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This study presents new soliton solutions for the nonlinear fractional twin-core couplers with Kerr law nonlinearity by employing the modifed extended tanh method with Riccati equation. The solutions are expressed in terms of some elementary functions including rational, trigonometric and hyperbolic types. Graphical demonstrations of the simulations are provided.

drtT, In this research a new fractional-order derivative is defined and applied to the fractional perturbed Chen–Lee–Liu nonlinear equation. Analytical soliton solutions are obtained by the modified exp(−(ξ ))-expansion function method (MEFM). Three new groups of bright soliton solutions, dark soliton solutions, periodic tan-type singular soliton solutions and rational singular soliton solutions are constructed with specific contains over the parameters of the nonlinear fractional perturbed Chen–Lee–Liu equation. Solutions are illustrated in 3D graphs, contour plots and 2D plots. Next, we compare our results with the results obtained previously for the nonlinear perturbed CCL. It is noticed that the solutions here introduced present some complex propagation features that appears for different time values in the 2D space graphs, these complex propagation phenomena are not observed in the integer-order nonlinear perturbed CCL equation or in the fractional case for other type of fractional derivatives previously introduced in the literature The fractionalorder derivative here introduced will be very useful for the description of some nonlinear dissipation, nonlinear dispersion and other complex effects in optical wave phenomena.

Systems of high-dimensional nonlinear ordinary differential equations play a significant role in Physics and applied sciences including big-data optimization, financial models, epidemic disease models. In this paper, we are concerned with numerical solutions of Bao’s system that is a 4-dimensional hyperchaotic system introduced by Bo-Cheng and Zhong (2008). We solve the Bao’s system with both the Crank-Nicolson and power series methods. Crank-Nicolson method is eventually evolved into a new system whose solution is presented in a quite neat algorithmic manner. By adding standard Brownian motion to each term in the model, we express the Bao’s system as a system of stochastic differential equations. We solve the stochastic system with an Euler-type approximate solution method. By adding noise and expressing time derivatives with Caputo-type fractional derivative, we study on synchronization and parameter estimation of the models. To the best of our knowledge, Bao’s system has not been numerically solved with the methods employed in this paper previously, and this paper considers fractional and stochastic Bao’s system for the first time in the history of research. Techniques employed by us in this paper may serve as a framework for solutions of many other systems of ordinary differential equations including Lorenz types and epidemic models.

A new local fractional-order derivative operator is introduced and the Lakshmanan–Porsezian–Daniel (LPD) model is interpreted via this operator. New analytical solutions to the LPD equation is presented by Jacobi elliptic functions and an anzätz method. The complex-valued LPD equation includes a nonlinear term which is considered from three diferent cases: Kerr, parabolic and anti-cubic law of nonlinearities. For each case, dark, bright, singular optical soliton solutions related with optical fbers are presented. Simulations representing behavior of these solutions for diferent parameter values are provided.

We are concerned with new form optical solitons to the non-linear Schrodinger equation (NLSE) by generalized anti-cubic non-linearity. Some new optical solitons containing dark, singular and periodic solitons in chirp-free and chirped form are constructed via modified extended tanh expansion method. These results are very useful in the field of optics. The obtained results are verified and also explain by drawing different graphs. Parametric conditions for the chirp-free and chirped shape soliton solutions are also mentioned. Computational results indicate the efficiency of the aforesaid analytical expansion scheme.

Applicability of undetermined coefficients methods to several fractional-stochastic models is investigated. These models are mostly generated by fractional-order derivative operators and include a fractional white noise term. Application of a polynomial chaos algorithm to stochastic Lotka-Volterra and Benney systems are also investigated. Fractional-stochastic equations considered in this paper are totally original systems which may serve as models for many scientific and engineering phenomena. It is pointed out that Galerkin type methods employed in this paper may be efficiently applied to fractional-order systems having uncertainty or a noise term.

We study on constructing new traveling wave solution of a system known as ion sound and Langmuir waves by using method of extended Jacobian elliptic function expansion and He’s semi-inverse technique. Some new solutions will be beneficial for researchers concerning with nonlinear physical phenomena. The proposed methods may serve as the framework for solutions of various equations in applied science. Graphical simulations are provided to illustrate the behavior of these solutions.

