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

Abstract

A 1-parameter initial boundary value problem (IBVP) for a linear homogeneous degenerate wave equation (JODEA, 28(1), 1 вЂ" 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..

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

“…Some eigenfunctions for the cases are shown in Figs. 3.1, 3.2, 3.3. In case 4, with the coefficients a 1 = a 2 = 4 (originated from c 1 = c 2 = 1, l = 2, and x 0 = 1 in (1.3)), the roots of (3.6) and the eigenfunctions (3.5) are reduced to…”

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

“…Separation of variables applied to the original IBVP (1.1) is known [2][3][4][5] to involve us into solving the following boundary-value problem…”

“…Nevertheless, any attempt to implement directly a numerical procedure to the IBVP involves a bulk of nested problems having relation to evaluating the flux (1.4) at the degeneracy, segment, where the flux degenerates. It was proved [2,4,5], using series solutions to the IBVP and to the degenerate wave equation alone, that the flux at the degeneracy segment does not vanish and is continuous. From this it immediately appears a problem to retain the above properties for the grid flux.…”

On boundary exact controllability of one‐dimensional wave equations with weak and strong interior degeneration

“…Remark It is clear that the proposed relaxation is only a matter of regularity of some solutions. We refer to the recent papers, 23–25 where the authors consider a particular case of the problem ()–() with for α ∈ [0, 2), and they show that this problem can admit many solutions, but only one of them satisfies transmission conditions ()–() and has a continuously differentiable flux at . As for the rest ones, they satisfy transmission conditions in the form ()–().…”