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1.
The purpose of this paper is to give sufficient conditions forall nontrivial solutions of the nonlinear differential equationx ' a(t)g(x) = 0 to be nonoscillatory. Here, g(x) satisfiesthe sign condition xg(x) > 0 if x 0, but is not assumedto be monotone increasing. This differential equation includesthe generalized Emden–Fowler equation as a special case.Our main result extends some nonoscillation theorems for thegeneralized Emden–Fowler equation. Proof is given by meansof some Liapunov functions and phase-plane analysis.  相似文献   

2.
The structural stability of constrained polynomial differentialsystems of the form a(x, y)x'+b(x, y)y'=f(x, y), c(x, y)x'+d(x,y)y'=g(x, y), under small perturbations of the coefficientsof the polynomial functions a, b, c, d, f and g is studied.These systems differ from ordinary differential equations at‘impasse points’ defined by adbc=0. Extensionsto this case of results for smooth constrained differentialsystems [7] and for ordinary polynomial differential systems[5] are achieved here. 1991 Mathematics Subject Classification34C35, 34D30.  相似文献   

3.
The relation between finite difference approximation and cubicspline solutions of a two-point boundary value problem for thedifferential equation y' +f(x)y'+g(x)y = r(x) has been consideredin a previous paper. The present paper extends the analysisto the integral equation formulation of the problem. It is shownthat an improvement in accuracy (local truncation error O(h6)rather than O(h4)) now results from a cubic spline approximationand that for the particular case f(x) 0 the resulting recurrencerelations have a form and accuracy similar to the well-knownNumerov formula. For this case also a formula with local truncationerror O(h8) is derived.  相似文献   

4.
We consider the observability of systems of the form = Ax +Nx, y = Fx, where A is a linear operator and N and F are nonlinear.We show that if the system is linearized about an equilibriumpoint xe and the linearized system is continuously initiallyobservable, then the nonlinear system is continuously initiallyobservable in some neighbourhood of xe. We then look at conditionsunder which solutions of the nonlinear system can be extendedfor all time and consider the problem of stabilizing the systemby feedback controls such that the solutions are eventuallyin the observability neighbourhood of xe. Finally, we applythese ideas to two systems: a wave equation and a diffusionequation with nonlinear perturbations and nonlinear observations.  相似文献   

5.
Existence of Periodic Solutions in Nonlinear Asymmetric Oscillations   总被引:1,自引:0,他引:1  
The existence of periodic solutions for the nonlinear asymmetricoscillator x' + x+ – rßx = h(t),(' =d/dt (is discussed, where , rß are positive constantssatisfying for some positive integer n N and h(t) L (0,2) is 2-periodic with x±= max {±x,0}. 2000 Mathematics Subject Classification34C10, 34C25.  相似文献   

6.
The main result ensures that the scalar problem x' = f(x),x(0) = x0, x'(0) = x1, has a nonconstant locally W2, 1 solutionif and only if there exists a nontrivial interval J such thatx0 J, for almost all y Jand Necessary and sufficient conditions for local and global uniquenessand for existence of periodic solutions are also established.  相似文献   

7.
On Hill's Equation with a Singular Complex-Valued Potential   总被引:3,自引:0,他引:3  
In this paper Hill's equation y' + qy = Ey, where q is a complex-valuedfunction with inverse square singularities, is studied. Resultson the dependence of solutions to initial value problems onthe parameter E and the initial point x0, on the structure ofthe conditional stability set, and on the asymptotic distributionof (semi-)periodic and Sturm-Liouville eigenvalues are obtained.It is proved that a certain subset of the set of Floquet solutionsis a line bundle on a certain analytic curve in C2. We establishnecessary and sufficient conditions for q to be algebro-geometric,that is, to be a stationary solution of some equation in theKorteweg-de Vries (KdV) hierarchy. To do this a distinctionbetween movable and immovable Dirichlet eigenvalues is employed.Finally, an example showing that the finite-band property doesnot imply that q is algebro-geometric is given. This is in contrastto the case where q is real and non-singular. 1991 MathematicsSubject Classification: 34L40, 14H60.  相似文献   

