Muhammet Kurulay

Solving the fractional nonlinear Klein-Gordon equation by means of the homotopy analysis method

In this paper, the homotopy analysis method is applied to obtain the solution of nonlinear fractional partial differential equations. The method has been successively provided for finding approximate analytical solutions of the fractional nonlinear Klein-Gordon equation. Different from all other analytic methods, it provides us with a simple way to adjust and control the convergence region of solution series by introducing an auxiliary parameter ħ. The analysis is accompanied by numerical examples. The algorithm described in this paper is expected to be further employed to solve similar nonlinear problems in fractional calculus.

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.

Some properties of the Mittag-Leffler functions and their relation with the Wright functions

This paper is a short description of our recent results on an important class of the so-called Mittag-Leffler functions, which became important as solutions of fractional order differential and integral equations, control systems and refined mathematical models of various physical, chemical, economical, management and bioengineering phenomena. We have studied the Mittag-Leffler functions as their typical representatives, including many interesting special cases that have already proven their usefulness in fractional calculus and its applications. We obtained a number of useful relationships between the Mittag-Leffler functions and the Wright functions. The Wright function plays an important role in the solution of a linear partial differential equation. The Wright function, which we denote by W(z;α,β), is so named in honor of Wright who introduced and investigated this function in a series of notes starting from 1933 in the framework of the asymptotic theory of partitions.MSC:33E12.

A Novel Method for Analytical Solutions of Fractional Partial Differential Equations

A new solution technique for analytical solutions of fractional partial differential equations (FPDEs) is presented. The solutions are expressed as a finite sum of a vector type functional. By employing MAPLE software, it is shown that the solutions might be extended to an arbitrary degree which makes the present method not only different from the others in the literature but also quite efficient. The method is applied to special Bagley-Torvik and Diethelm fractional differential equations as well as a more general fractional differential equation.

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 solution of the Bagley-Torvik equation by hybridizable discontinuous Galerkin methods

In this paper, we introduce a hybridizable discontinuous Galerkin method for numerically solving a boundary value problem associated with the Bagley-Torvik equation that arises in the study of the motion of a plate immersed in a Newtonian fluid. One of the main features of these methods is that they are efficiently implementable since it is possible to eliminate all internal degrees of freedom and obtain a global linear system that only involves unknowns at the element interfaces. We display the results of a series of numerical experiments to ascertain the performance of the method.

Approximate analytic solutions of the modified Kawahara equation with homotopy analysis method

In this paper, we applied the homotopy analysis method (HAM) to solve the modified Kawahara equation. Numerical results demonstrate that the methods provide efficient approaches to solving the modified Kawahara equation. It is shown that the method, with the help of symbolic computation, is very effective and powerful for discrete nonlinear evolution equations in mathematical physics.

Computational Solution of a Fractional Integro-Differential Equation

Although differential transform method (DTM) is a highly efficient technique in the approximate analytical solutions of fractional differential equations, applicability of this method to the system of fractional integro-differential equations in higher dimensions has not been studied in detail in the literature. The major goal of this paper is to investigate the applicability of this method to the system of two-dimensional fractional integral equations, in particular to the two-dimensional fractional integro-Volterra equations. We deal with two different types of systems of fractional integral equations having some initial conditions. Computational results indicate that the results obtained by DTM are quite close to the exact solutions, which proves the power of DTM in the solutions of these sorts of systems of fractional integral equations.

An efficient computer application of the sinc-Galerkin approximation for nonlinear boundary value problems

A powerful technique based on the sinc-Galerkin method is presented for obtaining numerical solutions of second-order nonlinear Dirichlet-type boundary value problems (BVPs). The method is based on approximating functions and their derivatives by using the Whittaker cardinal function. Without any numerical integration, the differential equation is reduced to a system of algebraic equations via new accurate explicit approximations of the inner products; therefore, the evaluation is based on solving a matrix system. The solution is obtained by constructing the nonlinear (or linear) matrix system using Maple and the accuracy is compared with the Newton method. The main aspect of the technique presented here is that the obtained solution is valid for various boundary conditions in both linear and nonlinear equations and it is not affected by any singularities that can occur in variable coefficients or a nonlinear part of the equation. This is a powerful side of the method when being compared to other models.

Efficient Variational Approaches for Deformable Registration of Images

Dirichlet, anisotropic, and Huber regularization terms are presented for efficient registration of deformable images. Image registration, an ill-posed optimization problem, is solved using a gradient-descent-based method and some fundamental theorems in calculus of variations. Euler-Lagrange equations with homogeneous Neumann boundary conditions are obtained. These equations are discretized by multigrid and finite difference numerical techniques. The method is applied to the registration of brain MR images of size 65 × 65. Computational results indicate that the presented method is quite fast and efficient in the registration of deformable medical images.