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Amiraliyev, Gabil M.; Cimen, Erkan; Amirali, Ilhame; Cakir, Musa

High-order finite difference technique for delay pseudo-parabolic equations. (English) Zbl 1366.65076

J. Comput. Appl. Math. 321, 1-7 (2017).
Summary: One dimensional initial-boundary delay pseudo-parabolic problem is being considered. To solve this problem numerically, we construct higher order difference method for approximation to the considered problem and obtain the error estimate for its solution. Based on the method of energy estimate the fully discrete scheme is shown to be convergent of order four in space and of order two in time. Numerical example is presented.

Cited in 16 Documents

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35K70 Ultraparabolic equations, pseudoparabolic equations, etc.

Keywords:

pseudo-parabolic equation; delay difference scheme; error estimate; convergence; initial boundary problem; numerical example
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References:

[1] Chen, P. J.; Gurtin, M. E., On a theory of heat conduction involving two temperatures, Z. Angew. Math. Phys., 19, 614-627 (1968) · Zbl 0159.15103
[2] Huilgol, R. A., A second order fluids of the differential type, Internat. J. Non-Linear Mech., 3, 471-482 (1968) · Zbl 0181.54004
[3] Barenblatt, G.; Entov, V.; Ryzhik, V., Theory of Fluid Flow Through Natural Rocks (1990), Kluwer: Kluwer Dordrecht · Zbl 0769.76001
[4] Yang, M., Analysis of second order finite volume element methods for pseudo-parabolic equations in three spatial dimensions, Appl. Math. Comput., 196, 94-104 (2008) · Zbl 1136.65087
[5] van Dujin, C. J.; Fan, Y.; Peletier, L. A.; Pop, I. S., Travelling wave solutions for degenerate pseudo-parabolic equations modelling two-phase flow in porous media, Nonlinear Anal. RWA, 14, 1361-1383 (2013) · Zbl 1261.35125
[6] Ford, W. H.; Ting, T. W., Uniform error estimates for difference approximations to nonlinear pseudo-parabolic partial differential equations, SIAM J. Numer. Anal., 15, 155-169 (1974) · Zbl 0244.65064
[7] Ewing, R. E., Time-stepping Galerkin methods for nonlinear Sobolev partial differential equations, SIAM J. Numer. Anal., 15, 1125-1150 (1978) · Zbl 0399.65083
[8] Amiraliyev, G. M., Difference method for the solution of one problem of the theory of dispersive waves, USSR Differ. Equ., 26, 2146-2154 (1990)
[9] Amiraliyev, G. M.; Mamedov, Y. D., Difference schemes on the uniform mesh for a singularly perturbed pseudo-parabolic equations, Tr. J. Math., 19, 207-222 (1995) · Zbl 0859.65088
[10] Sun, T.; Yang, D., The finite difference streamline diffusion methods for Sobolev equations with convection-dominated term, Appl. Math. Comput., 125, 325-345 (2002) · Zbl 1030.65095
[11] Amiraliyev, G. M.; Amiraliyeva, I. G., Difference schemes for the singularly perturbed Sobolev equations, (Proceedings of the International Conference on Difference Equations, Special Functions and Applications (2005), Elsevier: Elsevier Munich) · Zbl 1127.65057
[12] Amiraliyev, G. M.; Duru, H.; Amiraliyeva, I. G., A parameter-uniform numerical method for a Sobolev problem with initial layer, Numer. Algorithms, 44, 185-203 (2007) · Zbl 1125.65077
[13] Cuesta, C. M.; Pop, I. S., Numerical schemes for a pseudo-parabolic burgers equation: Discontinuous data and long-time behaviour, J. Comput. Appl. Math., 224, 269-283 (2009) · Zbl 1154.76036
[14] Fan, Y.; Pop, I. S., Equivalent formulations and numerical schemes for a class of pseudo-parabolic equations, J. Comput. Appl. Math., 246, 86-93 (2013) · Zbl 1452.35082
[15] Bouzani, A., Initial-boundary value problems for a class of pseudo-parabolic equations with integral boundary conditions, J. Math. Anal. Appl., 291, 371-386 (2004) · Zbl 1064.35095
[16] Ptashnyk, M., Nonlinear pseudo parabolic equations as singular limit of reaction-diffusion equations, Appl. Anal., 85, 1285-1299 (2006) · Zbl 1107.35061
[17] Fan, Y.; Pop, I. S., A class of pseudo-parabolic equations: existence, uniqueness of weak solutions, and error estimates for the Euler-implicit discretization, Math. Methods Appl. Sci., 34, 2329-2339 (2011) · Zbl 1235.35182
[18] Lightbourne, J. H.; Rankin, S. M., A partial functional differential equation of Sobolev type, J. Math. Anal. Appl., 93, 328-337 (1983) · Zbl 0519.35074
[19] Di, H.; Shang, Y., Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms, Discrete Contin. Dyn. Syst. Ser. B, 21, 781-801 (2016) · Zbl 1331.35204
[20] Okcu, P.; Amiraliyev, G. M., Error estimates for differential difference schemes to pseudo-parabolic initial-boundary value problem with delay, Math. Comput. Appl., 18, 283-292 (2013) · Zbl 1396.65141
[21] Amirali, I.; Amiraliyev, G. M.; Cakir, M.; Cimen, E., Explicit finite difference methods for the delay pseudo parabolic equations, Sci. World J., 2014, 7 (2014)
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