D
ans cette thèse, on a traité deux problèmes aux limites mal-posés. On a donné un déve- loppement théorique de la méthode de mollification appliquée à un problème de Cauchy elliptique dans un cadre abstrait, et la méthode des valeurs aux limites auxiliaire ap- pliquée à une équation de la chaleur non-classique. Des résultats de convergence et des estimations d’erreur ont été établis dans des classes de correction bien appropriées.C
omme perspectives, on projette d’étudier d’autres approches de régularisation appli- quées aux problèmes d’évolution mal posés avec des variables déviées, ainsi que leur mé- thodes d’approximation numérique.[1] L.S. Abdulkerimov, Regularization of an ill-posed Cauchy problem for evolution equations in a Banach space, Azerbaidzan. Gos. Univ. Ucen. Zap. Fiz. Mat., 1 (1977), 32-36 (MR0492645) (in Russian).
[2] G. Alessandrini L. Rondi, E. Rosset and S. Vessella,The stability for the Cauchy problem for elliptic equa- tions, Inverse Problems 25 (2009) 123004 (47pp).
[3] G. Alessandrini,Stable determination of a crack from boundary measurements, Proc. Roy. Soc. Edinburgh
Sect. A, (1993), 123(3) :497-516.
[4] D.D. Ang, R. Gorenflo, V.K.Le and D.D. Trong,Moment Theory and Some Inverse Problems in Potential Theory and Heat Conduction, Lecture Notes in Mathematics 1792, Springer-Verlag, Berlin, 2002.
[5] K.A. Ames, B. Straughan, Non-Standard and Improperly Posed Problems, Academic Press (1997). [6] L. Bourgeois,Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace’s
equation, Inverse Problems 21 (2005), No. 3, 1087-1104.
[7] L. Bourgeois,A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace’s equation,
Inverse Problems 22 (2006), 413-430.
[8] N. Boussetila, Etude de Problèmes Non Locaux et Régularisation de Problèmes Mal Posés en EDP, Thèse de Doctorta, U. BM-Annaba (2006).
[9] N. Boussetila, S. Hamida and F. Rebbani,Spectral Regularization Methods for an Abstract Ill-Posed Elliptic Problem, Abstract and Applied Analysis, Volume 2013, Article ID 947379, 11 pages.
[10] A. Bouzitouna1, N. Boussetila and F. Rebbani,Two regularization methods for a class of inverse boundary value problems of elliptic type, Boundary Value Problems 2013 2013 :178.
[11] A. Lakhdari and N. Boussetila, An iterative regularization method for an abstract ill-posed biparabolic problem, Boundary Value Problems (2015) 2015 :55.
[12] A. Benrabah, Etude d’un problème d’évolution non locale et régularisation d’un problème elliptique mal posé, Thèse de Doctorta, U. BM-Annaba (2011).
[13] L. Bourgeois and J. Dardé,A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data, Inverse Problems 26 (2010) 095016 (21pp).
[14] E.P. Belan, Dynamics of stationary structures in a parabolic problem with reflected spatial argument, Cybernetics and Systems Analysis, Vol. 46, No. 5, 2010, 772-783.
[15] M.Sh. Burlutskaya, A.P. Khromov,Substantiation of Fourier method in mixed problem with involution, Izv.
Bibligraphie
[16] M.Sh. Burlutskaya, A.P. Khromov,Mixed problem for simplest hyperbolic first order equations with involu- tion, Izv. Saratov. Univ. Mat. Mekh. Inform., 2014, Vol.14, Issue 1, 10-20.
[17] N. Dunford and J. Schwartz, Linear Operators, Part II, John Wiley and Sons, Inc., New York, 1967. [18] A.M. Denisov, E.V. Zakharov, A. V. Kalinin and V.V. Kalinin,Numerical Solution of the Inverse Electro-
cardiography Problem with the Use of the Tikhonov Regularization Method, Computational Mathematics
and Cybernetics, (2008), Vol. 32, No. 2, 61-68.
