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Lie group classifications and exact solutions for time-fractional Burgers equation  [PDF]
Guo-cheng Wu
Mathematics , 2010, DOI: 10.1088/0253-6102/55/6/23
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.
Note on the Convergence Analysis of Homotopy Perturbation Method for Fractional Partial Differential Equations  [PDF]
Asma Ali Elbeleze,Adem K?l??man,Bachok M. Taib
Abstract and Applied Analysis , 2014, DOI: 10.1155/2014/803902
Abstract: We apply the homotopy perturbation method to obtain the solution of partial differential equations of fractional order. This method is powerful tool to find exact and approximate solution of many linear and nonlinear partial differential equations of fractional order. Convergence of the method is proved and the convergence analysis is reliable enough to estimate the maximum absolute truncated error of the series solution. The fractional derivatives are described in the Caputo sense. Some examples are presented to verify convergence hypothesis and simplicity of the method. 1. Introduction Recently, the partial differential equations of fractional order have attracted much attention. This is mostly due to their frequent appearance in many applications in fluid mechanics, viscoelastic, biology, engineering, and physics [1, 2]. Most of partial differential equations of fractional order do not have exact analytical solution, so approximations and numerical techniques must be used. Some of these methods are series solution methods which include Adomain decomposition method [3], homotopy analysis method [4, 5], variationalmiteration method [6], and homotopy perturbation method [7–9]. The homotopy perturbation method [10] proposed by He in 1998. This method is useful tool for obtaining exact and approximate solution of linear and nonlinear partial differential equations of fractional order. There is no need for a small parameter or linearization, the solution procedure is very simple, and only few iterations lead to high accurate solutions which are valid for the all solution domains. The solution is expressed as the summation of an infinite series which is supposed to be convergent to the exact solution. This method has been used to solve effectively, easily, and accurately many types of fractional equations of linear and nonlinear problems with approximations. For example, [11] applied HPM to solve a class of initialboundary value problems of fractional partial differential equations over finite domain. [12] used HPM for solving the Klein-Gordon partial differential equations of fractional order. Furthermore, many authors applied HPM for solving and investigating linear and nonlinear partial differential equations of fractional ordering; see [13, 14]. For more details about homotopy perturbation method and its applications, we refer to [15, 16]. Our aim in this study is to extend the applications of HPM to obtain approximate solution of some partial differential equations of fractional order such as Burgers’ equation of fractional order and fractional
Solution of Fractional Partial Differential Equations in Fluid Mechanics by Extension of Some Iterative Method  [PDF]
A. A. Hemeda
Abstract and Applied Analysis , 2013, DOI: 10.1155/2013/717540
Abstract: An extension of the so-called new iterative method (NIM) has been used to handle linear and nonlinear fractional partial differential equations. The main property of the method lies in its flexibility and ability to solve nonlinear equations accurately and conveniently. Therefore, a general framework of the NIM is presented for analytical treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are described in the Caputo sense. Numerical illustrations that include the fractional wave equation, fractional Burgers equation, fractional KdV equation, fractional Klein-Gordon equation, and fractional Boussinesq-like equation are investigated to show the pertinent features of the technique. Comparison of the results obtained by the NIM with those obtained by both Adomian decomposition method (ADM) and the variational iteration method (VIM) reveals that the NIM is very effective and convenient. The basic idea described in this paper is expected to be further employed to solve other similar linear and nonlinear problems in fractional calculus. 1. Introduction Recent advances of fractional differential equations are stimulated by new examples of applications in fluid mechanics, viscoelasticity, mathematical biology, electrochemistry, and physics. For example, the nonlinear oscillation of earthquake can be modeled with fractional derivatives [1], and the fluid-dynamic traffic model with fractional derivatives [2] can eliminate the deficiency arising from the assumption of continuum traffic flow. Based on experimental data fractional partial differential equations for seepage flow in porous media are suggested in [3], and differential equations with fractional order have recently proved to be valuable tools to the modeling of many physical phenomena [4]. Different fractional partial differential equations have been studied and solved including the space-time-fractional diffusion-wave equation [5–7], the fractional advection-dispersion equation [8, 9], the fractional telegraph equation [10], the fractional KdV equation [11], and the linear inhomogeneous fractional partial differential equations [12]. The NIM [13–15] is a suitable approach to provide analytical approximation to linear and nonlinear problems and it is particularly valuable as tool for scientists and applied mathematicians, because it provides immediate and visible symbolic terms of analytical solutions, as well as numerical approximate solutions to both linear and nonlinear differential equations without linearization or discretization. The NIM, proposed
Fractional Method of Characteristics for Fractional Partial Differential Equations  [PDF]
Guo-cheng Wu
Physics , 2010,
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.
