Solutions to a Simplified Initial Boundary Value Problem for 1D Hyperbolic Equation with Interior Degeneracy

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.

In the current study we shall try to continuously match the one-sided solutions (1.3) of the first and fifth kinds (therefore, 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'. The current study is arranged as follows.
In Section 2 we: 1) give some preliminaries on SV in relation to the original IBVP in the case of weak degeneracy and based on the one-sided solutions of kinds 1, 5, both continuous and improving to have the continuous fluxes; 2) split the original IBVP posed in the space-time rectangle [0, T ] × [−1, +1] and describing the behavior of the continuous 'string', into the derived IBVP 2 posed in the left space-time rectangle [0, T ] × [−1, 0]) and the IBVP 1 posed in the right space-time rectangle [0, T ] × [0, +1]), describing respectively the behaviour of the left and the right parts of the 'string' separately in the case of strong degeneracy; 3) formulate the conditions for continuous matching the bounded solutions u 1,j (t, x; α) to the IBVP j and expressing the integrity of the 'string' and continuity of the flux; 4) apply the method of SV to find the unique bounded solutions u(t, x; α) to the IBVP in the case of weak degeneracy, continuous and having the continuous flux; 5) apply the method of SV to find families of bounded solution u 1,j (t, x; α) to the IBVP j in the case of strong degeneracy, having the continuous flux and depending on undetermined functions h j+2 (t; α) ∈ C 1 [0, T ] C 2 (0, T ]; 6) apply the continuity condition to the solutions u 1,j (t, x; α) to derive a linear integro-differential equation of convolution type with respect to the required functions h j+2 (t; α).
In Section 3 we solve the above integro-differential equation with respect to the difference h 3 (t; α) − h 4 (t; α) and show that one of the two functions can be chosen quite freely, that is, the bounded solutions to the IBVP of the resulting family are continuous and have the continuous fluxes.
In Section 4 we summarize the results obtained and some observations on the procedures applied.
In Section 2 we place some useful rules to calculate the coefficients of expansions in the series of the eigenfunctions used in Section 2.

Preliminaries to SV
Implementing SV to the IBVP (1.1) is essentially based on the following two assertions.
Proposition 2.1. Let the following incomplete 1-parameter boundary value problems be given

1)
then: 1) the eigenvalues and the eigenfunctions of the problems of the two kinds λ k,j,µ (α), Z k,j,µ (x; α) ≡ λ k,µ (α), Z k,µ (x; α) (marked with the first subscript k ∈ {1, 5}) are defined as follows Let the following composite 1-parameter boundary value problem be given then: 1) in the case of weak degeneracy, the eigenvalues and the eigenfunctions of the problem of the two kinds (marked with the first subscript k ∈ {1, 5}) are defined as follows where σ 2 k,µ and Z k,µ (x; α) are given in (2.2) of Proposition 2.1; 2) the eigenfunctions (2.7) of both kinds are orthogonal in L 2 (−1, +1), that is Proof. From Proposition 2.1 it follows that the functions X k,µ (x; α) (2.7) satisfy the differential equation of the boundary value problem, hence, we concentrate our efforts on calculating the one-sided values of X k,µ (x; α) and a(x; α) X k,µ (x; α) at the interior degeneracy location. Substituting the known power series [7] (2.9) into (2.5) obtains the series representations for the quantities of interest (2.11) The resulting series (2.10), (2.11) yield to the required values , α ∈ (0, 2) , and this completes the proof of the first part of the proposition. Orthogonality of the eigenfunctions of each kind directly follows from Proposition 2.1, therefore, our concern is orthogonality of the eigenfunctions of the different kinds, that can be quite easily proved, indeed, This completes the proof of the second part of the proposition.
Before implementing the method of SV, we make some notes. First, to build the eigenfunctions Z 5,µ (x; α), we use the Bessel functions of the first kind and order + , rather than the proper Neumann functions [7], to simplify our analysis of the IBVP. It means that the integer values of order − can not be considered, i. e., the values of α = 1, 3 2 , 5 3 , 7 4 , etc., produced by the values of m = 0, 1, 2, 3, etc.
Second, to guarantee uniform convergency of the expansions in series of the eigen-functions Z 1,µ (x; α), X k,µ (x; α), based on the Bessel functions J ∓ (s), we have to impose the following restriction [6,7] on the values of Third, to solve the IBVP (1.1) in the case of weak degeneracy, we apply the bounded eigenfunctions of Prop. 2.2.
Fourth, 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 (2.14) space-time rectangles and referred to as the IBVP 1 and the IBVP 2 respectively; 2) matching the solutions u 1 (t, x; α) and u 2 (t, x; α) to the above initial boundary value problems by imposing the condition of continuity at the degeneracy segment [0, T ]×{0} When applying the above procedure, we drop the subscript k, indicating the first kind of the solutions (1.3), the only one bounded in the case of strong degeneracy, therefore, the only remaining subscript is j.

