Publications (26)28.73 Total impact
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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 sstage 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. 
Article: Numerical solution of stochastic differential equations with additive noise by rungekutta methods
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ABSTRACT: In this paper we study the numerical treatment of Stochastic Differential Equations with additive noise and one dimensional Wiener process. We develop two, three and four stage RungeKutta methods which attain deterministic order up to four and stochastic order up to one and a half specially constructed for this class of problems. Numerical tests and comparisons with other known methods in the solution of various problems justify our effort, especially for our three stages methods. © 2009 European Society of Computational Methods in Sciences and Engineering.  [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 RungeKutta (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.  [Show abstract] [Hide abstract]
ABSTRACT: In the present work, we are concerned with the derivation of continuous RungeKuttaNyström methods for the numerical treatment of secondorder ordinary differential equations with Nyström methods for the numerical treatment of secondorder ordinary differential equations with periodic solutions. Numerical methods used for solving such problems are better to have the characteristics of high phaselag order. First we analyse the construction algorithm for a high phaselag order scaled extension of an explicit RungeKuttaNyström method. Using this procedure, we manage to construct a phaselag order 14 continuous extension of a popular nine stages 8(6) order ERKN pair. In the literature, only phaselag 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 phaselag order is 14 and partly satisfy the dissipation order conditions. It has be seen that the property of phaselag is more important than the nonempty interval in constructing numerical methods for the solution of Schrödinger equation and related problems.13 Numerical results over some well known problems in physics and mechanics indicate the superiority of the new methods.  [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 fivestage fifthorder 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 Pstable, 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 RungeKutta 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 RungeKutta 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 RungeKutta 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.  [Show abstract] [Hide abstract]
ABSTRACT: The construction of a RungeKutta 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, costfree secondorder 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 RungeKutta pairs with respect to minimized truncation error co ecients, maximal phaselag 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 fthorder 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.  [Show abstract] [Hide abstract]
ABSTRACT: Explicit RungeKutta formula pairs of different orders of accuracy form a class of efficient algorithms for treating nonstiff ordinary differential equations. So far, several RungeKutta 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) RungeKutta 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) RungeKutta 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 RungeKutta (RK) or RungeKuttaNyström (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 secondorder 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 phaseerrors, 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.  [Show abstract] [Hide abstract]
ABSTRACT: We study the relative merits of the phaselag property of RungeKutta 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 phaselag order. From each family we choose a pair with the highest achievable phaselag order, optimized with respect to a measure of the magnitude of its truncation error coefficients. The new 5(4) algebraic order pairs are of phaselag order 8(4) and 8(6) and they are both nondissipative, while the 6(5) pair is dissipative and of phaselag 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 PhasenfehlerMerkmals von RungeKuttaPaaren und schlagen in der vorliegenden Arbeit neue direkte Paare von RungeKuttaMethoden 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 RungeKuttaNyström (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 RungeKuttaNyström 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.  [Show abstract] [Hide abstract]
ABSTRACT: RungeKuttaNystrom (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 RungeKuttaNystrom (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 RungeKutta 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 RungeKuttaVerfahren fr Anfangswertprobleme werden vorgeschlagen. Diese Interpolanten beruhen auf Werten der Lsung und ihrer Ableitung aus zwei aufeinanderfolgenden Integrationsschritten. Fr Verfahren der Ordnung 5 (nmlich RungeKuttaFehlberg 4(5), DormandPrince (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. 
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ABSTRACT: New RungeKuttaNyström 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 RungeKuttaNyström (SRKN) methods, are designed to be used with existing RungeKuttaNyström (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.  [Show abstract] [Hide abstract]
ABSTRACT: In this paper a general interpolant for the Explicit RungeKutta methods is proposed. These interpolants are based on second derivatives on meshpoints 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.  [Show abstract] [Hide abstract]
ABSTRACT: Our intention in this paper is to present the results obtained from a systematic comparative study of several highorder RungeKutta methods as applied in a composite problem of celestial dynamics, namely the magneticbinary problem.
Publication Stats
189  Citations  
28.73  Total Impact Points  
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Institutions

2009

Applied Physical Sciences
Groton, Connecticut, United States


19862003

National Technical University of Athens
 • School of Applied Mathematical and Physical Science
 • Department of Mathematics
Athínai, Attica, Greece
