Generalized Riccati Wick differential equation and applications
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Generalized Riccati Wick differential equation and applications Marwa Missaoui1 · Hafedh Rguigui2,3 · Slaheddine Wannes4
© Instituto de Matemática e Estatística da Universidade de São Paulo 2020
Abstract Using the Wick derivation operator and the Wick product of elements in a distribution space F∗𝜃 (S�ℂ ) , we introduce the generalized Riccati Wick differential equation as a distribution analogue of the classical Riccati differential equation. The solution of this new equation is given. Finally, we finish this paper by building some applications. Keywords Wick product · Wick derivation · Generalized Riccati Wick differential equation · Space of entire functions with 𝜃-exponential growth condition of minimal type
Communicated by Carlos Tomei. * Hafedh Rguigui [email protected] Marwa Missaoui [email protected] Slaheddine Wannes [email protected] 1
Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, Road of Soukra, 1171 3000 Sfax, Tunisia
2
Department of Mathematics, AL‑Qunfudhah University College, Umm Al-Qura University, KSA, Mecca, Saudi Arabia
3
High School of Sciences and Technology of Hammam Sousse, University of Sousse, Rue Lamine Abassi, 4011 Hammam Sousse, Tunisia
4
Department of Mathematics, Faculty of Sciences of Gabes, University of Gabes, 6072 Gabes, Tunisia
13
Vol.:(0123456789)
São Paulo Journal of Mathematical Sciences
1 Introduction The generalized Riccati equation is named after Jacopo Francesco Riccati, Italian mathematician (1676−1754) [21]. As soon as differential equations were invented, the Riccati differential equation was the first one to be investigated extensively since the end of the 17th century [22]. This equation appears widely in large variety of applications in engineering and applied science such as random processes, diffusion problems, optimal control etc., [1, 11, 12, 15]. It has the form
y� = a(x)y + b(x)y2 + c(x).
(1)
where a(x), b(x) and c(x) are continuous functions of x. In order to find the general solution for the equation (1), a particular solution is needed even though the equation is nonlinear. Since the usual product in (1) is not defined in the distribution case, we need to use a suitable product to introduce a distribution analogue of Eq. (1). The Wick product, denoted by ⋄ , was introduced by Hida and Ikeda [8], and it has been used extensively in the study of generalized Bernoulli Wick differential equations and white noise integral equations, see [2, 9, 10, 19] and references cited therein. This product permits us to extend the equation (1) in infinite dimensional distribution case, which will be called the generalized Riccati Wick differential equation, as follows:
D(𝛷) = A ⋄ 𝛷 + B ⋄ 𝛷⋄2 + H
(2)
D is a Wick derivation operator. where A, B, H ∈ The paper is organized as follows: In Sect. 2, we briefly recall some basic notations in quantum white noise calculus. Namely, we give definitions and properties of the test functions space of entire functions with 𝜃-exponential growth condition of minimal
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