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Nonlinear models and problems in applied sciences from differential quadrature to generalized collocation methods. (English) Zbl 0898.65074
Mathematical methods for nonlinear problems in applied sciences are investigated. The contents are based on a quadrature method proposed by R. Bellmann, B. G. Kashef and J. Casti [J. Comput. Phys. 10, 40-52 (1972; Zbl 0247.65061)], which leads to the so called generalized collocation method. First, a general description is given. Then recent developments concerning integro-differential equations, domain decomposition and stochastic problems are discussed. Finally, improvements in the algorithms and computation of error estimates are given.

MSC:
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65R20 Numerical methods for integral equations
45K05 Integro-partial differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
65N15 Error bounds for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
65C99 Probabilistic methods, stochastic differential equations
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[1] Bellomo, N.; Preziosi, L., Modelling, mathematical methods and scientific computation, (1995), CRC Press Boca Raton, PL · Zbl 0871.65001
[2] Bellman, R.; Casti, J., Differential quadrature and long term integration, J. math. anal. appl., 34, 235-238, (1971) · Zbl 0236.65020
[3] Bellman, R.; Kashef, B.; Casti, J., Differential quadrature: solution of nonlinear partial differential equations, J. comp. phys., 10, 40-52, (1972) · Zbl 0247.65061
[4] Bellman, R.; Adomian, G., Partial differential equations, (1985), Reidel Dordrecht
[5] Satofuka, A., A new explicit method for the solution of parabolic differential equations, (), 97-108
[6] Bellomo, N.; Flandoli, F., Stochastic partial differential equations in continuum physics, Mathl. comp. in simul., 31, 3-17, (1988) · Zbl 0706.60063
[7] Bellomo, N.; Brzezniak, Z.; de Socio, L.M., Nonlinear stochastic evolution problems in applied sciences, (1992), Kluwer Amsterdam · Zbl 0770.60061
[8] Longo, E.; Teppati, G.; Bellomo, N., Modeling and solution of stochastic inverse problems in mathematical physics, Computers math. applic., 32, 4, 65-81, (1996) · Zbl 0858.65104
[9] Bert, C.; Malik, M., Differential quadrature method in computational mechanics, ASME review, 49, 1-27, (1996)
[10] Bert, C.; Jang, S.; Stritz, A., Nonlinear bending analysis for orthotropic rectangular plates by the method of differential quadrature, Comp. mech., 5, 217-226, (1988)
[11] Bert, C.; Jang, S.; Stritz, A., Static and free vibrational analysis of beam and plates by the method of differential quadrature, Acta. mech., 102, 12-24, (1994)
[12] Bellomo, N.; Ridolfi, L., Solution of nonlinear initial-boundary value problems by sinc collocation-interpolation methods, Computers math. applic., 29, 4, 15-28, (1995) · Zbl 0822.65075
[13] Bonzani, I., Time evolution of random fields in stochastic continuum mechanics, Mathl. comput. modelling, 17, 6, 37-45, (1993) · Zbl 0770.60096
[14] Bonzani, I., Solution of nonlinear evolution problems by parallelized collocation interpolation methods, in differential quadrature methods, Computers math. applic., (1997), (to appear) · Zbl 0905.35046
[15] Preziosi, L.; deSocio, L., A nonlinear inverse phase transition problem for the heat equation, Math. models methods in appl. sci., 1, 167-182, (1991) · Zbl 0741.60061
[16] Preziosi, L.; Teppati, G.; Bellomo, N., Modeling and solution of stochastic inverse problems in mathematical physics, Mathl. comput. modelling, 16, 5, 37-51, (1992) · Zbl 0755.60102
[17] Stenger, F., Numerical methods based on wittaker cardinal or sinc functions, SIAM review, 23, 165-224, (1983) · Zbl 0461.65007
[18] Stenger, F., Numerical methods based on sinc and analytic functions, (1993), Springer Berlin · Zbl 0803.65141
[19] Lund, J.; Bowers, K., Sinc methods, (1992), SIAM Philadelphia, PA · Zbl 0753.65081
[20] Brezzi, F.; Fortin, M., Mixed and hybrid finite element methods, (1991), Springer Berlin · Zbl 0788.73002
[21] Pantuso, D.; Bathe, K.J., A four node quadrilateral mixed-interpolated element for solids and fluids, Math. models methods in appl. sci., 5, 1113-1128, (1995) · Zbl 0848.73068
[22] Gottelieb, D.; Orszag, S., Numerical analysis of spectral methods, (1977), SIAM Philadelphia, PA
[23] Ringhofer, C., On the convergence of spectral methods for the wiegner-Poisson problem, Math. models methods in appl. sci., 2, 91-112, (1992)
[24] Desvilletes, L.; Mischeler, S., About splitting algorithm for Boltzmann and B.G.K. equations, Math. models methods in appl. sci., 6, 1079-1102, (1996) · Zbl 0876.35088
[25] Lifits, S.A.; Reutskiy, S.Yu.; Tirozzi, B., A quasi-Trefftz-type spectral methods for initial value problem with moving boundary, Math. models methods in appl. sci., 7, 385-404, (1997) · Zbl 0912.65082
[26] S.A. Lifits, S.Yu. Reutskiy and B. Tirozzi, Modelling Math. Methods in Appl. Sci. (to appear).
[27] Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A., Spectral methods in fluid dynamics, (1989), Springer Berlin
[28] Lybeck, J.; Bowers, K., Domain decomposition methods in conjunction with sinc methods for Poisson’s equation, Numeric. meth. for partial diff. egs., 12, 461-486, (1996) · Zbl 0857.65128
[29] Corlenzoli, C.; Zanolli, P., Domain decomposition approximation to a generalized Stokes problem by spectral methods, Math. models meth. appl. sci., 1, 4, 501-515, (1991) · Zbl 0758.35061
[30] Canuto, C.; Russo, A., A viscous-inviscid coupling under mixed boundary conditions, Math. models meth. appl. sci., 2, 4, 461-482, (1992) · Zbl 0769.76067
[31] Canuto, C.; Russo, A., On the elliptic-hyperbolic coupling: the advection-diffusion equation via the χ-formulation, Math. models meth. appl. sci., 3, 145-170, (1993) · Zbl 0773.76066
[32] Shishkin, G.; Vabishchevich, P., Parallel domain decomposition methods with overlapping of subdomains for parabolic problems, Math. models meth. appl. sci., 6, 1169-1186, (1996)
[33] Quarteroni, A., Domain decomposition methods using spectral methods, Surveys math. ind., 1, 75-115, (1991)
[34] Back, J.; Blackwell, B.; Clair, C.St., Inverse heat conduction, (1985), Wiley London
[35] Antonelli, D.; Romano, D.; Bellomo, N., Computation of the thermal field in tools during machining processes, Mathl. comput. modelling, 21, 8, 53-68, (1995) · Zbl 0824.65139
[36] I. Bonzani, Domain decomposition and discretization of continuous mathematical models, Computers Math. Applic. (to appear). · Zbl 0972.65071
[37] Jager, E.; Segel, L., On the distribution of dominance in a population of interacting anonymous organisms, SIAM J. appl. math., 52, 1442-1468, (1992) · Zbl 0759.92011
[38] Arlotti, L.; Bellomo, N., Population dynamics with stochastic intersections, Transp. theory statist. phys., 24, 431-443, (1995) · Zbl 0838.92018
[39] Bellomo, N.; Le Tallec, P.; Perthame, B., Nonlinear Boltzmann equation solutions and applications to fluid dynamics, ASME review, 49, 1-27, (1996)
[40] Preziosi, L.; Longo, E.; Bellomo, N., The semicontinuous Boltzmann equation: towards a model for fluid dynamic applications, Math. models meth. appl. sci., 3, 65-94, (1993) · Zbl 0770.76057
[41] B. Firmani, L. Guerri and L. Preziosi, Tumors immune system competition with medically induced activation and disactivation, Math. Models Meth. Appl. Sci. (to appear). · Zbl 0934.92016
[42] Markov, K.; Kolev, M., On the adsorption problem for random dispersions, Math. models meth. appl. sci., 6, 755-772, (1996) · Zbl 0818.60073
[43] Capinski, M.; Cutland, N., Nonstandard methods for stochastic fluid mechanics, (1995), World Scientific Singapore · Zbl 0824.76003
[44] Adomian, G.; Malakian, K., Stochastic analysis, Mathl. modelling, 1, 3, 211-235, (1980) · Zbl 0523.60058
[45] Zeider, E., Applied functional analysis, (1995), Springer Berlin
[46] Preziosi, L., The theory of deformable porous media and its application to composite material manufacturing, Surv. math. ind., 6, 167-214, (1996) · Zbl 0858.73055
[47] Daubechies, I., ()
[48] Meyer, Y., Wavelets algorithms and applications, (1993), SIAM Philadelphia, PA
[49] Masson, R., Biorthogonal spline wavelets on the interval for the solution of boundary value problems, Math. models meth. appl. sci., 6, 749-792, (1996)
[50] Canuto, C.; Cravero, I., A wavelet-based adaptative finite element method for advection-diffusion equations, Math. models meth. appl. sci., 7, 265-290, (1997) · Zbl 0872.65099
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