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We investigate solving the inverse coefficient issue for a hyperbolic equation under overspecified condition of general integral type. Crank–Nicolson finite difference method (FDM) combined with the trapezoidal rule quadrature has been used for direct problem. While, the inverse problem was reformulated as nonlinear regularized least-square optimization problem with simple bound and solved efficiently by MATLAB subroutine lsqnonlin from optimization toolbox. Since problem under investigation is generally ill-posed, small error in the input data lead to huge error in the output, then Tikhonov's regularization technique is applied to obtain a regularized stable results. [ABSTRACT FROM AUTHOR]

Maximum-entropy moment closures for describing non-equilibrium rare fied gaseous flow shave previously been shown to provide accurate and computationally efficient descriptions of transition-regime flows. Unfortunately, for high-order variants of these closures above second order in velocity space, there are no analytical closures for the systems of hyperbolic partial differential equations (PDEs) which govern the transport ofthe macroscopic moment quantities and instead approximate closures have been sought. In this study, abi-Gaussian approximation for then umber density function (NDF) is considered both for approximating the NDF and closing moment fluxes of the resulting fourth-order 14-moment maximum-entropy closure associated with fully three-dimensional kinetic theory. Prior investigations of the bi-Gaussian approximation applied to simplified one-dimensional univariate kinetic theory has yielded excellent results when compared to the actual maximum-entropy solutions as well a similar interpolative-based maximum-entropy-based (IBME) closure. In the one-dimensional univariate case, the bi-Gaussian closure is equivalent to the so-called extended quadrature method of moments (EQMOM) with anormal or Gaussian kernel basis function. A potential benefit of the bi-Gaussian approach proposed herein is that an essentially closed-form analytical expression results for the NDF. In this study, the extension of the bi-Gaussian closure to the multi-dimensional case is considered and compared to the equivalent multi-dimensional IBME closure. The approximate form for the NDF and closing fluxes interms of the relevant moments are derived and the validity and hyperbolicity of the closure for the space of realizable predicted moments are all explored and compared to those of the IBME closure. It is shown that the bi-Gaussian closure in the multi-dimensional case unfortunately suffers from several de ficiencies: firstly, the valid region of realizable moment space for the bi-Gaussian closure is a small subset of the full realizable 14-moment space; and secondly, the closure and moment equation eigen structure for solutions associated with zero heat fluxbe come undefined. The findings here in suggest that the proposed bi-Gaussian closure may not be agood choice for practical multi-dimensional rare fied flow predictions despite the promising results exhibited in the one-dimensional case. [ABSTRACT FROM AUTHOR]