Legendre-Chebyshev pseudo-spectral method for the diffusion equation with non-classical boundary conditions

Chattouh, Abdeldjalil; Saoudi, Khaled

doi:10.2478/mjpaa-2020-0023

Abstract

The present paper is devoted to the numerical approximation for the diffusion equation subject to non-local boundary conditions. For the space discretization, we apply the Legendre-Chebyshev pseudospectral method, so that, the problem under consideration is reduced to a system of ODEs which can be solved by the second order Crank-Nicolson schema. Optimal error estimates for the semi-discrete scheme are derived in L²-norm. Numerical tests are included to demonstrate the effectiveness of the proposed method.

References

[1] A. Borhanifar, S. Shahmorad, A matrix formulated algorithm for solving parabolic equations with nonlocal boundary conditions, Num. Algo., Vol. 74, No. 4 (2017), 12031221.10.1007/s11075-016-0192-x
Search in Google Scholar Back to article
[2] A. Bouziani, N. Merazga, S. Benamira, Galerkin method applied to a parabolic evolution problem with nonlocal boundary conditions, Nonlin. Anal.: TMA, Vol. 69, No. 5-6 (2008), 1515-1524.10.1016/j.na.2007.07.008
Search in Google Scholar Back to article
[3] J.R. Cannon, The solution of the heat equation subject to the specification of energy, Quar. Appl. Math., Vol. 21 (1963), 155-160.
Search in Google Scholar Back to article
[4] C. Canuto, M.Y. Hussaini, A. Quarteroni, Jr. Thomas, Spectral methods in fluid dynamics, Springer Science & Business Media, NewYork, (2012).
Search in Google Scholar Back to article
[5] V. Capasso, K. Kunisch, A reactiondiffusion system arising in modeling man-environment diseases, Quart. Appl. Math., Vol. 40, No. 3 (1988), 431-449.
Search in Google Scholar Back to article
[6] R. Čiupaila, M. Sapavogas, K. Pupalaigė, M-matrices and Convergence of Finite Difference Scheme for Parabolic Equation with an Integral Boundary Condition, Math. Mod. Analysis, Vol. 25, No. 2 (2020), 167-183.
Search in Google Scholar Back to article
[7] J.H. Cushman, H. Xu, F. Deng, Nonlocal reactive transport with physical and chemical heterogeneity: Localization error, Water Resour. Res., Vol 31, No. 9 (1995), 2219-2237.
Search in Google Scholar Back to article
[8] W.A. Day, A decreasing property of solutions of parabolic equations with applications to thermoelasticity, Quart. Appl. Math., Vol. 40, No. 4 (1983), 319-330.
Search in Google Scholar Back to article
[9] W.A. Day, Parabolic equations and thermodynamics, Quart. Appl. Math., Vol. 50, No. 3 (1992), 523-533.
Search in Google Scholar Back to article
[10] M. Dehghan, Second-order schemes for a boundary value problem with Neumann’s boundary conditions, J. Comput. Appl. Math., Vol. 138, No. 1 (2002),173-184.
Search in Google Scholar Back to article
[11] M. Dehghan, M. Shamsi: Numerical solution of two-dimensional parabolic equation subject to nonstandard boundary specifications using the pseudospectral legendre method, Numer. Methods Partial. Differ. Equ., Vol. 22, No. 6 (2006), 12551266.10.1002/num.20150
Search in Google Scholar Back to article
[12] G. Fairweather, J.C. Lpez-Marcos, Galerkin methods for a semilinear parabolic problem with nonlocal boundary conditions, Adv. Comput. Math., Vol. 6 (1996), 243262.10.1007/BF02127706
Search in Google Scholar Back to article
[13] A. Golbabai, M. JavidiA numerical solution for non-classical parabolic problem based on Chebyshev spectral collocation, Appl. Math. Comput., Vol. 190, No. 1 (2007), 179-185.
Search in Google Scholar Back to article
[14] D. Gotlieb, S.A. Orszag, Numerical analysis of spectral methods: theory and applications, SIAM, Philadelphia, (1977).
Search in Google Scholar Back to article
[15] L. Hu, L. Ma, J. Shen, Efficient Spectral-Galerkin Method and Analysis for Elliptic PDEs with Non-local Boundary Conditions, J. Sci. Comput., Vol. 88, No. 2 (2016), 417-437.
Search in Google Scholar Back to article
[16] H. Ma, ChebyshevLegendre spectral viscosity method for nonlinear conservation laws, SIAM J. Numer. Anal., Vol. 35 (1998), 869892.10.1137/S0036142995293900
Search in Google Scholar Back to article
[17] J. Martin-Vaquero, Two-level fourth-order explicit schemes for diffusion equations subject to boundary integral specifications, Chaos Solit. Fract., Vol. 42 (2009), 2364-2372.
Search in Google Scholar Back to article
[18] J. Shen, T. Tang, L.L. Wang, Spectral methods: algorithms, analysis and applications, Springer Science & Business Media, Berlin, (2011).
Search in Google Scholar Back to article
[19] M. Slodička, Semilinear parabolic problems with nonlocal Dirichlet boundary conditions, J. Comput. App. Math., Vol. 19, No. 5 (2011), 705-716.
Search in Google Scholar Back to article
[20] M. Slodička, S. Dehilis, A nonlinear parabolic equation with a nonlocal boundary term, J. Comput. App. Math., Vol. 233, No. 12 (2010), 3130-3138.
Search in Google Scholar Back to article
[21] M. Slodička, S. Dehilis, A numerical approach for a semilinear parabolic equation with a nonlocal boundary condition, J. Comput. App. Math., Vol. 231, No. 2 (2009), 715-724.
Search in Google Scholar Back to article
[22] Z. Sun, A second-order accurate finite difference scheme for a class of nonlocal parabolic equations with natural boundary conditions, J. Comput. Appl. Math., Vol. 76, No. 4 (1996), 137-146.
Search in Google Scholar Back to article
[23] E. Tohidi, F. Toutounian, Numerical solution of time-dependent diffusion equations with nonlocal boundary conditions via a fast matrix approach, J. Egy. Math. Soc., Vol. 24, No. 1 (2016), 86-91.
Search in Google Scholar Back to article
[24] T. Zhao, C. Li, Z. Zang and Y. Wu, Chebyshev-Legendre pseudo-spectral method for the generalised Burgers-Fisher equation, App. Math. Mod., Vol. 36, No. 3 (2012), 1046-1056.
Search in Google Scholar Back to article

Legendre-Chebyshev pseudo-spectral method for the diffusion equation with non-classical boundary conditions

Abstract

Paradigm

My account