On the numerical solution of direct and inverse problems for the heat equation in a semi-infinite region
We consider the initial boundary value problem for the heat equation in a region with infinite and finite boundaries (direct problem) and the related problem to reconstruct the finite boundary from Cauchy data on the infinite boundary (inverse problem). The numerical solution of the direct problem i...
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| Vydáno v: | Journal of computational and applied mathematics Ročník 108; číslo 1; s. 41 - 55 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Amsterdam
Elsevier B.V
15.08.1999
Elsevier |
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| ISSN: | 0377-0427, 1879-1778 |
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| Abstract | We consider the initial boundary value problem for the heat equation in a region with infinite and finite boundaries (direct problem) and the related problem to reconstruct the finite boundary from Cauchy data on the infinite boundary (inverse problem). The numerical solution of the direct problem is realized by a boundary integral equation method. For an approximate solution of the inverse problem we use a regularized Newton method based on numerical approach for the direct problem. Numerical examples illustrating our results are presented. |
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| AbstractList | We consider the initial boundary value problem for the heat equation in a region with infinite and finite boundaries (direct problem) and the related problem to reconstruct the finite boundary from Cauchy data on the infinite boundary (inverse problem). The numerical solution of the direct problem is realized by a boundary integral equation method. For an approximate solution of the inverse problem we use a regularized Newton method based on numerical approach for the direct problem. Numerical examples illustrating our results are presented. |
| Author | Chapko, Roman |
| Author_xml | – sequence: 1 givenname: Roman surname: Chapko fullname: Chapko, Roman email: chapko@franko.lviv.ua organization: Department of Applied Mathematics and Computer Science, Lviv University, 290602 Lviv, Ukraine |
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| Cites_doi | 10.1007/978-3-662-03537-5 10.1137/S0036139994277828 10.1007/978-3-642-61527-6 10.1007/BF01385870 10.1216/jiea/1181075987 10.1007/BF01210400 10.1090/qam/998101 10.1142/9789812798886_0010 10.1088/0266-5611/14/4/006 10.1007/978-3-642-97146-4 10.1007/BF01385694 10.1007/978-3-642-48860-3 10.1002/mma.1670160203 10.1088/0266-5611/6/6/003 10.1007/978-1-4612-2706-9 |
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| Keywords | Trigonometric interpolation Initial boundary value problem Heat equation Semi-infinite region Inverse boundary problem Green's function Integral equation Collocation method Newton method Regularization Sobolev space Initial value problem Green function Uniqueness theorem Partial differential equation Convergence Inverse problem Direct problem Interpolation Boundary value problem |
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| References | Hsiao, Saranen (BIB15) 1993; 16 A.J. Nowak, BEM approach to inverse thermal problems, in: D.B. Ingham, L.C. Wrobel (Eds.), Boundary Integral Formulations for Inverse Analysis, Computational Mechanics Publications, Southampton, 1997, pp. 259–298. Lubich, Schneider (BIB20) 1992; 63 Bryan, Caudill (BIB5) 1996; 56 N.N. Lebedev, Special Functions and Their Applications, Prentice-Hall, Englewood Cliffs, 1965. C.A. Brebbia, J.C.F. Telles, L.C. Wrobel, Boundary Element Techniques, Springer, New York, 1984. D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd Edition, Springer, Berlin, 1998. R. Kress, Linear Integral Equations, Springer, Berlin, 1989. R. Dautray, J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 1, Physical Origins and Classical Methods, Springer, Berlin, 1990. Costabel (BIB10) 1990; 13 Kress, Sloan (BIB17) 1993; 66 A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, 1964. Banks, Kojima, Winfree (BIB2) 1990; 6 R.B. Guenther, J.W. