Search

Variable‐order fractional calculus operators can be used as a valuable tool to simulate many nonlinear models with a memory property in fractional calculus applications. In this paper, we developed a novel predictor–corrector method for solving numerically nonlinear differential equations with variable‐order Liouville–Caputo‐type fractional derivatives. First, we found an inversion integral formula that is equivalent to the studied problem, and then we used it to formulate our numerical algorithm. Numerical approximate solutions of some variable‐order fractional models have been presented to display the efficiency and accuracy of the proposed algorithm. The influence of the variable‐order on the dynamic behavior of the considered problems is described. [ABSTRACT FROM AUTHOR]

In this study, two predictor–corrector schemes are developed for the numerical solution of variable exponent fractional‐type integral equations. The proposed schemes considered a Volterra‐type integral equation and then employed Trapezoidal and Simpson's 1/3 rules to obtain numerically approximate solutions. Some test problems are examined to demonstrate the performance, validity, and effectiveness of the proposed schemes. Comparisons are made between the proposed schemes to highlight the accuracy and convergence of these techniques. As an application, the effect of the variable‐order exponent on the dynamics of the studied systems is discussed. [ABSTRACT FROM AUTHOR]

In this article, a powerful modification of the optimized decomposition method is presented for dealing with nonlinear fractional differential equations, where the fractional derivatives are described in the Caputo sense. The new modification, the Legendre‐based approach of the optimized decomposition method, is suggested by employing the Legendre orthogonal polynomials to construct an optimal decomposition of the series solutions. An illustrative comparison between the Legendre‐based approaches of the Adomian decomposition method and the optimized decomposition method is performed to highlight the features and the computational efficiency of the suggested approach. The resulting numerical approximations demonstrate that the approximate solutions of the proposed approach have better accuracy and convergence. [ABSTRACT FROM AUTHOR]

In this article, the most recent version of an optimal homotopy analysis method (HAM), called linearization‐based approach of HAM or simply LHAM, has been applied to obtain a numerical solution of one of the principal nonlinear fractional‐order hyperbolic problems known as the time‐fractional hyperbolic partial differential equation. Such method is constructed based on employing Taylor series linearization method in order to design an optimal auxiliary linear operator with its corresponding optimal initial guessing. These two optimum contributors will accelerate the convergence of series solutions for the problem at hand. Several numerical comparisons have revealed the efficiency of the proposed method in obtaining a numerical solution of the problem rather than that solution presented by using the standard HAM. All theoretical findings in this work have been verified numerically using MATLAB software package. [ABSTRACT FROM AUTHOR]

In this paper, a novel approach, namely, the linearization‐based approach of homotopy analysis method, to analytically treat non‐linear time‐fractional PDEs is proposed. The presented approach suggests a new optimized structure of the homotopy series solution based on a linear approximation of the non‐linear problem. A comparative study between the proposed approach and standard homotopy analysis approach is illustrated by solving two examples involving non‐linear time‐fractional parabolic PDEs. The performed numerical simulations demonstrate that the linearization‐based approach reduces the computational complexity and improves the performance of the homotopy analysis method. [ABSTRACT FROM AUTHOR]

In this paper, the approximate analytical solutions of Lotka-Volterra model with fractional derivative have been obtained by using hybrid analytic approach. This approach is amalgamation of homotopy analysis method, Laplace transform, and homotopy polynomials. First, we present an alternative framework of the method that can be used simply and effectively to handle nonlinear problems arising in several physical phenomena. Then, existence and uniqueness of solutions for the fractional Lotka-Volterra equations are discussed. We also carry out a detailed analysis on the stability of equilibrium. Further, we have derived the approximate solutions of predator and prey populations for different particular cases by using initial values. The numerical simulations of the result are depicted through different graphical representations showing that this hybrid analytic method is reliable and powerful method to solve linear and nonlinear fractional models arising in science and engineering. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper deals with constructing generalized 'fractional' power series representation for solutions of fractional order differential equations. We present a brief review of generalized Taylor's series and generalized differential transform methods. Then, we study the convergence of fractional power series. Our emphasis is to address the sufficient condition for convergence and to estimate the truncated error. Numerical simulations are performed to estimate maximum absolute truncated error when the generalized differential transform method is used to solve non-linear differential equations of fractional order. The study highlights the power of the generalized differential transform method as a tool in obtaining fractional power series solutions for differential equations of fractional order. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, a new adaption of homotopy analysis method is presented to handle nonlinear problems. The proposed approach is capable of reducing the size of calculations and easily overcome the difficulty arising in calculating complicated integrals. Furthermore, the homotopy polynomials that decompose the nonlinear term of the problem as a series of polynomials are introduced. Then, an algorithm of calculating such polynomials, which makes the solution procedure more straightforward and more effective, is constructed. Numerical examples are examined to highlight the significant features of the developed techniques. The algorithms described in this paper are expected to be further employed to solve nonlinear problems in mathematical physics. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]