A Reliable Treatment of Homotopy Perturbation Method for Solving the Nonlinear Klein-Gordon Equation of Arbitrary (Fractional) Orders
A. M. A. El-Sayed,A. Elsaid,D. Hammad
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/581481
Abstract: The reliable treatment of homotopy perturbation method (HPM) is applied to solve the Klein-Gordon partial differential equation of arbitrary (fractional) orders. This algorithm overcomes the difficulty that arises in calculating complicated integrals when solving nonlinear equations. Some numerical examples are presented to illustrate the efficiency of this technique.
Series Solutions of Time-Fractional PDEs by Homotopy Analysis Method  [PDF]
O. Abdulaziz,I. Hashim,A. Saif
International Journal of Differential Equations , 2008, DOI: 10.1155/2008/686512
Abstract: The homotopy analysis method (HAM) is applied to solve linear and nonlinear fractional partial differential equations (fPDEs). The fractional derivatives are described by Caputo's sense. Series solutions of the fPDEs are obtained. A convergence theorem for the series solution is also given. The test examples, which include a variable coefficient, inhomogeneous and hyperbolic-type equations, demonstrate the capability of HAM for nonlinear fPDEs.
Using Homo-Separation of Variables for Solving Systems of Nonlinear Fractional Partial Differential Equations  [PDF]
Abdolamir Karbalaie,Hamed Hamid Muhammed,Bjorn-Erik Erlandsson
International Journal of Mathematics and Mathematical Sciences , 2013, DOI: 10.1155/2013/421378
Abstract: A new method proposed and coined by the authors as the homo-separation of variables method is utilized to solve systems of linear and nonlinear fractional partial differential equations (FPDEs). The new method is a combination of two well-established mathematical methods, namely, the homotopy perturbation method (HPM) and the separation of variables method. When compared to existing analytical and numerical methods, the method resulting from our approach shows that it is capable of simplifying the target problem at hand and reducing the computational load that is required to solve it, considerably. The efficiency and usefulness of this new general-purpose method is verified by several examples, where different systems of linear and nonlinear FPDEs are solved. 1. Introduction In the recent years, it has turned out that many phenomena in various technical and scientific fields can be described very successfully by using fractional calculus. In particular, fractional calculus can be employed to solve many problems within the biomedical research field and get better results. Such a practical application of fractional order models is to use these models to improve the behavior and efficiency of bioelectrodes. The importance of this application is based on the fact that bioelectrodes are usually needed to be used for all types of biopotential recording and signal measurement purposes such as electroencephalography (EEG), electrocardiography (ECG), and electromyography (EMG) [1–3]. Another promising biomedical application field is proposed by Arafa et al. [4] where a fractional-order model of HIV-1 infection of CD4 T cells is introduced. Other examples of applications of fractional calculus in life sciences and technology can be found in botanics [5], biology [6], rheology [7], and elastography [8]. Numerous analytical methods have been presented in the literature to solve FPDEs, such as the fractional Greens function method [9], the Fourier transform method [2], the Sumudu transform method [10], the Laplace transform method, and the Mellin transform method [11]. Some numerical methods have also widely been used to solve systems of FPDEs, such as the variational iteration method [12], the Adomian decomposition method [2], the homotopy perturbation method [13] and the homotopy analysis method [14]. Some of these methods use specific transformations and others give the solution as a series which converges to the exact solution. In addition, some numerical methods use a combination of utilizing specific transformations and obtaining series which converge to the
Computing the ground and first excited states of the fractional Schrodinger equation in an infinite potential well  [PDF]
Siwei Duo,Yanzhi Zhang
Physics , 2014,
Abstract: In this paper, we numerically study the ground and first excited states of the fractional Schrodinger equation in an infinite potential well. Due to the non-locality of the fractional Laplacian, it is challenging to find the eigenvalues and eigenfunctions of the fractional Schrodinger equation either analytically or numerically. We first introduce a fractional gradient flow with discrete normalization and then discretize it by using the trapezoidal type quadrature rule in space and the semi-implicit Euler method in time. Our method can be used to compute the ground and first excited states not only in the linear cases but also in the nonlinear cases. Moreover, it can be generalized to solve the fractional partial differential equations (PDEs) with Riesz fractional derivatives in space. Our numerical results suggest that the eigenfunctions of the fractional Schrodinger equation in an infinite potential well are significantly different from those of the standard (non-fractional) Schrodinger equation. In addition, we find that the strong nonlocal interactions represented by the fractional Laplacian can lead to a large scattering of particles inside of the potential well. Compared to the ground states, the scattering of particles in the first excited states is larger. Furthermore, boundary layers emerge in the ground states and additionally inner layers exist in the first excited states of the fractional nonlinear Schrodinger equation.
