DEVELOPMENT OF NEW METHODS FOR THE SOLUTION OF DIFFERENTIAL EQUATIONS BY THE METHOD OF LIE SERIES

1969 ◽  
Author(s):  
W. Groebner ◽  
K. H. Kestlunger ◽  
H. Reitberger ◽  
R. Saely ◽  
G. Wanner
Keyword(s):  
Differential Equations ◽  
Lie Series ◽  
New Methods

scholarly journals Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 174
Author(s):  
Janez Urevc ◽  
Miroslav Halilovič
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Integration ◽  
Integral Form ◽  
Collocation Methods ◽  
Nonlinear Odes ◽  
New Class ◽  
New Methods ◽  
Runge Kutta

In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration of ordinary differential equations (ODEs) is presented. Its derivation is based on the integral form of the differential equation. The approach enables enhancing the accuracy of the established collocation Runge–Kutta methods while retaining the same number of stages. We demonstrate that, with the proposed approach, the Gauss–Legendre and Lobatto IIIA methods can be derived and that their accuracy can be improved for the same number of method coefficients. We expressed the methods in the form of tables similar to Butcher tableaus. The performance of the new methods is investigated on some well-known stiff, oscillatory, and nonlinear ODEs from the literature.

scholarly journals V. On the calculus of symbols, with applications to the theory of differential equations

1861 ◽  
Vol 151 ◽  
pp. 69-82 ◽  
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Variable Coefficients ◽  
Linear Differential Equations ◽  
New Methods ◽  
Methods Of Calculation ◽  
Great Utility ◽  
Long Time ◽  
Constant Coefficients ◽  
Linear Differential

The calculus of generating functions, discovered by Laplace, was, as is well known, highly instrumental in calling the attention of mathematicians to the analogy which exists between differentials and powers. This analogy was perceived at length to involve an essential identity, and several analysts devoted themselves to the improvement of the new methods of calculation which were thus called into existence. For a long time the modes of combination assumed to exist between different classes of symbols were those of ordinary algebra; and this sufficed for investigations respecting functions of differential coefficients and constants, and consequently for the integration of linear differential equations, with constant coefficients. The laws of combination of ordinary algebraical symbols may be divided into the commutative and distributive laws; and the number of symbols in the higher branches of mathematics, which are commutative with respect to one another, is very small. It became then necessary to invent an algebra of non-commutative symbols. This important step was effected by Professor Boole, for certain classes of symbols, in his well-known and beautiful memoir published in the Transactions of this Society for the year 1844, and the object of the paper which I have now the honour to lay before the Society is to perfect and develope the methods there employed. For this purpose I have constructed systems of multiplication and division for functions of non-commutative symbols, subject to the same laws of combination as those assumed in Professor Boole’s memoir, and I thus arrive at equations of great utility in the integration of linear differential equations with variable coefficients.

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

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Kasim Hussain ◽  
Fudziah Ismail ◽  
Norazak Senu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Order Conditions ◽  
Numerical Results ◽  
Type Method ◽  
First Order ◽  
New Methods ◽  
Runge Kutta ◽  
Fifth Order

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.

scholarly journals New 4(3) Pairs Diagonally Implicit Runge-Kutta-Nyström Method for Periodic IVPs

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Norazak Senu ◽  
Mohamed Suleiman ◽  
Fudziah Ismail ◽  
Norihan Md Arifin
Keyword(s):  
Differential Equations ◽  
Truncation Error ◽  
Second Order ◽  
Phase Lag ◽  
Initial Value ◽  
Third Order ◽  
Second Order Differential Equations ◽  
New Methods ◽  
Oscillating Solutions ◽  
Runge Kutta

New 4(3) pairs Diagonally Implicit Runge-Kutta-Nyström (DIRKN) methods with reduced phase-lag are developed for the integration of initial value problems for second-order ordinary differential equations possessing oscillating solutions. Two DIRKN pairs which are three- and four-stage with high order of dispersion embedded with the third-order formula for the estimation of the local truncation error. These new methods are more efficient when compared with current methods of similar type and with the L-stable Runge-Kutta pair derived by Butcher and Chen (2000) for the numerical integration of second-order differential equations with periodic solutions.

A Block Implicit Advanced Step-point (BIAS) Algorithm for Stiff Differential Systems

2006 ◽  
Vol 2 (1-2) ◽  
pp. 51-58 ◽  
Author(s):  
G. Psihoyios
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Detailed Account ◽  
Differential Systems ◽  
Stiff Ordinary Differential Equations ◽  
New Methods ◽  
Future Paper ◽  
Near Future

In this paper we propose a new block implicit multistep algorithm, which has been constructed with the aim to successfully tackle the numerical solution of stiff ordinary differential equations. In order to maximise the chances of producing a good new scheme, the first mandatory step is to ensure it incorporates very good theoretical properties. A rather detailed account of the new methods is presented and the properties of the algorithm will be discussed in some detail. In the near future we will also have the opportunity to evaluate the implementation potential of the scheme in a future paper.

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