Singular Cauchy problem for the general Euler-Poisson-Darboux equation: General Euler-Poisson-Darboux equation

E.L. Shishkina

doi:10.1515/math-2018-0005

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Singular Cauchy problem for the general Euler-Poisson-Darboux equation

General Euler-Poisson-Darboux equation

E.L. Shishkina

Veröffentlicht/Copyright: 8. Februar 2018

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$Open Mathematics$

Aus der Zeitschrift Open Mathematics Band 16 Heft 1

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.

Keywords: Bessel operator; Euler-Poisson-Darboux equation; Singular Cauchy problem

MSC 2010: 26A33; 44A15

1 Introduction

The classical Euler-Poisson-Darboux equation has the form

∑i=1n∂2u∂xi2=∂2u∂t2+kt∂u∂t,u=u(x,t),x∈Rn,t>0,−∞<k<∞.(1)

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

(Bk)t=∂2∂t2+kt∂∂t.

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

∑i=1n∂2u∂xi2+γixi∂u∂xi=∂2u∂t2+kt∂u∂t,0<γi,i=1,...,n,k>0(2)

with the initials conditions

u(x,0)=f(x),∂u∂t|t=0=0.(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 [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

u(x,0)=0,limt→0tk∂u∂t=φ(x).(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).

Let

R+n={x=(x1,…,xn)∈Rn,x1>0,…,xn>0}

and Ω is open set in ℝ_n which is symmetric correspondingly to each hyperplane x_i=0, i=1, …, n, Ω₊ = Ω ∩ R+n and Ω₊ = Ω ∩ R¯+n where

R¯+n={x=(x1,…,xn)∈Rn,x1≥0,…,xn≥0}.

We have Ω₊ ⊆ R+n and Ω₊ ⊆ R¯+n. Consider the set 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_i for all i = 1, …, n are continuous up to the x_i=0. Class Cevm(Ω¯+) consists of functions from C^m(Ω₊) such that ∂2k+1f∂xi2k+1|x=0=0 for all non-negative integers k≤m−12 and all x_i, i = 1, …, n (see [1], p. 21). A multi-index γ=(γ₁, …, γ_n) consists of fixed positive numbers γ_i > 0, i=1, …, n and |γ|=γ₁+…+γ_n.

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

(△γ)xu=(Bk)tu,−∞<k<∞,u=uk(x,t),x∈R+n,t>0,(5)

where

(△γ)x=△γ=∑i=1n(Bγ1)xi=∑i=1n∂2∂xi2+γixi∂∂x,(6)

(Bk)t=∂2∂t2+kt∂∂t,k∈R.

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

uk=t1−ku2−k,(7)

utk=tuk+2.(8)

Proof

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

wt=(k−1)tk−2v+tk−1vt=k−1tw+tk−1vt,wtt=(k−1)(k−2)tk−3v+(k−1)tk−2vt+(k−1)tk−2vt+tk−1vtt==(k−1)(k−2)t2w+2(k−1)tk−2vt+tk−1vtt,2−ktwt=−(k−1)(k−2)t2w+(2−k)tk−2vt,wtt+2−ktwt=2(k−1)tk−2vt+tk−1vtt+(2−k)tk−2vt=tk−1vtt+ktvt

wtt+2−ktwt=tk−1vtt+ktvt.(9)

If w = t^k−1v satisfies the equation

Δγw=wtt+2−ktwt,

then using (9) we get

tk−1Δγv=tk−1vtt+ktvt

which means that v satisfies the equation

Δγv=vtt+ktvt.

Denoting w = u^2−k we obtain (7).

Now we prove the (8). Let tw = v_t, v = u^k. We obtain

wt=−1t2vt+1tvtt,wtt=2t3vt−2t2vtt+1tvttt.

We find now k+2twt:

k+2twt=−k+2t3vt+k+2t2vtt.

Then we get

wtt+k+2twt=2t3vt−2t2vtt+1tvttt−k+2t3vt+k+2t2vtt==1tvttt−kt3vt+kt2vtt=1tvttt−kt2vt+ktvtt=1t∂∂tvtt+ktvt

wtt+k+2twt=1t∂∂tvtt+ktvt.(10)

□

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 ∂2u∂t2+kt∂u∂t−△u=0.

