Abstract
A 1-parameter initial boundary value problem (IBVP) for a linear homogeneous degenerate 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.
Highlights
Introduction and Setting of the ProblemThe current study is a sequel to [2] and deals with the following 1-parameter simplified initial boundary value problem (IBVP) for the degenerate wave equation in the space-time rectangle [0, T ] × [−1, +1] ∂2u(t, x; α) ∂ ∂u(t, x; α) ∂t2 = a(x; α)∂x u(t, −1; α) = h2(t; α), u(t, +1; α) = h1(t; α) ∂u(0, x; α) = ∗u∗(x; α)∂t u(0, x; α) u∗
In the case of weak degeneracy we reduce solving the IBVP (1.1) to the following two-step procedure: 1) solving the derived initial boundary value problems
In the current section our concern is the bounded solution to the IBVP in the case of weak degeneracy
Summary
The current study is a sequel to [2] and deals with the following 1-parameter simplified initial boundary value problem (IBVP) for the degenerate wave equation in the space-time rectangle [0, T ] × [−1, +1]. In the current study we shall try to continuously match the one-sided solutions (1.3) of the first and fifth kinds (the subscript k takes values {1, 5}) to find bounded solutions to the IBVP (1.1) using the method of separation of variables (SV) and implying an analogy of the required solutions with a continuous imaginary ‘string’.
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