×

A Galerkin method for two-dimensional hyperbolic integro-differential equation with purely integral conditions. (English) Zbl 1410.65376

Summary: The present paper is devoted to the solution for two-dimensional hyperbolic integro-differential equations subject to purely integral conditions. First, it is demonstrated the existence and uniqueness of the solution under certain conditions. For the numerical approach, a Galerkin method based on least squares is proposed. The numerical examples illustrate the technique and show the efficiency of the proposed method.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
35R09 Integro-partial differential equations
65R20 Numerical methods for integral equations
45K05 Integro-partial differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bouziani, A.; Benouari, N.-E., Probleme mixte avec conditions integrales pour une classe d’equations paraboliques, Comptes Rendus de l’Académie des Sciences Série, 9, 1177-1182 (1995) · Zbl 0837.35057
[2] Allegretto, W.; Lin, Y.; Zhou, A., A box scheme for coupled systems resulting from microsensor thermistor problems, Dyn. Discr., Contin. Impuls. Syst., 5, 209-223 (1999) · Zbl 0979.78023
[3] Ashyralyev, A.; Yildirim, O., A note on the second order of accuracy stable difference schemes for the nonlocal boundary value hyperbolic problem, Abstr. Appl. Anal., 2012 (2012) · Zbl 1253.65123
[4] Bouziani, A., Mixed problem with boundary integral conditions for a certain parabolic equation, J. Appl. Math. Stoch. Anal., 9, 3, 323-330 (1996) · Zbl 0864.35049
[5] Bouziani, A., Solution forte d’un probleme mixte avec condition intégrale pour une classe d’equations paraboliques, Maghreb Math.Rev., 6, 1-17 (1997)
[6] Bouziani, A., Solution forte d’un probleme mixte avec condition non locale pour une classe d’equations hyperboliques, Acad. R. Belg. Bull. Cl. Sci., 8, 53-70 (1997) · Zbl 1194.35239
[7] Capasso, V.; Kunish, K., A reaction-diffusion system arising in modelling man-environment diseases, Quart. Appl. Math., 46, 431-450 (1988) · Zbl 0704.35069
[8] Chrysafinos, K., Approximations of parabolic integro-differential equations using wavelet-galerkin compression techniques, BIT Numer. Math., 47, 3, 487-505 (2007) · Zbl 1126.65115
[9] Cushman, J. H.; Xu, H.; Deng, F., Nonlocal reactive transport with physical and chemical heterogeneity: localization errors, Water Resour. Res., 31, 9, 2219-2237 (1995)
[10] Dagan, G., The significance of heterogeneity of evolving scales to transport in porous formations, Water Resour. Res., 30, 3327-3336 (1994)
[11] Day, W. A., Parabolic equations and thermodynamics, Quart. Appl. Math., 50, 523-533 (1992) · Zbl 0794.35069
[12] Larsson, S.; Thomée, V.; Wahlbin, L. B., Numerical solution of parabolic integro-differential equations by the discontinuous galerkin method, Math. Comput., 67, 221, 45-71 (1998) · Zbl 0896.65090
[13] Martín-Vaquero, J., Two-level fourth-order explicit schemes for diffusion equations subject to boundary integral specifications, Chaos, Solitons Fract., 42, 4, 2364-2372 (2009) · Zbl 1198.65004
[14] Martín-Vaquero, J., Polynomial-based mean weighted residuals methods for elliptic problems with nonlocal boundary conditions in the rectangle, Nonlinear Anal.: Model. Control, 19, 3, 448-459 (2014) · Zbl 1314.65153
[15] Martín-Vaquero, J.; Queiruga-Dios, A.; Encinas, A. H., Numerical algorithms for diffusion-reaction problems with non-classical conditions, Appl. Math. Comput., 218, 9, 5487-5492 (2012) · Zbl 1244.65126
[16] Martín-Vaquero, J.; Wade, B. A., On efficient numerical methods for an initial-boundary value problem with nonlocal boundary conditions, Appl. Math. Model., 36, 8, 3411-3418 (2012) · Zbl 1252.65167
[18] Merad, A.; Bouziani, A.; Araci, S., Existence and uniqueness of a solution for pseudohyperbolic equation with nonlocal boundary condition, Appl. Math. Inf. Sci., 9, 2, 1-7 (2015)
[19] Merad, A.; Bouziani, A.; Ozel, C.; Kiliçman, A., On solvability of the integrodifferential hyperbolic equation with purely nonlocal conditions, Acta Math. Sci., 35, 3, 601-609 (2015) · Zbl 1340.65322
[20] Sapagovas, M.; Čiupaila, R.; Jokšienė, Ž.; Meškauskas, T., Computational experiment for stability analysis of difference schemes with nonlocal conditions, Informatica, 24, 275-290 (2013) · Zbl 1360.65216
[21] Shen, J.; Tang, T., Spectral and High-order Methods with Applications (2006), Science Press · Zbl 1234.65005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.