On the Numerical Solution of a Semilinear Sobolev Equation Subject to Nonlocal Dirichlet Boundary Condition

Year 2020, Volume: 3 Issue: 1, 11 - 18, 15.12.2020

Abdeldjalil Chattouh , Khaled Saoudi

Abstract

A semilinear pseudo-parabolic equation ∂t(u − ∆u) − ∆u = f(∇u) with a Dirichlet-type integral boundary condition is investigated in this contribution. Using the Rothe method which is based on a semi-discretization of the problem under consideration with respect to the time variable, we prove the existence and uniqueness of a solution in a weak sense. For the spatial discretization, a suitable approach based on Legendre spectral-method is presented. Two numerical examples are included to examine the effectiveness and accuracy of the proposed approach.

Keywords

Nonlocal boundary condition, Rothe method, Sobolev equation, Spectral method

References

• 1 A. Bouziani, N. Merazga, and S. Benamira, Galerkin method applied to a parabolic evolution problem with nonlocal boundary conditions, Nonlin. Anal. 69 (2008), 1515–1524.
• 2 S. Cohn, K. Pfabe, and J. Redepenning, A similarity solution to a problem in nonlinear ion transport with a nonlocal condition, Math. Models Methods Appl. Sci. 9(3) (1999), 445–461.
• 3 W.A. Day, Extensions of a property of the heat equation to linear thermoelasticity and other theories, Q. Appl. Math. 40 (1982), 319–330.
• 4 W.A. Day, A decreasing property of solutions of parabolic equations with applications to thermoelasticity, Q. Appl. Math. 41 (1983), 468–475.
• 5 A. Hasanov, B. Pektas, and S. Hasanoglu, An analysis of nonlinear ion transport model including diffusion and migration, J. Math. Chem. 46(4) (2009), 1188–1202.
• 6 L. Hu, L. Ma and J. Shen, Efficient spectral-Galerkin method and analysis for elliptic PDEs with non-local boundary conditions, J. Sci. Compu. 68(2) (2016), 417–437.
• 7 A. Guezane-Lakoud, D. Belakroum , Time-discretization schema for an integrodifferential Sobolev type equation with integral conditions, App. Math. Compu. 212 (2012), 4695–4702.
• 8 J. Kacur, Method of Rothe in Evolution Equations, Teubner Texte zur Mathematik., Teubner, Leipzig, 1985.
• 9 A. Merad, A. Bouziani and S. Araci, Existence and uniqueness for a solution of pseudohyperbolic equation with nonlocal noundary condition, Appl. Math. Inf. Sci. 9(4) (2015), 1855–1861.
• 10 M. Slodicka and S. Dehilis, A numerical approach for a semilinear parabolic equation with a nonlocal boundary condition, J. Comput. Appl. Math. 231 (2009), 715–724.
• 11 M. Slodicka and S. Dehilis, A nonlinear parabolic equation with a nonlocal boundary term, J. Comput. Appl. Math. 233(12) (2010), 3130–3138.
• 12 M. Slodicka, Semilinear parabolic problems with nonlocal Dirichlet boundary conditions, Inverse. Prob. Sci. Eng. 19(5) (2011), 705–716.
• 13 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)(2012), 1046–1056.