Emrah Yilmaz

Reconstruction of potential function and its derivatives for Sturm–Liouville problem with eigenvalues in boundary condition

The purpose of this article is solving inverse nodal problem for Sturm–Liouville equation with a boundary condition depending on spectral parameter. Taking into account Law and Chens method, we construct the potential function q and its derivatives by using nodal data. We give several lemmas in order to complete proof of the main theorem. Especially, we obtain an explicit formula for potential function and its derivatives from the nodal data by a pointwise limit.

Reconstruction of the Potential Function and its Derivatives for the Diffusion Operator

We solve the inverse nodal problem for the diffusion operator. In particular, we obtain a reconstruction of the potential function and its derivatives using only nodal data. Results are a generalization of Law’s and Yang’s works.

Spectral theory of Dirac system on time scales

Boundary value problem on time scales is fairly a new subject and has great importance in mathematics. In this study, we explore an eigenvalue problem for Dirac system with separated boundary conditions on an arbitrary time scale . Recent results in the literature about spectral theory of classical Dirac system as orthogonality of eigenfunctions and simplicity of the eigenvalues are generalized and improved. Furthermore, some eigenfunction estimates of the first and second canonic forms for Dirac system are established on . These results will provide a significant contribution to the development of the spectral theory on time scales.

Uniform Statistical Convergence on Time Scales

We will introduce the concept of - and -uniform density of a set and - and -uniform statistical convergence on an arbitrary time scale. However, we will define -uniform Cauchy function on a time scale. Furthermore, some relations about these new notions are also obtained.

P-Laplacian Dirac system on time scales

ABSTRACT The -Laplacian type Dirac systems are nonlinear generalizations of the classical Dirac systems. They can be observed as a bridge between nonlinear systems and linear systems. The purpose of this study is to consider -Laplacian Dirac boundary value problem on an arbitrary time scale to get forceful results by examining some spectral properties of this problem on time scales. Interesting enough, the -Laplacian type Dirac boundary value problem exhibits the classical Dirac problem on time scales. Moreover, we prove Picones identity for -Laplacian type Dirac system which is an important tool to prove oscillation criteria on time scales. It generalizes a classical and well-known theorem for to general case

Solving inverse Sturm-Liouville problem with separated boundary conditions by using two different input data

ABSTRACT In this study, we consider Sturm–Liouville (SL) equation under the separated boundary conditions on a finite interval. We get the approximate solutions of inverse SL problem by using different input data as eigenvalues and nodes (zeros of eigenfunctions), separately, and calculate the computed errors related to the obtained approximate solutions. To calculate the approximate solutions, we use Chebyshev interpolation technique by applying first kind of Chebyshev polynomials. Eventually, the numerical results are presented by providing some examples.

Statistical convergence of multiple sequences on a product time scale

Abstract In this study, we extend the concepts and fundamental results on statistical convergence from a single time scale to any product time scale. Various characterizations about these new notions are also obtained.