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On an initial boundary-value problem for 1D hyperbolic equation with interior degeneracy: Series solutions with the continuously differentiable fluxes

Borsch VL, Kogut PI, Leugering G (2020)

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Publication Type: Journal article

Publication year: 2020

Journal

Book Volume: 28

Pages Range: 1-42

Journal Issue: 1

DOI: 10.15421/142001

Abstract

A 1-parameter initial boundary value problem for the linear homogeneous degenerate wave equation utt(t, x; α)−(a(x; α) ux(t, x; α))x = 0 (JODEA, 27(2), 29 - 44), where: 1) (t, x) ∈ [0, T] ×[−l, +l]; 2) the weight function a(x; α): a) a0^xc^α, 0 6 |x| 6 c; b) a0, c 6 |x| 6 l; c) a0 is a constant reference value; and 3) the parameter α ∈ (0, +∞); is considered. Using a string analogy, the IBVP can be treated as an attempt to set an initially fixed 'string' in motion, the left end of the 'string' being fixed, whereas the right end being forced to move. It has been proved, using the methods of Frobenius and separation of variables, that: 1) there exist 6 series solutions u(t, x; α), (t, x) ∈ [0, T]×[−c, +c], of the degenerate wave equation; 2) the only series solution, having continuous and continuously differentiable flux f(a, u) = −aux, reads u(t, x; α) = Uα0(t) + Uα1(t)|x|^θ + Uα2(t)|x|^2θ +..., where a) θ = 2 − α is a derived parameter; b) the coefficient functions obey the following linear recurrence relations: Uα,µ⁰⁰−1(t) = µθ [(µ−1) θ + 1] c^−αa0 Uα,µ(t), µ∈N. It has been revealed that a nonlinear change of the independent variables (t, x) → (τ, ξ) transforms: 1) the degenerate wave equation to the wave equation υττ − υξξ = ξρ, or rewritten as the balance law πτ + ϕξ = ρ, where π = υτ, −ϕ(υ; α) = υξ + ξρ, ρ(υ; α) = θ ξ2 α υ having: a) no singularity in its principal part (due to inflation of the degeneracy), and b) the only series solution of the form υ(τ, ξ; α) = Vα0(τ) + Vα1(τ) ξ² + Vα2(τ) ξ⁴ +... (out of 5 existing and found similarly to those of the degenerate wave equation), leading to the continuous and continuously differentiable regularized flux ϕ(υ; α) and the continuous regularized source term ρ(υ; α), where υ(τ, ξ; α) = υ(τ, ξ; α) − υ(τ, 0; α); 2) the IBVP for the degenerate wave equation to the IBVP for the transformed wave equation. It has been shown, that if α∈(0, 2): 1) the above results are valid; 2) the state of being fixed for the 'string' is not necessary for (t, x) ∈ [0, T] ×[−l, 0], that is a traveling wave could pass the degeneracy and excite vibrations of the 'string' between its fixed end and the point of degeneracy.

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APA:

Borsch, V.L., Kogut, P.I., & Leugering, G. (2020). On an initial boundary-value problem for 1D hyperbolic equation with interior degeneracy: Series solutions with the continuously differentiable fluxes. Journal of Optimization, Differential Equations and Their Applications, 28(1), 1-42. https://doi.org/10.15421/142001

MLA:

Borsch, Vladimir L., Peter I. Kogut, and Günter Leugering. "On an initial boundary-value problem for 1D hyperbolic equation with interior degeneracy: Series solutions with the continuously differentiable fluxes." Journal of Optimization, Differential Equations and Their Applications 28.1 (2020): 1-42.

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