Mean Square Convergent Non-Standard Numerical Schemes for Linear Random Differential Equations with Delay
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Título: | Mean Square Convergent Non-Standard Numerical Schemes for Linear Random Differential Equations with Delay |
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Autor/es: | Calatayud, Julia | Cortés, Juan Carlos | Jornet, Marc | Rodríguez, Francisco |
Grupo/s de investigación o GITE: | Análisis de Datos y Modelización de Procesos en Biología y Geociencias |
Centro, Departamento o Servicio: | Universidad de Alicante. Departamento de Matemática Aplicada |
Palabras clave: | Delay random differential equation | Non-standard finite difference method | Mean square convergence |
Área/s de conocimiento: | Matemática Aplicada |
Fecha de publicación: | 24-ago-2020 |
Editor: | MDPI |
Cita bibliográfica: | Calatayud J, Cortés JC, Jornet M, Rodríguez F. Mean Square Convergent Non-Standard Numerical Schemes for Linear Random Differential Equations with Delay. Mathematics. 2020; 8(9):1417. https://doi.org/10.3390/math8091417 |
Resumen: | 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. |
Patrocinador/es: | This work has been supported by the Spanish Ministerio de Economía, Industria y Competitividad (MINECO), the Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017–89664–P. |
URI: | http://hdl.handle.net/10045/108719 |
ISSN: | 2227-7390 |
DOI: | 10.3390/math8091417 |
Idioma: | eng |
Tipo: | info:eu-repo/semantics/article |
Derechos: | © 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). |
Revisión científica: | si |
Versión del editor: | https://doi.org/10.3390/math8091417 |
Aparece en las colecciones: | INV - MODDE - Artículos de Revistas |
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