[19] Dinh Nho Hào,A mollification method for ill-posed problems, Numer. Math., 68 (1994), pp. 469-506.
[20] D.N.Hào, N.V. Duc and D. Lesnic,A non-local boundary value problem method for the Cauchy problem for elliptic equations, Inverse Problems 25(2009) : 055002(27pp).
[21] H.W. Engel, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic, (2000). [22] L. Eldén and V. Simoncini,A numerical solution of a Cauchy problem for an elliptic equation by Krylov
subspaces, Inverse Problems 25 (2009) 065002 (22pp).
[23] H. O. Fattorini, The Cauchy Problem, Encyclopedia of Mathematics and its Applications, 18, Cam- bridge University Press 1983.
[24] K.S. Fayazov,A certain ill-posed cauchy problem for first-order and second-order differential equations with
operator coefficients, Siberian Mathematical Journal, (1994) Vol. 35, No. 3, 631-635.
[25] X.L.Feng, L. Eldén and C.L. Fu,A quasi-boundary-value method for the Cauchy problem for elliptic equa- tions with nonhomogeneous Neumann data, J. Inv. Ill-Posed Problems 18 (2010), 617-645.
[26] I.C. Gohberg, M.G. Krein, Introduction to theory of linear non selfadjoint operators, AMS (1963). [27] J. Hadamard,Lecture note on Cauchy’s problem in linear partial differential equations,Yale Uni Press, New
Haven, 1923.
[28] J. R. Higgins. Completeness and basis properties of sets of special functions, Cambridge University Press (1977).
[29] S.G. Krein and Ju.I. Petunin,Scales of Banach spaces, Uspehi Mat. Nauk, 21 (1966), 89-168.
[30] R. Kress, Linear Integral Equations, vol. 82 of Applied Mathematical Sciences, Springer (1989). [31] V.A. Kozlov, V.G. Maz’ya, On iterative procedure for solving ill-posed boundary value problems that
preserve differential equations, Leningrad Math J., (1990) 1, 1207-1228.
[32] P.G. Kaup, F. Santosa,Nondestructive evaluation of corrosion damage using electrostatic measurements, J.
Nondestr. Eval., (1995) 14(3), 127-136.
[33] S.I. Kabanikhin and M. Schieck,Impact of conditional stability : convergence rates for general linear regu- larization methods, J. Inverse Ill-Posed Probl., (2008) 16(3) :267-282.
[34] T.Sh. Kalmenov and A.Sh. Shaldanbaev,Necessary and Sufficient Condition for the Existence of a Strong
Solution for a Parabolic Equation in Reverse Time, Vestn. Kazakh. Nar. Univ., (2007), no. 2 (53), 58-72.
[35] T.Sh. Kalmenov and U.A. Iskakova, A Criterion for the Strong Solvability of the Mixed Cauchy Problem for the Laplace Equation, Doklady Mathematics, 2007, Vol. 75, No. 3, 370-373.
[36] T.Sh. Kalmenov and U.A. Iskakova,Criterion for the Strong Solvability of the Mixed Cauchy Problem for the Laplace Equation, Differential Equations, (2009), Vol. 45, No. 10, 1460-1466.
[37] R. Lattès, J.-L. Lions, The method of quasi-reversibility. Applications to partial differential equations, Elsevier, New York (1969).
[38] M.M. Lavrentev, V.G. Romanov and G.P. Shishatskii, Ill-posed Problems in Mathematical Physics and Analysis, Providence, RI : American Mathematical Society 1986.
[39] M.M. Lavrentiev and L.Ya. Saveliev, Operator theory and ill-posed problems, Inverse and Ill-Posed Problems Series. VSP, Leiden Boston (2006).
[41] D.A. Murio, The mollification method and the numerical solution of ill-posed problems, Wiley- Interscience Publication (1993).
[42] L.E. Payne,Bounds in the Cauchy problem for the Laplace equation, Arch. Rational Mech. Anal., (1960)
5:35-45.
[43] L. E. Payne, Improperly Posed Problems in Partial Differential Equations, Philadelphia, PA : Society for Industrial and Applied Mathematics, 1975.
