[Show abstract][Hide abstract] ABSTRACT: Fuzzy differential equations (FDEs) generalize the concept of crisp initial value problems. In this article, we deal with the numerical solution of FDEs. The notion of convergence of a numerical method is defined and a category of problems which is more general than the one already found in the numerical analysis literature is solved. Efficient s-stage Runge–Kutta methods are used for the numerical solution of these problems and the convergence of the methods is proved. Several examples comparing these methods with the previously developed Euler method are displayed.
[Show abstract][Hide abstract] ABSTRACT: We present families of explicit Runge–Kutta Methods for the numerical treatment of Stochastic Differential Equations with additive noise and one dimensional Wiener process. We study methods with two, three and four stages attaining deterministic order up to four and stochastic orders one and one and a half. The methods are tested in the solution of various problems and are compared with known other methods. The results modify our effort.
[Show abstract][Hide abstract] ABSTRACT: We present the equations of condition up to sixth order for Runge-Kutta (RK) methods, when integrating scalar autonomous problems. Two RK pairs of orders 5(4) are derived. The first at a cost of only five stages per step, while the other having an extremely small principal truncation error. Numerical tests show the superiority of the new pairs over traditional ones.
International Journal of Computer Mathematics 02/2003; 80(2):201-209. DOI:10.1080/00207160304669 · 0.82 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In the present work, we are concerned with the derivation of continuous Runge-Kutta-Nyström methods for the numerical treatment of second-order ordinary differential equations with Nyström methods for the numerical treatment of second-order ordinary differential equations with periodic solutions. Numerical methods used for solving such problems are better to have the characteristics of high phase-lag order. First we analyse the construction algorithm for a high phase-lag order scaled extension of an explicit Runge-Kutta-Nyström method. Using this procedure, we manage to construct a phase-lag order 14 continuous extension of a popular nine stages 8(6) order ERKN pair. In the literature, only phase-lag order 12 continuous extension of nine stage 8(6) ERKN pairs can be found, so the proposed scaling method has the higher, until now, dispersion order. Numerical tests for the proposed methods are done over various test problems.
[Show abstract][Hide abstract] ABSTRACT: New explicit hybrid Numerov type methods are presented in this paper. These efficient methods are constructed using a new approach, where we do not need the use of the intermediate high accuracy interpolatory nodes, since only the Taylor expansion of the internal points is needed. The methods share sixth algebraic order at a cost of five stages per step while their phase-lag order is 14 and partly satisfy the dissipation order conditions. It has be seen that the property of phase-lag is more important than the nonempty interval in constructing numerical methods for the solution of Schrödinger equation and related problems.1-3 Numerical results over some well known problems in physics and mechanics indicate the superiority of the new methods.
International Journal of Modern Physics C 06/2001; 12(5):657-666. DOI:10.1142/S0129183101001869 · 1.26 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In this work we dial with the treatment of second order retarded differential equations with periodic solutions by explicit Runge–Kutta–Nyström methods. In the past such methods have not been studied for this class of problems. We refer to the underline theory and study the behavior of various methods proposed in the literature when coupled with Hermite interpolants. Among them we consider methods having the characteristic of phase–lag order. Then we consider continuous extensions of the methods to treat the retarded part of the problem. Finally we construct scaled extensions and high order interpolants for RKN pairs which have better characteristics compared to analogous methods proposed in the literature. In all cases numerical tests and comparisons are done over various test problems.
[Show abstract][Hide abstract] ABSTRACT: A five-stage fifth-order singly diagonally implicit Runge–Kutta–Nyström method for the integration of second order differential
equations possessing an oscillatory solution, is presented in this article. This method is P-stable, which is recommended
for problems with a theoretical solution consisting of a periodic part of moderate frequency with a high frequency oscillation
with small amplitude superimposed. It also attains an order which is one higher than existing methods of this type. Numerical
comparisons with existing methods of this type show its clear advantage.
[Show abstract][Hide abstract] ABSTRACT: Computer assisted derivation and improved techniques have led to effective explicit Runge-Kutta methods of higher order. These methods become inefficient when the step size must be reduced often to produce approximations at specified points. Considerable effort has been devoted to providing Runge-Kutta methods with an interpolation capability, so that approximations can be produced inexpensively at intermediate points of a successful step. New high order Hermite interpolants for two well known embedded Runge-Kutta methods of orders 7 and 8 are presented. These interpolants are constructed using values from two successive integration steps, are locally of O(h ) or O(h ), and require only one or four extra function evaluations per step respectively.
