Embedded explicit Runge–Kutta type methods for directly solving special third order differential equations y 000 ¼ f ðx; yÞ (original) (raw)
Related papers
Applied Mathematics and Computation, 2014
In this paper three pairs of embedded Runge-Kutta type methods for directly solving special third order ordinary differential equations (ODEs) of the form y 000 ¼ f ðx; yÞ denoted as RKD methods are presented. The first is the RKD4(3) pair which is third order embedded in fourth-order method has the property first same as last (FSAL) whereby the last row of the coefficient matrix is equal to the vector output. The second method is the RKD5(4) pair followed by the RKD6(5) pair. The methods are derived with the strategies such that the higher order methods are very accurate and the lower order methods will give the best error estimates. Variables stepsize codes are developed based on the methods and used to solve a set of special third order problems. Numerical results are compared with the existing embedded Runge-Kutta pairs which require the problems to be reduced into a system of first order ODEs. Numerical results have clearly shown the advantage and the efficiency of the new RKD pairs.
Mathematical Problems in Engineering, 2015
We present two pairs of embedded Runge-Kutta type methods for direct solution of fourth-order ordinary differential equations (ODEs) of the formy(iv)=f(x,y)denoted as RKFD methods. The first pair, which we will call RKFD5(4), has orders 5 and 4, and the second one has orders 6 and 5 and we will call it RKFD6(5). The techniques used in the derivation of the methods are that the higher order methods are very precise and the lower order methods give the best error estimate. Based on these pairs, we have developed variable step codes and we have used them to solve a set of special fourth-order problems. Numerical results show the robustness and the efficiency of the new RKFD pairs as compared with the well-known embedded Runge-Kutta pairs in the scientific literature after reducing the problems into a system of first-order ordinary differential equations (ODEs) and solving them.
Runge-Kutta Type Methods for Directly Solving Special Fourth-Order Ordinary Differential Equations
Mathematical Problems in Engineering, 2015
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 Pair of Diagonally Implicit Runge-Kutta Method for Solving Ordinary Differential Equations
Sains …, 2010
Improvements over embedded diagonally implicit Runge-Kutta pair of order four in five are presented. Method of higher stage order with a zero first row and the last row of the coefficient matrix is identical to the vector output is given. The stability aspect of it is also looked into and a standard test problems are solved using the method. Numerical results are tabulated and compared with the existing method.
A new four-stage sixth order Runge-Kutta method for direct integration of special third order ordinary differential equations (ODEs) is constructed. The method is proven to be zero-stable. Stability polynomial of the method for linear special third order ODE is given. A set of test problems consisting of ordinary differential equations is tested upon. The problems are solved using the new method and numerical comparisons are made when the same problems are reduced to a first order system of ODEs and solved using the existing Runge-Kutta methods of different orders. Numerical results have clearly shown the advantage and the efficiency of the new method in terms of accuracy and computational time. [Mohammed Mechee , Fudziah Ismail , Zailan Siri , and Norazak. Senu. A Four-Stage Sixth-Order RKD Method for Directly Solving Special Third-Order Ordinary Differential Equations. Life Sci J 2014;11(3):399-404] (ISSN:1097-8135). http://www.lifesciencesite.com. 57
International Journal of Mathematical, Engineering and Management Sciences
The Runge-Kutta method is a one step method with multiple stages, the number of stages determine order of method. The method can be applied to work out on differential equation of the type’s explicit, implicit, partial and delay differential equation etc. The present paper describes a review on recent computational techniques for solving differential equations using Runge-Kutta algorithm of various order. This survey includes the summary of the articles of last decade till recent years based on third; fourth; fifth and sixth order Runge-Kutta methods. Along with this a combination of these methods and various other type of Runge-Kutta algorithm based articles are included. Comparisons of methods with own critical comments as remarks have been included.
2013
A new four-stage sixth order Runge-Kutta method for direct integration of special third order ordinary differential equations (ODEs) is constructed. The method is proven to be zero-stable. Stability polynomial of the method for linear special third order ODE is given. A set of test problems consisting of ordinary differential equations is tested upon. The problems are solved using the new method and numerical comparisons are made when the same problems are reduced to a first order system of ODEs and solved using the existing Runge-Kutta methods of different orders. Numerical results have clearly shown the advantage and the efficiency of the new method in terms of accuracy and computational time. [Mohammed Mechee , Fudziah Ismail , Zailan Siri , and Norazak. Senu. A Four-Stage Sixth-Order RKD Method for Directly Solving Special Third-Order Ordinary Differential Equations. Life Sci J 2014;11(3):399-404] (ISSN:1097-8135). http://www.lifesciencesite.com. 57
Reformulation on Modified Runge-Kutta Third Order Methods for Solving Initial Value Problems
Abstract – Ordinary Differential equation with Initial Value Problems (IVP) frequently arise in many physical problems. Numerical methods are widely used for solving the problems especially in case of numerical simulation. Several numerical methods are available in the literature for solving IVP. Runge-Kutta (which is actually Arithmetic Mean (AM) based method) is one of the best commonly used numerical approaches for solving the IVP. Recently Evans[1] proposed Geometric Mean (GM) based Runge-Kutta third order method and Wazwaz [2] proposed Harmonic Mean (HM) based Runge-Kutta third order method for solving IVP. Also Yanti et al. [3] proposed the linear combination of AM, HM and GM based Runge-Kutta third order method. We extensively perform several experiments on those approaches to find robustness of the approaches. Theoretically as well as experimentally we observe that GM based and Linear combination of AM, GM and HM based approaches are not applicable for all kinds of problems. To overcome some of these drawbacks we propose modified formulas correspond to those AM and linear combination of AM, GM, HM based methods. Experimentally it is shown that the proposed modified methods are most robust and able to solve the IVP efficiently.
A Three-stage Fifth-order Runge-Kutta Method for Directly Solving Special Third
2013
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.
Journal of Pure & Applied Sciences
this article, an explicit Runge-Kutta-Nystr m method of sixth order for solving directly second-order ordinary differential equations is constructed. The stability property of the new method is discussed. Numerical illustrations are presented to show the efficiency of the new method by comparing it with other existing Runge-Kutta-Nystr m and Runge-Kutta methods in the scientific literature.