# Abstract

In this paper we obtain the solution of the singular Cauchy problem for the Euler-Poisson-Darboux equation when differential Bessel operator acts by each variable.

## 1 Introduction

The classical Euler-Poisson-Darboux equation has the form

The operator acting by *t* in (1) is called the **Bessel operator**. For the Bessel operator we use the notation (see. [1], p. 3)

The Euler-Poisson-Darboux equation for *n* = 1 appears in Euler’s work (see [2], p. 227). Further Euler’s case of (1) was studied by Poisson in [3], Riemann in [4] and Darboux in [5] (for the history of this issue see also in [6], p. 532 and [7], p. 527). The generalization of it was studied in [8]. When *n* ≥ 1 the equation (1) was considered, for example, in [9, 10]. The Euler-Poisson-Darboux equation appears in different physics and mechanics problems (see [11, 12, 13, 14, 15]). In [16] (see also [17], p. 243) and in [18] there were different approaches to the solution of the Cauchy problem for the general Euler-Poisson-Darboux equation

with the initials conditions

The Cauchy problem with the nonequal to zero first derivative by *t* of *u* for the (2) (and for (1)) is incorrect. However, if we use the special type of the initial conditions containing the nonequal to zero first derivative by *t* of *u* then such Cauchy problem for the (2) will by solvable. Following [17] and [19] we will use the term singular Cauchy problem in this case. The abstract Euler-Poisson-Darboux equation (when in the left hand of (2) an arbitrary closed linear operator is presented) was studied in [20, 21, 22].

In this article we consider the solution of the problem (2)-(3) when −∞ < *k* < +∞ and its properties. Besides this, we get the formula for the connection of solution of the problem (2)-(3) and solution of a simpler problem. Also using the solution of the problem (2)-(3) we obtain solution of the singular Cauchy problem for the equation (2) when *k* < 1 with the conditions

## 2 Property of general Euler-Poisson-Darboux equations’ solutions

In this section we give some necessary definitions and obtain two fundamental recursion formulas for solution of (2).

Let

and *Ω* is open set in ℝ_{n} which is symmetric correspondingly to each hyperplane *x _{i}*=0,

*i*=1, …,

*n*,

*Ω*

_{+}=

*Ω*∩

*Ω*

_{+}=

*Ω*∩

We have *Ω*_{+} ⊆
*Ω*_{+} ⊆
*C ^{m}*(

*Ω*

_{+}),

*m*≥ 1, consisting of differentiable functions on

*Ω*

_{+}by order

*m*. Let

*C*(

^{m}*Ω*

_{+}) be the set of functions from

*C*(

^{m}*Ω*

_{+}) such that all their derivatives by

*x*for all

_{i}*i*= 1, …,

*n*are continuous up to the

*x*=0. Class

_{i}*C*(

^{m}*Ω*

_{+}) such that

*x*,

_{i}*i*= 1, …,

*n*(see [1], p. 21). A multi-index

*γ*=(

*γ*

_{1}, …,

*γ*) consists of fixed positive numbers

_{n}*γ*> 0,

_{i}*i*=1, …,

*n*and |

*γ*|=

*γ*

_{1}+…+

*γ*.

_{n}We consider the multidimensional Euler-Poisson-Darboux equation wherein the Bessel operator acts in each of the variables:

where

Equation (5) we will call **the general Euler-Poisson-Darboux equation**.

## Statement 2.1

*Let u ^{k}* =

*u*(

^{k}*x*,

*t*)

*denote the solution of*(5)

*when the next two fundamental recursion formulas hold*

## Proof

Following [23] we prove (7). Putting *w* = *t*^{k−1}*v*, *v* = *u ^{k}* we have

or

If *w* = *t*^{k−1}*v* satisfies the equation

then using (9) we get

which means that *v* satisfies the equation

Denoting *w* = *u*^{2−k} we obtain (7).

Now we prove the (8). Let *tw* = *v _{t}*,

*v*=

*u*. We obtain

^{k}We find now

Then we get

or

□

Recursion formulas (7) and (8) allow us to obtain, from a solution *u _{k}* of equation (5), the solutions of the same equation with the parameter

*k*+2 and 2 −

*k*, respectively. Both formulas are proved for Euler-Poisson-Darboux equation

## 3 Weighted spherical mean and the first Cauchy problem for the general Euler-Poisson-Darboux equation

Here we present the solutions of the problem (2)-(3) for different values of *k* for which we obtain solution of (2)-(4) in the next section, and get formula for the connection of solution of problem (2)-(3) and solution of simpler problem when *k* = 0 in (2).

In
*γ*:

where each

The below-considered weighted spherical mean generated by a multidimensional generalized translation ^{γ}*T ^{t}* has the form (see [25])

where

(see [26], p. 20, formula (1.2.5) in which we should put *N*=*n*). Construction of a multidimensional generalized translation and the weighted spherical mean are transmutation operators (see [27]).

Theorems 3.1-3.4 have been proved in [28]. We give formulations of these theorems here because they will be needed in the next section.

