Abstract
This paper considers the Cauchy problem of a semi-linear elliptic equation and uses a generalized Tikhonov-type regularization method to overcome its ill-posedness. The existence, uniqueness, and stability for regularized solution are proven. Under an a priori bound assumption for exact solution, we derive the convergence estimate of Hölder type for this method. An application of this method to the Cauchy problem of Helmholtz equation is discussed, and we investigate the stability and convergence estimates for different wave numbers. Finally, an iterative scheme is constructed to calculate the regularization solution, numerical results show that this method is stable and feasible.
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References
Belgacem, F.B.: Why is the Cauchy problem severely ill-posed? Inverse Prob. 23, 823 (2007)
Beskos, D.E.: Boundary element method in dynamic analysis: Part II (1986-1996). ASME Appl. Mech. Rev. 50, 149–197 (1997)
Chen, J.T., Wong, F.C.: Dual formulation of multiple reciprocity method for the acoustic mode of a cavity with a thin partition. J. Sound Vib. 217, 75–95 (1998)
Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems, Volume 375 of Mathematics and Its Applications. Kluwer Academic Publishers Group, Dordrecht (1996)
Evans, L.C.: Partial Diferential Equations, American Mathematical Society, vol. 19 (1998)
Feng, X.L., Ning, W.T., Qian, Z.: A quasi-boundary-value method for a Cauchy problem of an elliptic equation in multiple dimensions. Inverse Prob. Sci. Eng. 22(7), 1045–1061 (2014)
Fokas, A.S., Pelloni, B.: The dirichlet-to-neumann map for the elliptic sine-Gordon equation. Nonlinearity 25(4), 1011–1031 (2012)
Gutshabash, E.S., Lipovskii, V.D.: Boundary value problem for the two-dimensional elliptic sine-Gordon equation and its applications to the theory of the stationary josephson effect. J. Math. Sci. 68, 197–201 (1994)
Hào, D.N., Duc, N.V., Lesnic, D.: A non-local boundary value problem method for the Cauchy problem for elliptic equations. Inverse Prob. 25, 055002 (2009)
Hào, D.N., Van, T.D., Gorenflo, R.: Towards the Cauchy problem for the Laplace equation. Partial differential equations pp. 111 (1992)
Isakov, V: Inverse Problems for Partial Differential Equations. Springer, Berlin (2006)
Khoa, V.A., Truong, M.T., Duy, N.H.: A general kernel-based regularization method for coupled elliptic sine-Gordon equations with Gevrey regularity. Comput. Phys. Commun. 183(8), 1813–1821 (2015)
Khoa, V.A., Truong, M.T., Duy, N.H., Tuan, N.H.: The Cauchy problem of coupled elliptic sine-Gordon equations with noise: Analysis of a general kernel-based regularization and reliable tools of computing. Computers & Mathematics with Applications 73(1), 141–162 (2017)
Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problems, Volume 120 of Applied 2 Sciences. Springer, New York (1996)
M.M. Lavrentiev, Romanov, V.G., Shishatski, S.P.: Ill-posed problems of mathematical physics and analysis, volume 64 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI. Translated from the Russian by J.R. Schulenberger, Translation edited by Lev J. Leifman (1986)
Tautenhahn, U.: Optimal stable solution of Cauchy problems of elliptic equations. Journal for Analysis and its Applications 15(4), 961–984 (1996)
Tran, Q.V., Kirane, M., Nguyen, H.T., Nguyen, V.T.: Analysis and numerical simulation of the three-dimensional Cauchy problem for quasi-linear elliptic equations. J. Math. Anal. Appl. 446(1), 470–492 (2017)
Trong, D.D., Quan, P.H., Tuan, N.H.: A quasi-boundary value method for regularizing nonlinear ill-posed problems. Electronic Journal of Differential Equations 2009(109), 1–16 (2009)
Tuan, N.H., Thang, L.D., Khoa, V.A.: A modified integral equation method of the nonlinear elliptic equation with globally and locally lipschitz source. Appl. Math. Comput. 265, 245–265 (2015)
Tuan, N.H., Thang, L.D., Khoa, V.A., Tran, T.: On an inverse boundary value problem of a nonlinear elliptic equation in three dimensions. J. Math. Anal. Appl. 426, 1232–1261 (2015)
Tuan, N.H., Thang, L.D., Lesnic, D.: A new general filter regularization method for Cauchy problems for elliptic equations with a locally Lipschitz nonlinear source. J. Math. Anal Appl. 434, 1376–1393 (2016)
Tuan, N.H., Thang, L.D., Trong, D.D., Khoa, V.A.: Approximation of mild solutions of the linear and nonlinear elliptic equations. Inverse Prob. Sci. Eng. 23(7), 1237–1266 (2015)
Tuan, N.H., Trong, D.D.: A nonlinear parabolic equation backward in time Regularization with new error estimates. Nonlinear Anal. Theory Methods Appl. 73 (6), 1842–1852 (2010)
Tuan, N.H., Trong, D.D., Quan, P.H.: A note on a Cauchy problem for the Laplace equation Regularization and error estimates. Appl. Math. Comput. 217, 2913–2922 (2010)
Zhang, H.W., Wang, R.H.: Modified boundary Tikhonov-type regularization method for the Cauchy problem of a semi-linear elliptic equation. Inverse Prob. Sci. Eng. 24(7), 1249–1265 (2016)
Zhang, H.W., Wei, T.: A Fourier truncated regularization method for a Cauchy problem of a semi-linear elliptic equation. Journal of Inverse and Ill-posed Problems 22(2), 143–168 (2014)
Zhang, H.W., Zhang, X.J.: Filtering function method for the Cauchy problem of a semi-linear elliptic equation. Journal of Applied Mathematics and Physics 3(2), 1599–1609 (2015)
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The authors would like to thank the reviewers for their constructive comments and valuable suggestions that improve the quality of our paper.
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The work was supported by the YSRP (2017KJ33), NSF of China (11761004, 11371181), and SRP (2017SXKY05) at North Minzu University.
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Zhang, H., Zhang, X. Generalized Tikhonov-type regularization method for the Cauchy problem of a semi-linear elliptic equation. Numer Algor 81, 833–851 (2019). https://doi.org/10.1007/s11075-018-0573-4
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DOI: https://doi.org/10.1007/s11075-018-0573-4