International Journal of Computer Mathematics 01/1997; 65(3):273-291. DOI:10.1080/00207169708804616 · 0.82 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The construction of a Runge-Kutta pair of order 5(4) with the minimal number of stages requires the solution of a nonlinear system of 25 order conditions in 27 unknowns. We dene a new family of pairs which in- cludes pairs using 6 function evaluations per integration step as well as pairs which additionally use the rst function evaluation from the next step. This is achieved by making use of Kutta's simplifying assumption on the original system of the order conditions, i.e., that all the internal nodes of a method contributing to the estimation of the endpoint solution provide, at these nodes, cost-free second-order approximations to the true solution of any dierential equation. In both cases the solution of the resulting system of nonlinear equa- tions is completely classied and described in terms of ve free parameters. Optimal Runge-Kutta pairs with respect to minimized truncation error co- ecients, maximal phase-lag order and various stability characteristics are presented. These pairs were selected under the assumption that they are used in Local Extrapolation Mode (the propagated solution of a problem is the one provided by the fth-order formula of the pair). Numerical results obtained by testing the new pairs over a standard set of test problems suggest a signicant improvement in eciency when using a specic pair of the new family with minimized truncation error coecients, instead of some other existing pairs.
Mathematics of Computation 07/1996; 65(215):1165-1181. DOI:10.1090/S0025-5718-96-00718-1 · 1.49 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Explicit Runge-Kutta formula pairs of different orders of accuracy form a class of efficient algorithms for treating nonstiff ordinary differential equations. So far, several Runge-Kutta pairs of order 6(5) have appeared in the literature. These pairs use 8 function evaluations per step and belong to certain families of solutions of a set of 54 nonlinear algebraic equations in 44 or 45 coefficients, depending on the use of the FSAL (first stage as last) device. These equations form a set of necessary and sufficient conditions that a 6(5) Runge-Kutta pair must satisfy. The solution of the latter is achieved by employing various types of simplifying assumptions. In this paper we make use of the fact that all these families of pairs satisfy a common set of simplifying assumptions. Using only these simplifying assumptions we define a new family of 6(5) Runge-Kutta pairs. Its main characteristic, which is also a property that no other known family shares, is that all of its nodes (except the last one, which equals 1) are free parameters of the resulting solution. A search has been carried out among the pairs of the new family and two nearly optimum pairs, with respect to accuracy and stability characteristics, have been constructed. The new pairs, as is exhibited by several numerical examples, compare favorably with all other currently known similar pairs.
[Show abstract][Hide abstract] ABSTRACT: An explicit Runge-Kutta (RK) or Runge-Kutta-Nyström (RKN) method, for the numerical approximation of the initial value problem, can be expanded by the addition of a “dense” formula which provides solutions at points within or outside the normal step intervals. In this paper, we are concerned with the construction of continuous extensions for RK and RKN methods, intended to approximate first- and second-order differential equations, respectively. First we derive the required equations of conditions that the coefficients of these extensions have to satisfy in order to produce reduced phase-errors, when applied to a linear homogeneous test equation. Moreover some particular continuous extensions of an explicit 6(5) RK and 8(6) RKN pair, respectively, are proposed and tested numerically.
Journal of Computational and Applied Mathematics 04/1996; 69(1-69):1-11. DOI:10.1016/0377-0427(95)00027-5 · 1.27 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We study the relative merits of the phase-lag property of Runge-Kutta pairs and we propose new explicit embedded pairs for the numerical solution of first order differential systems with periodical solution. We analyze two families of 5(4) paris and one family of 6(5) pairs with respect to the attainable phase-lag order. From each family we choose a pair with the highest achievable phase-lag order, optimized with respect to a measure of the magnitude of its truncation error coefficients. The new 5(4) algebraic order pairs are of phase-lag order 8(4) and 8(6) and they are both non-dissipative, while the 6(5) pair is dissipative and of phase-lag order 10(6). The new pairs exhibit an improved performance, in comparison with other currently known general and special purpose methods, when they are applied to semidiscretized hyperbolic equations and problems describing free and weakly forced oscillations.Wir studieren die relativen Vorteile des Phasenfehler-Merkmals von Runge-Kutta-Paaren und schlagen in der vorliegenden Arbeit neue direkte Paare von Runge-Kutta-Methoden fr die numerische Lsung von Differentialgleichungssystemen erster Ordnung mit periodischen Lsungen vor. Wir betrachten Familien von Paaren der algebraischen Ordnung 5(4) bzw. 6(5) bezglich der erreichbaren Ordnung des Phasenfehlers. Von jeder Familie whlen wir ein Paar mit der hchsten Ordnung des Phasenfehlers aus, mit einer Optimierung der Koeffizienten des Abbrechfehlers. Die neuen Paare der algebraischen Ordung 5(4) haben Phasenfehler der Ordnung 8(4) und 8(6). Das Paar der algebraischen Ordnung 6(5) hat die Phasenordnung 10(6). Die neuen Paare von Methoden sind effektiver als andere bekannte Methoden, wenn sie auf semidiskrete hyperbolische Gleichungen und Problemen, die freie und schwachgedmpfte Schwingungen beschreiben, angewandt werden.
