Solutions to a Simplified Initial Boundary Value Problem for 1D Hyperbolic Equation with Interior Degeneracy

A 1-parameter initial boundary value problem (IBVP) for a linear homogeneousdegenerate wave equation (JODEA, 28(1), 1) in a space-time rectangle is considered. The origin of degeneracy is the power law coefficient function with respect to the spatial distance to the symmetry line of the rectangle, the exponent being the only parameter of the problem, ranging in (0,1) and (1,2) and producing the weak and strong degeneracy respectively. In the case of weak degeneracy separation of variables is used in the rectangle to obtain the unique bounded continuous solution to the IBVP, having the continuous flux. In the case of strong degeneracy the IBVP splits into the two derived IBVPs posed respectively in left and right half-rectangles and solved separately using separation of variables. Continuous matching of the obtained left and right families of bounded solutions to the IBVPs results in a linear integro-differential equation of convolution type. The Laplace transformation is used to solve the equation and obtain a family of bounded solutions to the IBVP, having the continuous flux and depending on one undetermined function.A 1-parameter initial boundary value problem (IBVP) for a linear homogeneousdegenerate wave equation (JODEA, 28(1), 1 â“ 42) in a space-time rectangle is considered. The origin of degeneracy is the power law coefficient function with respect to the spatial distance to the symmetry line of the rectangle, the exponent being the only parameter of the problem, ranging in (0,1) and (1,2) and producing the weak and strong degeneracy respectively. In the case of weak degeneracy separation of variables is used in the rectangle to obtain the unique bounded continuous solution to the IBVP, having the continuous flux. In the case of strong degeneracy the IBVP splits into the two derived IBVPs posed respectively in left and right half-rectangles and solved separately using separation of variables. Continuous matching of the obtained left and right families of bounded solutions to the IBVPs results in a linear integro-differential equation of convolution type. The Laplace transformation is used to solve the equation and obtain a family of bounded solutions to the IBVP, having the continuous flux and depending on one undetermined function

Variational Approach for the Reconstruction of Damaged Optical Satellite Images Through Their Co-Registration with Synthetic Aperture Radar

In this paper the problem of reconstruction of damaged multi-band opticalimages is studied in the case where we have no information about brightness of suchimages in the damage region. Mostly motivated by the crop field monitoring problem,we propose a new variational approach for exact reconstruction of damaged multi-bandimages using results of their co-registration with Synthetic Aperture Radar (SAR) imagesof the same regions. We discuss the consistency of the proposed problem, give the schemefor its regularization, derive the corresponding optimality system, and describe in detailthe algorithm for the practical implementation of the reconstruction procedure.In this paper the problem of reconstruction of damaged multi-band opticalimages is studied in the case where we have no information about brightness of suchimages in the damage region. Mostly motivated by the crop field monitoring problem,we propose a new variational approach for exact reconstruction of damaged multi-bandimages using results of their co-registration with Synthetic Aperture Radar (SAR) imagesof the same regions. We discuss the consistency of the proposed problem, give the schemefor its regularization, derive the corresponding optimality system, and describe in detailthe algorithm for the practical implementation of the reconstruction procedure

On Indirect Approach to the Solvability of Quasi-Linear Dirichlet Elliptic Boundary Value Problem with BMO-Anisotropic p-Laplacian

We study here Dirichlet boundary value problem for a quasi-linear elliptic equation with anisotropic p-Laplace operator in its principle part and L^1-control in coefficient of the low-order term. As characteristic feature of such problem  is a specification of the matrix of anisotropy A=A^{sym}+A^{skew} in BMO-space. Since we cannot expect to have a solution of the state equation in the classical Sobolev space W^{1,p}_0(\Omega), we specify a suitable functional class in which we look for solutions and prove existence of weak solutions in the sense of Minty using a non standard approximation procedure and compactness arguments in variable spaces

How Can We Manage Repairing a Broken Finite Vibrating String? Formulations of the Problem

Three approaches to solve a well-known IBVP posed for vibrating composite string with piece-wise constant properties have been applied. The main issue of the stufy is the number of the matching conditions to be imposed for the solution to the IBVP to be obtained

Can a Finite Degenerate ‘String’ Hear Itself The Exact Solution to a Simplified IBVP

The separation of variables based solution to a simplified (compared to that published earlier in JODEA, 28 (1) (2020), 1 – 42) initial boundary value problem for a 1D linear degenerate wave equation, posed in a space-time rectangle, has been presented in a fully complete form. Degeneracy of the equation is due to vanishing its coefficient in an interior point of the spatial segment being the side of the rectangle. For the sake of convenience, the solution is interpreted as a vibrating ‘string’. The solution obtained in the case of weak degeneracy is smooth and bounded, whereas that in the case of strong degeneracy is piece-wise smooth, piece-wise continuous and unbounded in a neighborhood of the point of degeneracy, nevertheless being satisfied some regularity conditions, including square-integrability. In both cases the travelling waves pass through the point of degeneracy, and this phenomenon is referred to as an ability of the ‘string’ to hear itself. The total energy of the ‘string’ is shown to conserve in both cases of degeneracy, provided the ends of the ‘string’ are fixed, though the above vibrating ‘string’ analogy fails in the case of strong degeneracy. The total energy conservation implies the uniqueness of the solution to the problem in both cases of degenerac

Can a Finite Degenerate ‘String’ Hear Itself? Numerical Solutions to a Simplified IBVP

Some discrete models for a simplified (compared to that published earlier in JODEA, 28 (1) (2020), 1 – 42) initial boundary value problem for a 1D linear degenerate wave equation, posed in a space-time rectangle and solved earlier exactly (JODEA, 30 (1) (2022), 89 – 121), have been considered. It has been demonstrated that the correct evaluation of the degenerate grid flux can be possible

On approximate solutions to the Neumann elliptic boundary value problem with non-linearity of exponential type

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ON EXISTENCE OF BOUNDED FEASIBLE SOLUTIONS TO NEUMANN BOUNDARY CONTROL PROBLEM FOR p-LAPLACE EQUATION WITH EXPONENTIAL TYPE OF NONLINEARITY

We study an optimal control problem for mixed Dirichlet-Neumann boundary value problem for the strongly non-linear elliptic equation with p-Laplace operator and L1-nonlinearity in its right-hand side. A distribution u acting on a part of boundary of open domain is taken as a boundary control. The optimal control problem is to minimize the discrepancy between a given distribution yd 2 L2( ) and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for any admissible control. After dening a suitable functional class in which we look for solutions and assuming that this problem admits at least one feasible solution, we prove the existence of optimal pairs. We derive also conditions when the set of feasible solutions has a nonempty intersection with the space of bounded distributions L1( )