TAILIEUCHUNG - On the solution of an inverse Sturm–Liouville problem with a delay and eigenparameter-dependent boundary conditions
In this paper, a boundary value problem consisting of a delay differential equation of the Sturm–Liouville type with eigenparameter-dependent boundary conditions is investigated. The asymptotic behavior of eigenvalues is studied and the parameter of delay is determined by eigenvalues. | Turk J Math (2018) 42: 3090 – 3100 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article On the solution of an inverse Sturm–Liouville problem with a delay and eigenparameter-dependent boundary conditions Seyfollah MOSAZADEH∗, Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran Received: • Accepted/Published Online: • Final Version: Abstract: In this paper, a boundary value problem consisting of a delay differential equation of the Sturm–Liouville type with eigenparameter-dependent boundary conditions is investigated. The asymptotic behavior of eigenvalues is studied and the parameter of delay is determined by eigenvalues. Then we obtain the connection between the potential function and the canonical form of the characteristic function. Key words: Delay differential equation, characteristic function, eigenvalues, inverse problem 1. Introduction Delay differential equations have multiple applications in science and engineering and are used as models for a variety of phenomena in physics, chemistry, technology, life sciences, etc. Therefore, this field of differential equations may be of interest for applied mathematics, multidisciplinary audiences, computational scientists, and engineers [1,3,6,13]. Moreover, boundary value problems with eigenparameters in boundary conditions appear in such problems of mathematical physics or mathematical chemistry [12,14]. In this paper, we consider the boundary value problem L := L(q(x), α) consisting of the following second-order differential equation of Sturm–Liouville type, y ′′ (x) + q(x)y(α(x − a)) + λ2 y(x) = 0, (1) on the finite interval [a, b] , together with the following boundary conditions, which depend on the spectral parameter λ > 0 : y ′ (a) + λr1 y(a) = 0, ′ y (b) − λ y(b) = 0, r2 y(h(x, α)) = y(a)ψ(h(x, α)), h(x, α) 0. Under the hypothesis of Lemma 1 we .
đang nạp các trang xem trước
