The Numerical Analysis of Implicit Runge-Kutta Methods for a Certain Nonlinear Integro-Differential Equation

1990 ◽  
Vol 54 (189) ◽  
pp. 155 ◽  
Author(s):  
Yuan Wei ◽  
Tang Tao
Keyword(s):  
Differential Equation ◽  
Numerical Analysis ◽  
Integro Differential Equation ◽  
Runge Kutta

THE DYNAMICS OF RUNGE–KUTTA METHODS

1992 ◽  
Vol 02 (03) ◽  
pp. 427-449 ◽  
Author(s):  
JULYAN H. E. CARTWRIGHT ◽  
ORESTE PIRO
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Numerical Analysis ◽  
Dynamical Systems ◽  
Systems Theory ◽  
Continuous Time ◽  
Time System ◽  
Integration Schemes ◽  
Runge Kutta ◽  
Continuous Time System

The first step in investigating the dynamics of a continuous-time system described by an ordinary differential equation is to integrate to obtain trajectories. In this paper, we attempt to elucidate the dynamics of the most commonly used family of numerical integration schemes, Runge–Kutta methods, by the application of the techniques of dynamical systems theory to the maps produced in the numerical analysis.

scholarly journals Numerical analysis of a method for a partial integro-differential equation model in regulatory gene networks

2018 ◽  
Vol 28 (10) ◽  
pp. 2069-2095 ◽  
Author(s):  
Antonio A. Alonso ◽  
Rodolfo Bermejo ◽  
Manuel Pájaro ◽  
Carlos Vázquez
Keyword(s):  
Differential Equation ◽  
Numerical Analysis ◽  
Gene Networks ◽  
Regulatory Networks ◽  
Regulatory Gene ◽  
Spatial Dimension ◽  
Equation Model ◽  
Lagrangian Method ◽  
Second Order ◽  
Integro Differential Equation

In this paper, we propose a semi-Lagrangian Runge–Kutta method to approximate the solution of a multidimensional partial integro-differential equation (PIDE) model for regulatory networks involving multiple genes with self- and cross-regulations. For the first time in the literature, we address the numerical analysis of a semi-Lagrangian method for a PIDE model without second-order derivative terms. From this analysis, we obtain second-order convergence in time and space. Moreover, some examples with analytical solution in one spatial dimension illustrate the theoretical results, while others in higher dimensions show the expected behavior of the solution. Finally, the scalability of the method and the comparison with a previously proposed first-order semi-Lagrangian method are discussed.

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