On an Initial Boundary-Value Problem for 1d Hyperbolic Equation With Interior Degeneracy: Series Solutions With the Continuously Differentiable Fluxes

Abstract

A 1-parameter initial boundary value problem for the linear homogeneous degenerate wave equation u tt (t, x; α)−(a(x; α) u x (t, x; α)) x = 0 (JODEA, 27(2), 29 -44),

The Exact Bounded Solution to an Initial Boundary Value Problem for 1D Hyperbolic Equation with Interior Degeneracy. I. Separation of Variables

“…1. Using the previously obtained results [2] concerning the particular solutions of the degenerate wave equation being under consideration, we succeeded in splitting the original IBVP for the degenerate wave equation into three auxiliary IBVPs involving the undetermined functions h 1 (t; α) and h 2 (t; α).…”

“…The right-hand side of the nonhomogeneous degenerate wave equation is expanded into the Fourier series [9] as follows 2) where the coefficients of the expansion…”

“…The current study is a sequel to our previous publications [1,2] on the subject dealing with the following 1-parameter initial boundary value problem (IBVP) for the degenerate wave equation in the space-time rectangle [0, α ∈ (0, 2), a * c α = 1, x = cx, and all the (dependent and independent) variables are non-dimensional. One should refer to [2] to find out more details on the problem formulation.…”

“…We also used the following dependent variables Here we remind in brief the main results of [2], used in the current study. I.…”

“…t, x; α) ∂t2 = ∂ ∂x a(x; α)∂u(t, x; α) ∂x , (t, x) ∈ (0, T )×(−1, +1) , u(t, −1; α) = h 3 (t) , t ∈ [0, T ] , u(t, +1; α) = h 0 (t) , t ∈ [0, T ] , u(0, x; α) = 0 , x ∈ [−1, +1] , ∂u(0, x; α) ∂t = 0 , x ∈ (−1, +1) , (1.1) where h 0 , h 3 ∈ C 2 ([0, T ])are known control functions, and the 1-parameter x-dependent coefficient function is defined as follows a(x; α) = a * |x| α ≡ |x| α , 0 |x| c ,…”