The classical Euler-Poisson-Darboux equation has the form (1)
The Euler-Poisson-Darboux equation for n = 1 appears in Euler’s work (see , p. 227). Further Euler’s case of (1) was studied by Poisson in , Riemann in  and Darboux in  (for the history of this issue see also in , p. 532 and , p. 527). The generalization of it was studied in . 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  (see also , p. 243) and in  there were different approaches to the solution of the Cauchy problem for the general Euler-Poisson-Darboux equation (2)
with the initials conditions (3)
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  and  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 (4)
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).
and Ω is open set in ℝn which is symmetric correspondingly to each hyperplane xi=0, i=1, …, n, Ω+ = Ω ∩ and Ω+ = Ω ∩ where
We have Ω+ ⊆ and Ω+ ⊆ . Consider the set Cm(Ω+), m ≥ 1, consisting of differentiable functions on Ω+ by order m. Let Cm(Ω+) be the set of functions from Cm(Ω+) such that all their derivatives by xi for all i = 1, …, n are continuous up to the xi=0. Class consists of functions from Cm(Ω+) such that for all non-negative integers and all xi, i = 1, …, n (see , p. 21). A multi-index γ=(γ1, …, γn) consists of fixed positive numbers γi > 0, i=1, …, n and |γ|=γ1+…+γn.
We consider the multidimensional Euler-Poisson-Darboux equation wherein the Bessel operator acts in each of the variables: (5)
Equation (5) we will call the general Euler-Poisson-Darboux equation.
Let uk = uk (x,t) denote the solution of (5) when the next two fundamental recursion formulas hold (7) (8)
If w = tk−1 v satisfies the equation
then using (9) we get
which means that v satisfies the equation
Denoting w = u2−k we obtain (7).
Now we prove the (8). Let tw = vt, v = uk. We obtain
We find now :
Then we get
or (10) □
Recursion formulas (7) and (8) allow us to obtain, from a solution uk 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 we will use multidimensional generalized translation corresponding to multi-index γ:
where each is defined by the formula (see )
The below-considered weighted spherical mean generated by a multidimensional generalized translation γ Tt has the form (see ) (11)
where and the coefficient is computed by the formula (12)
The weighted spherical mean of f ∈ satisfies the general equation Euler–Poisson–Darboux equation (13)
and the conditions (14)
This theorem has been proved in ).
We give theorems on the solution of the Cauchy problem for the general Euler–Poisson–Darboux equation for the remaining values of k. (15) (16)
Using weighted spherical mean we can write (18)
where m is a minimum integer such that is the solution of the Cauchy problem (20) (21)
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].
Let k > 0. The twice continuously differentiable on solution u=uk(x,t) of the Cauchy problem (24) (25)
such that is connected with the twice continuously differentiable on solution w=w(x,t) of the Cauchy problem (26) (27)
such that wxi(x1, …, xi−1,0,xi+1, …, xn,t) = 0, i = 1, …, n by formula (28)
where is transmutation Poisson operator (see ) acting by α
where ξ = α t. Further integrating by parts we obtain
For we have
Let k > 0 then is the Riemann-Liouville left-sided fractional integral of the order (see , p. 33):
Thus we have unique representation of (see , p. 44, theorem 24)
4 The second Cauchy problem for the general Euler-Poisson-Darboux equation
If then the solution v = vk(x,t) of (29) (30)
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+2q ≥ n+|γ| − 1.
Let q ≥ 0 be the smallest positive integer number such that 2−k+2q ≥ n+|γ| − 1 i.e. and let v2−k+2q(x,t) be a solution of (29) when we take 2 − k+2q instead of k such that (31)
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 (32)
Taking into account formula (33) we obtain vk(x,0) = 0 and
Now we obtain the representation of vk through the integral. Using formula (18) we get
If 2−k+2q = n+|γ|−1 then and □
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About the article
Published Online: 2018-02-08
Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 23–31, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0005.
© 2018 Shishkina, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0