×

zbMATH — the first resource for mathematics

Finite element methods for parabolic and hyperbolic partial integro- differential equations. (English) Zbl 0657.65142
For parabolic integro-partial differential equations of the following forms \[ D_ tu-\nabla \cdot (a(x,t,u)\nabla u)=\int^{t}_{0}f(x,t,s,u(x,s),\nabla u(x,s))ds,\quad (x,t)\in \Omega \times (0,T],\quad and \]
\[ C(x,t,u)D_ tu- D_{xx}u=\int^{t}_{0}f(x,t,s,u(x,s),D_ xu(x,s))ds,\quad (x,t)\in (0,1)\times (0,T], \] the authors formulate and investigate in detail several discrete-time Galerkin methods and discrete-time collocation methods, respectively, which are proved to provide approximations of second order in time and of optimal order of accuracy in space. Besides it they briefly describe Galerkin and collocation methods for hyperbolic integro-partial differential equations and give corresponding error estimates without proof.
Reviewer: V.Kamen

MSC:
65R20 Numerical methods for integral equations
45K05 Integro-partial differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Engler, H., On some parabolic integro-differential equations: existence and asymptotics of solutions, (), 161-167
[2] Habetler, G.T.; Schiffman, R.L., A finite difference method for analyzing the compression of poro-viscoelastic media, Comput., 6, 342-348, (1970) · Zbl 0295.73036
[3] Pao, C.V., Bifurcation analysis of a nonlinear diffusion system in reactor dynamics, Appl. analysis, 9, 107-119, (1979) · Zbl 0404.45009
[4] Pao, C.V., Solution of a nonlinear integrodifferential system arising in nuclear reactor dynamics, J. math. analysis applic., 48, 470-492, (1974) · Zbl 0293.45016
[5] Pachpatte, B.G., On a nonlinear diffusion system arising in reactor dynamics, J. math. analysis applic., 94, 501-508, (1983) · Zbl 0524.35055
[6] Hlavacek, I., On the existence and uniqueness of solution of the Cauchy problem for linear integro-differential equations with operator coefficients, Appl. mat., 16, 64-79, (1971) · Zbl 0217.15801
[7] Aquistapace, P., Existence and maximal time regularity for linear parabolic integrodifferential equations, J. integral eqns, 10, 5-43, (1985)
[8] Lunardi, A.; Sinestrari, E., Cα-regularity for non-autonomous linear integrodifferential equations of parabolic type, J. diff. eqns, 63, 88-116, (1986) · Zbl 0596.45019
[9] Aizicovici, S., Time-dependent Volterra integrodifferential equations, J. integral eqns, 10, 45-60, (1985) · Zbl 0587.45017
[10] Di Blasio, G., Nonautonomous integrodifferential equations in Lp spaces, J. integral eqns, 10, 111-121, (1985) · Zbl 0585.45006
[11] Lunardi, A.; Sinestrari, E., Existence in the large and stability for nonlinear Volterra equations, J. integral eqns, 10, 213-239, (1985) · Zbl 0587.45016
[12] Lunardi, A., Laplace transform methods in integrodifferential equations, J. integral eqns, 10, 185-211, (1985) · Zbl 0587.45015
[13] Sinestrari, E., Time-dependent integrodifferential equations in Banach spaces, (), 939-946
[14] Sinestrari, E., Continuous interpolation spaces and spatial regularity in nonlinear Volterra integrodifferential equations, J. integral eqns, 5, 287-308, (1983) · Zbl 0519.45013
[15] Mangeron, D.; Krivashein, L.E., New methods for numerical calculation for the solutions of various integrodifferential systems of interest in mathematical physics, I, Rev. roum. sci. techn. ser. mec. appl., 9, 1195-1221, (1964) · Zbl 0261.65081
[16] Nohel, J.A., Nonlinear Volterra equations for heat flow in materials with memory, (), 3-82
[17] Raynal, M.L., On some nonlinear problems of diffusion, (), 251-266
[18] Teo, K.L., Existence and uniqueness of solutions of systems governed by second order quasilinear integro-partial differential equations of parabolic type, Aequationes math., 20, 133-148, (1980) · Zbl 0438.45009
[19] Ugowski, H., On integro-differential equations of parabolic and elliptic type, Annls Pol. math., 22, 255-275, (1970) · Zbl 0192.45802
[20] Ugowski, H., On integro-differential equations of parabolic type, Annls Pol. math., 25, 9-22, (1971) · Zbl 0217.43502
[21] Ugowski, H., Some theorems on the estimate and existence of solutions of integro-differential equations of parabolic type, Annls Pol. math., 25, 311-323, (1972) · Zbl 0229.45009
[22] Ugowski, H., On a certain non-linear initial-boundary value problem for integro-differential equations of parabolic type, Annls Pol. math., 28, 249-259, (1973) · Zbl 0243.45014
[23] Heard, M.L., A class of hyperbolic Volterra integrodifferential equations, Nonlinear analysis, 8, 79-93, (1984) · Zbl 0535.45007
[24] Douglas, J.; Jones, B.F., Numerical methods for integro-differential equations of parabolic and hyperbolic types, Num. math., 4, 96-102, (1962) · Zbl 0135.18002
[25] Budak, B.M.; Pavlov, A.R., A difference method of solving boundary-value problems for a quasi-linear integro-differential equation of parabolic type, Soviet math. dokl., 14, 565-569, (1973) · Zbl 0301.65074
[26] Rektorys, K., The method of discretization in time, (1982), D. Reidel Dordrecht, Holland · Zbl 0613.65122
[27] Rektorys, K., Solution of mixed boundary value problems by the method of discretization in time, (), 132-145
[28] D’jakanov, E.G., On the stability of difference schemes for some nonstationary problems, (), 63-87
[29] Oden, J.T.; Armstrong, W.H., Analysis of nonlinear, dynamic coupled thermoviscoelasticity problems by the finite element method, Computers and structures, 1, 603-621, (1971)
[30] Greenwell-Yanik, C.E.; Fairweather, G., Analyses of spline collocation methods for parabolic and hyperbolic problems in the two space variables, SIAM J. numer. analysis, 23, 282-296, (1986) · Zbl 0595.65122
[31] Fairweather, G., Finite element Galerkin methods for differential equations, (1978), Marcel Dekker New York · Zbl 0372.65044
[32] Wheeler, M.F., A priori L^{2} error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. numer. analysis, 10, 723-759, (1973) · Zbl 0232.35060
[33] Dupont, T.; Fairweather, G.; Johnson, J.P., Three-level Galerkin methods for parabolic equations, SIAM J. numer. analysis, 11, 392-409, (1974) · Zbl 0313.65107
[34] Lees, M., An extrapolated Crank-Nicolson difference scheme for quasilinear parabolic equations, (), 193-201 · Zbl 0178.18504
[35] Greenwell, C.E., Finite element methods for partial integro-differential equations, ()
[36] Dendy, J.E., Galerkin’s method for some highly nonlinear problems, SIAM J. numer. analysis, 14, 327-347, (1977) · Zbl 0365.65065
[37] Douglas, J.; Dupont, T., Collocation methods for parabolic equations in a single space variable, ()
[38] Douglas, J.; Dupont, T., A finite element collocation method for quasilinear parabolic equations, Maths comput., 27, 17-28, (1973) · Zbl 0256.65050
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.