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations, Prentice-Hall, Englewood Cliffs, 1988. Chapko, Kress, Yoon (BIB8) 1998; 14 R. Chapko, R. Kress, On a quadrature method for a logarithmic integral equation of the first kind, in: R.P. Agarwal (Ed.), Contributions in Numerical Mathematics, World Scientific, Singapore, 1993, pp. 127–140. Banks, Kojima (BIB1) 1989; 47 Z.S. Berezhanskaya, I.V. Lyudkevich, Numerical solution of a spatial boundary value problem for the heat equation by integral equations, Vychisl. Prikl. Mat. 41 (1980) 3–7 (in Russian). V. Galazyuk, R. Chapko, The Chebyshev-Laguerre transformation and integral equations methods for initial boundary value problems for the telegraph equation, Dokl. Akad. Nauk UkrSSR 8 (1990) 11–14 (in Russian). Chapko, Kress (BIB7) 1997; 9 F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer, Berlin, 1993. O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, AMS Publications, Providence, 1968. 10.1016/S0377-0427(99)00099-0_BIB6 Hsiao (10.1016/S0377-0427(99)00099-0_BIB15) 1993; 16 10.1016/S0377-0427(99)00099-0_BIB9 Chapko (10.1016/S0377-0427(99)00099-0_BIB8) 1998; 14 Costabel (10.1016/S0377-0427(99)00099-0_BIB10) 1990; 13 10.1016/S0377-0427(99)00099-0_BIB3 10.1016/S0377-0427(99)00099-0_BIB4 Kress (10.1016/S0377-0427(99)00099-0_BIB17) 1993; 66 Lubich (10.1016/S0377-0427(99)00099-0_BIB20) 1992; 63 Chapko (10.1016/S0377-0427(99)00099-0_BIB7) 1997; 9 10.1016/S0377-0427(99)00099-0_BIB19 10.1016/S0377-0427(99)00099-0_BIB18 10.1016/S0377-0427(99)00099-0_BIB16 Banks (10.1016/S0377-0427(99)00099-0_BIB1) 1989; 47 10.1016/S0377-0427(99)00099-0_BIB14 Banks (10.1016/S0377-0427(99)00099-0_BIB2) 1990; 6 10.1016/S0377-0427(99)00099-0_BIB13 10.1016/S0377-0427(99)00099-0_BIB12 10.1016/S0377-0427(99)00099-0_BIB11 10.1016/S0377-0427(99)00099-0_BIB22 Bryan (10.1016/S0377-0427(99)00099-0_BIB5) 1996; 56 10.1016/S0377-0427(99)00099-0_BIB21 |
| References_xml | – reference: R.B. Guenther, J.W. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations, Prentice-Hall, Englewood Cliffs, 1988. – reference: V. Galazyuk, R. Chapko, The Chebyshev-Laguerre transformation and integral equations methods for initial boundary value problems for the telegraph equation, Dokl. Akad. Nauk UkrSSR 8 (1990) 11–14 (in Russian). – reference: N.N. Lebedev, Special Functions and Their Applications, Prentice-Hall, Englewood Cliffs, 1965. – reference: R. Chapko, R. Kress, On a quadrature method for a logarithmic integral equation of the first kind, in: R.P. Agarwal (Ed.), Contributions in Numerical Mathematics, World Scientific, Singapore, 1993, pp. 127–140. – volume: 6 start-page: 897 year: 1990 end-page: 921 ident: BIB2 article-title: Boundary estimation problems arising in thermal tomography publication-title: Inverse Problems – volume: 63 start-page: 455 year: 1992 end-page: 481 ident: BIB20 article-title: Time discretizations of parabolic boundary integral equations publication-title: Numer. Math. – volume: 16 start-page: 87 year: 1993 end-page: 114 ident: BIB15 article-title: Boundary integral solution of the two-dimensional heat equation publication-title: Math. Meth. Appl. Sci. – reference: O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, AMS Publications, Providence, 1968. – volume: 56 start-page: 715 year: 1996 end-page: 735 ident: BIB5 article-title: An inverse problem in thermal imaging publication-title: SIAM J. Appl. Math. – reference: F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer, Berlin, 1993. – reference: C.A. Brebbia, J.C.F. Telles, L.C. Wrobel, Boundary Element Techniques, Springer, New York, 1984. – reference: D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd Edition, Springer, Berlin, 1998. – volume: 13 start-page: 498 year: 1990 end-page: 552 ident: BIB10 article-title: Boundary integral operators for the heat equation publication-title: Integral Equations Operator Theory – reference: R. Dautray, J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 1, Physical Origins and Classical Methods, Springer, Berlin, 1990. – reference: Z.S. Berezhanskaya, I.V. Lyudkevich, Numerical solution of a spatial boundary value problem for the heat equation by integral equations, Vychisl. Prikl. Mat. 41 (1980) 3–7 (in Russian). – reference: R. Kress, Linear Integral Equations, Springer, Berlin, 1989. – volume: 14 start-page: 853 year: 1998 end-page: 867 ident: BIB8 article-title: On the numerical solution of an inverse boundary value problem for the heat equation publication-title: Inverse Problems – volume: 66 start-page: 199 year: 1993 end-page: 214 ident: BIB17 article-title: On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation publication-title: Numer. Math. – volume: 9 start-page: 47 year: 1997 end-page: 69 ident: BIB7 article-title: Rothe's method for heat equation and boundary integral equations publication-title: J. Integral Equations Appl. – reference: A.J. Nowak, BEM approach to inverse thermal problems, in: D.B. Ingham, L.C. Wrobel (Eds.), Boundary Integral Formulations for Inverse Analysis, Computational Mechanics Publications, Southampton, 1997, pp. 259–298. – volume: 47 start-page: 273 year: 1989 end-page: 293 ident: BIB1 article-title: Boundary shape identification problems in two-dimensional domains related to thermal testing of materials publication-title: Quart. J. Appl. Math. – reference: A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, 1964. – ident: 10.1016/S0377-0427(99)00099-0_BIB9 doi: 10.1007/978-3-662-03537-5 – volume: 56 start-page: 715 year: 1996 ident: 10.1016/S0377-0427(99)00099-0_BIB5 article-title: An inverse problem in thermal imaging publication-title: SIAM J. Appl. Math. doi: 10.1137/S0036139994277828 – ident: 10.1016/S0377-0427(99)00099-0_BIB21 – ident: 10.1016/S0377-0427(99)00099-0_BIB11 doi: 10.1007/978-3-642-61527-6 – volume: 63 start-page: 455 year: 1992 ident: 10.1016/S0377-0427(99)00099-0_BIB20 article-title: Time discretizations of parabolic boundary integral equations publication-title: Numer. Math. doi: 10.1007/BF01385870 – ident: 10.1016/S0377-0427(99)00099-0_BIB18 – volume: 9 start-page: 47 year: 1997 ident: 10.1016/S0377-0427(99)00099-0_BIB7 article-title: Rothe's method for heat equation and boundary integral equations publication-title: J. Integral Equations Appl. doi: 10.1216/jiea/1181075987 – volume: 13 start-page: 498 year: 1990 ident: 10.1016/S0377-0427(99)00099-0_BIB10 article-title: Boundary integral operators for the heat equation publication-title: Integral Equations Operator Theory doi: 10.1007/BF01210400 – volume: 47 start-page: 273 year: 1989 ident: 10.1016/S0377-0427(99)00099-0_BIB1 article-title: Boundary shape identification problems in two-dimensional domains related to thermal testing of materials publication-title: Quart. J. Appl. Math. doi: 10.1090/qam/998101 – ident: 10.1016/S0377-0427(99)00099-0_BIB6 doi: 10.1142/9789812798886_0010 – ident: 10.1016/S0377-0427(99)00099-0_BIB3 – volume: 14 start-page: 853 year: 1998 ident: 10.1016/S0377-0427(99)00099-0_BIB8 article-title: On the numerical solution of an inverse boundary value problem for the heat equation publication-title: Inverse Problems doi: 10.1088/0266-5611/14/4/006 – ident: 10.1016/S0377-0427(99)00099-0_BIB16 doi: 10.1007/978-3-642-97146-4 – volume: 66 start-page: 199 year: 1993 ident: 10.1016/S0377-0427(99)00099-0_BIB17 article-title: On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation publication-title: Numer. Math. doi: 10.1007/BF01385694 – ident: 10.1016/S0377-0427(99)00099-0_BIB19 – ident: 10.1016/S0377-0427(99)00099-0_BIB12 – ident: 10.1016/S0377-0427(99)00099-0_BIB13 – ident: 10.1016/S0377-0427(99)00099-0_BIB14 – ident: 10.1016/S0377-0427(99)00099-0_BIB4 doi: 10.1007/978-3-642-48860-3 – volume: 16 start-page: 87 year: 1993 ident: 10.1016/S0377-0427(99)00099-0_BIB15 article-title: Boundary integral solution of the two-dimensional heat equation publication-title: Math. Meth. Appl. Sci. doi: 10.1002/mma.1670160203 – volume: 6 start-page: 897 year: 1990 ident: 10.1016/S0377-0427(99)00099-0_BIB2 article-title: Boundary estimation problems arising in thermal tomography publication-title: Inverse Problems doi: 10.1088/0266-5611/6/6/003 – ident: 10.1016/S0377-0427(99)00099-0_BIB22 doi: 10.1007/978-1-4612-2706-9 |
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| SubjectTerms | Collocation method Exact sciences and technology Green's function Heat equation Initial boundary value problem Integral equation Inverse boundary problem Mathematical analysis Mathematics Newton method Numerical analysis Numerical analysis. Scientific computation Ordinary differential equations Partial differential equations Partial differential equations, initial value problems and time-dependant initial-boundary value problems Regularization Sciences and techniques of general use Semi-infinite region Trigonometric interpolation |
| Title | On the numerical solution of direct and inverse problems for the heat equation in a semi-infinite region |
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