Existing Runge-Kutta Methods Research Articles

Improved Runge-Kutta Method for Oscillatory Problem Solution Using Trigonometric Fitting Approach

This paper provides a four-stage Trigonometrically Fitted Improved Runge-Kutta (TFIRK4) method of four orders to solve oscillatory problems, which contains an oscillatory character in the solutions. Compared to the traditional Runge-Kutta method, the Improved Runge-Kutta (IRK) method is a natural two-step method requiring fewer steps. The suggested method extends the fourth-order Improved Runge-Kutta (IRK4) method with trigonometric calculations. This approach is intended to integrate problems with particular initial value problems (IVPs) using the set functions and for trigonometrically fitted. To improve the method's accuracy, the problem primary frequency is used. The novel method is more accurate than the conventional Runge-Kutta method and IRK4. Several test problems for the system of first-order ordinary differential equations carry out numerically to demonstrate the effectiveness of this approach. The computational studies show that the TFIRK4 approach is more efficient than the existing Runge-Kutta methods.

Exponentially-fitted forth-order explicit modified Runge-Kutta type method for solving third-order ODEs

In this paper exponentially-fitted explicit modified Runge-Kutta type method denoted as EFMRKT for solving y′″(x) = f(x, y, y′) is derived. The idea presented is based on the Simos and Berghe approach which exactly integrates initial value problems whose solutions are linear combinations of the set functions ewx and e−wx with w ∈ R the principal frequency of the problem. We developed the new EFMRKT three-stage fourth-order method called EFRKTG4 for solving third-order initial value problems. The numerical results indicate that EFRKTG4 method is more efficient than existing Runge-Kutta methods.

Trigonometrically-fitted Higher Order Two Derivative Runge-Kutta Method for Solving Orbital and Related Periodical IVPs

In this paper, a trigonometrically-fitted two derivative Runge-Kutta method (TFTDRK) of high algebraic order for the numerical integration of first order Initial Value Problems (IVPs) which possesses oscillatory solutions is constructed. Using the trigonometrically-fitted property, a sixth order four stage Two Derivative Runge-Kutta (TDRK) method is designed. The numerical experiments are carried out with the comparison with other existing Runge-Kutta methods (RK) to show the accuracy and efficiency of the derived methods.

Fourth-Order Improved Runge–Kutta Method for Directly Solving Special Third-Order Ordinary Differential Equations

In this paper, fourth-order improved Runge–Kutta method (IRKD) for directly solving a special third-order ordinary differential equation is constructed. The fourth-order IRKD method has a lower number of function evaluations compared with the fourth-order Runge–Kutta method. The stability polynomial of the method is given. Numerical comparisons are also performed using the existing Runge–Kutta method after reducing the problems into a system of first-order equations and solving them, and direct RKD method for solving special third-order ordinary differential equations. Numerical examples are presented to illustrate the efficiency and the accuracy of the new method in terms of number of function evaluations as well as max absolute error.

Explicit Direct Integrators of RK Type for Solving Special Fifth-Order Ordinary Differential Equations

The main contribution of this paper is the development of direct explicit methods of Runge-Kutta (RK) type for solving special fifth-Order Ordinary Differential Equations (ODEs). For this purpose, we have generalized RKD and RKT methods of special third and fourth-order ODEs. Using Taylor expansion, we have derived the algebraic equations of algebraic equations of order conditions for the proposed RKM integrators up to the eighth order. Based on these conditions, two RKM methods of orders five and six with three and four-stage are derived. Numerical implementation shows that the new methods agree well with existing RK methods, but requires less function evaluations. This is so due to the fact that RKM methods are direct; hence, they save considerable amount of computational time.

Two Derivative Runge-Kutta Method with FSAL Property for the Solution of First Order Initial Value Problems

A new Two Derivative Runge-Kutta method (TDRK) based on First Same as Last (FSAL) technique for the numerical solution of first order Initial Value Problems (IVPs) is derived. We present a fourth order three stages TDRK method designed using the FSAL property. The stability of the new method is analyzed. The numerical experiments are carried out to show the efficiency of our methods in comparison with other existing Runge-Kutta methods (RK).

