Method of fundamental solutions for a Cauchy problem of the Laplace equation in a half-plane. (English) Zbl 1524.35781
Summary: This paper is to provide an analysis of an ill-posed Cauchy problem in a half-plane. This problem is novel since the Cauchy data on the accessible boundary is given, whilst the additional temperature is involved on a line. The Dirichlet boundary condition on part of the boundary is an essential condition in the physical meaning. Then we use a redefined method of fundamental solutions (MFS) to determine the temperature and the normal heat flux on the inaccessible boundary. The present approach will give an ill-conditioned system, and this is a feature of the numerical method employed. In order to overcome the ill-posedness of the corresponding system, we use the Tikhonov regularization method combining Morozov’s discrepancy principle to get a stable solution. At last, four numerical examples, including a smooth boundary, a boundary with a corner, and a boundary with a jump, are given to show the effectiveness of the present approach.
MSC:
| 35R35 | Free boundary problems for PDEs |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 35J08 | Green’s functions for elliptic equations |
| 35J25 | Boundary value problems for second-order elliptic equations |
References:
| [1] | Bukhgeim, A.L., Cheng, J., Yamamoto, M.: Stability for an inverse boundary problem of determining a part of a boundary. Inverse Probl. 15(4), 1021-1032 (1999) · Zbl 0934.35202 |
| [2] | Chapko, R., Johansson, B.T.: An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite domains. Inverse Probl. Imaging 2, 317-333 (2008) · Zbl 1180.35545 |
| [3] | Chen, B., Ma, F., Guo, Y.: Time domain scattering and inverse scattering problems in a locally perturbed half-plane. Appl. Anal. 96(8), 1-23 (2017) · Zbl 1431.65151 |
| [4] | Chen, J.T., Chen, K.H.: Analytical study and numerical experiments for Laplace equation with overspecified boundary conditions. Appl. Math. Model. 22, 703-725 (1998) · Zbl 1428.74109 |
| [5] | Chen, J.T., Yang, J.L., Li, Y.T., Chang, Y.L.: Formulation of the MFS for the two-dimensional Laplace equation with an added constant and constraint. Eng. Anal. Bound. Elem. 46, 96-107 (2014) · Zbl 1297.65194 |
| [6] | Chen, L.Y., Chen, J.T., Hong, H.K., Chen, C.H.: Application of Cesaro mean and the L-curve for the deconvolution problem. Soil Dyn. Earthq. Eng. 14, 361-373 (1995) |
| [7] | Chen, W., Fu, Z.: Boundary particle method for inverse Cauchy problems of inhomogeneous Helmholtz equations. J. Mar. Sci. Technol. 17(3), 157-163 (2009) |
| [8] | Fairweather, G., Karageorghis, A.: The method of fundamental solutions for elliptic boundary value problems. Adv. Comput. Math. 9(1-2), 69-95 (1998) · Zbl 0922.65074 |
| [9] | Fu, Z.J., Xi, Q., Ling, L., Cao, C.Y.: Numerical investigation on the effect of tumor on the thermal behavior inside the skin tissue. Int. J. Heat Mass Transf. 108, 1154-1163 (2017) |
| [10] | Gu, M.H., Young, D.L., Fan, C.M.: The method of fundamental solutions for one-dimensional wave equations. J. Mar. Sci. Technol. 11(3), 185-208 (2009) |
| [11] | Gu, Y., He, X., Chen, W., Zhang, C.: Analysis of three-dimensional anisotropic heat conduction problems on thin domains using an advanced boundary element method. Comput. Math. Appl. 75, 33-44 (2018) · Zbl 1416.80006 |
| [12] | Hansen, P.C.: Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev. 34, 561-580 (1992) · Zbl 0770.65026 |
| [13] | Hon, Y.C., Wei, T.: Backus-Gilbert algorithm for the Cauchy problem of the Laplace equation. Inverse Probl. 17(2), 261-271 (2001) · Zbl 0980.35167 |
| [14] | Hong, H.K., Chen, J.T.: Derivations of integral equations of elasticity. J. Eng. Mech. ASCE 114, 1028-1044 (1998) |
| [15] | Hsiao, G.C., Wendland, W.L.: Boundary Integral Equations. Springer, Berlin (2008) · Zbl 1157.65066 |
| [16] | Karageorghis, A., Lesnic, D., Marin, L.: A survey of applications of the MFS to inverse problems. Inverse Probl. Sci. Eng. 19(3), 309-336 (2011) · Zbl 1220.65157 |
| [17] | Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problems, 2nd edn. Springer, New York (2011) · Zbl 1213.35004 |
| [18] | Lee, J.Y., Yoon, J.R.: A numerical method for Cauchy problem using singular value decomposition. Commun. Korean Math. Soc. 16(3), 487-508 (2001) · Zbl 1101.65314 |
| [19] | Li, J., Chen, W.: A modified singular boundary method for three-dimensional high frequency acoustic wave problems. Appl. Math. Model. 54, 189-201 (2018) · Zbl 1480.74151 |