8.
On a model of viscoelastic rod in unilateral contact with a rigid wall   总被引:1,自引:0,他引:1  
** Corresponding author. Email: atanackovic{at}uns.ns.ac.yu We study translatory motion of a body to which a viscoelasticrod with the constitutive equation with fractional derivativesis attached. The body with a rod impacts against a rigid wall.It is shown that the problem is described with a coupled systemof differential equations having integer and fractional derivativeshaving the form x(2) = –f; f + af() = x + bx(), x(0) =0, x(1)(0) = 1. The unique solvability in S'+ is proved andinterpretation of solutions is given. Also, some a priori estimatesof the solution are given. In particular, we showed that restrictionson coefficients that follow from the second law of thermodynamicsimply that the velocity after the impact is smaller than thevelocity before the impact.  相似文献   

9.
We investigate the location and nature of the spectrum of thefourth-order self-adjoint equation (p0 y')'+(p1 y')'+qy=zwy subject to certain asymptotic assumptions on the coefficients.The main tools are the theory of asymptotic integration andthe Titchmarsh–Weyl M-matrix. Asymptotic integration yieldsasymptotic formulae for the solutions of the differential equationwhich are then used to derive properties of the M-matrix. Thecharacterisation of spectral properties in terms of the boundarybehaviour of M leads to the desired results.  相似文献   

10.
We consider the existence of unique absolutely continuous solutionsfor x' = p(t)f(x) + p(t)h(t), t 0, x(0) = 0, where p, f, andh are positive almost everywhere, but none of them needs becontinuous or monotone. Moreover, p and f can be unbounded aroundzero. Our uniqueness results are not based on assumptions onthe differences f(x) – f(y), as it is usual in most uniquenessresults, and they are new even when p, f, and h are continuous.  相似文献   

11.
* Presently at Deparment of Mathematics, Indian Institute of Technology, Madras, India. The optimum Runge-Kutta method of a particular order is theone whose truncation error is minimum. In this paper, we havederived optimum Runge-Kutta mehtods of 0(hm+4), 0(hm+5) and0(hm+6) for m = 0(1)8, which can be directly used for solvingthe second order differential equation yn = f(x, y, y'). Thesemethods are based on a transformation similar to that of Fehlbergand require two, three and four evaluations of f(x, y, y') respectively,for each step. The numercial solutions of one example obtainedwith these methods are given. It has been assumed that f(x,y, y')is sufficiently differentiable in the entire region ofintegration.  相似文献   

12.
New efficient embedded Runge-Kutta-Nystrom processes of orders8(6) and 12(10) are presented for the numerical solution ofthe special second-order differential equation y'(x) = f[x,y(x)]. Test results indicate their improved efficiency relativeto other RKN formulae in current use.  相似文献   

13.
We study non-negative solutions of the porous medium equationwith a source and a nonlinear flux boundary condition, ut =(um)xx + up in (0, ), x (0, T); – (um)x (0, t) = uq (0,t) for t (0, T); u (x, 0) = u0 (x) in (0, ), where m > 1,p, q > 0 are parameters. For every fixed m we prove thatthere are two critical curves in the (p, q-plane: (i) the criticalexistence curve, separating the region where every solutionis global from the region where there exist blowing-up solutions,and (ii) the Fujita curve, separating a region of parametersin which all solutions blow up from a region where both globalin time solutions and blowing-up solutions exist. In the caseof blow up we find the blow-up rates, the blow-up sets and theblow-up profiles, showing that there is a phenomenon of asymptoticsimplification. If 2q < p + m the asymptotics are governedby the source term. On the other hand, if 2q > p + m theevolution close to blow up is ruled by the boundary flux. If2q = p + m both terms are of the same order.  相似文献   

