Aydin Secer

A Note on Fractional Order Derivatives and Table of Fractional Derivatives of Some Special Functions

The purpose of this note is to present the different fractional order derivatives definition that are commonly used in the literature on one hand and to present a table of fractional order derivatives of some functions in Riemann-Liouville sense On other the hand. We present some advantages and disadvantages of these fractional derivatives. And finally we propose alternative fractional derivative definition.

The Time-Fractional Coupled-Korteweg-de-Vries Equations

We put into practice a relatively new analytical technique, the homotopy decomposition method, for solving the nonlinear fractional coupled-Korteweg-de-Vries equations. Numerical solutions are given, and some properties exhibit reasonable dependence on the fractional-order derivatives’ values. The fractional derivatives are described in the Caputo sense. The reliability of HDM and the reduction in computations give HDM a wider applicability. In addition, the calculations involved in HDM are very simple and straightforward. It is demonstrated that HDM is a powerful and efficient tool for FPDEs. It was also demonstrated that HDM is more efficient than the adomian decomposition method (ADM), variational iteration method (VIM), homotopy analysis method (HAM), and homotopy perturbation method (HPM).

On Generalized Fractional Kinetic Equations Involving Generalized Bessel Function of the First Kind

We develop a new and further generalized form of the fractional kinetic equation involving generalized Bessel function of the first kind. The manifold generality of the generalized Bessel function of the first kind is discussed in terms of the solution of the fractional kinetic equation in the paper. The results obtained here are quite general in nature and capable of yielding a very large number of known and (presumably) new results.

Efficient solutions of systems of fractional PDEs by the differential transform method

In this paper we obtain approximate analytical solutions of systems of nonlinear fractional partial differential equations (FPDEs) by using the two-dimensional differential transform method (DTM). DTM is a numerical solution technique that is based on the Taylor series expansion which constructs an analytical solution in the form of a polynomial. The traditional higher order Taylor series method requires symbolic computation. However, DTM obtains a polynomial series solution by means of an iterative procedure. The fractional derivatives are described in the Caputo fractional derivative sense. The solutions are obtained in the form of rapidly convergent infinite series with easily computable terms. DTM is compared with some other numerical methods. Computational results reveal that DTM is a highly effective scheme for obtaining approximate analytical solutions of systems of linear and nonlinear FPDEs and offers significant advantages over other numerical methods in terms of its straightforward applicability, computational efficiency, and accuracy.

Sinc-Galerkin method for approximate solutions of fractional order boundary value problems

In this paper we present an approximate solution of a fractional order two-point boundary value problem (FBVP). We use the sinc-Galerkin method that has almost not been employed for the fractional order differential equations. We expand the solution function in a finite series in terms of composite translated sinc functions and some unknown coefficients. These coefficients are determined by writing the original FBVP as a bilinear form with respect to some base functions. The bilinear forms are expressed by some appropriate integrals. These integrals are approximately solved by sinc quadrature rule where a conformal map and its inverse are evaluated at sinc grid points. Obtained results are presented as two new theorems. In order to illustrate the applicability and accuracy of the present method, the method is applied to some specific examples, and simulations of the approximate solutions are provided. The results are compared with the ones obtained by the Cubic splines. Because there are only a few studies regarding the application of sinc-type methods to fractional order differential equations, this study is going to be a totally new contribution and highly useful for the researchers in fractional calculus area of scientific research.

The sinc-Galerkin method and its applications on singular Dirichlet-type boundary value problems

The application of the sinc-Galerkin method to an approximate solution of second-order singular Dirichlet-type boundary value problems were discussed in this study. The method is based on approximating functions and their derivatives by using the Whittaker cardinal function. The differential equation is reduced to a system of algebraic equations via new accurate explicit approximations of the inner products without any numerical integration which is needed to solve matrix system. This study shows that the sinc-Galerkin method is a very effective and powerful tool in solving such problems numerically. At the end of the paper, the method was tested on several examples with second-order Dirichlet-type boundary value problems.

A Novel Matching of MR Images Using Gabor Wavelets

Abstract This study introduces a hybrid method for deformable matching of Magnetic resonance (MR) images by utilizing the advantages of both wavelets and variational calculus. Image matching problem is expressed as an optimal control problem and discretization of the resulting Euler-Lagrange equations is written in terms of the system of linear equations in the form of Au = f, where u is the image displacement field. Implementation of the algorithm exploits Gabor wavelet energy maps of MR images. The proposed algorithm provides an efficient MR matching technique. Experimental results proved that the method can match MR images better than the only variational or only wavelet-based methods.

Approximate analytic solution of fractional heat-like and wave-like equations with variable coefficients using the differential transforms method

This paper uses the differential transform method (DTM) to obtain analytical solutions of fractional heat- and wave-like equations with variable coefficients. The time fractional heat-like and wave-like equations with variable coefficients were obtained by replacing a first-order and a second-order time derivative by a fractional derivative of order 0<α<2. The approach mainly rests on the DTM which is one of the approximate methods. The method can easily be applied to many problems and is capable of reducing the size of computational work. Some examples are presented to show the efficiency and simplicity of the method.

An efficient algorithm for solving fractional differential equations with boundary conditions

Abstract In this paper, a sinc-collocation method is described to determine the approximate solution of fractional order boundary value problem (FBVP). The results obtained are presented as two new theorems. The fractional derivatives are defined in the Caputo sense, which is often used in fractional calculus. In order to demonstrate the efficiency and capacity of the present method, it is applied to some FBVP with variable coefficients. Obtained results are compared to exact solutions as well as Cubic Spline solutions. The comparisons can be used to conclude that sinc-collocation method is powerful and promising method for determining the approximate solutions of FBVPs in different types of scenarios.

Convexity of Certain

and Applied Analysis 3 where f(z) is analytic in a simply connected region of the zplane containing the origin and the q-binomial function (z − tq)δ−1 is given by (z − tq) δ−1 = z δ−1 1 Φ0 [q −δ+1 ; −; q, tq δ z ] . (19) The series 1 Φ0[δ; −; q, z] is single valued when | arg(z)| < π and |z| < 1 (see for details [2], pp. 104–106); therefore, the function (z − tq) δ−1 in (18) is single valued when | arg(−tq/z)| < π, |tq/z| < 1, and | arg(z)| < π. Definition 2 (fractional q-derivative operator). The fractional q-derivative operator D q,z of a function f(z) of order δ is defined by D δ q,z f (z) ≡ Dq,z I 1−δ q,z f (z) = 1 Γq (1 − δ)