Abstract
We introduce the concept of very weak solution to a Cauchy problem for elliptic equations. The Cauchy problem is regularized by a well-posed non-local boundary value problem whose solution is also understood in a very weak sense. A stable finite difference scheme is suggested for solving the non-local boundary value problem and then applied to stabilizing the Cauchy problem. Some numerical examples are presented for showing the efficiency of the method.
Funding statement: The work by Dinh Nho Hào was partially supported by Vietnam Academy of Science and Technology (VAST) under the Grant VAST.HTQT.NGA.09/17-18, the work by S. I. Kabanikhin and M. A. Shishlenin was partially supported by RFBR (Grant 17-51-540004, 16-29-15120, 16-01-00755). Part of this work has been done during Dinh Nho Hào’s stay at Vietnam Institute for Advanced Study in Mathematics.
References
[1] C. Bardos, Hadamard et les équations aux dérivées partielles, Matapli 103 (2014), 21–32. Search in Google Scholar
[2] A. V. Bicadze and A. A. Samarskiĭ, Some elementary generalizations of linear elliptic boundary value problems, Dokl. Akad. Nauk SSSR 185 (1969), 739–740. Search in Google Scholar
[3] V. B. Glasko, E. A. Mudretsova and V. N. Strakhov, Inverse problems in gravimetry and magnetometry, Ill-Posed Problems in the Natural Sciences (in Russian), Adv. Sci. Tech. USSR Math. Mech. Ser., Moscow State University, Moscow (1987), 89–102. Search in Google Scholar
[4] J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Dover Publications, New York, 1953. 10.1063/1.3061337Search in Google Scholar
[5] 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), no. 5, Article ID 055002. 10.1088/0266-5611/25/5/055002Search in Google Scholar
[6] D. N. Hào, T. D. ú’c Vân and R. Gorenflo, Towards the Cauchy problem for the Laplace equation, Partial Differential Equations. Part 1 (Warsaw 1990), Banach Center Publ. 27, Polish Academy of Sciences, Warsaw (1992), 111–128. 10.4064/-27-1-111-128Search in Google Scholar
[7] V. A. Il’in, On solvability of mixed problems for hyperbolic and parabolic equations (in Russian), Uspekhi Mat. Nauk 15(92) (1960), no. 2, 97–154; translated in Russian Math. Surveys 15 (1960) no. 2, 85–142. 10.1070/RM1960v015n02ABEH004217Search in Google Scholar
[8] V. A. Il’in and I. A. Shishmarev, On the connection between the classical and the generalized solution to Dirichlet’s problem and to the problem of eigenvalues (in Russian), Dokl. Akad. Nauk SSSR 126 (1959), 1176–1179. Search in Google Scholar
[9] V. A. Il’in and I. A. Shishmarev, On the equivalence of the systems of generalized and classical eigenfunctions (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 24 (1960), 757–774. Search in Google Scholar
[10] V. Isakov, Inverse Problems for Partial Differential Equations, 2nd ed., Appl. Math. Sci. 127, Springer, New York, 2006. Search in Google Scholar
[11] S. I. Kabanikhin, Definitions and examples of inverse and ill-posed problems, J. Inverse Ill-Posed Probl. 16 (2008), no. 4, 317–357. 10.1515/JIIP.2008.019Search in Google Scholar
[12] S. I. Kabanikhin, Y. S. Gasimov, D. B. Nurseitov, M. A. Shishlenin, B. B. Sholpanbaev and S. Kasenov, Regularization of the continuation problem for elliptic equations, J. Inverse Ill-Posed Probl. 21 (2013), no. 6, 871–884. 10.1515/jip-2013-0041Search in Google Scholar
[13] S. I. Kabanikhin and A. L. Karchevsky, Optimizational method for solving the Cauchy problem for an elliptic equation, J. Inverse Ill-Posed Probl. 3 (1995), no. 1, 21–46. 10.1515/jiip.1995.3.1.21Search in Google Scholar
[14] S. I. Kabanikhin, D. B. Nurseitov, M. A. Shishlenin and B. B. Sholpanbaev, Inverse problems for the ground penetrating radar, J. Inverse Ill-Posed Probl. 21 (2013), no. 6, 885–892. 10.1515/jip-2013-0057Search in Google Scholar
[15] S. I. Kabanikhin and M. A. Shishlenin, Regularization of the decision prolongation problem for parabolic and elliptic equations from border part, Eurasian J. Math. Comp. Appl. 2 (2014), no. 2, 81–91. 10.32523/2306-6172-2014-2-2-81-91Search in Google Scholar
[16] S. I. Kabanikhin, M. A. Shishlenin, D. B. Nurseitov, A. T. Nurseitova and S. E. Kasenov, Comparative analysis of methods for regularizing an initial boundary value problem for the Helmholtz equation, J. Appl. Math. 2014 (2014), Article ID 786326. 10.1155/2014/786326Search in Google Scholar
