Direct integrators of Runge–Kutta type for special third-order ordinary differential equations

2013 ◽  
Vol 74 ◽  
pp. 128-150 ◽  
Author(s):  
Xiong You ◽  
Zhaoxia Chen
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Third Order ◽  
Runge Kutta

scholarly journals INTEGRATION FOR SPECIAL THIRD-ORDER ORDINARY DIFFERENTIAL EQUATIONS USING IMPROVED RUNGE-KUTTA DIRECT METHOD

2015 ◽  
Vol 34 (2) ◽  
pp. 172-179 ◽  
Author(s):  
Kasim Abbas Hussain ◽  
Fudziah Ismail ◽  
Norazak Senu ◽  
Faranak Rabiei ◽  
Rabha Ibrahim
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Direct Method ◽  
Third Order ◽  
Runge Kutta

Boundary-Value Problems for Third-Order Lipschitz Ordinary Differential Equations

2014 ◽  
Vol 58 (1) ◽  
pp. 183-197 ◽  
Author(s):  
John R. Graef ◽  
Johnny Henderson ◽  
Rodrica Luca ◽  
Yu Tian
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Point Boundary ◽  
Third Order ◽  
Optimal Length ◽  
The Third

AbstractFor the third-order differential equationy′″ = ƒ(t, y, y′, y″), where, questions involving ‘uniqueness implies uniqueness’, ‘uniqueness implies existence’ and ‘optimal length subintervals of (a, b) on which solutions are unique’ are studied for a class of two-point boundary-value problems.

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.

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