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1.
This paper presents approximate analytical solutions for systems of fractional differential equations using the differential transform method. The fractional derivatives are described in the Caputo sense. The application of differential transform method, developed for differential equations of integer order, is extended to derive approximate analytical solutions of systems of fractional differential equations. The solutions of our model equations are calculated in the form of convergent series with easily computable components. Some examples are solved as illustrations, using symbolic computation. The numerical results show that the approach is easy to implement and accurate when applied to systems of fractional differential equations. The method introduces a promising tool for solving many linear and nonlinear fractional differential equations. 相似文献
2.
The article, being a continuation of the first one [A.A. Kilbas and J.J. Trujillo (2001). Differential equations of fractional order. Methods, results and problems, I. Applicable Analysis , 78 (1-2), 153-192.], deals with the so-called differential equations of fractional order in which an unknown function is contained under the operation of a derivative of fractional order. The methods and the results in the theory of such fractional differential equations are presented including the Dirichlet-type problem for ordinary fractional differential equations, studying such equations in spaces of generalized functions, partial fractional differential equations and more general abstract equations, and treatment of numerical methods for ordinary and partial fractional differential equations. Problems and new trends of research are discussed. 相似文献
3.
The fractional stochastic differential equations have wide applications in various fields of science and engineering. This paper addresses the issue of existence of mild solutions for a class of fractional stochastic differential equations with impulses in Hilbert spaces. Using fractional calculations, fixed point technique, stochastic analysis theory and methods adopted directly from deterministic fractional equations, new set of sufficient conditions are formulated and proved for the existence of mild solutions for the fractional impulsive stochastic differential equation with infinite delay. Further, we study the existence of solutions for fractional stochastic semilinear differential equations with nonlocal conditions. Examples are provided to illustrate the obtained theory. 相似文献
4.
Exact solutions of nonlinear time fractional partial differential equations by sub‐equation method
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Ahmet Bekir Esin Aksoy Adem C. Cevikel 《Mathematical Methods in the Applied Sciences》2015,38(13):2779-2784
In this article, the sub‐equation method is presented for finding the exact solutions of a nonlinear fractional partial differential equations. For this, the fractional complex transformation method has been used to convert fractional‐order partial differential equation to ordinary differential equation. The fractional derivatives are described in Jumarie's the modified Riemann–Liouville sense. We apply to this method for the nonlinear time fractional differential equations. With the aid of symbolic computation, a variety of exact solutions for them are obtained. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
5.
Hongmei Zhang Fawang Liu 《高等学校计算数学学报(英文版)》2007,16(2):181-192
In this paper, the space-time Riesz fractional partial differential equations with periodic conditions are considered. The equations are obtained from the integral partial differential equation by replacing the time derivative with a Caputo fractional derivative and the space derivative with Riesz potential. The fundamental solutions of the space Riesz fractional partial differential equation (SRFPDE) and the space-time Riesz fractional partial differential equation (STRFPDE) are discussed, respectively. Using methods of Fourier series expansion and Laplace transform, we derive the explicit expressions of the fundamental solutions for the SRFPDE and the STRFPDE, respectively. 相似文献
6.
Zaid M. Odibat Sunil Kumar Nabil Shawagfeh Ahmed Alsaedi Tasawar Hayat 《Mathematical Methods in the Applied Sciences》2017,40(1):40-48
This paper deals with constructing generalized ‘fractional’ power series representation for solutions of fractional order differential equations. We present a brief review of generalized Taylor's series and generalized differential transform methods. Then, we study the convergence of fractional power series. Our emphasis is to address the sufficient condition for convergence and to estimate the truncated error. Numerical simulations are performed to estimate maximum absolute truncated error when the generalized differential transform method is used to solve non‐linear differential equations of fractional order. The study highlights the power of the generalized differential transform method as a tool in obtaining fractional power series solutions for differential equations of fractional order. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
7.
