
Previous Article
Identification of obstacles using only the scattered Pwaves or the scattered Swaves
 IPI Home
 This Issue

Next Article
Small volume asymptotics for anisotropic elastic inclusions
On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach
1.  Faculty of Applied Mathematics and Computer Science, Ivan Franko National University of Lviv, 79000 Lviv, Ukraine 
2.  School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT 
References:
[1] 
A.P. Calderón, Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math., 80 (1958), 1636. 
[2] 
F. Cakoni and R. Kress, Integral equations for inverse problems in corrosion detection from partial Cauchy data, Inverse Probl. Imaging, 1 (2007), 229245. doi: 10.3934/ipi.2007.1.229. 
[3] 
F. Cakoni, R. Kress and C. Schuft, Integral equations for shape and impedance reconstruction in corrosion detection, Inverse Problems, 26 (2010), 095012, 24 pp. 
[4] 
H. Cao, M. V. Klibanov and S. V. Pereverzev, A Carleman estimate and the balancing principle in the quasireversibility method for solving the Cauchy problem for the Laplace equation, Inverse Problems, 25 (2009), 035005, 21 pp. 
[5] 
T. Carleman, Sur un problème d'unicité pur les systèmes d'équations aux dérivées partielles à deux variables indépendantes, (French) Ark. Mat., Astr. Fys., 26 (1939), 9 pp. 
[6] 
R. Chapko and B. T. Johansson, An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semiinfinite domains, Inverse Probl. Imaging, 2 (2008), 317333. 
[7] 
R. Chapko and R. Kress, A hybrid method for inverse boundary value problems in potential theory, J. Inverse IllPosed Probl., 13 (2005), 2740. doi: 10.1515/1569394053583711. 
[8] 
J. Cheng, Y. C. Hon, T. Wei and M. Yamamoto, Numerical computation of a Cauchy problem for Laplace's equation, ZAMM Z. Angew. Math. Mech., 81 (2001), 665674. doi: 10.1002/15214001(200110)81:10<665::AIDZAMM665>3.0.CO;2V. 
[9] 
X. Escriva, T. N. Baranger and N. H. Tlatli, Leak identification in porous media by solving the Cauchy problem, C. R. Mecanique, 335 (2007), 401406. doi: 10.1016/j.crme.2007.04.001. 
[10] 
Dinh Nho Hào, "Methods for Inverse Heat Conduction Problems," Habilitationsschrift, University of Siegen, Siegen, 1996, Methoden und Verfahren der Mathematischen Physik, 43, Peter Lang, Frankfurt am Main, 1998. 
[11] 
Dinh Nho Hào, B. T. Johansson, D. Lesnic and Pham Minh Hien, A variational method and approximations of a Cauchy problem for elliptic equations, J. Algorithms Comput. Technol., 4 (2010), 89119. doi: 10.1260/17483018.4.1.89. 
[12] 
M. Hanke and M. Bruhl, Recent progress in electrical impedance tomography. Special section on imaging, Inverse Problems, 19 (2003), S65S90. doi: 10.1088/02665611/19/6/055. 
[13] 
P. C. Hansen, The Lcurve and its use in the numerical treatment of inverse problems, in "Computational Inverse Problems in Electrocardiology" (ed. P. Johnston), WIT Press, Southampton, (2001), 119142. 
[14] 
B. He, Y. Wang and D. Wu, Estimating cortical potentials from scalp EEG's in a realistically shaped inhomogeneous head model by means of the boundary element method, IEEE Trans. Biomedical Engn., 46 (1999), 12641268. doi: 10.1109/10.790505. 
[15] 
J. Helsing and B. T. Johansson, Fast reconstruction of harmonic functions from Cauchy data using integral equation techniques, Inverse Probl. Sci. Engn., 18 (2010), 381399. doi: 10.1080/17415971003624322. 
[16] 
D. Holder, "Electrical Impedance Tomography: Methods, History and Applications," Institute of Physics, Bristol, 2005. 
[17] 
Y. C. Hon and T. Wei, A meshless scheme for solving inverse problems of Laplace equation, in "Recent Development in Theories & Numerics," World Sci. Publ., River Edge, NJ, (2003), 291300. 
[18] 
V. A. Kozlov and V. G. Maz'ya, Iterative procedures for solving illposed boundary value problems that preserve differential equations, Algebra i Analiz, 1 (1989), 144170; Translation in Leningrad Math. J., 1 (1990), 12071228. 
