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1.
A family of two-step fourth order methods, which requires two function evaluations per step, is derived fory=f(x,y). We then show the existence of a sub-family of these methods which when applied toy=–k
2
y,k real, areP-stable. 相似文献
2.
《Journal of Computational and Applied Mathematics》1986,15(2):213-223
This paper deals with a class of symmetric (hybrid) two-step fourth order P-stable methods for the numerical solution of special second order initial value problems. Such methods were proposed independently by Cash [1] and Chawla [3] and normally require three function evaluations per step. The purpose of this paper is to point out that there are some values of the (free) parameters available in the methods proposed which can reduce this work; we study two classes of such methods. The first is the class of ‘economical’ methods (see Definition 3.1) which reduce this work to two function evaluations per step, and the second is the class of ‘efficient’ methods (see Definition 3.2) which reduce this work with respect to implementation for nonlinear problems. We report numerical experiments to illustrate the order, acuracy and implementational aspects of these two classes of methods. 相似文献
3.
A family of two step difference schemes of the fourth order has been developed for linear ODEs of the second order. Stability
properties for such schemes are discussed and results of numerical tests are given. It is shown how the proposed technique
can be extended to non-linear ODEs of second order. 相似文献
4.
E. Hairer 《Numerische Mathematik》1979,32(4):373-379
We use the concept of order stars (see [1]) to prove and generalize a recent result of Dahlquist [2] on unconditionally stable linear multistep methods for second order differential equations. Furthermore a result of Lambert-Watson [3] is generalized to the multistage case. Finally we present unconditionally stable Nyström methods of order 2s (s=1,2, ...) and an unconditionally stable modification of Numerov's method. The starting point of this paper was a discussion with G. Wanner and S.P. Nørsett. The author is very grateful to them. 相似文献
5.
M. M. Chawla 《BIT Numerical Mathematics》1983,23(4):541-542
We report a modification of Noumerov's method which produces a family of unconditionally stable fourth order methods fory''=f(t, y). 相似文献
6.
It is well known that the numerical solution of stiff stochastic ordinary differential equations leads to a step size reduction when explicit methods are used. This has led to a plethora of implicit or semi-implicit methods with a wide variety of stability properties. However, for stiff stochastic problems in which the eigenvalues of a drift term lie near the negative real axis, such as those arising from stochastic partial differential equations, explicit methods with extended stability regions can be very effective. In the present paper our aim is to derive explicit Runge–Kutta schemes for non-commutative Stratonovich stochastic differential equations, which are of weak order two and which have large stability regions. This will be achieved by the use of a technique in Chebyshev methods for ordinary differential equations. 相似文献
7.
8.
A. Iserles 《BIT Numerical Mathematics》1981,21(1):80-96
A family of two-stepA-stable methods of maximal order for the numerical solution of ordinary differential systems is developed. If these methods are applied to the stiff, large systems which originate from linear parabolic differential equations they yield a large, sparse set of linear algebraic equations of special form. This set is considerably easier to solve than the algebraic equations which are obtained when using diagonal Obrechkoff methods, which are one-step,A-stable and of maximal order 相似文献
9.
10.
Said R. Grace Ravi P. Agarwal John R. Graef 《Journal of Applied Mathematics and Computing》2009,30(1-2):75-88
The authors investigate the oscillatory behavior of all solutions of the fourth order functional differential equations $\frac{d^{3}}{dt^{3}}(a(t)(\frac{dx(t)}{dt})^{\alpha})+q(t)f(x[g(t)])=0$ and $\frac{d^{3}}{dt^{3}}(a(t)(\frac{dx(t)}{dt})^{\alpha})=q(t)f(x[g(t)])+p(t)h(x[\sigma(t)])$ in the case where ∫ ∞ a ?1/α (s)ds<∞. The results are illustrated with examples. 相似文献
11.
Summary In this paper a new approach involving the use of two signum functions together with a suitably chosen Lyapunov function is
employed to investigate the boundedness property of solutions of two special cases of(1.3). This approach makes for considerable reduction in the conditions imposed on f, g in an earlier paper[1].
Entrata in Redazione il 25 ottobre 1970. 相似文献
12.
A general class of linear two-step schemes for solving stochastic differential equations is presented. Necessary and sufficient conditions on its parameters to obtain mean square order 1.5 are derived. Then the linear stability of the schemes is investigated. In particular, among others, the stability regions of generalizations of the classical two-step schemes Adams-Bashford, Adams-Moulton, and BDF are obtained. 相似文献
13.
Hypoelliptic second order differential equations 总被引:34,自引:0,他引:34
Lars Hörmander 《Acta Mathematica》1967,119(1):147-171
14.
《Communications in Nonlinear Science & Numerical Simulation》2014,19(8):2653-2659
Meleshko presented a new method for reducing third order autonomous ordinary differential equations (ODEs) to Lie linearizable second order ODEs. We extended his work by reducing fourth order autonomous ODEs to second and third order linearizable ODEs and then applying the Ibragimov and Meleshko linearization test for the obtained ODEs. The application of the algorithm to several ODEs is also presented. 相似文献
15.
A new class of two-step Runge-Kutta methods for the numerical solution of ordinary differential equations is proposed. These methods are obtained using the collocation approach by relaxing some of the collocation conditions to obtain methods with desirable stability properties. Local error estimation for these methods is also discussed. 相似文献
16.
By using the bilinear form and the Leray-Schauder principle, we prove the new anti-periodic existence results for second order differential equations. 相似文献
17.
18.
H. B. Thompson 《Applicable analysis》2013,92(1-2):27-43
Explicit upper and lower bounds are constructed for a functional defined on the difference of solutions to a class of nonlinear hyperbolic problems modelling string vibrations. Such bounds indicate how solutions are affected by the input data over a finite time interval. The logarithmic convexity method is used to obtain two second order ordinary differential inequalities for the appropriate functional. These inequalities are then related to two systems of first order equations whose solutions are the desired bounds. 相似文献
19.
Gary D Jones 《Journal of Mathematical Analysis and Applications》1978,64(3):620-628
We define various ways in which linear-quadratic control problems can be robust and show that there are conditions which are simultaneously necessary and sufficient for robust problems to have a solution when such conditions do not exist for nonrobust problems. 相似文献
20.
In this paper, we deal with oscillatory and asymptotic properties of solutions of a fourth order sub-linear differential equation with the oscillatory operator. We establish conditions for the nonexistence of positive and bounded solutions and an oscillation criterion. 相似文献