Xinxiu Li 《Communications in Nonlinear Science & Numerical Simulation》2012,17(10):3934-3946
Physical processes with memory and hereditary properties can be best described by fractional differential equations due to the memory effect of fractional derivatives. For that reason reliable and efficient techniques for the solution of fractional differential equations are needed. Our aim is to generalize the wavelet collocation method to fractional differential equations using cubic B-spline wavelet. Analytical expressions of fractional derivatives in Caputo sense for cubic B-spline functions are presented. The main characteristic of the approach is that it converts such problems into a system of algebraic equations which is suitable for computer programming. It not only simplifies the problem but also speeds up the computation. Numerical results demonstrate the validity and applicability of the method to solve fractional differential equation. 相似文献
8.
9.
Fulai Chen 《Nonlinear Analysis: Real World Applications》2012,13(1):287-298
We present some results for the global attractivity of solutions for fractional differential equations involving Riemann-Liouville fractional calculus. The results are obtained by employing Krasnoselskii’s fixed point theorem. Similar results for fractional differential equations involving Caputo fractional derivative are also obtained by using the classical Schauder’s fixed point theorem. Several examples are given to illustrate our main results. 相似文献
10.
In this article, we implement relatively new analytical techniques, the variational iteration method and the Adomian decomposition method, for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in the Caputo sense. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. Numerical results show that the two approaches are easy to implement and accurate when applied to partial differential equations of fractional order. 相似文献
11.
Safar Irandoust-Pakchin Somayeh Abdi-Mazraeh Ali Khani 《Computational Mathematics and Mathematical Physics》2017,57(12):2047-2056
In this paper, a variable-order fractional derivative nonlinear cable equation is considered. It is commonly accepted that fractional differential equations play an important role in the explanation of many physical phenomena. For this reason we need a reliable and efficient technique for the solution of fractional differential equations. This paper deals with the numerical solution of class of fractional partial differential equation with variable coefficient of fractional differential equation in various continues functions of spatial and time orders. Our main aim is to generalize the Chebyshev cardinal operational matrix to the fractional calculus. Finally, illustrative examples are included to demonstrate the validity and applicability of the presented technique. 相似文献
12.
Fahimeh Saberi Zafarghandi Maryam Mohammadi Robert Schaback 《Mathematical Methods in the Applied Sciences》2019,42(11):3877-3899
The paper provides the fractional integrals and derivatives of the Riemann‐Liouville and Caputo type for the five kinds of radial basis functions, including the Powers, Gaussian, Multiquadric, Matérn, and Thin‐plate splines, in one dimension. It allows to use high‐order numerical methods for solving fractional differential equations. The results are tested by solving two test problems. The first test case focuses on the discretization of the fractional differential operator while the second considers the solution of a fractional order differential equation. 相似文献
13.
Yuanguo Zhu 《Mathematical Methods in the Applied Sciences》2015,38(15):3359-3368
The concept of uncertain fractional differential equation is introduced, and solutions of several uncertain fractional differential equations are presented. This kind of equation is a counterpart of stochastic fractional differential equation. By the proposed concept, an interest rate model is considered, and the price of a zero‐coupon bond is obtained. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
14.
We analyze self-similar solutions to a nonlinear fractional diffusion equation and fractional Burgers/Korteweg–deVries equation in one spatial variable. By using Lie-group scaling transformation, we determined the similarity solutions. After the introduction of the similarity variables, both problems are reduced to ordinary nonlinear fractional differential equations. In two special cases exact solutions to the ordinary fractional differential equation, which is derived from the diffusion equation, are presented. In several other cases the ordinary fractional differential equations are solved numerically, for several values of governing parameters. In formulating the numerical procedure, we use special representation of a fractional derivative that is recently obtained. 相似文献
15.
Hang Xu 《Communications in Nonlinear Science & Numerical Simulation》2009,14(5):1978-1983
In this paper, we apply the homotopy analysis method (HAM) to solve the fractional Volterra’s model for population growth of a species in a closed system. This technique is extended to give solutions for nonlinear fractional integro–differential equations. The whole HAM solution procedure for nonlinear fractional differential equations is established. Further, the accurate analytical approximations are obtained for the first time, which are valid and convergent for all time t. This indicates the validity and great potential of the homotopy analysis method for solving nonlinear fractional integro–differential equations. 相似文献
16.