[44] A. Pazy, Semigroups of linear operators and application to partial differential equations, Springer- Verlag, 1983.
[45] Z. Qian, C.L. Fu. X.T. Xiong,Fourth-order modified method for the Cauchy problem for the Laplace equation,
J. Comput. Appl. Math., 192 (2006) 205-218.
[46] Z. Qian, C.L. Fu and Z.P. Li,Two regularization methods for a Cauchy problem for the Laplace equation, J.
Math. Anal. Appl., (2008) 338(1), 479-489.
[47] A. Qian, J. Mao, and L. Liu, A Spectral Regularization Method for a Cauchy Problem of the Mo- dified Helmholtz Equation, Boundary Value Problems, Volume 2010, Article ID 212056, 13 pages,
doi :10.1155/2010/212056.
[48] A.V. Razgulin, The Problem of Control of a Two-Dimensional Transformation of Spatial Arguments in a Parabolic Functional-Differential Equation, Differential Equations, 2006, vol. 42, no. 8, 1140-1155.
[49] A.V. Razgulin, T.E. Romanenko,Rotating Waves in Parabolic Functional-Differential Equations with Rota-
tion of Spatial Argument and Time Delay, Comput. Math. Math. Phys., 2013, vol. 53, no. 11, 1626-1643.
[50] A.A. Samarskii, P.N. Vabishchevich, Numerical Methods for Solving Inverse Problems of Mathematical Physics, Inverse and Ill-Posed Problems Series, Walter de Gruyter, Berlin. New York, 2007.
[51] K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Springer 2012.
[52] M A. Sadybekov and A.M. Sarsenbi, Criterion for the Basis Property of the Eigenfunction System of a Multiple Differentiation Operator with an Involution, Differential Equations, 2012, Vol. 48, No. 8, 1112-
1118.
[53] A.N. Tikhonov and V.Y. Arsenin, Solution of Ill-posed Problems, Winston & Sons, Washington, DC, (1977).
[54] U. Tautenhahn,Optimal stable solution of Cauchy problems for elliptic equations, Z. Anal. Anwendungen
(1996) 15, no. 4, 961-984.
[55] V. I. Voitinskii,Density of Cauchy initial data for solutions of elliptic equations, Mat. Sbornik Math., Tom
85127(1971), No. 1, 131-139.
[56] P.N. Vabishchevich and A.Yu. Denisenko,Regularization of nonstationary problems for elliptic equations,
J. Eng. Phys. Thermophys, (1993) 65 1195-1199.
[57] E.M. Varfolomeev,On Some Properties Of Elliptic And Parabolic Functional Differential Operators Arising
In Nonlinear Optics, Journal of Mathematical Sciences, Vol. 153, No. 5, 2008, 649-682.
[58] J. Wiener, Generalized Solutions of Functional Differential Equations, World Scientific Publishing (1993).
[59] T. Wei, H.H. Qin and H.W. Zhang,Convergence Estimates for Some Regularization Methods to Solve a Cauchy Problem of the Laplace Equation, Numer. Math. Theor. Meth. Appl., (2011) Vol. 4, No. 4, 459-
477.
[60] H. Zhang, Modified quasi-boundary value method for cauchy problems of elliptic equations with variable coefficients, Electronic Journal of Differential Equations, Vol. 2011 (2011), No. 106, 1-10.
[61] H. Zhang and T. Wei,An improved non-local boundary value problem method for a cauchy problem of the Laplace equation, Numer Algor., (2012) 59, 249-269.
[62] H.W. Zhang, T. Wei, Two iterative methods for a Cauchy problem of the elliptic equation with variable coefficients in a strip region, Numer Algor (2014) 65 :875-892.
[63] H. Zhang,Modified Tikhonov Method for Cauchy Problem of Elliptic Equation with Variable Coefficients, American Journal of Computational Mathematics (2014) 4, 213-222.
Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 947379,11pages http://dx.doi.org/10.1155/2013/947379