In this paper, linear second-order differential equations of Sturm-Liouville type having a finite number of singularities and turning points in a finite interval are investigated. First, we obtain the dual equations associated with the Sturm-Liouville equation. Then, we prove the uniqueness theorem for the solutions of dual initial value problems.

The Sturm–Liouville differential equation is an important tool for physics, applied mathematics, and other fields of engineering science has wide applications in quantum mechanics, classical wave phenomena. In this paper, we investigate the coupled hybrid version equation. Indeed, study existence solutions with multi-point boundary condition. Furthermore, integral We give application some examp...

This is the first of three papers on the geometric and combinatorial characterization of global Sturm attractors which consist of a single closed 3-ball. The underlying scalar PDE is parabolic, ut = uxx + f(x, u, ux) , on the unit interval 0 < x < 1 with Neumann boundary conditions. Equilibria are assumed to be hyperbolic. The geometric description is in terms of the regular cell complex define...

We have formalised the Sturm-Tarski theorem (also referred as the Tarski theorem): Given polynomials p, q ∈ R[x], the Sturm-Tarski theorem computes the sum of the signs of q over the roots of p by calculating some remainder sequences. Note, the better-known Sturm theorem is an instance of the Sturm-Tarski theorem when q = 1. The proof follows the classic book by Basu et al. [1] and Cyril Cohen’...

We establish the results on the existence of three positive solutions to Sturm-Liouville boundary value problems of the singular fractional differential equation with the nonlinearity depending on D 0+u ⎪⎨ ⎪⎩ D0+u(t)+ f (t,u(t),D μ 0+ u(t)) = 0, t ∈ (0,1),1 < α < 2, a limt→0 I2−α 0+ u(t)−b limt→0 [ I2−α 0+ u(t) ]′ = ∫ 1 0 g(s,u(s),D μ 0+u(s))ds, c D 0+u(1)+du(1) = ∫ 1 0 h(s,u(s),D μ 0+u(s))ds. ...

where the potentials are given functions. Under various boundary conditions, Sturm and Liouville established that solutions of problem (1) can exist only for particular values of the real parameter λ, which is called an eigenvalue. Relevant examples of linear Sturm-Liouville problems are the Bessel equation and the Legendre equation. The classical Sturm-Liouville theory does not depend upon the...

Based on a Morse-Smale structure, we study planar global attractors Af of the scalar reaction-advection-diffusion equation ut = uxx + f(x, u, ux) in one space dimension. We assume Neumann boundary conditions on the unit interval, dissipativeness of f , and hyperbolicity of equilibria. We call Af Sturm attractor because our results strongly rely on nonlinear nodal properties of Sturm type. Plana...