In this paper, we consider the analytical solutions of multi-term time–space fractional advection–diffusion equations with mixed boundary conditions on a finite domain. The technique of spectral representation of the fractional Laplacian operator is used to convert the multi-term time–space fractional advection–diffusion equations into multi-term time fractional ordinary differential equations. By applying Luchko’s theorem to the resulting fractional ordinary differential equations, the desired analytical solutions are obtained. Our results are applied to derive the analytical solutions of some special cases to demonstrate their practical applications. 相似文献
17.
In this paper, a novel approach is proposed for generating multi-wing chaotic attractors from the fractional linear differential system via nonlinear state feedback controller equipped with a duality-symmetric multi-segment quadratic function. The main idea is to design a proper nonlinear state feedback controller by using four construction criterions from a fundamental fractional differential nominal linear system, so that the controlled fractional differential system can generate multi-wing chaotic attractors. It is the first time in the literature to report the multi-wing chaotic attractors from an uncoupled fractional differential system. Furthermore, some basic dynamical analysis and numerical simulations are also given, confirming the effectiveness of the proposed method. 相似文献
18.
Stability analysis of nonlinear fractional differential systems has been an open problem since the 1990s of the last century. Apparently, Lyapunov’s second method seems to be invalid for nonlinear fractional differential systems (equations). In this paper, we are concerned with this open problem and have solved it partly. Based on Lyapunov’s second method, a novel stability criterion for a class of nonlinear fractional differential system is derived. Our result is simple, global and theoretically rigorous. The conditions to guarantee the stability of the nonlinear fractional differential system are convenient for testing. Compared with the stability criteria in the literature, our criterion is straightforward and suitable for application. Several examples are provided to illustrate the applications of our result. 相似文献
19.
S. Saha Ray 《Computational Mathematics and Mathematical Physics》2013,53(12):1870-1881
In this paper, the modified fractional reduced differential transform method (MFRDTM) has been proposed and it is implemented for solving fractional KdV (Korteweg-de Vries) equations. The fractional derivatives are described in the Caputo sense. In this paper, the reduced differential transform method is modified to be easily employed to solve wide kinds of nonlinear fractional differential equations. In this new approach, the nonlinear term is replaced by its Adomian polynomials. Thus the nonlinear initial-value problem can be easily solved with less computational effort. In order to show the power and effectiveness of the present modified method and to illustrate the pertinent features of the solutions, several fractional KdV equations with different types of nonlinearities are considered. The results reveal that the proposed method is very effective and simple for obtaining approximate solutions of fractional KdV equations. 相似文献
20.
Yi‐Fei Pu Patrick Siarry Ji‐Liu Zhou Ni Zhang 《Mathematical Methods in the Applied Sciences》2014,37(12):1784-1806
Traditional integer‐order partial differential equation based image denoising approach can easily lead edge and complex texture detail blur, thus its denoising effect for texture image is always not well. To solve the problem, we propose to implement a fractional partial differential equation (FPDE) based denoising model for texture image by applying a novel mathematical method—fractional calculus to image processing from the view of system evolution. Previous studies show that fractional calculus has some unique properties that it can nonlinearly enhance complex texture detail in digital image processing, which is obvious different with integer‐order differential calculus. The goal of the modeling is to overcome the problems of the existed denoising approaches by utilizing the aforementioned properties of fractional differential calculus. Using classic definition and property of fractional differential calculus, we extend integer‐order steepest descent approach to fractional field to implement fractional steepest descent approach. Then, based on the earlier fractional formulas, a FPDE based multiscale denoising model for texture image is proposed and further analyze optimal parameters value for FPDE based denoising model. The experimental results prove that the ability for preserving high‐frequency edge and complex texture information of the proposed fractional denoising model are obviously superior to traditional integral based algorithms, as for texture detail rich images. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献