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1.
Bäcklund transformation of fractional Riccati equation and its applications to the space–time FDEs 下载免费PDF全文
In this paper, the Bäcklund transformation of fractional Riccati equation is presented to establish traveling wave solutions for two nonlinear space–time fractional differential equations in the sense of modified Riemann–Liouville derivatives, namely, the space–time fractional generalized reaction duffing equation and the space–time fractional diffusion reaction equation with cubic nonlinearity. The proposed method is effective and convenient for solving nonlinear evolution equations with fractional order. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
2.
王明建 《数学的实践与认识》2006,36(7):382-386
通过对一般Riccati方程进行初等变换,使之变为特殊的Riccati方程,然后利用公式、观察实验,或利用二阶微分方程的特解,或利用一阶微分方程组的特解等方法,求得这些Riccati方程的特解. 相似文献
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4.
R Vasudevan 《Journal of Mathematical Analysis and Applications》1974,45(3):629-638
The usual Matrix Riccati equation can be related to the transport of a beam of particles through a medium of finite length with conservative type of interactions (R. Bellman, S. Ueno, and R. Vasudevan, On the matrix Riccati equation of transport processes, University of Southern California Technical Report No. 72-13). Varying the physical picture of the interactions in the medium, we arrive at extensions of the Riccati equations with higher order nonlinear terms. Using the physical picture to guide our intuition, we achieve a linearization of these generalized Riccati equations in this paper, and this procedure will assist numerical computation of the solution of such nonlinear equations in a substantial manner. 相似文献
5.
屈英 《数学的实践与认识》2011,41(7)
研究三阶中立型分布时滞微分方程(r(t)[x(t)+p(t)x(r(t))]″)′+∫_a~b q(t,ξ)f(x[g(t,ξ)])dσ(ξ)=0的振动性.利用广义Riccati变换和积分平均技巧,建立了保证此方程一切解振动或者收敛到零的若干新的充分条件. 相似文献
6.
In seeking a generalized Rodrigues' formula solution to second-order linear ordinary differential equations a new Riccati equation is encountered and solved, leading to a novel development of the theory of orthogonal polynomials. Additional benefits of this approach are the automatic attainment of second solutions, the solution of the inhomogeneous form, and a means of determining the eigenvalues of fundamental classes of second-order linear differential equations. The results of this analysis will be of interest to a broad spectrum of readers and should be of pedagogic as well as theoretical value. 相似文献
7.
运用广义Riccati变换给出时标上二阶非线性中立型动力学方程振动的充分条件,进一步研究了具扰动项的动力学方程解的性态.所得结论推广和改进了已知文献的部分结果. 相似文献
8.
《Communications in Nonlinear Science & Numerical Simulation》2014,19(11):4037-4041
It is known that the simplest equation method is applied for finding exact solutions of autonomous nonlinear differential equations. In this paper we extend this method for finding exact solutions of non-autonomous nonlinear differential equations (DEs). We applied the generalized approach to look for exact special solutions of three Painlevé equations. As ODE of lower order than Painlevé equations the Riccati equation is taken. The obtained exact special solutions are expressed in terms of the special functions defined by linear ODEs of the second order. 相似文献
9.
C. Tudor 《Mathematische Nachrichten》1990,145(1):135-149
We study the stochastic regulator problem in Hilbert spaces for systems governed by linear stochastic differential equations with retarded controls and with state and control dependent noise. We use integral Riccati equations and no reference to a Riccati differential equation or to the Ito formula is made. 相似文献
10.
Alexei V. Penskoi Pavel Winternitz 《Journal of Mathematical Analysis and Applications》2004,294(2):533-547
An ordinary differential equation is said to have a superposition formula if its general solution can be expressed as a function of a finite number of particular solution. Nonlinear ODE's with superposition formulas include matrix Riccati equations. Here we shall describe discretizations of Riccati equations that preserve the superposition formulas. The approach is general enough to include q-derivatives and standard discrete derivatives. 相似文献
11.
Jing Wang 《Applied mathematics and computation》2010,217(4):1652-1657
In this letter, a new Riccati equation expansion method is presented for constructing exact travelling-wave solutions of nonlinear partial differential equations. The main idea of this method is to take full advantage of the solutions of the Riccati equation to construct exact travelling-wave solutions of nonlinear partial differential equations. As a result, some more generalized solutions, which contain triangular periodic solutions, exp function solutions and the soliton-like solutions, are obtained. 相似文献
12.
An effective characterization is given for a class of generalized nonlinear diffusion equations with power law dependent terms. Further, a new auxiliary equation ansatz is derived. Consequently, new exact traveling wave trigonometric function, solitary-like and Weierstrass elliptic solutions to a subclass are obtained by means of an auxiliary equation method and a generalized Riccati equation expansion method. 相似文献
13.