14.
On the Local and Superlinear Convergence of Quasi-Newton Methods   总被引:13,自引:0,他引:13  
This paper presents a local convergence analysis for severalwell-known quasi-Newton methods when used, without line searches,in an iteration of the form to solve for x* such that Fx* = 0. The basic idea behind theproofs is that under certain reasonable conditions on xo, Fand xo, the errors in the sequence of approximations {Hk} toF'(x*)–1 can be shown to be of bounded deterioration inthat these errors, while not ensured to decrease, can increaseonly in a controlled way. Despite the fact that Hk is not shownto approach F'(x*)–1, the methods considered, includingthose based on the single-rank Broyden and double-rank Davidon-Fletcher-Powellformulae, generate locally Q-superlinearly convergent sequences{xk}.  相似文献   

15.
Let C be a genus 2 algebraic curve defined by an equation ofthe form y2 = x(x2 – 1)(xa)(x – 1/a). Asis well known, the five accessory parameters for such an equationcan all be expressed in terms of a and the accessory parameter b corresponding to a. The main result of the paper is thatif a' = 1 – a2, which in general yields a non-isomorphiccurve C', then b'a'(a'2 – 1) = – – ba(a2– 1). This is proven by it being shown how the uniformizing functionfrom the unit disk to C' can be explicitly described in termsof the uniformizing function for C.  相似文献   

16.
Two-point Pad? approximants are used to calculate tight upperand lower bounds on the quantity <?, f> associated withKirkwood-Riseman integral equations (1+yL)?=f, which arise inthe diffusion theory of flexible macromolecules. The self-adjointoperator L is an integral operator on –1 x 1, with weaklysingular kernel |xx'|–?, and the two specificcases (i) f = 1, (ii) f = x2 are studied. In case (i) directbounds on <?, 1> are obtained; this quantity is inverselyproportional to the translational diffusion constant. In case(ii) bounds on <?, 1 > are found by a new technique involvingcombinations of bounds for the three cases f = 1, f = x2 andf = bx2?b–1. Various types of Pade and related approximantsare compared, using the information <f, Lnf>, n = –2,–1, 0, 1, 2, 3 and (an upper bound on L) for severalvalues of the positive parameter y. Pad?-approximant-generating trial vectors are investigated anda convergence theorem is established. The vector consistingof an optimum linear combination of L–1f, f and Lf isfound to be an accurate approximation to a numerical solutionin case (ii), for all values of y and x. Specific analyticalexpressions are derived for the approximate solutions.  相似文献   

17.
Four extended generalized hexagons related to the simple groupsG2(2)', PSU4(3), HJ and Suz are characterized by the conditionthat any triple of points {x, y, z} is a clique of the pointgraph not in a circle of the extended hexagon if and only ifthe distance of y and z in the residue at x is 3.  相似文献   

18.
Let C'(x) denote the number of integers n x such that thereis no non-abelian group of order n, but there exists a non-cyclicgroup of order n. Here it is shown that C' (x) , x , where denotes Euler's constant.  相似文献   

19.
Let B and C be continuous nonlinear operators in a Banach space.Assume that B satisfies a dissipativeness condition, and thatthe range of C is relatively compact. Some sufficient conditionsare given for the existence of solutions of the equation u'(t)= B(t, u(t)) + C(t, u(t)).  相似文献   

20.
Let G be a simply connected domain in the t plane (t = x + iy),bounded by the three straight lines x = 0, y = 0, x = 1 anda Jordan arc with cartesian equation y = (x). Also, let g bethe function which maps conformally a rectangle R onto G, sothat the four corners of R are mapped onto those of G. In thispaper we show that the method considered in 1982 by Challis& Burley for determining approximations to g is equivalentto a special case of the well-known method of Garrick for themapping of doubly connected domains. Hence, by using resultsalready available in the literature, we provide some theoreticaljustification for the method of Challis & Burley.  相似文献   

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