B?cklund Transformation of Fractional Riccati Equation and Infinite Sequence Solutions of Nonlinear Fractional PDEs  [PDF]
Bin Lu
Abstract and Applied Analysis , 2014, DOI: 10.1155/2014/572052
Abstract: The B?cklund transformation of fractional Riccati equation with nonlinear superposition principle of solutions is employed to establish the infinite sequence solutions of nonlinear fractional partial differential equations in the sense of modified Riemann-Liouville derivative. To illustrate the reliability of the method, some examples are provided. 1. Introduction Recently, nonlinear fractional differential equations increasingly are used to describe nonlinear phenomena in fluid mechanics, biology, engineering, physics, and other areas of science [1–3]. Much efforts have been spent in recent years to develop various techniques to deal with fractional differential equations. However, for the nonlinear differential equations including fractional calculus, the analytical or numerical results are usually difficult to be obtained. It is therefore needed to find a proper method to solve the problem of nonlinear differential equations containing fractional calculus. In the past, several methods have been formulated, such as Adomian decomposition method [4, 5], variational iteration method [6, 7], homotopy perturbation method [8, 9], differential transform method [10, 11], and fractional subequation method [12–14]. S. Zhang and H.-Q. Zhang [12] first proposed a new direct method called fractional subequation method in solving nonlinear time fractional biological population model and ( )-dimensional space-time fractional Fokas equation, based on the homogeneous balance principle and Jumarie’s modified Riemann-Liouville derivative. In this paper, based on the B?cklund transformation technique and the known seed solutions, we will devise effective way for solving fractional partial differential equations. It will be shown that the use of the B?cklund transformation allows us to obtain new exact solutions from the known seed solutions. 2. B?cklund Transformation of the Fractional Riccati Equation and Nonlinear Superposition Principle Firstly, we give some definitions and properties of the modified Riemann-Liouville derivative [15] which are used in this paper. Assume that , denote a continuous (but not necessarily differentiable) function, and let denote a constant discretization span. Jumarie defined the fractional derivative in the limit form where This definition is close to the standard definition of the derivative (calculus for beginners), and as a direct result, the th derivative of a constant, , is zero. An alternative, which is strictly equivalent to (1) is as follows: Some properties of the fractional modified Riemann-Liouville derivative that were
Matrix approach to discrete fractional calculus II: partial fractional differential equations  [PDF]
Igor Podlubny,Aleksei V. Chechkin,Tomas Skovranek,YangQuan Chen,Blas M. Vinagre Jara
Physics , 2008, DOI: 10.1016/j.jcp.2009.01.014
Abstract: A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation. The suggested method is the development of Podlubny's matrix approach (Fractional Calculus and Applied Analysis, vol. 3, no. 4, 2000, 359--386). Four examples of numerical solution of fractional diffusion equation with various combinations of time/space fractional derivatives (integer/integer, fractional/integer, integer/fractional, and fractional/fractional) with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach. The fifth example illustrates that the method can be equally simply used for fractional differential equations with delays. A set of MATLAB routines for the implementation of the method as well as sample code used to solve the examples have been developed.
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