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 New Method for Parameter Sensitivity Analysis of Lorenz 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

Stability, Synchronization Control and Numerical Solution of Fractional Shimizu–Morioka Dynamical System

In this paper we concern with asymptotic stability, synchronization control and 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 conditions 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

Stability, Synchronization Control and Numerical Solution of Fractional Shimizu–Morioka Dynamical System

Abstract: In this paper we concern with asymptotic stability, synchronization control and 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 conditions 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

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

Numerical Solution of Fractional Benney Equation

Abstract: 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 applications. 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 equation 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

Numerical Solution of Fractional Benney 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 applications. 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 equation 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

NEW SOLUTIONS OF HYPERBOLIC TELEGRAPH EQUATION

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