Aytekin Eryılmaz

The Approximate Solutions of Fredholm Integrodifferential-Difference Equations with Variable Coefficients via Homotopy Analysis Method

Numerical solutions of linear and nonlinear integrodifferential-difference equations are presented using homotopy analysis method. The aim of the paper is to present an efficient numerical procedure for solving linear and nonlinear integrodifferential-difference equations. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments and performed on the computer algebraic system.

Buckling of Euler Columns with a Continuous Elastic Restraint via Homotopy Analysis Method

Homotopy Analysis Method (HAM) is applied to find the critical buckling load of the Euler columns with continuous elastic restraints. HAM has been successfully applied to many linear and nonlinear, ordinary and partial, differential equations, integral equations, and difference equations. In this study, we presented the application of HAM to the critical buckling loads for Euler columns with five different support cases continuous elastic restraints. The results are compared with the analytic solutions.

A uniformly valid approximation algorithm for nonlinear ordinary singular perturbation problems with boundary layer solutions.

This paper is concerned with two-point boundary value problems for singularly perturbed nonlinear ordinary differential equations. The case when the solution only has one boundary layer is examined. An efficient method so called Successive Complementary Expansion Method (SCEM) is used to obtain uniformly valid approximations to this kind of solutions. Four test problems are considered to check the efficiency and accuracy of the proposed method. The numerical results are found in good agreement with exact and existing solutions in literature. The results confirm that SCEM has a superiority over other existing methods in terms of easy-applicability and effectiveness.

The Approximate Solutions of First Order Difference Equations with Constant and Variable Coefficients

In this paper a numerical method namely Homotopy Analysis Method (HAM) is presented to solve linear difference equations with variable coefficients under the mixed conditions. In addition, three numerical examples are given to make clear and illustrate the features and capabilities of the presented method. Then numerical results obtained by HAM are compared with the exact solutions.

Numerical solutions of linear and nonlinear Lane-Emden type equations by using magnus expansion method

In this paper, we study on the numerical solutions of linear and nonlinear singular initial value problems of Lane-Emden type with different orders of Magnus expansion method (MEM). The proposed method is different from other numerical techniques as it is based on lie groups and lie algebras. To show the efficiency of the method, we present approximate and exact values and also L∞ error norms with detailed tables and figures.

Successive Complementary Expansion Method for Solving Troesch’s Problem as a Singular Perturbation Problem

A simple and efficient method that is called Successive Complementary Expansion Method (SCEM) is applied for approximation to an unstable two-point boundary value problem which is known as Troesch’s problem. In this approach, Troesch’s problem is considered as a singular perturbation problem. We convert the hyperbolic-type nonlinearity into a polynomial-type nonlinearity using an appropriate transformation, and then we use a basic zoom transformation for the boundary layer and finally obtain a nonlinear ordinary differential equation that contains SCEM complementary approximation. We see that SCEM gives highly accurate approximations to the solution of Troesch’s problem for various parameter values. Moreover, the results are compared with Adomian Decomposition Method (ADM) and Homotopy Perturbation Method (HPM) by using tables.

Spectral analysis of dissipative fractional Sturm–Liouville operators

Abstract We study fractional Sturm–Liouville operators. We give some basic definitions and properties of fractional calculus. Using the method of Pavlov [31, 30, 32], we prove a theorem on the completeness of the system of eigenvectors and associated vectors of dissipative fractional Sturm–Liouville operators.

Livšic’s theorem for q-Sturm—Liouville operators

In this paper, we study dissipative q-Sturm—Liouville operators in Weyl’s limit circle case. We describe all maximal dissipative, maximal accretive, self adjoint extensions of q-Sturm—Liouville operators. Using Livsic’s theorems, we prove a theorem on completeness of the system of eigenvectors and associated vectors of the dissipative q-Sturm—Liouville operators.

On Critical Buckling Loads Of Euler Columns With Elastic End Restraints

In recent years, a great number of analytical approximate solution techniques have been introduced to find a solution to the nonlinear problems that arised in applied sciences. One of these methods is the homotopy analysis method (HAM). HAM has been successfully applied to various kinds of nonlinear differential equations. In this paper, HAM is applied to find buckling loads of Euler columns with elastic end restraints. The critical buckling loads obtained by using HAM are compared with the exact analytic solutions in the literature. Perfect match of the results veries that HAM can be used as an efficient, powerfull and accurate tool for buckling analysis of Euler columns with elastic end restraints. Keywords: Homotopy analysis method (HAM); Euler column; Buckling load; Elastic restraint DOI: 10.17350/HJSE19030000026 Full Text:

Magnus expansion method for solving singularly perturbed turning point problems having boundary layers

In this paper, Magnus expansion method which is based on Lie groups and Lie algebras is presented to solve singularly perturbed boundary value problems having boundary layers. Numerical results are obtained by using different step sizes and small e values which makes the problem stiff or nonstiff. In addition, maximum errors norms with L∞ are tabulated in detail to show efficiency of the method.