On the Telegrapher’s Equation with Three Space Variables in Non-Rectangular Coordinates

This article provides a closed form solution to the telegrapher’s equation with three space variables defined on a subset of a sphere within two radii, two azimuthal angles and one polar angle. The Dirichlet problem for general boundary conditions is solved in detail, on the basis of which Neumann and Robin conditions are easily handled. The solution to the simpler problem in cylindrical coordinates is also provided. Ways to efficiently implement the formulae are explained. Minor adjustments result in solutions to the wave equation and to the heat equation on the same domain as well, since the latter are particular cases of the more general telegrapher’s equation.


Introduction
Hyperbolic partial differential equations (PDEs) are among the three main classes of PDEs in mathematical physics. The most extensively studied hyperbolic PDE is the wave equation. Another major hyperbolic PDE is the telegrapher's equation. In rectangular coordinates, its canonical form is the following: , , , , where x is a vector of n space variables, n ∆ is the n-dimensional Laplacian operator, Λ is a "sufficiently" smooth function and 1 2 3 , , κ κ κ are three posi-tive constants.
A good introduction to the fundamental properties and main applications of this PDE can be found in Zauderer [1] or Ockendon et al. [2].
Among all the major second order linear PDEs of mathematical physics, the telegrapher's equation is noticeable for its generality. Indeed, all the most important non-stationary linear equations can be retrieved from (1). In particular, the wave equation is a special case of the telegrapher's equation obtained by taking ( ) 1 3 , 0 t κ κ = = Λ = x . The Klein-Gordon equation for a function ( ) , w t x is obtained through the following simple relation: An obvious parameterization of the second-order time derivative in (1) would also enable to recover the parabolic heat equation. For more general background on PDEs related to Equation (1), one can refer to [3].
A number of initial boundary value problems (IBVPs) involving PDE (1) have already been analytically solved, especially in rectangular coordinates. Certainly the most comprehensive compilation of known analytical formulae is the handbook by Polyanin and Nazaikinskii [4], which collects and updates all the results from mathematical physics disseminated in previous handbooks such as [5] and [6]. It appears that not all IBVPs associated with the three-dimensional telegrapher's equation have had their exact solutions derived in the literature. This can often be explained by the fact that there already exist solution techniques applied to other non-stationary equations that can readily be extended to the multidimensional telegrapher's equation. However, it ought to be noticed that, in some cases, such an extension entails non-trivial computational issues. This is especially true when it comes to non-rectangular coordinates. On spherical domains, the most general known formulae can cope with boundary conditions with respect to the radial variable, but they cannot handle functional dependency on the boundaries of both the radial and the angular variables. In other words, if we denote the radius by r, the azimuthal (or longitudinal) angle by φ and the polar (or latitudinal) angle by θ , then the only exact solutions to the telegrapher's equation currently available are those that apply to the following domain [4]: Such a restriction rules out problems that require functional dependency on the boundaries of the variables φ or θ as well as on the boundary of r. Of course, one can always resort to the usual numerical techniques of approximation, but, in this case, it is not easy to design efficient and accurate finite difference schemes, due to the dimension of the problem and to the complexity of the telegrapher's equation in a spherical domain. The main contribution of this paper is to provide a closed form solution for a class of IBVPs involving the linear telegrapher's equation with three space variables and general boundary condi- The domain in (4) allows for partial ranges in both angular variables. As such, it includes subsets of the sphere with many more shapes than the domain in (3), so that more flexible models can be considered, with a greater number of physical specifications.
The linear and separable nature of the PDE under consideration enables the application of classical methods of separation of variables and eigenfunction expansions. One of the main advantages of this approach is that it easily allows to handle non-homogeneity in the equation or in the boundary conditions, once the homogeneous solution has been obtained. However, the solution process involves a non-standard generalized Fourier expansion of the first order initial condition, as well as computational difficulties associated with the integration of associated Legendre functions with non-integral parameters, which may explain why there is currently no available formula in the literature.
It must be emphasized that the telegrapher's equation is the focus of this article due to its high degree of generality, not because a specific physical application is studied. Thus, it is expected that the analytical results in this article can be fruitfully applied to parabolic problems-after adjustment of the initial conditions, as well as to hyperbolic problems that do not model the same physical phenomena as those initially described by the telegrapher's equation.
This paper is organized as follows. In Section 2, the main result is stated under the title "Result 1", i.e. the solution of the three-dimensional telegrapher's equation on a spherical domain with boundary conditions on all the space variables. The rest of Section 2 is dedicated to the proof of Result 1, which derives from the combination of two lemmas, along with comments on the numerical implementation of the formula. Section 3 presents two corollaries that handle non-homogeneity of the equation and of the boundary conditions, and then states, under the title "Result 2", the solution of the three-dimensional telegrapher's equation on a cylindrical domain with boundary conditions on all the space variables.

