zbMATH — the first resource for mathematics

Modeling and solution of stochastic inverse problems in mathematical physics. (English) Zbl 0755.60102
Summary: This paper deals with modelling aspects and solution techniques of inverse type problems for nonlinear mathematical models of continuum physics (and in general of applied sciences) defined by nonlinear partial differential equations with random coefficients. The paper deals with initial-boundary value problems with random initial and/or boundary conditions with special attention to the analysis of nonlinearities and representation of the random fields generated by the solution of the mathematical problem.

60K40 Other physical applications of random processes
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
60H99 Stochastic analysis
Full Text: DOI
[1] Courant, R.; Hilbert, D., Methods of mathematical physics, (1953), Interscience New York · Zbl 0729.00007
[2] Tikhonov, A.N.; Samarskii, A.A., Equations of mathematical physics, (1972), Pergamon Press Oxford, Nauka, Moscow, English translation: · Zbl 0044.09302
[3] Lions, J.L.; Magenes, E., Problemes aux limites non homogenes et applications, Vol. 1-3, (1970), Dunod Paris · Zbl 0197.06701
[4] Kampee de Feriet, J., Integrales aleatoires de l’equation de la chaleur dans une barre infinie, C. R. acad. sci. Paris, 240, 710-712, (1955) · Zbl 0064.13306
[5] Kampee de Feriet, J., Random solutions of partial differential equations, (), 199-208
[6] Malishev, I., An inverse source problem for the heat equation, J. math. anal. appl., 142, 206-218, (1989)
[7] Markowski, A., Development and application of ill-posed problems in USSR, Appl. mech. review, 41, 247-256, (1988)
[8] Repaci, A., On the solution of a class of inverse evolution problems by a Bellman-Adomian method, Appl. math. letters, 2, 2, 151-153, (1989) · Zbl 0709.35100
[9] Repaci, A., Bellman-Adomian solution of nonlinear inverse problems in continuum physics, J. math. anal. appl., 143, 57-65, (1989) · Zbl 0693.35160
[10] Bellman, R.; Kashef, B.G.; Casti, J., Differential quadratures: A technique for the rapid solution of nonlinear partial differential equations, J. comp. phys., 10, 40-52, (1972) · Zbl 0247.65061
[11] Bellomo, N.; de Socio, L.M.; Monaco, R., The random heat equation: solution by the stochastic adaptative interpolation method, Comp. math. appl., 16, 9, 759-766, (1988) · Zbl 0661.60082
[12] Bellomo, N.; Flandoli, F., Stochastic partial differential equations in continuum physics, Math. comp. simul., 31, 3-17, (1989) · Zbl 0706.60063
[13] I. Bonzani, Time evolution of random fields in stochastic continuum mechanics, Comp. Math. Modelling (to appear). · Zbl 0770.60096
[14] Monaco, R.; Repaci, A.; Bellomo, N., Analysis of stochastic systems in continuum physics, Struct. safety, 8, 93-101, (1990)
[15] Preziosi, L.; de Socio, L.M., Nonlinear inverse phase transition problems for the heat equation, M3 AS:, math. models meth. appl. sci., 1, (1991) · Zbl 0741.60061
[16] Bellomo, N.; Brzezniak, Z.; de Socio, L.M., Nonlinear stochastic evolution problems in applied sciences, (1992), Kluwer Amsterdam · Zbl 0770.60061
[17] Demidovich, B.; Maron, I., Elements de calcul numerique, (1979), Mir Moscow, French Translation
[18] Soong, T.T., Random differential equations in science and engineering, (1973), Academic Press New York · Zbl 0348.60081
[19] Adomian, G., Stochastic systems, (1983), Academic Press New York · Zbl 0504.60066
[20] Bellomo, N.; Riganti, R., Nonlinear stochastic systems in physics and mechanics, (1987), World Sci London, New Jersey, Singapore · Zbl 0623.60084
[21] Bellomo, N., Book review, Foundation of physics, 19, 443-448, (1989)
[22] Bonzani, I., On a class of nonlinear stochastic dynamical systems: analysis of the transient behaviour, J. math. anal. appl., 128, 39-50, (1987) · Zbl 0626.60061
[23] Cherruault, Y., Convergence of Adomian’s method, Cybernetics, 18, 31-38, (1989) · Zbl 0697.65051
[24] Repaci, A., Nonlinear dynamical systems: on the accuracy of Adomian’s decomposition method, Appl. math. lett., 3, 4, 35-39, (1990) · Zbl 0719.93041
[25] Adomian, G.; Malakian, K., Stochastic analysis, Math. modelling, 1, 3, 211-235, (1983) · Zbl 0523.60058
[26] Soize, C., Steady-state solution of Fokker-Planck equation in higher dimension, Prob. eng. mech., 3, 196-206, (1988)
[27] Temam, R., Sur la stabilite et la convercence de la methode des pas fractionnaires, () · Zbl 0174.45804
[28] Beck, J.V.; Blackwell, B.; Clair, C.R.St., Inverse heat conduction, (1985), Wiley London
[29] Ramm, A.G., Multi-dimensional inverse problems and completeness of products of solutions to PDE, J. math. anal. appl., 134, 211-253, (1988) · Zbl 0632.35076
[30] Ramm, A.G., Numerical methods for solving 3D inverse problems of geophysics, J. math. anal. appl., 136, 352-356, (1988) · Zbl 0663.65133
[31] Ivanov, A.V.; Leonenko, N.N., Statistical analysis of random fields, (1979), Kluver Amsterdam · Zbl 0721.62097
[32] Ramm, A.G., Random fields estimation theory, (1989), Pitmann London · Zbl 0712.47042
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.