On Mellin transforms of solutions of differential equation $$\chi ^{(n)}(x)+\gamma _{n}x\chi (x)=0$$
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On Mellin transforms of solutions of differential equation (n) (x) + n x(x) = 0 Hassan Askari1 · Alireza Ansari1 Received: 23 August 2018 / Revised: 23 August 2018 / Accepted: 5 October 2020 © Springer Nature Switzerland AG 2020
Abstract In this paper, for n = 2, 3, . . . , we consider the differential equation
χ
(n)
(x) + γn xχ (x) = 0,
γn = (−1)k , γn = −1,
n = 2k, n = 2k + 1,
and find the linear independent solutions in terms of the higher-order Airy functions (n = 2k) and the higher-order Lévy stable functions (n = 2k + 1). The integral representations of solutions are presented and their Mellin transforms are also given. Keywords Airy functions · Lévy stable functions · Laplace path · Mellin transform Mathematics Subject Classification 33C10 · 35C15 · 33E20
1 Introduction In the literature, for the higher-order heat equation ∂u(x, t) ∂ n u(x, t) = γn , u(x, 0) = δ(x), t > 0, x ∈ R, ∂t ∂xn
B
(1-1)
Alireza Ansari [email protected] Hassan Askari [email protected]
1
Department of Applied Mathematics, Faculty of Mathematical Sciences, Shahrekord University, P.O. Box 115, Shahrekord, Iran 0123456789().: V,-vol
57
Page 2 of 24
H. Askari, A. Ansari
where δ(x) is the Dirac delta function, one can derive the following integral solution u(x, t) =
1 2π
∞
−∞
exp −ir x + γn (−ir )n t dr , n ∈ {2, 3, 4, . . .}, γn ∈ C. (1-2)
This solution (depend on the values of n) plays an important role in engineering and applied mathematics and is reduced to the higher-order (generalized) Airy function Ai2k+1 (x) [1–7] 1 Ai2k+1 (x) = π
∞
0
r 2k+1 dr , cos r x + 2k + 1
(1-3)
and the Lévy stable function [8–13] 1 L(x) = π
∞
e−r cos(r x)dr , k ∈ N. 2k
(1-4)
0
For the even values of n, the solution (1-2) is appeared in the probabilistic distributions and the stochastic theory [14,15]. See also [16–28] for other mathematical aspects of this solution. For the odd values of n, the mathematical properties of solution (1-2) in the probabilistic distributions was discussed in [29,30]. For this case, see also other works [17,31–36]. In this paper, from another point of view, we intend to derive the higherorder Airy function and the higher-order Lévy stable function from the following differential equation χ (n) (x) + γn xχ (x) = 0,
γn = (−1)k , n = 2k, γn = −1, n = 2k + 1,
(1-5)
and get other linear independent solutions with the associated integral representations. In this sense, we organize the paper as follows. In Sect. 2, for the case n = 2k, we consider the Laplace integral as the formal solution of differential (2-1) in the complex plane and obtain all integral representations of distinct solutions. We construct the complex and real basis for these solutions and name them as the complex and real higher-order Airy functions. The Mellin transforms of these functions are presented and some special cases of the Mellin transforms are also discussed. In Sect. 3, for the case n = 2k + 1, we obtain all integral representations of distinct solutions and construct the complex and
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