Finite element methods for first order systems with applications to mixed equations (Mathematical Sciences)

Issued as Annual progress report and Final report, Project no. G-37-63

An unconditionally stable nonstandard finite difference method applied to a mathematical model of HIV infection

We formulate and analyze an unconditionally stable nonstandard finite difference method for a mathematical model of HIV transmission dynamics. The dynamics of this model are studied using the qualitative theory of dynamical systems. These qualitative features of the continuous model are preserved by the numerical method that we propose in this paper. This method also preserves the positivity of the solution, which is one of the essential requirements when modeling epidemic diseases. Robust numerical results confirming theoretical investigations are provided. Comparisons are also made with the other conventional approaches that are routinely used for such problems.IS

Gaussian solitary waves and compactons in Fermi-Pasta-Ulam lattices with Hertzian potentials

We consider a class of fully-nonlinear Fermi-Pasta-Ulam (FPU) lattices, consisting of a chain of particles coupled by fractional power nonlinearities of order $\alpha >1$. This class of systems incorporates a classical Hertzian model describing acoustic wave propagation in chains of touching beads in the absence of precompression. We analyze the propagation of localized waves when $\alpha$ is close to unity. Solutions varying slowly in space and time are searched with an appropriate scaling, and two asymptotic models of the chain of particles are derived consistently. The first one is a logarithmic KdV equation, and possesses linearly orbitally stable Gaussian solitary wave solutions. The second model consists of a generalized KdV equation with H\"older-continuous fractional power nonlinearity and admits compacton solutions, i.e. solitary waves with compact support. When $\alpha \rightarrow 1^+$, we numerically establish the asymptotically Gaussian shape of exact FPU solitary waves with near-sonic speed, and analytically check the pointwise convergence of compactons towards the limiting Gaussian profile

Models of Delay Differential Equations

This book gathers a number of selected contributions aimed at providing a balanced picture of the main research lines in the realm of delay differential equations and their applications to mathematical modelling. The contributions have been carefully selected so that they cover interesting theoretical and practical analysis performed in the deterministic and the stochastic settings. The reader will find a complete overview of recent advances in ordinary and partial delay differential equations with applications in other multidisciplinary areas such as Finance, Epidemiology or Engineerin

Mathematical modelling of virus RSV: qualitative properties, numerical solutions and validation for the case of the region of Valencia

El objetivo de esta memoria se centra en primer lugar en la modelización del comportamiento de enfermedades estacionales mediante sistemas de ecuaciones diferenciales y en el estudio de las propiedades dinámicas tales como positividad, periocidad, estabilidad de las soluciones analíticas y la construcción de esquemas numéricos para las aproximaciones de las soluciones numéricas de sistemas de ecuaciones diferenciales de primer orden no lineales, los cuales modelan el comportamiento de enfermedades infecciosas estacionales tales como la transmisión del virus Respiratory Syncytial Virus (RSV). Se generalizan dos modelos matemáticos de enfermedades estacionales y se demuestran que tiene soluciones periódicas usando un Teorema de Coincidencia de Jean Mawhin. Para corroborar los resultados analíticos, se desarrollan esquemas numéricos usando las técnicas de diferencias finitas no estándar desarrolladas por Ronald Michens y el método de la transformada diferencial, los cuales permiten reproducir el comportamiento dinámico de las soluciones analíticas, tales como positividad y periocidad. Finalmente, las simulaciones numéricas se realizan usando los esquemas implementados y parámetros deducidos de datos clínicos De La Región de Valencia de personas infectadas con el virus RSV. Se confrontan con las que arrojan los métodos de Euler, Runge Kutta y la rutina de ODE45 de Matlab, verificándose mejores aproximaciones para tamaños de paso mayor a los que usan normalmente estos esquemas tradicionales.Arenas Tawil, AJ. (2009). Mathematical modelling of virus RSV: qualitative properties, numerical solutions and validation for the case of the region of Valencia [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/8316Palanci

Nonstandard finite difference schemes for general linear delay differential systems