Implementing SV to the IBVP
In the current section our concern is the bounded solution to the IBVP in the case of weak degeneracy. The required solution is assumed to have the following representation where: a) the function v(t, x; α) is also required; b) the function w(t, x; α) is given as follows c) the smooth blending functions φ 1 (x; α), φ 2 (x; α) satisfy the following boundary and regularity conditions, respectively and b) reformulation of the IBVP into the following one with respect to v(t, x; α) where the right-hand side of the above degenerate wave equation reads Then the initial functions (2.22) and the right-hand side (2.24) are expanded into the series where the functions X k,µ (x; α) are defined in Prop. 2.2 and the coefficients are calculated directly by integration Assuming that the ansatz for the required solution to the initial boundary value problem (2.23) to be as follows we obtain the Cauchy problems with respect to the desired coefficient functions of the ansatz (2.29) The resulting expressions for the coefficients, after applying some trivial trigonometric manipulations, can be presented in the convolution form as follows or, shortly, as Finally, the representation (2.18) yields to the required unique bounded solution to the IBVP (2.31) The above procedure can be readily interpreted in terms of decomposition of the functions v(t, x; α), * v(x; α), * * v(x; α), w(t, x; α), and g(t, x; α) into their even and odd parts, for example where both parts are defined as follows leading to decomposition of the initial boundary value problem (2.23) into the derived problems Applying SV to the above problems yields to the bounded solutions in the form of the following series where the coefficient functions O e,µ (t; α) and O o,µ (t; α) are evidently the solutions to respectively the same Cauchy problems (2.29). And eventually, using the representations (2.33), (2.32) and (2.18), the same unique bounded solution to the IBVP can be found again.
Calculating the flux of the obtained solution to the IBVP proves that the following condition holds The required solutions to the IBVP j in the case of strong degeneracy are assumed to have the following representation where: a) the functions v j (t, x; α) are required; b) the functions w j (t, x; α) are given as follows c) the smooth blending functions φ j (x; α), φ j+2 (x; α) satisfy the following boundary and regularity conditions, respectively: v(0, x; α) and combining (2.36) -(2.39) yields to: a) the initial conditions for v j (t, x; α) and b) reformulation of the IBVP j into the following auxiliary IBVP a j with respect to the functions v j (t, x; α) where the right-hand sides of the above degenerate wave equations being expanded due to (2.37), read as follows (2.45) Then the initial functions (2.42) and the right-hand sides (2.45) are expanded into the series where the coefficients are determined straightforwardly by integration. The expanded forms of the coefficients in (2.47) are (2.50) And now substituting the ansatz for the solutions into the IBVP a j obtains the Cauchy problems for the coefficient functions (2.52) The resulting expressions for the coefficients can be readily presented in the convolution form as follows (2.53) Finally, the representation (2.36) obtains the required solutions to the IBVP j (2.55) Calculating the fluxes of the obtained solutions u j (t, x; α) proves that the following condition holds yet before matching the solutions, due to: a) Prop. 2.1 and b) the regularity conditions (2.40) and (2.41) imposed on the blending functions φ j (x; α) and φ j+2 (x; α) (or, shortly, due to continuous differentiability of the functions w j (t, x; α) (2.37)).

