Analytical Solution of a Dynamic System Containing Fractional Derivative of Order One-Half by Adomian Decomposition Method

Ray, S. Saha; Poddar, Banibrata; Bera, Rasajit Kumar

doi:10.1115/1.1839184

Cited by 62 publications

(32 citation statements)

References 43 publications

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“…This has also been obtained by Saha Ray, Poddar and Bera [19] for α = 1/2 using Adomian decomposition (with a slightly different selection for L, R, and N).…”

Section: Examplesmentioning

confidence: 90%

“…However, as mentioned in the introduction, it has also been used by several authors for fractional differential equations [17][18][19][20][21][22][23]. As it will be also used in this paper, it is presented in the followings.…”

Section: Adomian Decomposition Methodsmentioning

confidence: 99%

“…For a special case, Saha Ray, Poddar and Bera [19] have proven that the Adomian solution converges to the analytical solution. Recently, Hu, Luo and Lu [24] have shown that this is true for any linear fractional differential equation.…”

Section: Introductionmentioning

confidence: 99%

“…Essentially, it approximates the solution of a non-linear differential equation with a series of functions. The method 2 A c c e p t e d m a n u s c r i p t is getting into use for the solution of fractional differential equations [17][18][19][20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

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Efficient solution of a vibration equation involving fractional derivatives

Pálfalvi

2010

International Journal of Non-Linear Mechanics

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Fractional order (or, shortly, fractional) derivatives are used in viscoelasticity since the late 1980's, and they grow more and more popular nowadays.However, their efficient numerical calculation is nontrivial, because, unlike integer-order derivatives, they require evaluation of history integrals in every time step. Several authors tried to overcome this difficulty, either by simplifying these integrals or by avoiding them. In this paper, the Adomian decomposition method is applied on a fractionally damped mechanical oscillator for a sine excitation, and the analytical solution of the problem is found. Also, a series expansion is derived which proves very efficient for calculations of transients of fractional vibration systems. Numerical examples are included.

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“…This has also been obtained by Saha Ray, Poddar and Bera [19] for α = 1/2 using Adomian decomposition (with a slightly different selection for L, R, and N).…”

Section: Examplesmentioning

confidence: 90%

Section: Adomian Decomposition Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Efficient solution of a vibration equation involving fractional derivatives

Pálfalvi

2010

International Journal of Non-Linear Mechanics

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“…Moreover, we have the advantage of a single global method for solving ordinary or partial differential equations as well as many types of other equations. Recently, the solution of fractional differential equation has been obtained through ADM by the researchers [26][27][28]. The application of ADM for the solution of nonlinear fractional differential equations has also been established by Shawagfeh, Saha Ray and Bera [27,28].…”

Section: Introductionmentioning

confidence: 99%

Application of Adomian’s Decomposition Method for the Analytical Solution of Space Fractional Diffusion Equation

Safari¹,

Danesh²

2011

APM

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Spatially fractional order diffusion equations are generalizations of classical diffusion equations which are increasingly used in modeling practical super diffusive problems in fluid flow, finance and others areas of application. This paper presents the analytical solutions of the space fractional diffusion equations by Adomian's decomposition method (ADM). By using initial conditions, the explicit solutions of the equations have been presented in the closed form. Two examples, the first one is one-dimensional and the second one is two-dimensional fractional diffusion equation, are presented to show the application of the present techniques. The present method performs extremely well in terms of efficiency and simplicity.

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A novel method for travelling wave solutions of fractional Whitham–Broer–Kaup, fractional modified Boussinesq and fractional approximate long wave equations in shallow water

Ray

2014

Math Methods in App Sciences

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In this paper, the analytical approximate traveling wave solutions of Whitham-Broer-Kaup (WBK) equations, which contain blow-up solutions and periodic solutions, have been obtained by using the coupled fractional reduced differential transform method. By using this method, the solutions were calculated in the form of a generalized Taylor series with easily computable components. The convergence of the method as applied to the WBK equations is illustrated numerically as well as analytically. By using the present method, we can solve many linear and nonlinear coupled fractional differential equations. The results justify that the proposed method is also very efficient, effective and simple for obtaining approximate solutions of fractional coupled modified Boussinesq and fractional approximate long wave equations. Numerical solutions are presented graphically to show the reliability and efficiency of the method. Moreover, the results are compared with those obtained by the Adomian decomposition method (ADM) and variational iteration method (VIM), revealing that the present method is superior to others.where˛,ˇ.0 <˛,ˇÄ 1/ are the order of the Caputo fractional time derivatives respectively and t 0. In WBK equations (1.1), the field of horizontal velocity is represented by u D u. x, t/, v D v.x, t/, which is the height that deviate from equilibrium position of liquid, and the constants a, b are represented in different diffusion power [14] Brief description of fractional calculusThe fractional calculus involves different definitions of the fractional operators, namely Riemann-Liouville fractional derivative, Caputo derivative, Riesz derivative and Grunwald-Letnikov fractional derivative [3]. The fractional calculus has gained considerable importance during the past decades mainly because of its applications in diverse fields of science and engineering. For the purpose of this paper, the Caputo's definition of fractional derivative will be used, taking advantage of Caputo's approach that the initial conditions for fractional differential equations with Caputo's derivatives take on the traditional form as for integer-order differential equations. .ˇ ˛C 1/ ,ˇ>˛ 1 (2.2.3)Similar to integer-order differentiation, the Caputo's derivative is linear. 1353S. SAHA RAY where and ı are constants and satisfies the so-called Leibnitz's rule.if f . / is continuous in OE0, t and g. / has continuous derivatives of sufficient number of times in OE0, t.Lemma 2.3 Let Re.˛/ > 0 and let n D OERe.˛/ C 1 for˛… N 0 D f0, 1, 2, : : :g; n D˛for˛2 N 0 . If f .t/ 2 AC n OEa, b (the space of functions f .t/, which are absolutely continuous and possess continuous derivatives up to order n 1 on OEa, b/ or f .t/ 2 C n OEa, b (the space of functions f .t/, which are n times continuously differentiable on OEa, b/, thenaccording to the definition of the Caputo derivative equation (2.2.1), C Dt J˛f .t/ D J n ˛Dn J˛f .t/ D J n ˛J˛ n f .t/, by the property J˛ n f .t/ D D n J˛f .t/ D f .t/ Thus, we derive Equation (2.3.1). Again, according to the definiti...

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Analytical Solution of a Dynamic System Containing Fractional Derivative of Order One-Half by Adomian Decomposition Method

Cited by 62 publications

References 43 publications

Efficient solution of a vibration equation involving fractional derivatives

Efficient solution of a vibration equation involving fractional derivatives

Application of Adomian’s Decomposition Method for the Analytical Solution of Space Fractional Diffusion Equation

A novel method for travelling wave solutions of fractional Whitham–Broer–Kaup, fractional modified Boussinesq and fractional approximate long wave equations in shallow water

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