Decomposition method for solving fractional Riccati differential equations

Momani, Shaher; Shawagfeh, Nabil

doi:10.1016/j.amc.2006.05.008

Cited by 171 publications

(96 citation statements)

References 17 publications

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“…According to (9), the governing equation can be deduced from the zero-order deformation (7). Define the vector…”

Section: Analysis Of the Homotopy Analysis Methodsmentioning

confidence: 99%

“…Author of [5] found an approximate solution of a nonlinear equation with Riemann-Liouville's fractional derivatives by He's variational iterational method. Several methods have been used to solve fractional partial differential equations, such as Adomian's decomposition method (ADM) [6,7], Fourier transform method [8], Laplace transform method [2,3,9], and so on. Some 0932-0784 / 10 / 1100-0935 $ 06.00 c 2010 Verlag der Zeitschrift für Naturforschung, Tübingen · http://znaturforsch.com fundamental works on various aspects of the fractional calculus are given by Abbasbandy [5], Al-Khaled and Momani [10], Caputo [11], Debnath [12], Diethelm et al [13], Jafari and Seifi [14,15], Hayat et al [16], Khan and Hayat [17], Kilbas and Trujillo [18], Kiryakova [19], Oldham and Spanier [20], Ray and Bera [21], Shawagfeh [22], Song and Zhang [23,24], Xu and Jie [25], Momani and his co-authors [26 -28], etc.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Key words: Nonlinear Differential-Difference Equations; Exp-Function Method; N-Soliton Solutions

Dehghan

Manafian

Saadatmandi

2010

Zeitschrift Für Naturforschung A

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In this paper, the homotopy analysis method is applied to solve linear fractional problems. Based on this method, a scheme is developed to obtain approximation solution of fractional wave, Burgers, Korteweg-de Vries (KdV), KdV-Burgers, and Klein-Gordon equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So the homotopy analysis method for partial differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.

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“…According to (9), the governing equation can be deduced from the zero-order deformation (7). Define the vector…”

Section: Analysis Of the Homotopy Analysis Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Key words: Nonlinear Differential-Difference Equations; Exp-Function Method; N-Soliton Solutions

Dehghan

Manafian

Saadatmandi

2010

Zeitschrift Für Naturforschung A

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“…In this paper, we establish a clear procedure for solving the proposed equations via HWCM; comparisons are made between HWCM and existing methods [13,20,24] with the exact solutions. The results reveal that the proposed method is very effective and simple and can be applied to other nonlinear problems.…”

Section: Introductionmentioning

confidence: 99%

“…[14]) much more general values of the order α appear in the equations and therefore one needs to consider numerical and analytical methods to solve differential equations of arbitrary order. Decomposition method is presented for solving fractional Riccati differential equations in [20].…”

Section: Introductionmentioning

confidence: 99%

Haar Wavelet Collocation Method for Solving Riccati and Fractional Riccati Differential Equations

Shiralashetti

Deshi

2016

BMSA

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Abstract. In this paper, numerical solutions of Riccati and fractional Riccati differential equations are obtained by the Haar wavelet collocation method. An operational matrix of integration based on the Haar wavelet is established, and the procedure for applying the matrix to solve these equations. The fundamental idea of Haar wavelet method is to convert the proposed differential equations into a group of non-linear algebraic equations. The accuracy of approximate solution can be further improved by increasing the level of resolution and an error analysis is computed. The examples are given to demonstrate the fast and flexibility of the method. The results obtained are in good agreement with the exact in comparison with existing ones and it is shown that the technique introduced here is robust, easy to apply and is not only enough accurate but also quite stable.

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“…The FDEs are increasingly used to model many problems in mathematical physics, including electromagnetic, fluid flow, diffusion, quantum mechanics, damping laws, viscoelasticity and other applications (Oldham and Spanier, 1974;Miller and Ross, 1993;Young, 1995;Mainardi, 1997;Luchko and Gorenflo, 1998;He, 1999;Podlubny, 1999;Caputo, 1967;Shawagfeh, 1999;Shawagfeh, 2002;Momani and Shawagfeh, 2002). Exact analytical solutions of FDEs are sometimes too complicated to be solved by conventional techniques due to the computational complexities of fractional calculus involving them.…”

Section: Introductionmentioning

confidence: 99%

An Expansion Iterative Technique for Handling Fractional Differential Equations Using Fractional Power Series Scheme

Abu-Gdairi¹,

Al‐Smadi²,

Gumah³

2015

Journal of Mathematics and Statistics

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In this study, we present a new analytical numerical technique for solving a class of time Fractional Differential Equations (FDEs) with variable coefficients based on the generalized Taylor series formula in the Caputo sense. This method provided the solution in the form of a rapidly convergent power series under a multiple fractional differentiability with easily computable components. An efficacious experiment is given to guarantee the procedure, to illustrate the theoretical statements of the present technique and to show its potentiality, generality and superiority for solving wide range of FDEs. The results reveal that the method is easy to implement, very effective, fully compatible with the complexity of such problems, straightforward and simple.

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Decomposition method for solving fractional Riccati differential equations

Cited by 171 publications

References 17 publications

Key words: Nonlinear Differential-Difference Equations; Exp-Function Method; N-Soliton Solutions

Key words: Nonlinear Differential-Difference Equations; Exp-Function Method; N-Soliton Solutions

Haar Wavelet Collocation Method for Solving Riccati and Fractional Riccati Differential Equations

An Expansion Iterative Technique for Handling Fractional Differential Equations Using Fractional Power Series Scheme

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