Matching the Solutions to the IBVP 1 and the IBVP 2
To implement matching the obtained one-sided solutions u 1 (t, x; α) (2.54) and u 2 (t, x; α) (2.55), we will follow the procedure: 1) substitute the above solutions into the matching condition (2.17), as follows 2) replace the values Z 1,µ (0; α) with the pre-derived formula (2.12) 3) account for the boundary conditions (2.38) and (2.39) imposed on the blending functions φ j (x; α) and φ j+2 (x; α), to obtain the following linear integro-differential equation of convolution type with respect to h 3 (t; α) and h 4 (t; α) The above representation of the matching condition (2.17) can be rewritten in the expanded form where the coefficient functions are defined by the following series

Finding the Images
To solve the integro-differential equation (2.59) of convolution type, we apply the Laplace transformation [3], producing for a function f (t), t ∈ [0, ∞), its transform as follows provided the original function f (t) satisfies the known sufficient conditions for the image function F (τ ) to exist. When applying the Laplace transformation we use: 1) the convolution theorem where the symbol 'middle dot' between the two images is used, where it is needed, for reminding about the origin of their multiplication; 2) the transforms of the control functions h j (t; α) and their second derivatives, accounting for the given initial conditions 3) the transforms of the required functions h 3 (t; α), h 4 (t; α) and their second derivatives, accounting for the prescribed initial conditions

Finding the Original Functions
We start from estimating applicability of some known approaches to invert the formula (3.7) and find the original function h(t; α) = h 3 (t; α) − h 4 (t; α). a) We could expect that rewriting the formula (3.7) as follows makes it possible to invoke the convolution theorem (3.2) and find the function h(t; α) provided both multipliers in (3.8) are the images.
The function Q 1 (τ ; α) has the same properties as both functions P j (τ ; α) have, therefore the function [1 + Q 1 (τ ; α)] −1 is not a transform, and this completes the proof of the second part of the proposition.
Although the first approach turnes out to be unsuccessful, nevertheless it follows from Proposition 3.1 that the right-hand side of the formula (3.7) is indeed the transform of the required function h(t; α).
b) The next approach is to invert the right-hand side of the formula (3.7) directly. Indeed, let the Laplace transform F (τ ) for an original function f (t) be given, then applying the inverse Laplace transformation [3], known also as the Bromwich integral, yields to the required original function where τ = ξ * is a vertical straight line lying to the right of all the singularities of F (τ ) (see Fig. 3.1, a). Practically, calculating the Bromwich integral is performed using the Cauchy residue theorem [5], but this approach implies that the singularities of the integrand are isolated and known (see Fig. 3.1, b). The functions P j (τ ; α) and Q 1 (τ ; α) have the same poles being removable singularities of the integrand and having no impact on calculating the integral; whereas finding all zeros of the function [1 + Q 1 (τ ; α)] generally implies some proper approximation [4] of the latter and results in a huge bulk of the computational work. Therefore, we do not reject calculating the Bromwich integral at all, but postpone applying this approach for a while. c) To implement the third approach, we: 1) recombine the terms in the series (3.5) and easily find the respective original functions where δ(t) is the Dirac delta function; 2) represent the denominator of the formula (3.7) as follows 3) rewrite the formula (3.7) as is usually done when solving integral equations of convolution type (3.14) 4) expand the 'fractional' term in (3.14) in the following power series provided that |Q 1 (τ ; α)| < 1 in a proper right half-plane of the τ -plane [3]; 5) invert the above power series in the form of the Neumann series [3] (3.16) or the sum of iterated kernels, whereq 1 (t; α) = C −1 αq1 (t; α) (3.11); 6) invert the terms in the brackets in (3.14) (3.17) 7) finally, invert the formula (3.7) by invoking the convolution theorem (3.2) (3.18)
2. In the case of weak degeneracy the bounded unique solution (2.31 are then matched to implement the other continuity condition nevertheless, the resulting matched family still retains one undetermined function.
where k ∈ N .