范恩贵 《高校应用数学学报(英文版)》2001,16(2):149-155
Abstract. A Riccati equation involving a parameter and symbolic computation are used to uni-formly construct the different forms of travelling wave solutions for nonlinear evolution equa-tions. It is shown that the sign of the parameter can be applied in judging the existence of vari-ous forms of travelling wave solutions. An efficiency of this method is demonstrated on some e-quations,which include Burgers-Huxley equation,Caudrey-Dodd-Gibbon-Kawada equation,gen-eralized Benjamin-Bona-Mahony equation and generalized Fisher equation. 相似文献
14.
Ahmed Sebbar 《Milan Journal of Mathematics》2016,84(2):317-347
We give different linear and nonlinear differential relations on Jacobi theta functions with more emphasis on the nonlinear differential equation of the third order of Jacobi. We present different points of view with a special attention to the role played by the second order linear differential equations, and their link to the Riccati equation and the Schwarzian equation. We also study an identity for theta functions resulting from the action of certain infinite order differential operators. 相似文献
15.
In this paper the Hamiltonian matrix formulation of the Riccati equation is used to derive the reduced-order pure-slow and pure-fast matrix differential Riccati equations of singularly perturbed systems. These pure-slow and pure-fast matrix differential Riccati equations are obtained by decoupling the singularly perturbed matrix differential Riccati equation of dimension n1+n2 into the pure-slow regular matrix differential Riccati equation of dimension n1 and the pure-fast stiff matrix differential Riccati equation of dimension n2. A formula is derived that produces the solution of the original singularly perturbed matrix differential Riccati equation in terms of solutions of the pure-slow and pure-fast reduced-order matrix differential Riccati equations and solutions of two reduced-order initial value problems. In addition to its theoretical importance, the main result of this paper can also be used to implement optimal filtering and control schemes for singularly perturbed linear time-invariant systems independently in pure-slow and pure-fast time scales. 相似文献
16.
Generalized method and its application in the higher-order nonlinear Schrodinger equation in nonlinear optical fibres 总被引:8,自引:0,他引:8
A generalized method, which is called the generally projective Riccati equation method, is presented to find more exact solutions of nonlinear differential equations based upon a coupled Riccati equation. As an application of the method, we choose the higher-order nonlinear Schrodinger equation to illustrate the method. As a result more new exact travelling wave solutions are found which include bright soliton solutions, dark soliton solution, new solitary waves, periodic solutions and rational solutions. The new method can be extended to other nonlinear differential equations in mathematical physics. 相似文献
17.
We give a formula for the Lipschitz constant in Thompson's part metric of any order-preserving flow on the interior of a (possibly infinite dimensional) closed convex pointed cone. This shows that in the special case of order-preserving flows, a general characterization of the contraction rate in Thompson's part metric, given by Nussbaum, leads to an explicit formula. As an application, we show that the flow of the generalized Riccati equation arising in stochastic linear quadratic control is a local contraction on the cone of positive definite matrices and characterize its Lipschitz constant by a matrix inequality. We also show that the same flow is no longer a contraction in other invariant Finsler metrics on this cone, including the standard invariant Riemannian metric. This is motivated by a series of contraction properties concerning the standard Riccati equation, established by Bougerol, Liverani, Wojtkowski, Lawson, Lee and Lim: we show that some of these properties do, and that some other do not, carry over to the generalized Riccati equation. 相似文献
18.
Mohamed Hayek 《Applied mathematics and computation》2011,218(6):2407-2420
Based on the simplest equation method, we propose exact and traveling-wave solutions for a nonlinear convection-diffusion-reaction equation with power law nonlinearity. Such equation can be considered as a generalization of the Fisher equation and other well-known convection-diffusion-reaction equations. Two important cases are considered. The case of density-independent diffusion and the case of density-dependent diffusion. When the parameters of the equation are constant, the Bernoulli equation is used as the simplest equation. This leads to new traveling-wave solutions. Moreover, some wavefront solutions can be derived from the traveling-wave ones. The case of time-dependent velocity in the convection term is studied also. We derive exact solutions of the equations by using the Riccati equation as simplest equation. The exact and traveling-wave solutions presented in this paper can be used to explain many biological and physical phenomena. 相似文献
19.
Interval oscillation criteria are established for a second-order functional dynamic equation of Emden-Fowler type with oscillatory potential by applying Riccati and generalized Riccati techniques. The results represent further improvements on those given even for differential and difference equations. Some examples are considered to illustrate the main results. 相似文献
20.
线性等式约束系统广义Riccati代数方程的求解* 总被引:1,自引:0,他引:1
本文基于定常离散LQ控制问题的动力学方程、价值泛函及系统的约束方程,根据极大值原理,给出了线性等式约束系统下的广义Riccati方程,进而对上述方程进行了深入的探讨,并给出了相应的数值例题。 相似文献