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We propose a regularization method based on the iterative conjugate gradient method for the solution of a Cauchy problem for the wave equation in one dimension. This linear but ill-posed Cauchy problem consists of finding the displacement and flux on a hostile and inaccessible part of the medium boundary from Cauchy data measurements of the same quantities on the remaining friendly and accessible part of the boundary. This inverse boundary value problem is recast as a least-squares minimization problem that is solved by using the conjugate gradient method whose iterations are stopped according to the discrepancy principle for obtaining stable reconstructions. The objective functional associated is proved Fréchet differentiable and a formula for its gradient is derived. The well-posed direct, adjoint and sensitivity problems present in the conjugate gradient method are solved by using a finite-difference method. Two numerical examples to illustrate the accuracy and stability of the proposed numerical procedure are thoroughly presented and discussed. [ABSTRACT FROM AUTHOR]

This paper is devoted to solve an inverse time-dependent source problem for a time-fractional diffusion-wave equation. The source function in the time-fractional diffusion-wave equation is assumed as a multiplication of a temporal function and a spatial function. We try to identify the time-dependent source term according to an additional solute concentration distribution measured at a point on the boundary or an inner point in the solution domain. Firstly, we discuss the continuity of the weak solution for the direct problem. Then, we transform the inverse problem into a first kind of Volterra integral equation and show its ill-posedness. We use a generalized Tikhonov regularization to solve the Volterra integral equation. The generalized cross-validation choice rule is applied to find a suitable regularization parameter. Lastly, we test three examples for the inverse time-dependent source problem and show the effectiveness of the proposed regularization method. [ABSTRACT FROM AUTHOR]

In this paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to the inverse time-dependent force function in the wave equation on regular and irregular domains. The SMRPI is developed for identifying the force function which satisfies in the wave equation subject to the integral overspecification over a portion of the spatial domain or to the overspecification at a point in the spatial domain. This method is based on erudite combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which play as basis functions in the frame of SMRPI. Since the problem is known to be ill-posed, Thikhonov regularization strategy is employed to solve effectively the discrete ill-posed resultant linear system. Three numerical examples are tested to show that numerical results are accurate for exact data and stable with noisy data. [ABSTRACT FROM PUBLISHER]

A non-linear wave equation with an unknown time-convolution kernel is considered. The missing kernel is recovered from an additional (space) integral measurement. The global in time existence, uniqueness as well as the regularity of a solution is addressed. A new numerical algorithm based on Rothe’s method is designed and error estimates are derived. [ABSTRACT FROM PUBLISHER]

The objective of this article is to develop a computational method, which combines an efficient high-order finite difference method for solving wave equation, auto differentiation tools and globally convergent optimization algorithms to estimate the unknown coefficient in a 1-D wave equation. The forward problem is solved by efficient and higher order finite difference methods while the gradient of the cost functional with respect to the coefficient is calculated by an adjoint code which is generated by an auto differentiation tool, such as Tangent linear and Adjoint Model Compiler. Three numerical examples are conducted to demonstrate the effectiveness, robustness and accuracy of the proposed computational method. [ABSTRACT FROM AUTHOR]