Main Result
The main result of this article is now stated as "Result 1".

Result 1
Solution to the Dirichlet problem for the telegrapher's equation with three space variables on a subset of a sphere located within two radii, two azimuthal angles and one polar angle.
be the solution to the following partial differential equation: on the domain: with the following boundary conditions: where 1 2 3 , , κ κ κ are three positive constants and f is a continuous function from D to  .
Let us denote: where the functions ( ) The numbers The coefficient mnp b is given by: where: Equating both sides of Equation (23) to 2 λ − , where 2 λ is a first separation constant, leads to the following two equations: The complex roots of the characteristic equation associated with the differential Equation (24) where 1 a and 2 a are arbitrary constants. Equating both sides of Equation (25) with a second separation constant 2 µ leads to the following two differential equations: Equation (27)  β is a third separation constant, results in the following two differential equations: The general solution to Equation (29) is given by: where 1 b and 2 b are arbitrary constants. The boundary condition (11) yields: θ is fixed and m β has already been determined, Equation (38) has to be solved for ν . A convenient way of doing this is to use the Mehler-Dirichlet representation for the associated Legendre function ( ) cos P µ ν θ , which is valid for any θ between zero and π : Some background on the formula (39) can be found in [8]. It is based on the fact that associated Legendre functions can have integral representations by means of contour and definite integrals. Formulae containing contour integrals have the most general character, but, for applications, representations by integrals along some segments of the real axis are more convenient. Of all such representations, the one in (39) is best suited to numerical computations as it is  rapidly become small, the distance between the seed of the root-searching algorithm and the exact value of the root at each new iteration is small too, and the algorithm is therefore efficient. Also, although the n ν 's can, in principle, take any value, only those values that are greater than −1/2 are actually needed, due to the following identity [8]: Other interesting properties of the roots of associated Legendre functions with non-integral parameters can be found in [9], where these functions are encoun-  There are no singularity points associated with the functions in the right hand side of (42), as the variable r is strictly positive. It is then easily seen, applying classical Sturm-Liouville theory [7], that the boundary condition (9) implies that, for each p ∈  , there is a real p-th root, denoted as p λ , of the following equation: ( ) ( ) ( ) ( )  which completes the proof of Lemma 1.
The next step consists in finding a way to satisfy the initial conditions (7) and (8).
First, the initial condition (7) immediately yields 0 mnp a = , so that we obtain: Therefore, considering r and θ as constants, (48) can be rewritten as: where: where m d is given by (55) and the coefficient np d remains to be determined.
The function ( ) 2 1 , ; ; F a b c z , also known as the Gauss hypergeometric function, admits the following series expansion [10]: This series representation is well-suited to our problem, since it converges for and 2 m n P β ν is given in (21).
A similar trigonometric series is derived by Bremer [11], who also develops a very efficient new algorithm of evaluation. As pointed out in [11], when the order and degree parameters are "small", i.e. typically close to zero or to the first few natural integers, the coefficients in the series decay rapidly as k increases.
The consequence is that only a small number of terms are required to achieve adequate convergence for all practical purposes. Likewise, even when the parameters are of "large" magnitude, i.e. typically greater than ten times some natural integer after the first few integers, the coefficients in the expansion decay rapidly with k provided that ϑ is sufficiently small. Since [ ] 0, ϑ ∈ π , the latter condition is guaranteed, so that the series expansion remains efficient in this range of parameters as well.
Therefore, (56) becomes: where m d is given by (55), n d is given by (63), and the coefficient p d is yet to be determined.

Extensions
that verifies the homogeneous boundary conditions stated in (7)- (11), is given by: is the Green's function, which is immediately derived from Result 1: which completes the statement of Corollary 1. Endowed with the Green's function, one can also obtain a formula for the Dirichlet problem with non-homogeneous boundary conditions.
Then, the function ( ) which ends the statement of Corollary 2. The theory underpinning the derivation of Corollary 1 and Corollary 2 is explained, e.g., in [12]. The role of the Green's function is elaborated on in [13]. The conditions listed in (80)-(85) are necessary requirements for the solution ( ) , , , u r t θ φ given in (86)-(93) to be unique. It should also be pointed out that Neumann or Robin boundary conditions can be easily handled in the same framework as the one used to solve the Dirichlet problem in Section 2. For example, if the radial boundary condition in (9) is replaced with the following: