Abstract
In this paper it is proved that the Cauchy problem and a boundary value problem for the Laplace equation is well-posed provided the data belong to a suitable function space. Explicit numerical procedures are described for approximating the solutions to these problems.
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References
Tikhonov, A.N., Arsenin, V.Y.: Solutions of ill-posed problems. Winston-Wiley, Washington, 1977.
Lavrentev, M.M., Romanov, V.G., Shishatskii, S.P.: Ill-posed problems in mathematical physics and analysis. Transl. of Math. Monographs, Vol. 64, AMS, Providence, Rhode Island, 1986.
Cannon, J.R., DuChateau, P.: Approximating the solution to the Cauchy problem for Laplace’s equation. SIAM J. Numer. Anal., 14, 3 (1977), 473–483.
Medeiros, L.A.: Remarks on a non-well posed problem. Proceedings of the Royal Society of Edinburgh, 102A (1986), 131–140.
Tran Due Van: On the pseudo differential operators with real analytic symbols and their applications. J. Fac. Sci. Univ. Tokyo, IA Mathematics, 36, 3 (1989), 803–825.
Dubinskii, Yu.A.: The algebra of pseudo differential operators with analytic symbols and its applications to mathematical physics. Russian Math. Surveys, 37 (1982), 109–153.
Dubinskii, Yu.A.: Sobolev spaces of infinite order and differential equations. Teubner-Texte zur Mathematik, Bd. 87, Leipzig, 1986.
Nikolskii, S.M.: Approximation of functions of several variables and imbedding theorems. Springer-Verlag, Berlin-Heidelberg-New York, 1975.
Trinh Ngoc Minh, Tran Due Van: Cauchy problems for systems of partial differential equations with a distinguished variable. Soviet Math. Dokl. 32, 2 (1985), 562–565.
Tran Due Van, Dinh Nho Hao, Trinh Ngoc Minh, Gorenflo, R.: On the Cauchy problems for systems of partial differential equations with a distinguished variable. To appear in J. “Numerical Functional Analysis and Optimization.”
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© 1992 Springer Fachmedien Wiesbaden
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Van, T.D., Hào, D.N., Gorenflo, R. (1992). Approximating the Solution to the Cauchy Problem and the Boundary Value for the Laplace Equation. In: Vogel, A., Sarwar, A.K.M., Gorenflo, R., Kounchev, O.I. (eds) Theory and Practice of Geophysical Data Inversion. Theory and Practice of Applied Geophysics, vol 5. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-322-89417-5_3
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DOI: https://doi.org/10.1007/978-3-322-89417-5_3
Publisher Name: Vieweg+Teubner Verlag, Wiesbaden
Print ISBN: 978-3-528-06454-9
Online ISBN: 978-3-322-89417-5
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