Localized Boundary-Domain Integro-partial Differential Formulations for Transient Scalar Transport Problems
Tóm tắt
Integro-partial differential equations, though very challenging can still be solved using conventional numerical techniques. Although these problems have been severally solved, it is still noticeable how little has been said about them in boundary element method (BEM) literature. A major reason for this is that for those problems where an encounter with the problem domain becomes a necessity, BEM’s inadequacies become highly apparent. Moreover for such situations, the fundamental solution is either not available in a cheaply computable form or a considerable numerical effort is required to handle this numerical challenge. Sometimes, the fundamental solutions may exist in a form that is highly non-local and lead to system of equations with a fully populated matrix that is cumbersome to handle numerically, especially for field problems. In the work reported herein, we adopt a fundamental solution of an auxiliary form of a governing partial differential equation coupled with the Green’s identity to discretize and localize an integro-partial differential transport equation by conversion into a boundary-domain form which is amenable to a hybrid boundary integral numerical formulation. It is observed that this numerical formulation is straightforward and yields accurate results when compared with those found in literature.
Tài liệu tham khảo
Barbeiro, S., Ferreira, J.A.: Integro-differential models for percutaneous drug absorption. Int. J. Comput. Math. 84, 451–467 (2007)
Branco, J.R., de Oliveira, F.J.A.: Numerical methods for the generalized Fisher–Kolmogorov–Petrovskii–Piskunov equation. Appl. Numer. Math. 57, 89–102 (2007)
Cattaneo, C.: Sulla Conduzione del calore atti del. Semin. Mat. Fis. Univ. Modena 3, 3–21 (1948)
Carini, A., Diligenti, M., Maier, G.: Boundary integral equation analysis in linear viscoelasticity variational and saddle point formulations. Comput. Mech. 8, 87–98 (1991)
Curtin, M.E., Pipkin, A.C.: A general theory of heat conduction with finite wave speeds. Arch. Ration. Mech. Anal. 31, 113–126 (1968)
Ferreira, J.A., de Oliveira, P.: Memory effects and random walks in reaction-transport systems. Appl. Anal. 86(1), 99–118 (2007)
Feng, X.F., Tian, Z.F.: Alternating group explicit method with exponential-type for the diffusion convection equation. Int. J. Comput. Math. 83, 765–775 (2006)
Galenko, P.K., Elder, K.R.: Marginal stability analysis of the phase field crystal model in one spatial dimension. Phys. Rev. B 83, 064113 (2011)
Gaul, L., Schanz, M.: BEM formulation in time domain for viscoelastic media based on analytical time integration. In: Brebbia, C., Dominguez, J., Paris, F. (eds.) Boundary Elements XIV, vol. II, pp. 223–234. Computational Mechanics Publications, Southampton (1992)
Hristov, J.: A note on the integral approach to the nonlinear heat conduction with Jeffrey’s fading memory. Therm. Sci. 17, 733–737 (2013)
Khuri, S.A., Sayfy, A.: A numerical approach for solving an extended Fisher–Kolmogorov–Petrovskii–Piskunov equation. J. Comput. App. Math. 233, 2081–2089 (2010)
Lubisch, C.: Convolution quadrature and discretized operational calculus. Numer. Math. 52, 129–145 (1988)
Mendez, V., Campos, D.: Population extinction and survival in a hostile environment. Phys. Rev. E 77, 022901 (2008)
Onyejekwe, O.O.: A Green element solution of the diffusion equation. In: 34th Heat Transfer and Fluid Mechanics Institute California State University Sacramento California, pp. 77–98 (1995)
Onyejekwe, O.O.: A Green element description of mass transfer in reacting systems. Numer. Heat Transf. B 30, 483–498 (1996)
Onyejekwe, O.O.: Green element solutions of nonlinear diffusion–reaction model. Comput. Chem. Eng. 26, 423–427 (2002)
Onyejekwe, O.O.: A note on Green element method discretization for Poisson equation in polar coordinate. Appl. Math. Lett. 19(8), 785–788 (2006)
Onyejekwe, O.O.: The effect of time stepping schemes on the accuracy of Green element formulation of unsteady transport. J. Appl. Math. Phys. 2, 621–633 (2014)
Schanz, M., Antes, H.: Application of operation quadrature methods in time domain boundary element methods. Meccanica 32(3), 179–186 (1997)
Ramos, J.I.: The application of finite difference and finite element methods to reaction-diffusion system in combustion. In: Taylor, C., Johnson, J.A., Smith, W.R. (eds.) Numerical Methods in Laminar and Turbulent Flow, pp. 1117–1127. Pineridge Press, Swansea (1983)
Shaw, R.P.: Green-functions for heterogeneous media potential problems. Eng. Anal. Bound. Elem. 13(3), 219–221 (1994)
Shaw, R.P., Manolis, G.D.: Elastic waves in one-dimensionally layered heterogeneous soil media. Adv. Earthq. 5, 215–246 (2000)
Taigbenu, A.E.: The Green element method. Int. J. Numer. Methods Eng. 38, 2241–2263 (1995)
Taigbenu, A.E., Onyejekwe, O.O.: Transient 1-D transport equation simulated by a mixed Green element formulation. Int. J. Numer. Methdos Fluids 25(4), 437–454 (1997)
Wolf, J.P., Dabre, G.R.: Time-domain boundary element method in viscoelasticity with application to a spherical cavity. Soil Dyn. Earthq. Eng. 5, 138–148 (1986)