scholarly journals Numerical solution of random differential equations: A mean square approach

2007 ◽  
Vol 45 (7-8) ◽  
pp. 757-765 ◽  
Author(s):  
J.C. Cortés ◽  
L. Jódar ◽  
L. Villafuerte
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Mean Square ◽  
Random Differential Equations

scholarly journals Mean Square Convergent Non-Standard Numerical Schemes for Linear Random Differential Equations with Delay

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1417 ◽  
Author(s):  
Julia Calatayud ◽  
Juan Carlos Cortés ◽  
Marc Jornet ◽  
Francisco Rodríguez
Keyword(s):  
Stochastic Process ◽  
Differential Equations ◽  
Random Variables ◽  
Mean Square ◽  
Discrete Delay ◽  
Numerical Schemes ◽  
Recent Contribution ◽  
Order Of Accuracy ◽  
Euler’S Method ◽  
Random Differential Equations

In this paper, we are concerned with the construction of numerical schemes for linear random differential equations with discrete delay. For the linear deterministic differential equation with discrete delay, a recent contribution proposed a family of non-standard finite difference (NSFD) methods from an exact numerical scheme on the whole domain. The family of NSFD schemes had increasing order of accuracy, was dynamically consistent, and possessed simple computational properties compared to the exact scheme. In the random setting, when the two equation coefficients are bounded random variables and the initial condition is a regular stochastic process, we prove that the randomized NSFD schemes converge in the mean square (m.s.) sense. M.s. convergence allows for approximating the expectation and the variance of the solution stochastic process. In practice, the NSFD scheme is applied with symbolic inputs, and afterward the statistics are explicitly computed by using the linearity of the expectation. This procedure permits retaining the increasing order of accuracy of the deterministic counterpart. Some numerical examples illustrate the approach. The theoretical m.s. convergence rate is supported numerically, even when the two equation coefficients are unbounded random variables. M.s. dynamic consistency is assessed numerically. A comparison with Euler’s method is performed. Finally, an example dealing with the time evolution of a photosynthetic bacterial population is presented.

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