Mehmet Ali Akinlar

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.

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.

Application of a finite element method for variational inequalities

In this paper we explore the application of a finite element method (FEM) to the inequality and Laplacian constrained variational optimization problems. First, we illustrate the connection between the optimization problem and elliptic variational inequalities; secondly, we prove the existence of the solution via the augmented Lagrangian multipliers method. A triangular type FEM is employed in the numerical calculations. Computational results indicate that the present finite element method is a highly efficient technique in these sorts of variational problems involving inequalities.AMS Subject Classification: 35J86, 26D10.

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.

Cancerous Lesion Detection from Nevoscope Skin Surface Images via Parametric Color Clustering

This paper describes a new approach to analyze the spectral information of the samples of skin tissue that are localized in the spatial plane of microscopic image for discrimination of three different skin cancerous lesion prognoses. First, a cancerous lesion image is segmented from the skin surface based on Otsu’s optimal histogram thresholding technique. This allows us to localize the abnormal area in the skin tissue that is affected most as compared to the surrounding cells that appear brighter in color. Color clusters of the segmented darker lesions are used to obtain the three-dimensional (3D) spectral distribution function in the (R, G, B) color space. The Maximum Likelihood (ML) parameter estimation is utilized for calculation of the mean vector and co-variance matrix of the Gaussian (or normal) density approximation of skin samples and with the Mahalonobis distance as similarity measure in the learning and the testing phases of the pattern recognition system.

Wavelet-Petrov-Galerkin Method for Numerical Solution of Boussinesq Equation

In this paper, we employ the Wavelet-Petrov-Galerkin method to obtain the numerical solutions of the nonlinear Boussinesq equation. Boussinesq equation has braod application areas at different branches of engineering and science including chemistry and physics. We first discretize the Boussinesq equation in terms of wavelet coefficients and scaling functions, secondly multiply the discrete equation with wavelet basis functions. Using connection coefficients we express the resulting equation as a matrix equation. One of the significant advantages of the present method is that it does not require a quadrature formula.

A computational method using multiresolution for volumetric data integration

In this paper, we present a new method for integration of 3-D medical data by utilizing the advantages of 3-D multiresolution analysis and techniques of variational calculus. We first express the data integration problem as a variational optimal control problem where we express the displacement field in terms of wavelet expansions and, secondly, we write the components of the displacement field in terms of wavelet coefficients. We solve this optimization problem with a blockwise descent algorithm. We demonstrate the registration of 3-D brain MR images in the size of 257×257×65 as an application of the present method. Experimental results indicate that the method can integrate 3-D MR images better than only variational or only wavelet-based methods.MSC: 68U10, 65D18, 65J05, 97N40.

Curvature-driven diffusion-based mathematical image registration models

This paper introduces several mathematical image registration models. Image registration, an ill-posed optimization problem, is formulated as the minimization of the sum of an image similarity metric and a regularization term. Curvature-driven diffusion-based techniques, in particular Perona-Malik, anisotropic diffusion, mean curvature motion (MCM), affine invariant MCM (AIMCM), are employed as a regularization term in this optimal control formulation. Adopting the steepest-descent marching with an artificial time t, Euler-Lagrange (EL) equations with homogeneous Neumann boundary conditions are obtained. These EL equations are approximately solved by the explicit Petrov-Galerkin scheme. The method is applied to the registration of brain MR images of size 257×257. Computational results indicate that all these regularization terms produce similarly good registration quality but that the cost associated with the AIMCM approach is, on average, less than that for the others.MSC:68U10, 65D18, 65J05, 97N40.