## Theorem 3.1

*The weighted spherical mean of f* ∈
*satisfies the general equation Euler–Poisson–Darboux equation*

*and the conditions*

This theorem has been proved in [25]).

We give theorems on the solution of the Cauchy problem for the general Euler–Poisson–Darboux equation for the remaining values of *k*.

## Theorem 3.2

*Let f* ∈
*Then for the case k* > *n*+|*γ* | − 1 *the solution of*(15)–(16)*is unique and given by*

*Using weighted spherical mean we can write*

## Theorem 3.3

*If**then the solution of*(15)–(16)*for k* < *n*+|*γ* | − 1, *k* ≠ −1,−3,−5,…

*where m is a minimum integer such that**is the solution of the Cauchy problem*

*The solution of*(15)–(16)*is unique for k* ≥ 0 *and not unique for negative k*.

## Theorem 3.4

*If f* ∈
*is B–polyharmonic of order**then one of the solutions of the Cauchy problem*(20)–(21)*for the k*=−1,−3,−5,… *is given by*

*The solution of*(15)–(16)*is not unique for negative k*.

The theorem 3.5 contains the explicit form of the transmutation operator for the solution. Definition, methods of construction and applications of the transmutation operators can be found in [27, 29, 30].

## Theorem 3.5

*Let k* > 0. *The twice continuously differentiable on**solution u*=*u ^{k}*(

*x*,

*t*)

*of the Cauchy problem*

*such that**is connected with the twice continuously differentiable on**solution w*=*w*(*x*,*t*) *of the Cauchy problem*

*such that w _{xi}*(

*x*

_{1}, …,

*x*

_{i−1},0,

*x*

_{i+1}, …,

*x*,

_{n}*t*) = 0,

*i*= 1, …,

*n by formula*

*where**is transmutation Poisson operator (see [24]) acting by α*

## Proof

The fact that the function *u ^{k}* defined by the equality (28) satisfies the conditions (31) is obvious. Let us show that

*u*defined by (28) satisfies (24)

^{k}where *ξ* = *α t*. Further integrating by parts we obtain

For

Finally,

Thus the function *u ^{k}* defined by equality (28) satisfies the problem (24)–(31).

Let us prove that from the relation (28) we can uniquely obtain a solution of the problem (26)–(27). By introducing new variables

Let *k* > 0 then

Thus we have unique representation of

or

□

## 4 The second Cauchy problem for the general Euler-Poisson-Darboux equation

In this section we obtain solution of (2)-(4).

## Theorem 4.1

*If**then the solution v* = *v ^{k}*(

*x*,

*t*)

*of*

*is given by*

*if n*+|*γ*|+*k is not an odd integer and*

*if n*+|*γ*|+*k is an odd integer*, *where q* ≥ 0 *is the smallest positive integer number such that* 2−*k*+2*q* ≥ *n*+|*γ*| − 1.

## Proof

Let *q* ≥ 0 be the smallest positive integer number such that 2−*k*+2*q* ≥ *n*+|*γ*| − 1 i.e.
*v*^{2−k+2q}(*x*,*t*) be a solution of (29) when we take 2 − *k*+2*q* instead of *k* such that

Then by property (7) we obtain that

is a solution of the equation

Further, applying *q*-times the formula (8) we obtain that

is a solution of the (29).

Let’s consider

We have shown that (32) satisfies the equation (29).

Now we will prove that *v ^{k}* satisfies the conditions (31). For

Taking into account formula (33) we obtain *v ^{k}*(

*x*,0) = 0 and

Now we obtain the representation of *v ^{k}* through the integral. Using formula (18) we get

If 2 − *k*+2*q* > *n*+|*γ*| − 1 then by applying (32) and (33) we write

If 2−*k*+2*q* = *n*+|*γ*|−1 then

□

### References

[1] Kipriyanov I.A., Singular Elliptic Boundary Value Problems, Moscow: Nauka, 1997Search in Google Scholar

[2] Euler L., Institutiones calculi integralis, Opera Omnia. Ser. 1. V. 13. Leipzig, Berlin, 1914, 1, 13, 212-230Search in Google Scholar

[3] Poisson S.D., Mémoire sur l’intégration des équations linéaires aux diffŕences partielles, J. de L’École Polytechechnique, 1823, 1, 19, 215-248Search in Google Scholar

[4] Riemann B., On the Propagation of Flat Waves of Finite Amplitude, Ouvres, OGIZ, Moscow-Leningrad, 1948, 376-395Search in Google Scholar

[5] Darboux G., Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal, vol. 2, 2nd edn, Gauthier-Villars, Paris, 1915Search in Google Scholar

[6] Mises R. von, The Mathematical Theory Of Compressible Fluid Flow, Academic Press, New York, 1958Search in Google Scholar

[7] Tsaldastani O., One-dimensional isentropic flow of fluid, In: Problems of Mechanics, Collection of Papers. R. von Mises and T. Karman (Eds.) [Russian translation], 1955, 519-552Search in Google Scholar

