluid. Dinamika. Sploshn. Sredy., 107:180–193, 1993.
E. Shivanian and A. Khodayari. Meshless local radial point interpolation (MLRPI) for generalized telegraph and heat diffusion equation with nonlocal boundary conditions. J. Theor. Appl. Mech., 55(2):571–582, 2017. https://doi.org/10.15632/jtam-pl.55.2.571
B. Shizgal. Spectral methods in chemistry and physics. Scientific Computation, Springer, Dordrecht, 2015.
M. Slodička. Semilinear parabolic problems with nonlocal Dirichlet boundary conditions. Inverse Probl. Sci. Eng., 19(05):705–716, 2011. https://doi.org/10.1080/17415977.2011.579608
M. Tatari and M. Dehghan. On the solution of the non-local parabolic partial differential equations via radial basis functions. Appl. Math. Model., 33(3):1729– 1738, 2009. https://doi.org/10.1016/j.apm.2008.03.006
S.A. Yousefi, M. Behroozifar and M. Dehghan. The operational matrices of Bernstein polynomials for solving the parabolic equation subject to specification of the mass. J. Comput. Appl. Math., 235(17):5272–5283, 2011. https://doi.org/10.1016/j.cam.2011.05.038
T. Zhao, C. Li, Z. Zang and Y. Wu. Chebyshev-Legendre pseudo-spectral method for the generalised Burgers-Fisher equation. Appl. Math. Model., 36(3):1046– 1056, 2012. https://doi.org/10.1016/j.apm.2011.07.059
S. Zhou and M. Cui. Approximate solution for a variable-coefficient semilinear heat equation with nonlocal boundary conditions. Int. J. Comput. Math., 86(12):2248–2258, 2009. https://doi.org/10.1080/00207160903229881