This paper deals with the construction of non‐standard finite difference methods for coupled linear delay differential systems in the general case of non‐commuting matrix coefficients. Based on an expression for the exact solution of the continuous initial value vector delay problem, a family of non‐standard numerical methods of increasing orders is defined. Numerical examples show that the new proposed numerical methods preserve the stability properties of the exact solutions. This work extends previous results for delay systems with commuting matrix coefficients, allowing the use of the new numerical non‐standard methods in more general problems.M.A.C. and F.R. acknowledge funding by Ministerio de Economía y Competitividad grant number CGL2017-89804-R

Workshop on Verification and Theorem Proving for Continuous Systems (NetCA Workshop 2005)

Oxford, UK, 26 August 200

Mean square convergent non-standard numerical schemes for linear random differential equations with delay

[EN] 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.This work has been supported by the Spanish Ministerio de Economia, Industria y Competitividad (MINECO), the Agencia Estatal de Investigacion (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-P.Calatayud, J.; Cortés, J.; Jornet, M.; Rodríguez, F. (2020). Mean square convergent non-standard numerical schemes for linear random differential equations with delay. Mathematics. 8(9):1-17. https://doi.org/10.3390/math8091417S11789Bocharov, G. A., & Rihan, F. A. (2000). Numerical modelling in biosciences using delay differential equations. Journal of Computational and Applied Mathematics, 125(1-2), 183-199. doi:10.1016/s0377-0427(00)00468-4Jackson, M., & Chen-Charpentier, B. M. (2017). Modeling plant virus propagation with delays. Journal of Computational and Applied Mathematics, 309, 611-621. doi:10.1016/j.cam.2016.04.024Chen-Charpentier, B. M., & Diakite, I. (2016). A mathematical model of bone remodeling with delays. Journal of Computational and Applied Mathematics, 291, 76-84. doi:10.1016/j.cam.2014.11.025Kyrychko, Y. N., & Hogan, S. J. (2010). On the Use of Delay Equations in Engineering Applications. Journal of Vibration and Control, 16(7-8), 943-960. doi:10.1177/1077546309341100Harding, L., & Neamţu, M. (2016). A Dynamic Model of Unemployment with Migration and Delayed Policy Intervention. Computational Economics, 51(3), 427-462. doi:10.1007/s10614-016-9610-3Mickens, R. E. (2005). Dynamic consistency: a fundamental principle for constructing nonstandard finite difference schemes for differential equations. Journal of Difference Equations and Applications, 11(7), 645-653. doi:10.1080/10236190412331334527Patidar, K. C. (2016). Nonstandard finite difference methods: recent trends and further developments. Journal of Difference Equations and Applications, 22(6), 817-849. doi:10.1080/10236198.2016.1144748García, M. A., Castro, M. A., Martín, J. A., & Rodríguez, F. (2018). Exact and nonstandard numerical schemes for linear delay differential models. Applied Mathematics and Computation, 338, 337-345. doi:10.1016/j.amc.2018.06.029Castro, M. Á., García, M. A., Martín, J. A., & Rodríguez, F. (2019). Exact and Nonstandard Finite Difference Schemes for Coupled Linear Delay Differential Systems. Mathematics, 7(11), 1038. doi:10.3390/math7111038Villafuerte, L., Braumann, C. A., Cortés, J.-C., & Jódar, L. (2010). Random differential operational calculus: Theory and applications. Computers & Mathematics with Applications, 59(1), 115-125. doi:10.1016/j.camwa.2009.08.061Cortés, J.-C., Jódar, L., Roselló, M.-D., & Villafuerte, L. (2012). Solving initial and two-point boundary value linear random differential equations: A mean square approach. Applied Mathematics and Computation, 219(4), 2204-2211. doi:10.1016/j.amc.2012.08.066Calatayud, J., Cortés, J.