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 R+n we will use multidimensional generalized translation corresponding to multi-index γ:

γTt=γ1Tx1t1...γnTxntn,

where each γiTxiτi is defined by the formula (see [24])

γiTxiτif(x)=Γγi+12Γγi2Γ12∫0πf(x1,...,xi−1,xi2+τi2−2xiτicos⁡αi,xi+1,...,xn)sinγi−1⁡αidαi.

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

Mfγ(x;r)=1|S1+(n)|γ∫S1+(n)γTxrθf(x)θγdS,(11)

where θγ=∏i=1nθiγi,S1+(n)={θ:|θ|=1,θ∈R+n} and the coefficient |S1+(n)|γ is computed by the formula

|S1+(n)|γ=∫S1+(n)∏i=1nxiγidS(y)=∏i=1nΓγi+122n−1Γn+|γ|2(12)

(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 ∈ Cev2satisfies the general equation Euler–Poisson–Darboux equation

(Δγ)xMfγ(x;t)=(Bk)tMfγ(x;t),k=n+|γ|−1(13)

and the conditions

Mfγ(x;0)=f,(Mfγ)t′(x;0)=0.(14)

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.

(Δγ)xu=(Bk)tu,u=uk(x,t),x∈R+n,t>0,(15)

uk(x,0)=f(x),utk(x,0)=0.(16)

Theorem 3.2

Let f ∈ Cev2. Then for the case k > n+|γ | − 1 the solution of(15)–(16)is unique and given by

uk(x,t)=2nΓk+12Γk−n−|γ|+12∏i=1nΓγi+12∫B1+(n)[γTtyf(x)](1−|y|2)k−n−|γ|−12yγdy.(17)

Using weighted spherical mean we can write

uk(x,t)=2t1−kΓk+12Γk−n−|γ|+12Γn+|γ|2∫0t(t2−r2)k−n−|γ|−12rn+|γ|−1Mfγ(x;r)dr.(18)

Theorem 3.3

Iff∈Cevn+|γ|−k2+2then the solution of(15)–(16)for k < n+|γ | − 1, k ≠ −1,−3,−5,…

uk(x,t)=t1−k∂t∂tm(tk+2m−1uk+2m(x,t)),(19)

where m is a minimum integer such thatm≥n+|γ|−k−12anduk+2m(x,t)is the solution of the Cauchy problem

(Bk+2m)tuk+2m(x,t)=(Δγ)xuk+2m(x,t),(20)

uk+2m(x,0)=f(x)(k+1)(k+3)...(k+2m−1),utk+2m(x,0)=0.(21)

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

Theorem 3.4

If f ∈ Cev1−kis B–polyharmonic of order1−k2then one of the solutions of the Cauchy problem(20)–(21)for the k=−1,−3,−5,… is given by

u−1(x,t)=f(x),(22)

uk(x,t)=f(x)+∑h=1−k+12Δγhf(k+1)...(k+2h−1)t2h2⋅4⋅....⋅2h,k=−3,−5,...(23)

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 onR+n+1solution u=u^k(x,t) of the Cauchy problem

(Δγ)xu=(Bk)tu,u=uk(x,t),x∈R+n,t>0,(24)

uk(x,0)=f(x),utk(x,0)=0(25)

such thatuxik(x1,...,xi−1,0,xi+1,...,xn,t)=0,i=1,...,nis connected with the twice continuously differentiable onR+nsolution w=w(x,t) of the Cauchy problem

(Δγ)xw=wtt,w=w(x,t),x∈R+n,t∈R,(26)

w(x,0)=f(x),wt(x,0)=0(27)

such that w_{x_i}(x₁, …, x_i−1,0,x_i+1, …, x_n,t) = 0, i = 1, …, n by formula

uk(x,t)=(P1k−12)αw(x,αt),(28)

where(Pτλ)αis transmutation Poisson operator (see [24]) acting by α

(Pτλ)αg(α)=2Γ(λ+1)πΓλ+121τ2λ∫0τg(α)[τ2−α2]λ−12dα.

Proof

The fact that the function u^k defined by the equality (28) satisfies the conditions (31) is obvious. Let us show that u^k defined by (28) satisfies (24)

(Δγ)xu=(P1k−12)α(Δγ)xw(x,αt)=(P1k−12)αwξξ(x,αt)==2Γk+12πΓk2∫01(Δγ)xw(x,αt)[1−α2]k2−1dα,

where ξ = α t. Further integrating by parts we obtain

∂uk∂t=2Γk+12πΓk2∫01αwξ(x,αt)[1−α2]k2−1dα==u=wξ(x,αt),dv=α[1−α2]k2−1dα,du=twξξ(x,αt)dα,v=−1k[1−α2]k2==2Γk+12πΓk2tk∫01wξξ(x,αt)[1−α2]k2dα==2Γk+12πΓk2tk∫01wξξ(x,αt)[1−α2]k2dα.

