Numerical solutions to an inverse problem for a non-linear Helmholtz equation

In this work, we develop numerical methods to solve forward and inverse wave problems for a nonlinear Helmholtz equation defined in a spherical shell between two concentric spheres centred at the origin. A spectral method is developed to solve the forward problem while a combination of a finite difference approximation and the least squares method are derived for the inverse problem. Numerical examples are given to verify the method. References R. Askey. Orthogonal polynomials and special functions. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, 1975. doi: 10.1137/1.9781611970470 G. Baruch, G. Fibich, and S. Tsynkov. High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension. Nonlinear Photonics. Optica Publishing Group, 2007. doi: 10.1364/np.2007.ntha6 G. Fibich and S. Tsynkov. High-Order Two-Way Artificial Boundary Conditions for Nonlinear Wave Propagation with Backscattering. J. Comput. Phys. 171 (2001), pp. 632–677. doi: 10.1006/jcph.2001.6800 G. Fibich and S. Tsynkov. Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions. J. Comput. Phys. 210 (2005), pp. 183–224. doi: 10.1016/j.jcp.2005.04.015 P. M. Morse and K. U. Ingard. Theoretical Acoustics. International Series in Pure and Applied Physics. McGraw-Hill Book Company, 1968 G. N. Watson. A treatise on the theory of Bessel functions. International Series in Pure and Applied Physics. Cambridge Mathematical Library, 1996. url: https://www.cambridge.org/au/universitypress/subjects/mathematics/real-and-complex-analysis/treatise-theory-bessel-functions-2nd-edition-1?format=PB&isbn=9780521483919