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An expansion result for a Sturm-Liouville eigenvalue problem with impulse

The paper is concerned with an eigenvalue problem for second order differential equations with impulse. Such a problem arises when the method of separation of variables applies to the heat conduction equation for two-layered composite. | Turk J Math 34 (2010) , 355 – 366. ¨ ITAK ˙ c TUB doi:10.3906/mat-0809-34 An expansion result for a Sturm-Liouville eigenvalue problem with impulse S ¸ erife Faydao˘glu and Gusein Sh. Guseinov Abstract The paper is concerned with an eigenvalue problem for second order differential equations with impulse. Such a problem arises when the method of separation of variables applies to the heat conduction equation for two-layered composite. The existence of a countably infinite set of eigenvalues and eigenfunctions is proved and a uniformly convergent expansion formula in the eigenfunctions is established. Key Words: Green’s function; Completely continuous operator; Impulse conditions; Eigenvalue; Eigenvector. 1. Introduction An equation for temperatures in a solid 0 ≤ x ≤ b composed of a layer 0 ≤ x 0. We shall assume that ρ(x), p(x), and q(x) are real-valued, p(x) is differentiable on [0, a) ∪ (a, b] , ρ(x), p (x), and q(x) are piecewise continuous on [0, a) ∪ (a, b] and ρ(x) > 0 , p(x) > 0 , q(x) ≥ 0 . In addition, it is assumed that there exist finite left-sided and right-sided limits ρ(a ± 0), p(a ± 0), and q(a ± 0), and that ρ(a ± 0) > 0 , p(a ± 0) > 0 . For solution u(x, t) of equation (1) we take at x = a interface conditions of the form u(a − 0, t) = αu(a + 0, t), ux(a − 0, t) = βux (a + 0, t), (2) in which α and β are given positive real numbers, and at the end faces x = 0 and x = b we take the zero temperature conditions u(0, t) = u(b, t) = 0. (3) AMS Mathematics Subject Classification: 34L10. 355 ˘ FAYDAOGLU, GUSEINOV The initial temperature of the composite is given by u(x, 0) = f(x), x ∈ [0, a) ∪ (a, b]. (4) Note that the conditions in (2) represent an impulse phenomenon at x = a (see [2, 3, 10, 14]). Let us look for a nontrivial solution of (1)–(3), ignoring the initial condition (4), which has the form u(x, t) = e−λt y(x), x ∈ [0, a) ∪ (a, b], (5) where λ is a complex constant and y(x) is a function independent of t (but, in general, .