For ∂2uk∂t2 we have

∂2uk∂t2=2Γk+12πΓk2∫01α2wξξ(x,αt)[1−α2]k2−1dα==2Γk+12πΓk2∫01(Δγ)xw(x,αt)α2[1−α2]k2−1dα.

Finally,

∂2uk∂t2+kt∂uk∂t=2Γk+12πΓk2∫01(Δγ)xw(x,αt)α2[1−α2]k2−1dα+∫01(Δγ)xw(x,αt)[1−α2]k2dα==2Γk+12πΓk2∫01(Δγ)xw(x,αt)[1−α2]k2−1dα=(Δγ)xuk.

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 αt=τ,t=y, we get

yk−12uk(x,y)=Γk+12πΓk2∫0yw(x,τ)τ(y−τ)k2−1dτ.

Let k > 0 then yk−12uk(x,y) is the Riemann-Liouville left-sided fractional integral of the order k2 (see [31], p. 33):

yk−12uk(x,y)=Γk+12πI0+k2w(x,τ)τ(y).

Thus we have unique representation of w(x,τ) (see [31], p. 44, theorem 24)

w(x,τ)=τπΓk+12D0+k2yk−12uk(x,y)(τ)

w(x,t)=2Γn−k2d2tdtn∫0tuk(x,z)zk(t2−z2)k2−n+1dz.

□

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

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

Theorem 4.1

Ifφ∈Cevn+|γ|+k−12then the solution v = v^k(x,t) of

(△γ)xv=(Bk)tv,0<γi,i=1,...,n,k<1,x∈R+n,t>0,(29)

vk(x,0)=0,limt→0tk∂v∂t=φ(x)(30)

is given by

vk(x,t)=Γ3−k2∏i=1nΓγi+12Γ2−k+2q−n−|γ|+122n+q(1−k)Γ3−k+2q2Γ2−k+2q21t∂∂tq××t1−k+2q∫B1+(n)[γTtyφ(x)](1−|y|2)2−k+2q−n−|γ|−12yγdy

if n+|γ|+k is not an odd integer and

vk(x,t)=2−qΓ3−k2(1−k)Γ3−k+2q21t∂∂tqtn+|γ|−2Mφγ(x;t).

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.

Proof

Let q ≥ 0 be the smallest positive integer number such that 2−k+2q ≥ n+|γ| − 1 i.e. q=n+|γ|+k−12 and let v^2−k+2q(x,t) be a solution of (29) when we take 2 − k+2q instead of k such that

v2−k+2q(x,0)=φ(x),vt2−k+2q(x,0)=0.(31)

Then by property (7) we obtain that

vk−2q=t1−k+2qv2−k+2q

is a solution of the equation

(△γ)xv=∂2v∂t2+k−2qt∂v∂t.

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

1t∂∂tqvk−2q=1t∂∂tq(t1−k+2qv2−k+2q)

is a solution of the (29).

Let’s consider

vk(x,t)=2−qΓ3−k2(1−k)Γ3−k+2q21t∂∂tq(t1−k+2qv2−k+2q).(32)

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

Now we will prove that v^k satisfies the conditions (31). For vk∈Cevq(Ω+) we have the formula (see [19], p.9)

1t∂∂tq(t1−k+2qv2−k+2q)=∑s=0q2q−sCqsΓ1−k2+q+1Γ1−k2+s+1t1−k+2s1t∂∂tsv2−k+2q.(33)

Taking into account formula (33) we obtain v^k(x,0) = 0 and

limt→0tkvtk(x,t)=2−qΓ3−k2(1−k)Γ3−k+2q2limt→0tk∂∂t1t∂∂tq(t1−k+2qv2−k+2q)==2−qΓ3−k2(1−k)Γ3−k+2q2limt→0tk∂∂t∑s=0q2q−sCqsΓ1−k2+q+1Γ1−k2+s+1t1−k+2s1t∂∂tsv2−k+2q==11−klimt→0tk∂∂tt1−kv2−k+2q=11−klimt→0tk(1−k)t−kv2−k+2q+t1−kvt2−k+2q==11−klimt→0(1−k)v2−k+2q+tvt2−k+2q=φ(x).

