The inverse nodal problem was initiated by McLaughlin [1], who proved that the Sturm-Liouville problem is uniquely determined by any dense subset of the nodal points. Some numerical schemes were given by Hald and McLaughlin [2] for the reconstruction of the potential. Recently Law, Yang and other authors have reconstructed the potential function and its derivatives of the Sturm-Liouville proble...

Inverse spectral problems consist in recovering operators from their spectral characteristics. Such problems play an important role in mathematics and have many applications in natural sciences (see, for example, [1 – 6]). In 1988, the inverse nodal problem was posed and solved for Sturm-Liouville problems by J. R. McLaughlin [7], who showed that the knowledge of a dense subset of nodal points ...

We construct a stochastic process, called the Liouville Brownian motion which we conjecture to be the scaling limit of random walks on large planar maps which are embedded in the euclidean plane or in the sphere in a conformal manner. Our construction works for all universality classes of planar maps satisfying γ < γc = 2. In particular, this includes the interesting case of γ = √ 8/3 which cor...

This paper gives constructive algorithms for the classical inverse Sturm-Liouville problem. It is shown that many of the formulations of this problem are equivalent to solving an overdetermined boundary value problem for a certain hyperbolic operator. Two methods of solving this latter problem are then provided, and numerical examples are presented.

We consider a boundary value problem involving a Riemann-Liouville fractional derivative of order α ∈ (3/2, 2) on the unit interval (0, 1). The standard Galerkin finite element approximation converges slowly due to the presence of singularity term xα−1 in the solution representation. In this work, we develop a simple technique, by transforming it into a second-order two-point boundary value pro...

One of the central problems of mathematics in the second half of the 19th century and at the beginning of the 20th century was the problem of uniformization of Riemann surfaces. The classics, Klein [1] and Poincaré [2], associated it with studying second-order ordinary differential equations with regular singular points. Poincaré proposed another approach to the uniformization problem [3]. It c...

In this paper, we study the inverse problem for Sturm Liouville with conformable fractional differential operators of order and finite number interior discontinuous conditions. For aim first, asymptotic formulas solutions, eigenvalues eigenfunctions are calculated. Then some uniqueness theorems proposed eigenvalue proved. Finally, Hald's theorem &#x0D; Sturm-Liouville is developed.

1 Professor and author of correspondence, Phone: +91 3222-283084, Fax: +91 3222 255303, Email: [email protected] ABSTRACT A numerical technique for the solution of a class of fractional optimal control problems has been proposed in this paper. The technique can used for problems defined both in terms of Riemann-Liouville and Caputo fractional derivatives. In this technique a Reflection Op...