[8] Rutkauskas S., Some boundary value problems for an analogue of the Euler–Poisson–Darboux equation, Differ. Uravn.,1984, 20, 1, 115–124Search in Google Scholar

[9] Weinstein A., Some applications of generalized axially symmetric potential theory to continuum mechanics, In: Papers of Intern. Symp. Applications of the Theory of Functions in Continuum Mechanics, Mechanics of Fluid and Gas, Mathematical Methods, Nauka, Moscow, 1965, 440–453Search in Google Scholar

[10] Olevskii M.N., The solution of the Dirichlet problem related to the equation

[11] Aksenov A.V., Periodic invariant solutions of equations of absolutely unstable media, Izv. AN. Mekh. Tverd. Tela, 1997, 2, 14-20Search in Google Scholar

[12] Aksenov A.V., Symmetries and the relations between the solutions of the class of Euler–Poisson– Darboux equations, Dokl. Ross. Akad. Nauk, 2001, 381, 2, 176-179Search in Google Scholar

[13] Vekua I.N., New Methods of Solving of Elliptic Equations, OGIZ, Gostekhizdat, Moscow– Leningrad, 1948Search in Google Scholar

[14] Dzhaiani G.V., The Euler–Poisson–Darboux Equation, Izd. Tbilisskogo Gos. Univ., Tbilisi, 1984Search in Google Scholar

[15] Zhdanov V.K., Trubnikov B.A., Quasigas Unstable Media, Nauka, Moscow, 1991Search in Google Scholar

[16] Fox D.N., The solution and Huygens’ principle for a singular Cauchy problem, J. Math. Mech., 1959, 8, 197-21910.1512/iumj.1959.8.58015Search in Google Scholar

[17] Carroll R.W., Showalter R.E., Singular and Degenerate Cauchy problems, N.Y.: Academic Press, 1976Search in Google Scholar

[18] Lyakhov L.N., Polovinkin I.P., Shishkina E.L., Formulas for the Solution of the Cauchy Problem for a Singular Wave Equation with Bessel Time Operator, Doklady Mathematics. 2014, 90, 3, 737-74210.1134/S106456241407028XSearch in Google Scholar

[19] Tersenov S.A., Introduction in the theory of equations degenerating on a boundary. USSR, Novosibirsk state university,1973Search in Google Scholar

[20] Glushak A.V., Pokruchin O.A., Criterion for the solvability of the Cauchy problem for an abstract Euler-Poisson-Darboux equation, Differential Equations, 52, 1, 2016, 39–5710.1134/S0012266116010043Search in Google Scholar

[21] Glushak A.V., Abstract Euler-Poisson-Darboux equation with nonlocal condition, Russian Mathematics, 60, 6, 2016, 21-2810.3103/S1066369X16060037Search in Google Scholar

[22] Glushak A.V., Popova V.A., Inverse problem for Euler-Poisson-Darboux abstract differential equation, Journal of Mathematical Sciences, 149, 4, 2008, 1453-146810.1007/s10958-008-0075-3Search in Google Scholar

[23] Weinstein A., On the wave equation and the equation of Euler-Poisson, Proceedings of Symposia in Applied Mathematics, V, Wave motion and vibration theory, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954, 137–14710.1090/psapm/005/0063544Search in Google Scholar

[24] Levitan B.M., Expansion in Fourier Series and Integrals with Bessel Functions, Uspekhi Mat. Nauk, 1951, 6, 2(42), 102–143Search in Google Scholar

[25] Lyakhov L.N., Polovinkin I.P., Shishkina E.L., On a Kipriyanov problem for a singular ultrahyperbolic equation, Differ. Equ., 2014, 50, 4, 513-52510.1134/S0012266114040090Search in Google Scholar

[26] Lyakhov L.N., Weight Spherical Functions and Riesz Potentials Generated by Generalized Shifts, Voronezh. Gos. Tekhn. Univ., Voronezh, 1997Search in Google Scholar

[27] Sitnik S.M. Transmutations and Applications: a survey, arXiv:1012.3741, 141, 2010Search in Google Scholar

[28] Shishkina E.L., Sitnik S.M., General form of the Euler-Poisson-Darboux equation and application of the transmutation method, arXiv:1707.04733v1, 28, 2017Search in Google Scholar

[29] Sitnik S.M., Transmutations and applications, Contemporary studies in mathematical analysis, Vladikavkaz, 2008, 226-293Search in Google Scholar

[30] Sitnik S.M., Factorization and estimates of the norms of Buschman-Erdelyi operators in weighted Lebesgue spaces, Soviet Mathematics Dokladi, 1992, 44, 2, 641-646Search in Google Scholar

[31] Samko S.G., Kilbas A.A., Marichev O.I., Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Sc. Publ., Amsterdam, 1993Search in Google Scholar

**Received:**2016-10-31

**Accepted:**2017-12-22

**Published Online:**2018-02-08

© 2018 Shishkina, published by De Gruyter

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.