Aydin Kurnaz

The solution of the Bagley–Torvik equation with the generalized Taylor collocation method

Abstract In this paper, the Bagley–Torvik equation, which has an important role in fractional calculus, is solved by generalizing the Taylor collocation method. The proposed method has a new algorithm for solving fractional differential equations. This new method has many advantages over variety of numerical approximations for solving fractional differential equations. To assess the effectiveness and preciseness of the method, results are compared with other numerical approaches. Since the Bagley–Torvik equation represents a general form of the fractional problems, its solution can give many ideas about the solution of similar problems in fractional differential equations.

n-Dimensional differential transformation method for solving PDEs

This paper presents the generalization of the differential transformation method to n-dimensional case in order to solve partial differential equations (PDEs). A distinctive practical feature of this method is its ability to solve especially nonlinear differential equations efficiently. We apply our results to a few initial boundary-value problems to illustrate the proposed method.

The differential transform approximation for the system of ordinary differential equations

We present a comparative study of the differential transformation for solving systems of linear or non-linear ordinary differential equations (ODEs). A remarkable practical feature of this method is its ability to solve the system of linear or non-linear differential equations efficiently. This method also enables us to control the truncation error by adjusting the step size used in the numerical scheme. We apply our results to some initial value problems to demonstrate the ability of the method to solve systems of differential equations.

A new analytical approximate method for the solution of fractional differential equations

A new analytical approximate method for the solution of fractional differential equations is presented. This method, which does not require any symbolic computation, is important as a tool for scientists and engineers because it provides an iterative procedure for obtaining the solution of both linear and non-linear fractional differential equations. The effectiveness of the proposed method is illustrated with some examples.

A new Fibonacci type collocation procedure for boundary value problems

In this study, we present a new procedure for the numerical solution of boundary value problems. This approach is mainly founded on the Fibonacci polynomial expansions, the so-called pseudospectral methods with the collocation method. The applicability and effectiveness of our proposed approach is shown by some illustrative examples. Then, the results indicate that this method is very effective and highly promising for linear differential equations defined on any subinterval of the real domain.MSC: 35A25.

Simple Correlations for Estimating the Energy Production of Turkey

Industrialization and population increases make the availability of potential energy in most areas a problem of great importance. Energy is considered to be a prime agent in the generation of wealth and also a significant factor in economic development. Most of the locations in Turkey have abundant energy resources, and energy utilization technologies can be profitably applied to these regions. In this study, we analyze the current status of Turkeys energy resources in terms of energy production and present simple corrections with high correlation coefficients for future projections. It is expected that this study will he helpful in developing highly applicable and productive planning for energy policies.

A new kind of double Chebyshev polynomial approximation on unbounded domains

In this study, a new solution scheme for the partial differential equations with variable coefficients defined on a large domain, especially including infinities, has been investigated. For this purpose, a spectral basis, called exponential Chebyshev (EC) polynomials, has been extended to a new kind of double Chebyshev polynomials. Many outstanding properties of those polynomials have been shown. The applicability and efficiency have been verified on an illustrative example.MSC:35A25.

Statistical Analysis of Solar Radiation Data Using Cubic Spline Functions

A cubic spline-type model has been developed to analyze the avarage daily total solar radiation data. This model has been found to perform significantly better than the other regression-type models. In this study, cubic spline functions have been used to analyze the solar radiation data of 5 years from 1994 to 1998 for G zmir. The reliability of the model has also been tested with a statistical hypothesis.

Adomian decomposition method by Gegenbauer and Jacobi polynomials

In this paper, orthogonal polynomials on [–1,1] interval are used to modify the Adomian decomposition method (ADM). Gegenbauer and Jacobi polynomials are employed to improve the ADM and compared with the method of using Chebyshev and Legendre polynomials. To show the efficiency of the developed method, some linear and nonlinear examples are solved by the proposed method, results are compared with other modifications of the ADM and the exact solutions of the problems.

An efficient approach for solving telegraph equation

In this study, a new numerical scheme for the solution of one dimensional telegraph equationis investigated. The approximate solutions are obtained in the form of the truncated Fibonacci type bivariate series. By means of the proposed method, quite effective results for the partial differential equations defined on the domain of Ω={(x,y):(x,y)∈[a,b]×[c,d]⊂ℝ×ℝ} can be obtained. An illustrative example is shown in order to clarify the findings of the approach. The results indicate that this method not only provide straightforward applicability and computational simplicity in the solution procedure, also achieve the validity.