[19] 
R. Kress, On the numerical solution of a hypersingular integral equation in scattering theory, J. Comp. Appl. Math., 61 (1995), 345360. doi: 10.1016/03770427(94)000737. 
[20] 
R. Kress and W. Rundell, Nonlinear integral equations and the iterative solution for an inverse boundary value problem, Inverse Problems, 21 (2005), 12071223. doi: 10.1088/02665611/21/4/002. 
[21] 
P. K. Kythe, "Green's Functions and Linear Differential Equations: Theory, Applications, and Computation," Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series, CRC Press, Boca Raton, FL, 2011. 
[22] 
M. M. Lavrentiev, "Some Improperly Posed Problems of Mathematical Physics," Springer Verlag, Berlin, 1967. 
[23] 
J. Y. Lee and J. R. Yoon, A numerical method for Cauchy problem using singular value decomposition, Comm. Korean Math. Soc. 16 (2001), 487508. 
[24] 
L. Marin, Relaxation procedures for an iterative MFS algorithm for twodimensional steadystate isotropic heat conduction Cauchy problems, Engng. Anal. Bound. Elem., 35 (2011), 415429. doi: 10.1016/j.enganabound.2010.07.011. 
[25] 
L. Marin and D. Lesnic, The method of fundamental solutions for the Cauchy problem associated with twodimensional Helmholtztype equations, Comput. & Structures, 83 (2005), 267278. doi: 10.1016/j.compstruc.2004.10.005. 
[26] 
L. E. Payne, "Improperly Posed Problems in Partial Differential Equations," Regional Conference Series in Applied Mathematics, No. 22, SIAM, Philadelphia, Pa., 1975. 
[27] 
T. Regińska, Regularization of discrete illposed problems, BIT, 44 (2004), 119133. doi: 10.1023/B:BITN.0000025090.68586.5e. 
[28] 
N. Tarchanov, "The Cauchy Problem for Solutions of Elliptic Equations," Mathematical Topics, 7, Akademie Verlag, Berlin, 1995. 
[29] 
A. Zeb, L. Elliott, D. B. Ingham and D. Lesnic, Solution of the Cauchy problem for Laplace equation, in "First UK Conference on Boundary Integral Methods" (eds. L. Elliott, D. B. Ingham and D. Lesnic), Leeds University Press, (1997), 297307. 
show all references
References:
[1] 
A.P. Calderón, Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math., 80 (1958), 1636. 
[2] 
F. Cakoni and R. Kress, Integral equations for inverse problems in corrosion detection from partial Cauchy data, Inverse Probl. Imaging, 1 (2007), 229245. doi: 10.3934/ipi.2007.1.229. 
[3] 
F. Cakoni, R. Kress and C. Schuft, Integral equations for shape and impedance reconstruction in corrosion detection, Inverse Problems, 26 (2010), 095012, 24 pp. 
[4] 
H. Cao, M. V. Klibanov and S. V. Pereverzev, A Carleman estimate and the balancing principle in the quasireversibility method for solving the Cauchy problem for the Laplace equation, Inverse Problems, 25 (2009), 035005, 21 pp. 
[5] 
T. Carleman, Sur un problème d'unicité pur les systèmes d'équations aux dérivées partielles à deux variables indépendantes, (French) Ark. Mat., Astr. Fys., 26 (1939), 9 pp. 
[6] 
R. Chapko and B. T. Johansson, An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semiinfinite domains, Inverse Probl. Imaging, 2 (2008), 317333. 
[7] 
R. Chapko and R. Kress, A hybrid method for inverse boundary value problems in potential theory, J. Inverse IllPosed Probl., 13 (2005), 2740. doi: 10.1515/1569394053583711. 
[8] 
J. Cheng, Y. C. Hon, T. Wei and M. Yamamoto, Numerical computation of a Cauchy problem for Laplace's equation, ZAMM Z. Angew. Math. Mech., 81 (2001), 665674. doi: 10.1002/15214001(200110)81:10<665::AIDZAMM665>3.0.CO;2V. 
[9] 
X. Escriva, T. N. Baranger and N. H. Tlatli, Leak identification in porous media by solving the Cauchy problem, C. R. Mecanique, 335 (2007), 401406. doi: 10.1016/j.crme.2007.04.001. 
