Accurate spectral collocation solutions to some Bratu’s type boundary value problems

We solve by Chebyshev spectral collocation some genuinely nonlinear Liouville-Bratu-Gelfand type, 1D and a 2D boundary value problems. The problems are formulated on the square domain [−1, 1] × [−1, 1] and the boundary condition attached is a homogeneous Dirichlet one. We pay a particular attention to the bifurcation branch on which a solution is searched and try to estimate empirically the attraction basin for each bifurcation variety. The first eigenvector approximating the corresponding the first eigenfunction of the linear problem is used as an initial guess in solving the nonlinear algebraic system of Chebyshev collocation to find the “small”solution. For the same value of the bifurcation parameter we use another initial guess, namely lowest basis function (1 point approximation), to find the “big”solution. The Newton-Kantorovich method solves very fast the nonlinear algebraic system in no more than eight iterations. Beyond being exact, the method is numerically stable, robust and easy to implement. Actually, the MATLAB code essentially contains three programming lines. It by far surpasses in simplicity and accuracy various methods used to solve some well-known problems. We end up by providing some numerical and graphical outcomes in order to underline the validity and the effectiveness of our method, i.e., norms of Newton updates in solving the algebraic systems and the decreasing rate of Chebyshev coefficients of solution.

Calin-Ioan Gheorghiu
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

Bratu’s problem; nonlinear eigenvalue problem; spectral collocation; accuracy, Chebfun.

https://arxiv.org/pdf/2011.13212

C.I. Gheorghiu, Accurate spectral collocation solutions to some Bratu’s type boundary value problems, arXiv:2011.13212

Google Scholar Profile

[1] Boyd, J.P., An Analytical and Numerical Study of the Two-Dimensional Bratu Equation. J. Sci. Comput. 1, 183-201 (1986)
[2] Boyd, J.P., Chebyshev polynomial expansions for simultaneous approximation of two branches of a function with application to the one-dimensional Bratu equation. Appl. Math. Comput. 143, 189-200 (2003)
[3] Boyd, J.P., Gheorghiu, C.I., All roots spectral methods: Constraints, floating point arithmetic and root exclusion. Applied Mathematics Letters 67, 28-32 (2017)
[4] Driscoll, T.A., Hale, N., Trefethen, L.N., Chebfun Guide. 1st Ed. For Chebfun version 5. (2014) https://www.chebfun.org.
[5] Gelfand, I.M., Some problems in the theory of quasi-linear equations. AMS Trans. Ser. 29, 295-381 (1963)
[6] Gheorghiu, C.I., Trif, D., The numerical approximation to positive solution for some reaction diffusion problems. PU. M. A. 11, 243-253 (2000)
[7] Gheorghiu, C.I., Trif, D., Direct and Indirect Approximations to Positive Solution for a Nonlinear Reaction-Diffusion Problem I. Direct (Variational) Rev. Anal. Numer. Theor. Approx.31 61-69 (2002)
[8] Gheorghiu, C.I., A third-order nonlinear BVP on the half-line. https://www.chebfun.org/examples/ode-nonlin/GulfStream.html
[9] Grindrod, P., The Theory and Applications of Reaction-Diffusion Equation, 2nd edn. Patterns and Waves. Clarendon Press, Oxford (1996)
[10] Haidvogel, D. B., Zang, T., The efficient solution of Poisson’s equation in two dimensions via Chebyshev approximation, J. Comp. Phys. 30, 167-180 (1979).
[11] Huang, S-Y., Wang, S-H., Proof of a Conjecture for the One-Dimensional Perturbed Gelfand Problem from Combustion Theory. Arch. Rational Mech. Anal. 222, 769-825 (2016). https://doi.org/: 10.1007/s00205-016-1011-1
[12] Kouibia, A., Pasadas, M., Akhrif, R., A variational method for solving two-dimensional Bratu’s problem. Numer. Algor. 84, 1589-1599 (2020). https://doi.org/10.1007/s11075-020-00957-y
[13] Kuttler, J. R., Sigillito, V. G., Eigenvalues of the Laplacian in Two Dimensions. SIAM Rev. 26, 163-193 (1984)
[14] Trefethen, L.N., Birkisson, A, Driscoll, T. A.,  Exploring ODEs. SIAM, Philadelphia (2018)
[15] Weideman, J.A.C., Reddy, S. C., A MATLAB Differentiation Matrix Suite. ACM T. Math. Software. 26, 465-519 (2000)