A new method to solve the boundary value problem arising in the study of scattering of two-dimensional surface water waves by a discontinuity in the surface boundary conditions is presented in this paper. The disconti...A new method to solve the boundary value problem arising in the study of scattering of two-dimensional surface water waves by a discontinuity in the surface boundary conditions is presented in this paper. The discontinuity arises due to the floating of two semi-infinite inertial surfaces of different surface densities. Applying Green＇s second identity to the potential functions and appropriate Green＇s functions, this problem is reduced to solving two coupled Fredholm integral equations with regular kernels. The solutions to these integral equations are used to determine the reflection and the transmission coefficients. The results for the reflection coefficient are presented graphically and are compared to those obtained earlier using other research methods. It is observed from the graphs that the results computed from the present analysis match exactly with the previous results.展开更多
The authors investigate an inverse problem of determining the radiative coefficient in a degenerate parabolic equation from the final overspecified data. Being different from other inverse coefficient problems in whic...The authors investigate an inverse problem of determining the radiative coefficient in a degenerate parabolic equation from the final overspecified data. Being different from other inverse coefficient problems in which the principle coefficients are assumed to be strictly positive definite, the mathematical model discussed in this paper belongs to the second order parabolic equations with non-negative characteristic form, namely, there exists a degeneracy on the lateral boundaries of the domain. Based on the optimal control framework, the problem is transformed into an optimization problem and the existence of the minimizer is established. After the necessary conditions which must be satisfied by the minimizer are deduced, the uniqueness and stability of the minimizer are proved. By minor modification of the cost functional and some a priori regularity conditions imposed on the forward operator, the convergence of the minimizer for the noisy input data is obtained in this paper. The results can be extended to more general degenerate parabolic equations.展开更多
基金Partially Supported by a DST Research Project to RG(No.SR/FTP/MS-020/2010)
文摘A new method to solve the boundary value problem arising in the study of scattering of two-dimensional surface water waves by a discontinuity in the surface boundary conditions is presented in this paper. The discontinuity arises due to the floating of two semi-infinite inertial surfaces of different surface densities. Applying Green＇s second identity to the potential functions and appropriate Green＇s functions, this problem is reduced to solving two coupled Fredholm integral equations with regular kernels. The solutions to these integral equations are used to determine the reflection and the transmission coefficients. The results for the reflection coefficient are presented graphically and are compared to those obtained earlier using other research methods. It is observed from the graphs that the results computed from the present analysis match exactly with the previous results.
基金supported by the National Natural Science Foundation of China(Nos.11061018,11261029)the Youth Foundation of Lanzhou Jiaotong University(No.2011028)+1 种基金the Long Yuan Young Creative Talents Support Program(No.252003)the Joint Funds of the Gansu Provincial Natural Science Foundation of China(No.1212RJZA043)
文摘The authors investigate an inverse problem of determining the radiative coefficient in a degenerate parabolic equation from the final overspecified data. Being different from other inverse coefficient problems in which the principle coefficients are assumed to be strictly positive definite, the mathematical model discussed in this paper belongs to the second order parabolic equations with non-negative characteristic form, namely, there exists a degeneracy on the lateral boundaries of the domain. Based on the optimal control framework, the problem is transformed into an optimization problem and the existence of the minimizer is established. After the necessary conditions which must be satisfied by the minimizer are deduced, the uniqueness and stability of the minimizer are proved. By minor modification of the cost functional and some a priori regularity conditions imposed on the forward operator, the convergence of the minimizer for the noisy input data is obtained in this paper. The results can be extended to more general degenerate parabolic equations.