[10] 
Dinh Nho Hào, "Methods for Inverse Heat Conduction Problems," Habilitationsschrift, University of Siegen, Siegen, 1996, Methoden und Verfahren der Mathematischen Physik, 43, Peter Lang, Frankfurt am Main, 1998. 
[11] 
Dinh Nho Hào, B. T. Johansson, D. Lesnic and Pham Minh Hien, A variational method and approximations of a Cauchy problem for elliptic equations, J. Algorithms Comput. Technol., 4 (2010), 89119. doi: 10.1260/17483018.4.1.89. 
[12] 
M. Hanke and M. Bruhl, Recent progress in electrical impedance tomography. Special section on imaging, Inverse Problems, 19 (2003), S65S90. doi: 10.1088/02665611/19/6/055. 
[13] 
P. C. Hansen, The Lcurve and its use in the numerical treatment of inverse problems, in "Computational Inverse Problems in Electrocardiology" (ed. P. Johnston), WIT Press, Southampton, (2001), 119142. 
[14] 
B. He, Y. Wang and D. Wu, Estimating cortical potentials from scalp EEG's in a realistically shaped inhomogeneous head model by means of the boundary element method, IEEE Trans. Biomedical Engn., 46 (1999), 12641268. doi: 10.1109/10.790505. 
[15] 
J. Helsing and B. T. Johansson, Fast reconstruction of harmonic functions from Cauchy data using integral equation techniques, Inverse Probl. Sci. Engn., 18 (2010), 381399. doi: 10.1080/17415971003624322. 
[16] 
D. Holder, "Electrical Impedance Tomography: Methods, History and Applications," Institute of Physics, Bristol, 2005. 
[17] 
Y. C. Hon and T. Wei, A meshless scheme for solving inverse problems of Laplace equation, in "Recent Development in Theories & Numerics," World Sci. Publ., River Edge, NJ, (2003), 291300. 
[18] 
V. A. Kozlov and V. G. Maz'ya, Iterative procedures for solving illposed boundary value problems that preserve differential equations, Algebra i Analiz, 1 (1989), 144170; Translation in Leningrad Math. J., 1 (1990), 12071228. 
[19] 
R. Kress, On the numerical solution of a hypersingular integral equation in scattering theory, J. Comp. Appl. Math., 61 (1995), 345360. doi: 10.1016/03770427(94)000737. 
[20] 
R. Kress and W. Rundell, Nonlinear integral equations and the iterative solution for an inverse boundary value problem, Inverse Problems, 21 (2005), 12071223. doi: 10.1088/02665611/21/4/002. 
[21] 
P. K. Kythe, "Green's Functions and Linear Differential Equations: Theory, Applications, and Computation," Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series, CRC Press, Boca Raton, FL, 2011. 
[22] 
M. M. Lavrentiev, "Some Improperly Posed Problems of Mathematical Physics," Springer Verlag, Berlin, 1967. 
[23] 
J. Y. Lee and J. R. Yoon, A numerical method for Cauchy problem using singular value decomposition, Comm. Korean Math. Soc. 16 (2001), 487508. 
[24] 
L. Marin, Relaxation procedures for an iterative MFS algorithm for twodimensional steadystate isotropic heat conduction Cauchy problems, Engng. Anal. Bound. Elem., 35 (2011), 415429. doi: 10.1016/j.enganabound.2010.07.011. 
[25] 
L. Marin and D. Lesnic, The method of fundamental solutions for the Cauchy problem associated with twodimensional Helmholtztype equations, Comput. & Structures, 83 (2005), 267278. doi: 10.1016/j.compstruc.2004.10.005. 
[26] 
L. E. Payne, "Improperly Posed Problems in Partial Differential Equations," Regional Conference Series in Applied Mathematics, No. 22, SIAM, Philadelphia, Pa., 1975. 
[27] 
T. Regińska, Regularization of discrete illposed problems, BIT, 44 (2004), 119133. doi: 10.1023/B:BITN.0000025090.68586.5e. 
[28] 
N. Tarchanov, "The Cauchy Problem for Solutions of Elliptic Equations," Mathematical Topics, 7, Akademie Verlag, Berlin, 1995. 
[29] 
A. Zeb, L. Elliott, D. B. Ingham and D. Lesnic, Solution of the Cauchy problem for Laplace equation, in "First UK Conference on Boundary Integral Methods" (eds. L. Elliott, D. B. Ingham and D. Lesnic), Leeds University Press, (1997), 297307. 