We present a new method based on unification of fictitious time integration (FTI) and group preserving (GP) methods. The GP method is applied in numerically discretized ordinary differential equations obtained from application of FTI method to a given partial differential equation (PDE). The algorithm is applied to hyperbolic telegraph equation and utilizes the Cayley transformation and the Pade approximations in the Minkowski space. It avoids unauthentic solutions and ghost fixed points which is one of the advantages of the present method over other related numerical methods in the literature. The technique is tested on three specific examples for various parameter values appearing in the telegraph equation and discretization steps. Such solutions of the telegraph equation are obtained first time in this paper. Illustrative figures are provided. Efficiency of the method is determined by an error analysis which is achieved by comparing numerical solutions with exact solutions.

In this paper we concern with asymptotic stability, synchronization control a nd numerical solution of incommensurate order fractional Shimizu-Morioka dynamical system. Firstly we prove the existence and uniqueness of the solutions via a new theorem. After finding steady-state points, we obtain necessary and sufficient co nditions for the asymptotic stability of the Shimizu-Morioka system. We also study the synchronization control where we employ master-slave synchronization scheme. Finally, employing Adams- Bashforth-Moulton's technique we solve the Shimizu-Morioka system numerically. To the best of our knowledge, there exist not any study about analysis of chaotic dynamics of fractional Shimizu-Morioka system in the literature. In this sense the present paper is going to be a totally new contribution and highly useful research for synthesis of a nonlinear system of fractional equations.

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.

In this paper, Homotopy Analysis Method (HAM) is applied to obtain approximate analytical solution of modified Kuramoto-Sivashinsky (KS) equation. HAM provides a simple way to adjust and control the convergence region of the series so- lution by introducing several parameters, namely, the auxiliary parameter, h, the auxiliary function, H(x;t), the initial guess, u0(x;t) and the auxiliary linear operator, L; as stated in (1). The obtained results show that HAM yields approximate analytical solutions which are quite close to the exact solution of KS equation, which proves the strength of HAM.

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.

In this paper we propose a new solution technique for numerical solution of fractional Benney equation, a fourth degree nonlinear fractional partial differential equation with broad range of ap plications. The method could be described as a hybrid technique which uses advantages of both wavelets and operational matrices. Having applied the present method, fractional Benney equation is converted into a matrix equation, which is easy to solve. To the best of our knowledge, the fractional Benney equation has not been solved with any numerical or analytical method in the literature. Solving this eq uation numerically and investigating the applicability of the wavelets on this problem is the main goal of this paper. Haar wavelets and Caputo type fractional derivatives are employed in the calculations. Computational results point the strength of Haar wavelets and feasibility of the present solution algorithm.

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.

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.

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.

A new method for parameter sensitivity analysis of Lorenz equations is presented. The sensitivity equations are derived based on the staggered methods. Experimental results indicate that it is possible to determine effects of parameters on model variables so that we can eliminate the less effective ones. Robustness can also be verified in some confidence intervals by simply looking at the corresponding phase portraits. This enables us to control the system. Although the stability properties of the Lorenz equations are studied extensively, to the best knowledge of the authors, the PSA of Lorenz equations has not been considered which is the main goal of this paper. WOS:000325000500001 Q2

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.

This paper describes a method that helps in detection of sun tanned or precancerous skin using gross-overall image segmentation and boundary tracing to localize three layers in human skin tissue. Skin cancer can become highly invasive and lethal if not treated at the earliest possible stage. The undertaken research herein aims to eliminate the need for a patient to undergo biopsy as a means of basic diagnosis for melanoma. When compared to the normal skin, the structure of abnormal skin has an irregular outer epidermal layer as well as the inner dermal layer. The current trend of non-invasive diagnosis analyses the entire mole. This paper detects skin cancer by processing the cross-section of the skin sample. There are three different skin samples considered to this end; namely, normal skin, sun tanned skin and precancerous skin. In this work, optimal histogram-based segmentation and boundary tracing are used as a means of classifying a sample tissue. Experimental results have shown that the Otsu thresholding is highly effective to assist separating the three layers of the skin sample leading to a highly robust discrimination of normal, sun tanned and melanoma type skin samples.

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.

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. WOS:000320393200001 Q2

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.

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.