| [20] | Li, J., Chen, W., Fu, Z., Qin, Q.: A regularized approach evaluating the near-boundary and boundary solutions for three-dimensional Helmholtz equation with wideband wavenumbers. Appl. Math. Lett. 91, 55-60 (2019) · Zbl 1411.35084 |
| [21] | Li, J., Chen, W., Gu, Y.: Error bounds of singular boundary method for potential problems. Numer. Methods Partial Differ. Equ. 33, 1987-2004 (2017) · Zbl 1390.65156 |
| [22] | Li, J., Qin, Q.H., Fu, Z.: A dual-level method of fundamental solutions for three-dimensional exterior high frequency acoustic problems. Appl. Math. Model. 63, 558-576 (2018) · Zbl 1480.76109 |
| [23] | Lin, J., Liu, C.S., Chen, W., Sun, L.: A novel Trefftz method for solving the multi-dimensional direct and Cauchy problems of Laplace equation in an arbitrary domain. J. Comput. Sci. 17(3), 275-302 (2018) |
| [24] | Lin, J., Zhang, C., Sun, L., Lu, J.: Simulation of seismic wave scattering by embedded cavities in an elastic half-plane using the novel singular boundary method. Adv. Appl. Math. Mech. 10(2), 322-342 (2018) · Zbl 1488.65719 |
| [25] | Liu, C.S.: A highly accurate MCTM for inverse Cauchy problems of Laplace equation in arbitrary plane domains. Comput. Model. Eng. Sci. 35(2), 91-111 (2008) · Zbl 1153.65359 |
| [26] | Liu, C.S., Wang, F., Gu, Y.: Trefftz energy method for solving the Cauchy problem of the Laplace equation. Appl. Math. Lett. 79, 187-195 (2018) · Zbl 1466.65168 |
| [27] | Marin, L.: An invariant method of fundamental solutions for two-dimensional steady-state anisotropic heat conduction problems. Int. J. Heat Mass Transf. 94, 449-464 (2016) |
| [28] | Marin, L., Cipu, C.: Non-iterative regularized MFS solution of inverse boundary value problems in linear elasticity: a numerical study. Appl. Math. Comput. 293, 265-286 (2017) · Zbl 1411.74063 |
| [29] | Polyanin, A.D.: Handbook of Linear Partial Differential Equations for Engineers and Scientists. Chapman & Hall/CRC, Boca Raton (2002) · Zbl 1027.35001 |
| [30] | Sun, Y.: Modified method of fundamental solutions for the Cauchy problem connected with the Laplace equation. Int. J. Comput. Math. 91(10), 2185-2198 (2014) · Zbl 1306.65259 |
| [31] | Sun, Y.: Indirect boundary integral equation method for the Cauchy problem of the Laplace equation. J. Sci. Comput. 71(2), 469-498 (2017) · Zbl 1387.31002 |
| [32] | Sun, Y., He, S.: A meshless method based on the method of fundamental solution for three-dimensional inverse heat conduction problems. Int. J. Heat Mass Transf. 108, 945-960 (2017) |
| [33] | Sun, Y., Ma, F., Zhang, D.: An integral equations method combined minimum norm solution for 3D elastostatics Cauchy problem. Comput. Methods Appl. Mech. Eng. 271, 231-252 (2014) · Zbl 1296.35187 |
| [34] | Sun, Y., Ma, F., Zhou, X.: An invariant method of fundamental solutions for the Cauchy problem in two-dimensional isotropic linear elasticity. J. Sci. Comput. 64(1), 197-215 (2015) · Zbl 06478158 |
| [35] | Sun, Y., Zhang, D., Ma, F.: A potential function method for the Cauchy problem of elliptic operators. J. Math. Anal. Appl. 395(1), 164-174 (2012) · Zbl 1254.65099 |
| [36] | Wang, F., Chen, W., Qu, W., Gu, Y.: A BEM formulation in conjunction with parametric equation approach for three-dimensional Cauchy problems of steady heat conduction. Eng. Anal. Bound. Elem. 63, 1-14 (2016) · Zbl 1403.80028 |
| [37] | Wang, F.J., Chen, W., Gu, Y.: Boundary element analysis of inverse heat conduction problems in 2D thin-walled structures. Int. J. Heat Mass Transf. 91, 1001-1009 (2015) |
| [38] | Wang, Z.: Multi-parameter Tikhonov regularization and model function approach to the damped Morozov principle for choosing regularization parameters. J. Comput. Appl. Math. 236(7), 1815-1832 (2012) · Zbl 1247.65072 |
| [39] | Wei, T., Hon, Y.C., Ling, L.: Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators. Eng. Anal. Bound. Elem. 31, 373-385 (2007) · Zbl 1195.65206 |
| [40] | Wei, T., Zhou, D.: Convergence analysis for the Cauchy problem of Laplace’s equation by a regularized method of fundamental solutions. Adv. Comput. Math. 33, 491-510 (2010) · Zbl 1213.65136 |
| [41] | Zhang, H.W., Wei, T.: Two iterative methods for a Cauchy problem of the elliptic equation with variable coefficients in a strip region. Numer. Algorithms 65(4), 875-892 (2014) · Zbl 1304.65211 |
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