On the Solution of Local Fractional Differential Equations Using Local Fractional Laplace Variational Iteration Method
TL;DR: In this paper, the authors presented iteration formulae of a fractional space-time telegraph equation using the combination of fractional variational iteration method and local fractional Laplace transform.
Abstract: We present iteration formulae of a fractional space-time telegraph equation using the combination of fractional variational iteration method and local fractional Laplace transform.
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TL;DR: In this article, a new neural network approach was proposed to approximate the solution of fractional differential equations using a new approach of artificial neural network and the neural network was compared with the analytical solutions and the numerical simulations obtained through the Adams-Bashforth-Moulton method.
Abstract: In this paper, we approximate the solution of fractional differential equations using a new approach of artificial neural network. We consider fractional differential equations of variable-order with Mittag-Leffler kernel in Liouville–Caputo sense. With this new neural network approach, it is obtained an approximate solution of the fractional differential equation and this solution is optimized using the Levenberg–Marquardt algorithm. The neural network effectiveness and applicability were validated by solving different types of fractional differential equations, the Willamowski-Rossler oscillator and a multi-scroll system. The solution of the neural network was compared with the analytical solutions and the numerical simulations obtained through the Adams-Bashforth-Moulton method. To show the effectiveness of the proposed neural network different performance indices were calculated.
86 citations
TL;DR: In this article, the authors presented an analysis based on the first integral method in order to construct exact solutions of the nonlinear fractional partial differential equations (FPDE) described by beta-derivative.
Abstract: In this paper, we present an analysis based on the first integral method in order to construct exact solutions of the nonlinear fractional partial differential equations (FPDE) described by beta-derivative. A general scheme to find the approximated solutions of the nonlinear FPDE is showed. The results obtained showed that the first integral method is an efficient technique for analytic treatment of nonlinear beta-derivative FPDE.
80 citations
TL;DR: In this article, the exact solutions obtained for the space-time conformable generalized Hirota-Satsuma-coupled KdV equation and coupled mKdV equations using the Atangana conformable deri...
Abstract: In this paper, we present the exact solutions obtained for the space–time conformable generalized Hirota–Satsuma-coupled KdV equation and coupled mKdV equation using the Atangana’s conformable deri...
63 citations
Cites methods from "On the Solution of Local Fractional..."
...In general, we can use direct and indirect implementation techniques for numerical approximation of the fractional operator; several numerical and analytical methods have been developed to obtain analytical and approximated solutions for these fractional operators, for instance, the differential transform method [14–16], the variational iteration method [17], Laplace transforms [18, 19], the homotopy perturbation method [20–22], fractional sub-equation method [23,24], exponential rational function method [25], extended trial equation method [26], expfunctionmethod [27–30],meshlessmethodof lines [31], (G’/G)-expansionmethod [32], He’s polynomials [33], Taylor’s expansions [34], and among others....
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TL;DR: In this article, the fractional cable equation can be changed into a system of integro-differential equations, and a full discrete numerical method for solving the system is studied, where in time axis the discontinuous Galerkin finite element method is used, and in spacial axis the GFE scheme is adopted.
29 citations
TL;DR: In this article, exact solutions of fractional physical differential equations were obtained via the natural transform method, and the fractional derivatives were described in the Caputo sense. But the results illustrate the power, efficiency, simplicity, and reliability of the proposed method.
20 citations
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Book•
02 Mar 2006
TL;DR: In this article, the authors present a method for solving Fractional Differential Equations (DFE) using Integral Transform Methods for Explicit Solutions to FractionAL Differentially Equations.
Abstract: 1. Preliminaries. 2. Fractional Integrals and Fractional Derivatives. 3. Ordinary Fractional Differential Equations. Existence and Uniqueness Theorems. 4. Methods for Explicitly solving Fractional Differential Equations. 5. Integral Transform Methods for Explicit Solutions to Fractional Differential Equations. 6. Partial Fractional Differential Equations. 7. Sequential Linear Differential Equations of Fractional Order. 8. Further Applications of Fractional Models. Bibliography Subject Index
11,492 citations
Book•
19 May 1993
TL;DR: The Riemann-Liouville Fractional Integral Integral Calculus as discussed by the authors is a fractional integral integral calculus with integral integral components, and the Weyl fractional calculus has integral components.
Abstract: Historical Survey The Modern Approach The Riemann-Liouville Fractional Integral The Riemann-Liouville Fractional Calculus Fractional Differential Equations Further Results Associated with Fractional Differential Equations The Weyl Fractional Calculus Some Historical Arguments.
7,643 citations
TL;DR: In this paper, a variational iteration method for non-linear problems is proposed, where the problems are initially approximated with possible unknowns and a correction functional is constructed by a general Lagrange multiplier, which can be identified optimally via the variational theory.
Abstract: In this paper, a new kind of analytical technique for a non-linear problem called the variational iteration method is described and used to give approximate solutions for some well-known non-linear problems. In this method, the problems are initially approximated with possible unknowns. Then a correction functional is constructed by a general Lagrange multiplier, which can be identified optimally via the variational theory. Being different from the other non-linear analytical methods, such as perturbation methods, this method does not depend on small parameters, such that it can find wide application in non-linear problems without linearization or small perturbations. Comparison with Adomian’s decomposition method reveals that the approximate solutions obtained by the proposed method converge to its exact solution faster than those of Adomian’s method.
2,371 citations
01 Jan 1980
455 citations