[17] O. A. Ladyženskaya, On solvability of the fundamental boundary problems for equations of parabolic and hyperbolic type (in Russian), Dokl. Akad. Nauk SSSR (N.S.) 97 (1954), 395–398. Search in Google Scholar
[18] O. A. Ladyženskaya, On non-stationary operator equations and their applications to linear problems of mathematical physics (in Russian), Mat. Sb. (N.S.) 45(87) (1958), 123–158. Search in Google Scholar
[19] O. A. Ladyžhenskaya, The Boundary Value Problems of Mathematical Physics, Appl. Math. Sci. 49, Springer, New York, 1985. 10.1007/978-1-4757-4317-3Search in Google Scholar
[20] E. M. Landis, On some properties of solutions of elliptic equations (in Russian), Dokl. Akad. Nauk SSSR (N.S.) 107 (1956), 640–643. Search in Google Scholar
[21] M. M. Lavrent’ev, On Cauchy’s problem for Laplace’s equation (in Russian), Dokl. Akad. Nauk SSSR (N.S.) 102 (1955), 205–206. Search in Google Scholar
[22] M. M. Lavrent’ev, On the Cauchy problem for Laplace equation (in Russian), Izv. Akad. Nauk SSSR. Ser. Mat. 120 (1956), 819–842. Search in Google Scholar
[23] M. M. Lavrent’ev, On the problem of Cauchy for linear elliptic equations of the second order (in Russian), Dokl. Akad. Nauk SSSR (N.S.) 112 (1957), 195–197. Search in Google Scholar
[24] M. M. Lavrent’ev, Some Improperly Posed Problems in Mathematical Physics, Springer, New York, 1967. 10.1007/978-3-642-88210-4Search in Google Scholar
[25] M. M. Lavrent’ev, V. G. Romanov and G. P. Shishatskii, Ill-Posed Problems in Mathematical Physics and Analysis, American Mathematical Society, Providence, 1986. 10.1090/mmono/064Search in Google Scholar
[26] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, New York, 1971. 10.1007/978-3-642-65024-6Search in Google Scholar
[27] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. II, Springer, New York, 1972. 10.1007/978-3-642-65161-8Search in Google Scholar
[28] L. E. Payne, Improperly Posed Problems in Partial Differential Equations, Society for Industrial and Applied Mathematics, Philadelphia, 1975. 10.1137/1.9781611970463Search in Google Scholar
[29] C. Pucci, Sui problemi di Cauchy non “ben posti”, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 18 (1955), 473–477. Search in Google Scholar
[30] C. Pucci, Discussione del problema di Cauchy per le equazioni di tipo ellittico, Ann. Mat. Pura Appl. (4) 46 (1958), 131–153. 10.1007/BF02412913Search in Google Scholar
[31] C. Pucci, Some Topics in Parabolic and Elliptic Equations, Lecture Ser. 36, University of Maryland, College Park, 1958. Search in Google Scholar
[32] A. A. Samarskii, The Theory of Difference Schemes, Monogr. Textb. Pure Appl. Math. 240, Marcel Dekker, New York, 2001. 10.1201/9780203908518Search in Google Scholar
[33] A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations. Vol. I: Direct Methods, Birkhäuser, Basel, 1989. 10.1007/978-3-0348-9272-8Search in Google Scholar
[34] A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations. Vol. II: Iterative Methods, Birkhäuser, Basel, 1989. 10.1007/978-3-0348-9272-8Search in Google Scholar
[35] J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, 2nd ed., Springer, New York, 1979. 10.1007/978-1-4757-5592-3Search in Google Scholar
[36] A. N. Tihonov, On the solution of ill-posed problems and the method of regularization (in Russian), Dokl. Akad. Nauk SSSR 151 (1963), 501–504. Search in Google Scholar
[37] A. N. Tikhonov, On the stability of inverse problems (in Russian), C. R. (Doklady) Acad. Sci. URSS (N.S.) 39 (1943), 176–179. Search in Google Scholar
[38] A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-posed Problems, John Wiley & Sons, New York, 1977. Search in Google Scholar
[39] P. N. Vabishchevich, Numerical solution of nonlocal elliptic problems (in Russian), Izv. Vyssh. Uchebn. Zaved. Mat. (1983), no. 5, 13–19. Search in Google Scholar
[40] P. N. Vabishchevich and A. Y. Denisenko, Regularization of nonstationary problems for elliptic equations, Inzh.-Fiz. Zh. 65 (1993), no. 6, 690–694. 10.1007/BF00861941Search in Google Scholar
[41] P. N. Vabishchevich and P. A. Pulatov, A method of numerical solution of the Cauchy problem for elliptic equations (in Russian), Vestnik Moskov. Univ. Ser. XV Vychisl. Mat. Kibernet. (1984), no. 2, 3–8. Search in Google Scholar
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