ON AN INITIAL BOUNDARY-VALUE PROBLEM FOR 1D HYPERBOLIC EQUATION WITH INTERIOR DEGENERACY: SERIES SOLUTIONS WITH THE CONTINUOUSLY DIFFERENTIABLE FLUXES
ON AN INITIAL BOUNDARY-VALUE PROBLEM FOR 1D HYPERBOLIC EQUATION WITH INTERIOR DEGENERACY: SERIES SOLUTIONS WITH THE CONTINUOUSLY DIFFERENTIABLE FLUXES
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DOI：
10.15421/142oo1
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发表时间：
2020-04
期刊：
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通讯作者：
V. Borsch;P. Kogut;G. Leugering
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作者：
V. Borsch;P. Kogut;G. Leugering
研究了线性齐次退化波动方程Utt(t，x；α)−(a(x；α)Ux(t，x；α))x=0(jodea，27(2)，29-44)的单参数初边值问题，其中：1)(t，x)∈[0，T]×[−L，+L]；2)权函数a(x；α)：a)a0|x|c||α，06|x|6c；b)a0，c6|x|6 L；c)a0为常值参考值；3)考虑参数α∈(0，+∞)；。通过使用字符串类比，IBVP可以被视为试图让一根最初固定的“字符串”运动，该“字符串”的左端是固定的，而右端是被迫移动的。利用Frobenius和分离变量的方法证明了：1)退化波动方程存在6个级数解u(t，x；α)，(t，x)∈[0，T]×[−c，+c]，2)唯一具有连续可微通量f(a，u)=−AUX的级数解为u(t，x；α)=Uα，0(T)+Uα，1(T)|x|+Uα，2(T)|x|+x|+。。其中a)θ=2−α为派生参数，b)系数函数服从如下线性递推关系：U‘’α，μ−1(T)=μθ[(μ−1)θ+1]ca0 Uα，μ(T)，μ∈N。自变量(t，x)→(τ，ξ的非线性变换可将1)退化的波动方程转化为波动方程υττ−υξξ=ξρ，或改写为平衡律πτ+φξ=ρ，其中π=υτ，−φ(υ；α)=υξ+ξρ，ρ(υ；α)=αθυξ2，具有：a)其主体部分没有奇性(由于简并性的膨胀)，以及b)形式为υ(τ，ξ；α)=Vα，0(τ)+Vα，1(τ)ξ2+Vα，2(τ)ξ4+.。。得到了连续的、连续可微的正则化通量φ(̊υ；α)和连续的正则化源项ρ(̊υ；α)，其中υ̊(τ，ξ；α)=υ(τ，ξ；α)−υ(τ，0；α)；2)简并波动方程的IBVP到变换后的波动方程的IBVP。证明了：如果α∈(0，2)：1)上述结果成立；2)对于(t，x)∈[0，T]×[−L，0]来说，‘弦’不一定是固定的状态，即行波可以通过‘弦’的简并，并在其固定端和简并点之间激发其振动。
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 ∣∣∣x c ∣∣∣α, 06 |x|6 c; b) a0, c6 |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) = Uα,0(t) + Uα,1(t)|x| + Uα,2(t)|x| + . . ., where a) θ = 2− α is a derived parameter; b) the coefficient functions obey the following linear recurrence relations: U ′′ α,μ−1(t)=μθ [(μ−1) θ + 1] ca0 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)= α θ υ ξ2 , 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)=Vα,0(τ) + Vα,1(τ) ξ 2 + Vα,2(τ) ξ 4 + . . . (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 υ̊(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.