-C., Jornet, M., & Villafuerte, L. (2018). Random non-autonomous second order linear differential equations: mean square analytic solutions and their statistical properties. Advances in Difference Equations, 2018(1). doi:10.1186/s13662-018-1848-8Calatayud, J., Cortés, J.-C., & Jornet, M. (2019). Improving the Approximation of the First- and Second-Order Statistics of the Response Stochastic Process to the Random Legendre Differential Equation. Mediterranean Journal of Mathematics, 16(3). doi:10.1007/s00009-019-1338-6Licea, J. A., Villafuerte, L., & Chen-Charpentier, B. M. (2013). Analytic and numerical solutions of a Riccati differential equation with random coefficients. Journal of Computational and Applied Mathematics, 239, 208-219. doi:10.1016/j.cam.2012.09.040Burgos, C., Calatayud, J., Cortés, J.-C., & Villafuerte, L. (2018). Solving a class of random non-autonomous linear fractional differential equations by means of a generalized mean square convergent power series. Applied Mathematics Letters, 78, 95-104. doi:10.1016/j.aml.2017.11.009Calatayud, J., Cortés, J.-C., & Jornet, M. (2019). Random differential equations with discrete delay. Stochastic Analysis and Applications, 37(5), 699-707. doi:10.1080/07362994.2019.1608833Calatayud, J., Cortés, J.-C., & Jornet, M. (2019). $$\mathrm {L}^p$$-calculus Approach to the Random Autonomous Linear Differential Equation with Discrete Delay. Mediterranean Journal of Mathematics, 16(4). doi:10.1007/s00009-019-1370-6Cortés, J. C., & Jornet, M. (2020). Lp-Solution to the Random Linear Delay Differential Equation with a Stochastic Forcing Term. Mathematics, 8(6), 1013. doi:10.3390/math8061013Caraballo, T., Cortés, J.-C., & Navarro-Quiles, A. (2019). Applying the random variable transformation method to solve a class of random linear differential equation with discrete delay. Applied Mathematics and Computation, 356, 198-218. doi:10.1016/j.amc.2019.03.048Cortés, J. C., Jódar, L., & Villafuerte, L. (2007). Numerical solution of random differential equations: A mean square approach. Mathematical and Computer Modelling, 45(7-8), 757-765. doi:10.1016/j.mcm.2006.07.017Cortés, J. C., Jódar, L., & Villafuerte, L. (2007). Mean square numerical solution of random differential equations: Facts and possibilities. Computers & Mathematics with Applications, 53(7), 1098-1106. doi:10.1016/j.camwa.2006.05.030El-Tawil, M. A. (2005). The approximate solutions of some stochastic differential equations using transformations. Applied Mathematics and Computation, 164(1), 167-178. doi:10.1016/j.amc.2004.04.062Calatayud, J., Cortés, J.-C., Díaz, J. A., & Jornet, M. (2019). Density function of random differential equations via finite difference schemes: a theoretical analysis of a random diffusion-reaction Poisson-type problem. Stochastics, 92(4), 627-641. doi:10.1080/17442508.2019.1645849Calatayud, J., Cortés, J.-C., Díaz, J. A., & Jornet, M. (2020). Constructing reliable approximations of the probability density function to the random heat PDE via a finite difference scheme. Applied Numerical Mathematics, 151, 413-424. doi:10.1016/j.apnum.2020.01.012Burgos, C., Cortés, J.-C., Villafuerte, L., & Villanueva, R.-J. (2020). Mean square convergent numerical solutions of random fractional differential equations: Approximations of moments and density. Journal of Computational and Applied Mathematics, 378, 112925. doi:10.1016/j.cam.2020.112925Strand, J. . (1970). Random ordinary differential equations. Journal of Differential Equations, 7(3), 538-553. doi:10.1016/0022-0396(70)90100-2Buckwar, E. (2000). Introduction to the numerical analysis of stochastic delay differential equations. Journal of Computational and Applied Mathematics, 125(1-2), 297-307. doi:10.1016/s0377-0427(00)00475-1Antonio Dorini, F., & Sampaio, R. (2012). Some Results on the Random Wear Coefficient of the Archard Model. Journal of Applied Mechanics, 79(5). doi:10.1115/1.400645