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

v2−k+2q=2Γ3−k+2q2Γ3−k+2q−n−|γ|2Γn+|γ|2∫01(1−r2)1−k+2q−n−|γ|2rn+|γ|−1Mφγ(x;rt)dr.

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

vk=2−qΓ3−k2(1−k)Γ3−k+2q2∑s=0q2q−sCqsΓ1−k2+q+1Γ3−k2+st1−k+2s1t∂∂tsv2−k+2q==Γ3−k21−k∑s=0qCqst1−k+2s2sΓ3−k2+s1t∂∂tsv2−k+2q==Γ3−k+2q2Γ1−k2Γ3−k+2q−n−|γ|2Γn+|γ|2∑s=0qCqst1−k+2s2sΓ3−k2+s××∫01(1−r2)1−k+2q−n−|γ|2rn+|γ|−11t∂∂tsMφγ(x;rt)dr.

If 2−k+2q = n+|γ|−1 then v2−k+2q=Mφγ(x;t) and

vk=2−qΓ3−k2(1−k)Γ3−k+2q21t∂∂tqtn+|γ|−2Mfγ(x;t)==2−1−qΓ1−k2Γ3−k+2q2∑s=0q2q−sCqsΓ3−k2+qΓ3−k2+st1−k+2s1t∂∂tsMfγ(x;t)==∑s=0qCqsΓ1−k22s+1Γ3−k2+st1−k+2s1t∂∂tsMfγ(x;t).

□

References

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

[2] Euler L., Institutiones calculi integralis, Opera Omnia. Ser. 1. V. 13. Leipzig, Berlin, 1914, 1, 13, 212-230Suche 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, 215-248Suche in Google Scholar

[4] Riemann B., On the Propagation of Flat Waves of Finite Amplitude, Ouvres, OGIZ, Moscow-Leningrad, 1948, 376-395Suche 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, 1915Suche in Google Scholar

[6] Mises R. von, The Mathematical Theory Of Compressible Fluid Flow, Academic Press, New York, 1958Suche 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, 519-552Suche in Google Scholar

[8] Rutkauskas S., Some boundary value problems for an analogue of the Euler–Poisson–Darboux equation, Differ. Uravn.,1984, 20, 1, 115–124Suche 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, 440–453Suche in Google Scholar

[10] Olevskii M.N., The solution of the Dirichlet problem related to the equation Δ+pxn∂uxn=ρ for the semispheric domain, Dokl. AN SSSR, 1949, 64, 767-770Suche in Google Scholar

[11] Aksenov A.V., Periodic invariant solutions of equations of absolutely unstable media, Izv. AN. Mekh. Tverd. Tela, 1997, 2, 14-20Suche 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, 176-179Suche in Google Scholar

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

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

[15] Zhdanov V.K., Trubnikov B.A., Quasigas Unstable Media, Nauka, Moscow, 1991Suche 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.1959.8.58015Suche in Google Scholar

[17] Carroll R.W., Showalter R.E., Singular and Degenerate Cauchy problems, N.Y.: Academic Press, 1976Suche 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/S106456241407028XSuche in Google Scholar

[19] Tersenov S.A., Introduction in the theory of equations degenerating on a boundary. USSR, Novosibirsk state university,1973Suche 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/S0012266116010043Suche in Google Scholar

[21] Glushak A.V., Abstract Euler-Poisson-Darboux equation with nonlocal condition, Russian Mathematics, 60, 6, 2016, 21-2810.3103/S1066369X16060037Suche 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/s10958-008-0075-3Suche 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/005/0063544Suche in Google Scholar

[24] Levitan B.M., Expansion in Fourier Series and Integrals with Bessel Functions, Uspekhi Mat. Nauk, 1951, 6, 2(42), 102–143Suche 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/S0012266114040090Suche in Google Scholar

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

[27] Sitnik S.M. Transmutations and Applications: a survey, arXiv:1012.3741, 141, 2010Suche 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, 2017Suche in Google Scholar

[29] Sitnik S.M., Transmutations and applications, Contemporary studies in mathematical analysis, Vladikavkaz, 2008, 226-293Suche 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, 641-646Suche 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, 1993Suche in Google Scholar

Received: 2016-10-31

Accepted: 2017-12-22

Published Online: 2018-02-08

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https://doi.org/10.1515/math-2018-0005

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Bessel operator; Euler-Poisson-Darboux equation; Singular Cauchy problem

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