P-stable methods for periodic initial value problems of second order differential equations

A class ofP-stable finite difference methods is discussed for solving initial value problems of second order differential equations which have periodic solutions. The methods depend upon a parameterp>0, and reduce to the classical Störmer-Cowell methods forp=0. It is shown that whenp is chosen for linear problems as the square of the frequency of the periodic solution, the methods areP-stable and for some suitable choice ofp, they have extended finite interval of periodicity.     

Two-step fourth orderP-stable methods for second order differential equations

A family of symmetric (hybrid) two-step fourth order methods is derived fory'=f(x,y). We then show the existence of a sub-family of these methods which when applied toy'=– ² y, real, areP-stable. We also note that a general (order) symmetric two-step method isP-stable iff it is unconditionally stable.     

Two-step fourth orderP-stable methods for second order differential equations

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.     

Characterisation of a class of P-stable methods for differential equations of second order

Some theorems in a recent paper by Chawla and Neta give sufficient but not necessary conditions for P-stability within particular classes of numerical methods. All P-stable methods in the relevant families are identified here.     

P-stable Obrechkoff methods of arbitrary order for second-order differential equations

Following the ideas of Ananthakrishnaiah we develop a family of P-stable Obrechkoff methods of arbitrary even order. The coefficients of these methods follow from a recursive algorithm. It is also shown that the stability functions of the thus obtained methods can be expressed as Padé approximants of the exponential function with a complex argument. A numerical example is given to illustrate the performance of the methods.     

P-stable exponentially-fitted Obrechkoff methods of arbitrary order for second-order differential equations

We consider the construction of P-stable exponentially-fitted symmetric two-step Obrechkoff methods for solving second order differential equations related to an initial value problem. Our approach is based on two ideas: for the exponential fitting, we follow the ideas of Ixaru and Vanden Berghe; for the P-stability we introduce exponentially-fitted Padé approximants to the exponential function. By combining these two ideas, we are able to construct P-stable methods of arbitrary (even) order.     

Unconditionally stable methods for second order differential equations

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.     

A class of two-step P-stable methods for the accurate integration of second order periodic initial value problems

In this paper we consider a two parameter family of two-step methods for the accurate numerical integration of second order periodic initial value problems. By applying the methods to the test equation y″ + λ²y = 0, we determine the parameters α, β so that the phase-lag (frequency distortion) of the method is minimal. The resulting method is a P-stable method with a minimal phase-lag λ⁶h⁶/42000. The superiority of the method over the other P-stable methods is illustrated by a comparative study of the phase-lag errors and by illustrating with a numerical example.     

Unconditionally stable noumerov-type methods for second order differential equations

We report a modification of Noumerov's method which produces a family of unconditionally stable fourth order methods fory''=f(t, y).     

Construction of two-step Runge-Kutta methods of high order for ordinary differential equations

The construction of two-step Runge-Kutta methods of order p and stage order q=p with stability polynomial given in advance is described. This polynomial is chosen to have a large interval of absolute stability for explicit methods and to be A-stable and L-stable for implicit methods. After satisfying the order and stage order conditions the remaining free parameters are computed by minimizing the sum of squares of the difference between the stability function of the method and a given polynomial at a sufficiently large number of points in the complex plane. This revised version was published online in August 2006 with corrections to the Cover Date.     

Weak second order S-ROCK methods for Stratonovich stochastic differential equations

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.     

Reduction of fourth order ordinary differential equations to second and third order Lie linearizable forms

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.     

Boundedness theorems for some fourth order differential equations

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.     

Oscillation theorems for fourth order functional differential equations

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.     

Oscillation of fourth order sub-linear differential equations

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.     

Oscillation properties of fourth order differential equations

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.