Methods For Periodic Initial Value Problems Research Articles

A P-STABLE EIGHTEENTH-ORDER SIX-STEP METHOD FOR PERIODIC INITIAL VALUE PROBLEMS

In this paper we present a new kind of P-stable eighteenth-order six-step method for periodic initial-value problems. We add the fourth derivatives to our previous P-stable six-step method to increase the accuracy. We apply two classes of well-known problems to our new method and compare it with the previous methods. The numerical results show that the new method is much more stable, accurate and efficient than the previous methods.

A new kind of high-efficient and high-accurate P-stable Obrechkoff three-step method for periodic initial-value problems

In order to improve the efficiency and accuracy of the previous Obrechkoff method, in this paper we put forward a new kind of P-stable three-step Obrechkoff method of O ( h 10 ) for periodic initial-value problems. By using a new structure and an embedded high accurate first-order derivative formula, we can avoid time-consuming iterative calculation to obtain the high-order derivatives. By taking advantage of new trigonometrically-fitting scheme we can make both the main structure and the first-order derivative formula to be P-stable. We apply our new method to three periodic problems and compare it with the previous three Obrechkoff methods. Numerical results demonstrate that our new method is superior over the previous ones in accuracy, efficiency and stability.

P-stable linear symmetric multistep methods for periodic initial-value problems

In this paper we present a new kind of P-stable multistep methods for periodic initial-value problems. From the numerical results obtained by the new method to well-known periodic problems, show the superior efficiency, accuracy, stability of the method presented in this paper.

An improved trigonometrically fitted P-stable Obrechkoff method for periodic initial-value problems

In this paper we present an improved P-stable trigonometrically fitted Obrechkoff method with phase-lag (frequency distortion) infinity. Compared with the previous P-stable trigonometrically fitted Obrechkoff method developed by Simos, our new method is simpler in structure and more stable in computation. We have also improved the accuracy of the first-order derivative formula. From the numerical illustration presented, we can show that the new method is much more accurate than the previous methods.

A four-step trigonometric fitted P-stable Obrechkoff method for periodic initial-value problems

In this paper, we present a new P-stable Obrechkoff four-step method, which greatly improves the performance of our previous Obrechkoff four-step method and extends its application range. By trigonometric fitting, we extend the interval of periodicity of the previous four-step method from about H 2 ∼ 16 to infinity and at the same time, we keep all its advantage in the accuracy and efficiency. We have tested the new method by four well-known problems, (1) the test-equation; (2) Stiefel and Bettis problem; (3) Duffing equation without damping; and (4) Bessel equation. The numerical results show that the new method is more accurate than any previous method. It also has great advantage in stability and efficiency.

Block Runge-Kutta methods for periodic initial-value problems

Block Runge-Kutta methods with minimal phase-lag for first order periodic initialvalue problems are developed. It should be noted that the new methods are based on the Runge-Kutta methods of algebraic order three, and on a new error estimate introduced in this paper. The numerical results indicate that these new methods are efficient for the numerical solution of differential equations with periodic solutions, using variable step size.

A P-stable complete in phase Obrechkoff trigonometric fitted method for periodic initial-value problems

In this paper we derive a P-stable trigonometric fitted Obrechkoff method with phase-lag (frequency distortion) infinity. It is easy to see, from numerical results presented, that the new method is much more accurate than previous methods.

A low-order embedded Runge—Kutta method for periodic initial-value problems

An embedded Runge—Kutta—Fehlberg method is developed. It should be noted that this embedded method is procuced using the Runge—Kutta—Fehlberg method with algebraic order four to estimate a truncation phase-lag error of algebraic order three. The numerical results indicate that this new method is efficient for the numerical solution of differential equations with periodic solution, using variable stepsize.

A Sixth-order P-stable Symmetric Multistep Method for Periodic Initial-value Problems of Second-order Differential Equations

A Sixth-order <i>P</i>-stable Symmetric Multistep Method for Periodic Initial-value Problems of Second-order Differential Equations

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

A class of P-stable singlestep methods is discussed for solving initial-value problems involving second-order differential equations. The methods depend upon a parameter p>0 and reduce to the classical methods for p=0. A few choices of p have been discussed for which the methods are P-stable. Further, when p is chosen for linear problems as the square of the frequency of the periodic solution, the methods are P-stable. Numerical results for both linear and non-linear problems show that the P-stability is an important requirement for determining periodic numerical solutions of second-order differential equations.