[1] 
Noui Djaidja, Mostefa Nadir. Comparison between Taylor and perturbed method for Volterra integral equation of the first kind. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 487493. doi: 10.3934/naco.2020039 
[2] 
Roman Chapko, B. Tomas Johansson. An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semiinfinite regions. Inverse Problems and Imaging, 2008, 2 (3) : 317333. doi: 10.3934/ipi.2008.2.317 
[3] 
Dong Li. A regularizationfree approach to the CahnHilliard equation with logarithmic potentials. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 24532460. doi: 10.3934/dcds.2021198 
[4] 
Virginia Agostiniani, Rolando Magnanini. Symmetries in an overdetermined problem for the Green's function. Discrete and Continuous Dynamical Systems  S, 2011, 4 (4) : 791800. doi: 10.3934/dcdss.2011.4.791 
[5] 
David Hoff. Pointwise bounds for the Green's function for the NeumannLaplace operator in $ \text{R}^3 $. Kinetic and Related Models, 2022, 15 (4) : 535550. doi: 10.3934/krm.2021037 
[6] 
Nicolas Fournier. A new regularization possibility for the Boltzmann equation with soft potentials. Kinetic and Related Models, 2008, 1 (3) : 405414. doi: 10.3934/krm.2008.1.405 
[7] 
V. Varlamov, Yue Liu. Cauchy problem for the Ostrovsky equation. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 731753. doi: 10.3934/dcds.2004.10.731 
[8] 
Adrien Dekkers, Anna RozanovaPierrat. Cauchy problem for the Kuznetsov equation. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 277307. doi: 10.3934/dcds.2019012 
[9] 
Andreas Kirsch. An integral equation approach and the interior transmission problem for Maxwell's equations. Inverse Problems and Imaging, 2007, 1 (1) : 159179. doi: 10.3934/ipi.2007.1.159 
[10] 
Yonggang Zhao, Mingxin Wang. An integral equation involving Bessel potentials on half space. Communications on Pure and Applied Analysis, 2015, 14 (2) : 527548. doi: 10.3934/cpaa.2015.14.527 
[11] 
Rudong Zheng, Zhaoyang Yin. The Cauchy problem for a generalized Novikov equation. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 35033519. doi: 10.3934/dcds.2017149 
[12] 
Claudia Bucur. Some observations on the Green function for the ball in the fractional Laplace framework. Communications on Pure and Applied Analysis, 2016, 15 (2) : 657699. doi: 10.3934/cpaa.2016.15.657 
[13] 
Nadia Lekrine, ChaoJiang Xu. Gevrey regularizing effect of the Cauchy problem for noncutoff homogeneous Kac's equation. Kinetic and Related Models, 2009, 2 (4) : 647666. doi: 10.3934/krm.2009.2.647 
[14] 
JinMyong An, JinMyong Kim, KyuSong Chae. Continuous dependence of the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation in $H^{s} (\mathbb R^{n})$. Discrete and Continuous Dynamical Systems  B, 2021 doi: 10.3934/dcdsb.2021221 
[15] 
Peter Bella, Arianna Giunti. Green's function for elliptic systems: Moment bounds. Networks and Heterogeneous Media, 2018, 13 (1) : 155176. doi: 10.3934/nhm.2018007 
[16] 
Sungwon Cho. Alternative proof for the existence of Green's function. Communications on Pure and Applied Analysis, 2011, 10 (4) : 13071314. doi: 10.3934/cpaa.2011.10.1307 
[17] 
Richard A. Norton, G. R. W. Quispel. Discrete gradient methods for preserving a first integral of an ordinary differential equation. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 11471170. doi: 10.3934/dcds.2014.34.1147 
[18] 
Arturo de Pablo, Guillermo Reyes, Ariel Sánchez. The Cauchy problem for a nonhomogeneous heat equation with reaction. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 643662. doi: 10.3934/dcds.2013.33.643 
[19] 
Shaoyong Lai, Yong Hong Wu. The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation. Discrete and Continuous Dynamical Systems  B, 2003, 3 (3) : 401408. doi: 10.3934/dcdsb.2003.3.401 
[20] 
Guillermo Reyes, JuanLuis Vázquez. The Cauchy problem for the inhomogeneous porous medium equation. Networks and Heterogeneous Media, 2006, 1 (2) : 337351. doi: 10.3934/nhm.2006.1.337 
2020 Impact Factor: 1.639
Tools
Metrics
Other articles
by authors
[Back to Top]