Optimal policies of M(t)/M/c/c queues with two different levels of servers

This paper deals with optimal control points of M(t)/M/c/c queues with periodic arrival rates and two levels of the number of servers. We use the results of this model to build a Markov decision process (MDP). The problem arose from a case study in the Kelowna General Hospital (KGH). The KGH uses surge beds when the emergency room is overcrowded which results in having two levels for the number of the beds. The objective is to minimize a cost function. The findings of this work are not limited to the healthcare; They may be used in any stochastic system with fluctuation in arrival rates and/or two levels of the number of servers, i.e., call centers, transportation, and internet services. We model the situation and define a cost function which needs to be minimized. In order to find the cost function we need transient solutions of the M(t)/M/c/c queue. We modify the fourth-order Runge–Kutta to calculate the transient solutions and we obtain better solutions than the existing Runge–Kutta method. We show that the periodic variation of arrival rates makes the control policies time-dependent and periodic. We also study how fast the policies converge to a periodic pattern and obtain a criterion for independence of policies in two sequential cycles.

A SIXTH ORDER DIAGONALLY IMPLICIT SYMMETRIC AND SYMPLECTIC RUNGE-KUTTA METHOD FOR SOLVING HAMILTONIAN SYSTEMS

The paper is concerned with construction of symmetric and symplectic Runge-Kutta methods for Hamiltonian systems. Based on the symplectic and symmetrical properties, a sixth-order diagonally implicit symmetric and symplectic Runge-Kutta method with seven stages is presented, the proposed method proved to be P-stable. Numerical experiments with some Hamiltonian oscillatory problems are presented to show the proposed method is as competitive as the existing Runge-Kutta methods in scientic literature.

Runge-Kutta Type Methods for Directly Solving Special Fourth-Order Ordinary Differential Equations

A Runge-Kutta type method for directly solving special fourth-order ordinary differential equations (ODEs) which is denoted by RKFD method is constructed. The order conditions of RKFD method up to order five are derived; based on the order conditions, three-stage fourth- and fifth-order Runge-Kutta type methods are constructed. Zero-stability of the RKFD method is proven. Numerical results obtained are compared with the existing Runge-Kutta methods in the scientific literature after reducing the problems into a system of first-order ODEs and solving them. Numerical results are presented to illustrate the robustness and competency of the new methods in terms of accuracy and number of function evaluations.

Embedded explicit two-step Runge-Kutta-Nystrom for solving special second-order IVPs

A third-order three-stage explicit Two-step Runge-Kutta-Nystr¨om (TSRKN) method embedded into fourth-order three-stage TSRKN method is developed to solve special second-order initial value problems (IVPs) directly. The stability of the method is investigated. Numerical results are obtained by solving a standard set of test problems, which then reduced to first-order system when solved using Runge-Kutta (RK) method and comparison are made with existing RK method with same order using variable step-size. The results clearly showed the advantage and the efficiency of the new method.

A Three-Stage Fifth-Order Runge-Kutta Method for Directly Solving Special Third-Order Differential Equation with Application to Thin Film Flow Problem

In this paper, a three-stage fifth-order Runge-Kutta method for the integration of a special third-order ordinary differential equation (ODE) is constructed. The zero stability of the method is proven. The numerical study of a third-order ODE arising in thin film flow of viscous fluid in physics is discussed. The mathematical model of thin film flow has been solved using a new method and numerical comparisons are made when the same problem is reduced to a first-order system of equations which are solved using the existing Runge-Kutta methods. Numerical results have clearly shown the advantage and the efficiency of the new method.

Embedded diagonally implicit Runge–Kutta–Nystrom 4(3) pair for solving special second-order IVPs

In this paper, third-order 3-stage diagonally implicit Runge–Kutta–Nystrom method embedded in fourth-order 4-stage for solving special second-order initial value problems is constructed. The method has the property of minimized local truncation error as well as the last row of the coefficient matrix is equal to the vector output. The stability of the method is investigated and a standard set of test problems are tested upon and comparisons on the numerical results are made when the same set of test problems are reduced to first-order system and solved using existing Runge–Kutta method. The results clearly shown the advantage and the efficiency of the new method.