[Show abstract][Hide abstract] ABSTRACT: A Runge-Kutta-Nyström (RKN) formula becomes inefficient when the step size must be reduced often to produce answers at specified points. The last years an effort has been started to providing Runge-Kutta-Nyström methods with an interpolation capability. Then approximations can be produced on intermediate points of a successful step inexpensively. New high order Hermite type interpolants for (RKN) methods are presented. The interpolants which approximate the solution is of O(h 9) and C 2 while the interpolants which approximate the corresponding derivative is of O(h 8) and C1. These interpolants have been constructed in two ways, using values from one and two steps respectively.
International Journal of Computer Mathematics 01/1993; 47(3-4):209-217. DOI:10.1080/00207169308804178 · 0.82 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Runge-Kutta-Nystrom (RKN) codes for the solution of the initial value problem for the general second order differential system have been developed recently, although the methodology on which they are based was known many years ago. In this paper we try to examine the efficiency of several known general Runge-Kutta-Nystrom (GRKN) methods by posing some criteria of cost and accuracy. These methods supplied with the corresponding interpolants, have been applied to some problems of Celestial Dynamics. The results obtained show that these codes have a good response in the approximation of the solution of these problems.
[Show abstract][Hide abstract] ABSTRACT: New interpolants of the explicit Runge-Kutta method for the initial value problem are proposed. These interpolants are based on values of the solution and its derivative from two successive integration steps. In this paper, three interpolants withO(h
6) local error (l.e.), for the fifth order solution, of the methods Fehlberg 4(5) (RKF 4(5)), Dormand and Prince 5(4) (RKDP 5(4)) and Verner 5(6) (RKV 5(6)) without extra cost are derived. An interpolant withO(h
7) (l.e.) for the sixth order solution of the Verner's method with only one extra function evaluation per integration step is also constructed. The above advantages are obtained without any cost in the magnitude of the error.Neue Interpolanten fr explizite Runge-Kutta-Verfahren fr Anfangswertprobleme werden vorgeschlagen. Diese Interpolanten beruhen auf Werten der Lsung und ihrer Ableitung aus zwei aufeinanderfolgenden Integrationsschritten. Fr Verfahren der Ordnung 5 (nmlich Runge-Kutta-Fehlberg 4(5), Dormand-Prince (5(4) und Verner 5(6)) haben wir Interpolanten mit einem lokalen FehlerO(h
6) hergeleitet, die keine zustzlichen Funktionsauswertungen bentigen. Fr die Lsung der Ordnung 6 des Verfahrens von Verner erhalten wir mit einer zustzlichen Funktionsauswertung einen Interpolanten mitO(h
7)-Fehler. Es tritt dabei keine Aufblhung der Grenordnung der Fehlerkonstanten auf.
[Show abstract][Hide abstract] ABSTRACT: In this paper a general interpolant for the Explicit Runge-Kutta methods is proposed. These interpolants are based on second derivatives on mesh-points of the integration interval, and first derivatives on interior points of each step. These first derivatives can be produced using lower order interpolants. Here an interpolant with 0(h ) local truncation error for the fifth order solution used in RKF4(5) method is presented, with a cost of “about” one extra function evaluation per integration step.
International Journal of Computer Mathematics 01/1989; 31(1-2):105-113. DOI:10.1080/00207168908803792 · 0.82 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: New Runge-Kutta-Nyström algorithms are presented which determine an approximation of the solution and its derivative of the second order differential equation ÿ = f(x,y) at intermediate points of a given integration step, as well as at the end of each step. These new algorithms, called scaled Runge-Kutta-Nyström (SRKN) methods, are designed to be used with existing Runge-Kutta-Nyström (RKN) formulas, using the function evaluations of these methods as the core of the new system. Thus, for a slight increase of the cost, the solution may be generated within a successful step, improving so the efficiency of the existing RKN methods.
International Journal of Computer Mathematics 01/1989; 28(1-4):139-150. DOI:10.1080/00207168908803734 · 0.82 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Our intention in this paper is to present the results obtained from a systematic comparative study of several high-order Runge-Kutta methods as applied in a composite problem of celestial dynamics, namely the magnetic-binary problem.
Astrophysics and Space Science 08/1988; 147(2). DOI:10.1007/BF00645672 · 2.26 Impact Factor