Kamil Oruçoğlu

Fundamental Solutions of Some Linear Operator Equations and Applications

A concept of a fundamental solution is introduced for linear operator equations given in some functional spaces. In the case where this fundamental solution does not exist, the representation of the solution is found by the concept of a generalized fundamental solution, which is introduced for operators with nontrivial and generally infinite-dimensional kernels. The fundamental and generalized fundamental solutions are also investigated for a class of Fredholm-type operator equations. Some applications are given for one-dimensional generally nonlocal hyperbolic problems with trivial, finite- and infinite-dimensional kernels. The fundamental and generalized fundamental solutions of such problems are constructed as particular solutions of a system of integral equations or an integral equation. These fundamental solutions become meaningful in a general case when the coefficients are generally nonsmooth functions satisfying only some conditions such as p-integrablity and boundedness.

The Riemann Function for the Third-Order One-Dimensional Pseudoparabolic Equation

In this work, some classes of initial boundary-value problems are investigated in the Sobolev space for the third-order linear pseudoparabolic equation having, in general, nonsmooth coefficients. A new type of Riemann function concept is given for these problems, which is more natural than the classical Riemann-type function concept, and an integral form of the solutions of nonhomogeneous problems can be found more naturally using this concept.

Investigation of Numerical Solution for Fourth‐Order Nonlocal Problem by the Reproducing Kernel Method

This work investigates the approximate solution for fourth‐order multi‐point boundary value problem represented by linear integro‐differential equation involving nonlocal integral boundary conditions by using the reproducing kernel method (RKM). The investigated solution is represented in the form of a series with easily computable components in the reproducing kernel space. When the used algorithm for approximation is applied directly for the given original conditions, it can be very troublesome to compute the reproducing kernel of space. Therefore firstly, it is considered more appropriate conditions to be computed the kernel easily than original ones. Nextly, the original conditions are taken into account. Analysis is illustrated by a numerical example. The results demonstrate that the method is quite accurate and effective.

A Representative Solution to m-Order Linear Ordinary Differential Equation with Nonlocal Conditions by Green's Functional Concept

Abstract In this work, we investigate a linear completely nonhomogeneous nonlocal multipoint problem for an m-order ordinary differential equation with generally variable nonsmooth coefficients satisfying some general properties such as p-integrability and boundedness. A system of m + 1 integro-algebraic equations called the special adjoint system is constructed for this problem. Greens functional is a solution of this special adjoint system. Its first component corresponds to Greens function for the problem. The other components correspond to the unit effects of the conditions. A solution to the problem is an integral representation which is based on using this new Greens functional. Some illustrative implementations and comparisons are provided with some known results in order to demonstrate the advantages of the proposed approach.

Approximate Solution to a Multi-Point Boundary Value Problem Involving Nonlocal Integral Conditions by Reproducing Kernel Method

In this work, we investigate a sequence of approximations converging to the existing unique solution of a multi-point boundary value problem(BVP) given by a linear fourth-order ordinary differential equation with variable coeffcients involving nonlocal integral conditions by using reproducing kernel method(RKM). Obtaining the reproducing kernel of the reproducing kernel space by using the original conditions given directly by RKM may be troublesome and may introduce computational costs. Therefore, in these cases, initially considering more admissible conditions which will allow the reproducing kernel to be computed more easily than the original ones and then taking into account the original conditions lead us to satisfactory results. This analysis is illustrated by a numerical example. The results demonstrate that the method is still quite accurate and effective for the cases with both derivative and integral conditions even if the accuracy is less compared to the cases with just derivative conditions.

Double crack problem in nonlocal elasticity

In the present work, the problem of two collinear cracks in an isotropic, homogeneous elastic medium which is subjected to uniform anti-plane shear loading at infinity is investigated in the context of nonlocal theory of elasticity. Governing equations of the problem are obtained by employing the field equations of the nonlocal elasticity. By use of the boundary conditions, the solution of the problem is reduced first to the investigation of the triple integral equations and then to a singular Fredholm integral equation. Numerical calculations are carried out for different values of the chosen parameter. Then stress analyses are given for single and double cracks.

Existence and Uniqueness of Positive Solutions of Boundary-Value Problems for Fractional Differential Equations with -Laplacian Operator and Identities on the Some Special Polynomials

We consider the following boundary-value problem of nonlinear fractional differential equation with -Laplacian operator , , , , , where , are real numbers, are the standard Caputo fractional derivatives, , , , , , are parameters, and are continuous. By the properties of Green function and Schauder fixed point theorem, several existence and nonexistence results for positive solutions, in terms of the parameters and are obtained. The uniqueness of positive solution on the parameters and is also studied. In the final section of this paper, we derive not only new but also interesting identities related special polynomials by which Caputo fractional derivative.

Investigation of a Fourth‐Order Ordinary Differential Equation with a Four‐Point Boundary Conditions by a New Green’s Functional Concept

A boundary value problem given by multi‐point conditions is investigated for a fourth‐order differential equation. A system of five integro‐algebraic equations called as an adjoint system is introduced for this problem. A Green’s functional concept is introduced as a special solution of the adjoint system. This new type of Green’s function concept, which is more natural than the classical Green‐type function concept, and an integral form of the nonhomogeneous problems can be found more naturally.