THE EXACT BOUNDED SOLUTION TO AN INITIAL BOUNDARY VALUE PROBLEM FOR 1D HYPERBOLIC EQUATION WITH INTERIOR DEGENERACY. I. SEPARATION OF VARIABLES

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, 28(1), 1 – 42) in the space-time rectangle [0, T ]×[−1,+1], where a(x;α) vanishes as |x̄| in the subsegment [−c,+c] b [−1,+1], x = cx̄, and α ∈ (0, 2), is considered. The IBVP is splitted into three auxiliary IBVPs, involving two undetermined functions h1(t;α) and h2(t;α). The auxiliary IBVPs are solved using the method of separation of variables. The matching conditions to gain continuity of the solution u(t, x;α) to the IBVP and its flux are imposed on the solutions u1(t, x;α), u2(t, x;α), and u3(t, x;α) to the auxiliary IBVPs to derive a linear convolution integro-differential system with respect to h1(t;α) and h2(t;α).


Introduction and the problem formulation
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, T ] × [−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 , 1 , c |x| 1 , (1. 2) 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.
(1. 4) Imposing the constraints U ∓ α,µ (t) ≡ U α,µ (t) on the above coefficient functions we obtained the following recurrence relations 5) and the two-sided series solution The flux for the series solution (1.6) was proved to be f ∈ C (2,1) ([0, T ] × [−1, +1]) and is given as follows A shroud reader could notice that it was sufficient to equate the first two one-sided coefficient functions U ∓ α,1 (t) and U ∓ α,2 (t) of the series (1.3) to obtain the following series solution and its flux manifesting the set of required properties: . Nevertheless, the recurrence relations (1.4) improved in this way lead to the two-sided series solution (1.6) again. II. We used the standard ansatz of the method of separation of variables (SV) where δ = δ(α) = 1 − α, a * θ Ω = 2, λ is a free parameter, J (s) is the 1-parameter family of the Bessel functions of the first kind [7,10] defined as particular solutions to the following second order ordinary differential equation and having the following power series representation (1.11) Substituting the argument of J in (1.9) into the series (1.11) proves that the 2-parameter family (1.9) includes the same power terms |x| µθ as the twosided series solution (1.6) of the degenerate wave equation (for the details one should refer to the proof of Proposition 2.2 at p. 7). For comparison, we also refer to the recent paper [3], where the Sturm-Liouville problem associated with the degenerate diffusion operator u → − (|x| α u ) has been studied in details.
The current study is aimed at obtaining the 1-parameter family of the exact solutions u(t, x; α) to the IBVP (1.1) with continuous and continuously differentiable flux by the method of SV and is arranged as follows.
In Section 2 we give an outline of SV applied to the IBVP. Implementing SV reduces the original IBVP to three auxiliary ones, referred to as IBVP 1 , IBVP 2 , and IBVP 3 . The boundary conditions for the auxiliary problems involve two undetermined functions h 1 (t; α) and h 2 (t; α) used to match the solutions u 1 (t, x; α), u 2 (t, x; α), and u 3 (t, x; α) to the IBVP 1 , IBVP 2 , and IBVP 3 , and considered as a part of the required solution u(t, x; α).
Finally, in Sections 7, we treat the resulting matching equations of Sect. 6 as a linear convolution integro-differential system to find the functions h 1 (t; α) and h 2 (t; α), nevertheless we postpone solving the integro-differential system to our next publication on the subject. The auxiliary problems are referred to as IBVP 1 , IBVP 2 , and IBVP 3 and posed for the same homogeneous degenerate wave equation. The solutions u 1 (t, x; α), u 2 (t, x; α), and u 3 (t, x; α) to the auxiliary IBVPs satisfy the same zero initial conditions and the following boundary conditions

Reformulating and solving IBVP 1
The representations (2.15), (2.16) yields to the following reformulation of the auxiliary IBVP 1 with respect to v 1 (t, x; α) where the right-hand side of the nonhomogeneous degenerate wave equation is expanded into the Fourier series [9] as follows (3. 2) The coefficients of the above expansion are determined straightforwardly by integration we can easily pose the proper Cauchy problems for finding the functions O 1,µ (t; α) Since the 2-parameter family of particular solutions of the homogeneous ordinary differential equations where A 1,µ and B 1,µ are undetermined constants (the parameters), we try to find the 2-parameter family of particular solutions of the nonhomogeneous equations of the problems (3.6) following the above solutions as the ansatz where A 1,µ and B 1,µ are no longer constants but required t-dependent coefficient functions. Substituting the above representation into the ordinary differential equations of the problems (3.6) yields to the systems of linear nonhomogeneous algebraic equations with respect to the first derivatives of the required coefficient functions + cos ω µ t A 1,µ (t; α) + sin ω µ t B 1,µ (t; α) = 0 , − sin ω µ t A 1,µ (t; α) + cos ω µ t B 1,µ (t; α) = ω −1 µ g 1,µ (t; α) .
The determinants ∆ 1,µ = cos 2 ω µ t + sin 2 ω µ t ≡ 1 of the above systems prove the systems to be unconditionally on α solvable and their solutions to read A 1,µ (t; α) = − ω −1 µ sin ω µ t g 1,µ (t; α) , B 1,µ (t; α) = + ω −1 µ cos ω µ t g 1,µ (t; α) . After integration, we obtain where A • 1,µ and B • 1,µ are undetermined constants. We take zero values of the constants to satisfy the initial conditions of the Cauchy problems (3.6) and to find the required functions of the ansatz (3.5) as follows The above formulas are nothing but the convolutions between trigonometrical sines and the Fourier coefficients (3.3) of the right-hand side of the nonhomogeneous equation of the reformulated IBVP 1 (3.1), therefore, hereinafter, we use the following convenient notation for the solutions of the Cauchy problems (3.6) then the required solution to the reformulated IBVP 1 (3.1) reads as follows

Reformulating and solving IBVP 3
Reformulation of the auxiliary IBVP 3 with respect to v 3 (t, x; α) now is clear The right-hand side of the nonhomogeneous degenerate wave equation is expanded into the Fourier series [9] as follows where the coefficients of the expansion are determined similarly to those of Sect. 3 The required solution to the reformulated IBVP 3 (5.1) reads exactly as (3.8)
A very effective tool for solving such systems is known to be the Laplace transformation [4,5].