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Guo-cheng Wu
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ABSTRACT: Lie group method provides an efficient tool to solve nonlinear partial differential equations. This paper suggests a fractional Lie group method for fractional partial differential equations. A time-fractional Burgers equation is used as an example to illustrate the effectiveness of the Lie group method and some classes of exact solutions are obtained. Comment: 9 pp, accepted
11/2010;
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ABSTRACT: With the modified Riemann-Liouville fractional derivative, a fractional Tu formula is presented to investigate generalized Hamilton structure of fractional soliton equations. The obtained results can be reduced to the classical Hamilton hierachy of ordinary calculus. Comment: 12 pp
11/2010;
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Guo-cheng Wu
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ABSTRACT: Lie group method provides an efficient tool to solve a differential equation. This paper suggests a fractional partner for fractional partial differential equations using a fractional characteristic method. A space-time fractional diffusion equation is used as an example to illustrate the effectiveness of the Lie group method. Comment: 5 pages,in press
07/2010;
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Guo-cheng Wu
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ABSTRACT: Recently, fractional differential equations have been investigated via the famous variational iteration method. However, all the previous works avoid the term of fractional derivative and handle them as a restricted variation. In order to overcome such shortcomings, a fractional variational iteration method is proposed. The Lagrange multipliers can be identified explicitly based on fractional variational theory. Comment: 12 pages, 1 figures
07/2010;
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Guo-cheng Wu
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ABSTRACT: The method of characteristics has played a very important role in mathematical physics. Preciously, it was used to solve the initial value problem for partial differential equations of first order. In this paper, we propose a fractional method of characteristics and use it to solve some fractional partial differential equations. Comment: 8 pages
07/2010;
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ABSTRACT: A fractional Adomian decomposition method for fractional nonlinear
differential equations is proposed. The iteration procedure is based on
Jumarie's fractional derivative. An example is given to elucidate the solution
procedure, and the results are compared with the exact solution, revealing high
accuracy and efficiency.
06/2010;
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Guo-cheng Wu
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ABSTRACT: Fractional variational approach has gained much attention in recent years. There are famous fractional derivatives such as Caputo derivative, Riesz derivative and Riemann-Liouville derivative. Several versions of fractional variational principles are proposed. However, it becomes difficult to apply the existing fractional variational theories to fractional differential models, due to the definitions of fractional variational derivatives which not only contain the left fractional derivatives but also appear right ones. In this paper, a new definition of fractional variational derivative is introduced by using a modified Riemann-Liouville derivative and the fractional Euler-Lagrange principle is established for fractional partial differential equations. Comment: 8 pages
06/2010;
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ABSTRACT: Woven fabric is manifestly not a continuum and therefore Darcy’s law or its modifications, or any other differential models are invalid theoretically. A differential-difference model for air transport in discontinuous media is introduced using conservation of mass, conservation of energy, and the equation of state in discrete space and continuous time, capillary pressure is obtained by dimensional analysis.
Chaos, Solitons & Fractals.
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ABSTRACT: The paper introduces the Menger–Urysohn mathematical theory of dimensions and Cantorian manifolds. It is shown that this topological theory is the basis of El Naschie’s E-infinity Cantorian spacetime theory.
Chaos, Solitons & Fractals.
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ABSTRACT: This paper proposes three standard variational iteration algorithms for solving differential equations, integro-differential equations, fractional differential equations, fractal differential equations, differential-difference equations and fractional/fractal differential-difference equations. The physical interpretations of the fractional calculus and the fractal derivative are given and an application to discrete lattice equations is discussed. The paper then examines the acceleration of some iteration formulae with particular emphasis being placed on the exponential Padé approximant that is suggested for solitary solutions and the sinusoidal Padé approximant that is usually used for periodic and compacton solutions. The paper points out that there may not be any physical meaning to the exact solutions of many nonlinear equations and stresses the importance of searching for approximate solutions that satisfy both the equations and the appropriate initial/boundary conditions. The variational iteration method is particularly suitable for solving this kind of problems. Approximate initial/boundary conditions and point boundary initial/conditions are also discussed, with the variational iteration method being capable of recovering the correct initial/boundary conditions and finding the solutions simultaneously. =============================================== Contents 1. Introduction 2. Variational Iteration Algorithm-I 3. Variational Iteration Algorithm-II 4. Variational Iteration Algorithm-III 5. Variational Iteration Algorithms for Ordinary Differential Equations and Partial Differential Equations 6. Variational Iteration Algorithms for Fractional Differential Equations 7. Physical Understanding of Fractional Calculus 8. Variational Iteration Algorithms for Fractal Differential Equations 9. Physical Understanding of Fractal Differential Equations 10. Variational Iteration Algorithms for Differential-difference Equations 11. Physical Understanding of Differential-difference Equations 12. Variational Iteration Algorithms for Fractal-difference Equations and Fractional-difference Equations 13. Series Solutions, Exponential Padé Approximant and Sinusoidal Padé Approximant 14. Approximate Solutions vs Exact Solutions 15. Approximate Initial/Boundary Conditions and Point Boundary Conditions 16. Conclusions http://www.nonlinearscience.com/
Ji-Huan He.
