24,895 research outputs found

    Lp-solution to the random linear delay differential equation with stochastic forcing term

    Full text link
    [EN] This paper aims at extending a previous contribution dealing with the random autonomous-homogeneous linear differential equation with discrete delay tau > 0, by adding a random forcing term f(t) that varies with time: x'(t) = ax(t) + bx(t-tau) + f(t), t >= 0, with initial condition x(t) = g(t), -tau <= t <= 0. The coefficients a and b are assumed to be random variables, while the forcing term f(t) and the initial condition g(t) are stochastic processes on their respective time domains. The equation is regarded in the Lebesgue space L-p of random variables with finite p-th moment. The deterministic solution constructed with the method of steps and the method of variation of constants, which involves the delayed exponential function, is proved to be an L-p-solution, under certain assumptions on the random data. This proof requires the extension of the deterministic Leibniz's integral rule for differentiation to the random scenario. Finally, we also prove that, when the delay tau tends to 0, the random delay equation tends in L-p to a random equation with no delay. Numerical experiments illustrate how our methodology permits determining the main statistics of the solution process, thereby allowing for uncertainty quantification.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.Cortés, J.; Jornet, M. (2020). Lp-solution to the random linear delay differential equation with stochastic forcing term. Mathematics. 8(6):1-16. https://doi.org/10.3390/math8061013S11686Xiu, D., & Karniadakis, G. E. (2004). Supersensitivity due to uncertain boundary conditions. International Journal for Numerical Methods in Engineering, 61(12), 2114-2138. doi:10.1002/nme.1152Casabán, M.-C., Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., Roselló, M.-D., & Villanueva, R.-J. (2016). A comprehensive probabilistic solution of random SIS-type epidemiological models using the random variable transformation technique. Communications in Nonlinear Science and Numerical Simulation, 32, 199-210. doi:10.1016/j.cnsns.2015.08.009Strand, J. . (1970). Random ordinary differential equations. Journal of Differential Equations, 7(3), 538-553. doi:10.1016/0022-0396(70)90100-2Villafuerte, 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.2017.11.009Nouri, K., & Ranjbar, H. (2014). Mean Square Convergence of the Numerical Solution of Random Differential Equations. Mediterranean Journal of Mathematics, 12(3), 1123-1140. doi:10.1007/s00009-014-0452-8Calatayud, 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). Lp\mathrm {L}^p-calculus Approach to the Random Autonomous Linear Differential Equation with Discrete Delay. Mediterranean Journal of Mathematics, 16(4). doi:10.1007/s00009-019-1370-6Caraballo, 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.2019.03.048Zhou, T. (2014). A Stochastic Collocation Method for Delay Differential Equations with Random Input. Advances in Applied Mathematics and Mechanics, 6(4), 403-418. doi:10.4208/aamm.2012.m38Shi, W., & Zhang, C. (2017). Generalized polynomial chaos for nonlinear random delay differential equations. Applied Numerical Mathematics, 115, 16-31. doi:10.1016/j.apnum.2016.12.004Khusainov, D. Y., Ivanov, A. F., & Kovarzh, I. V. (2009). Solution of one heat equation with delay. Nonlinear Oscillations, 12(2), 260-282. doi:10.1007/s11072-009-0075-3Shaikhet, L. (2016). Stability of equilibrium states of a nonlinear delay differential equation with stochastic perturbations. International Journal of Robust and Nonlinear Control, 27(6), 915-924. doi:10.1002/rnc.3605Benhadri, M., & Zeghdoudi, H. (2018). Mean square asymptotic stability in nonlinear stochastic neutral Volterra-Levin equations with Poisson jumps and variable delays. Functiones et Approximatio Commentarii Mathematici, 58(2). doi:10.7169/facm/1657Santonja, F.-J., & Shaikhet, L. (2012). Analysing Social Epidemics by Delayed Stochastic Models. Discrete Dynamics in Nature and Society, 2012, 1-13. doi:10.1155/2012/530472Liu, L., & Caraballo, T. (2018). Analysis of a Stochastic 2D-Navier–Stokes Model with Infinite Delay. Journal of Dynamics and Differential Equations, 31(4), 2249-2274. doi:10.1007/s10884-018-9703-xLupulescu, V., & Abbas, U. (2011). Fuzzy delay differential equations. Fuzzy Optimization and Decision Making, 11(1), 99-111. doi:10.1007/s10700-011-9112-7Krapivsky, P. L., Luck, J. M., & Mallick, K. (2011). On stochastic differential equations with random delay. Journal of Statistical Mechanics: Theory and Experiment, 2011(10), P10008. doi:10.1088/1742-5468/2011/10/p10008GARRIDO-ATIENZA, M. J., OGROWSKY, A., & SCHMALFUSS, B. (2011). RANDOM DIFFERENTIAL EQUATIONS WITH RANDOM DELAYS. Stochastics and Dynamics, 11(02n03), 369-388. doi:10.1142/s0219493711003358Cortés, J.-C., Villafuerte, L., & Burgos, C. (2017). A Mean Square Chain Rule and its Application in Solving the Random Chebyshev Differential Equation. Mediterranean Journal of Mathematics, 14(1). doi:10.1007/s00009-017-0853-6Cortés, 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.2006.07.017Braumann, C. A., Cortés, J.-C., Jódar, L., & Villafuerte, L. (2018). On the random gamma function: Theory and computing. Journal of Computational and Applied Mathematics, 335, 142-155. doi:10.1016/j.cam.2017.11.045Khusainov, D. Y., & Pokojovy, M. (2015). Solving the Linear 1D Thermoelasticity Equations with Pure Delay. International Journal of Mathematics and Mathematical Sciences, 2015, 1-11. doi:10.1155/2015/47926

    Optimisation of Mobile Communication Networks - OMCO NET

    Get PDF
    The mini conference “Optimisation of Mobile Communication Networks” focuses on advanced methods for search and optimisation applied to wireless communication networks. It is sponsored by Research & Enterprise Fund Southampton Solent University. The conference strives to widen knowledge on advanced search methods capable of optimisation of wireless communications networks. The aim is to provide a forum for exchange of recent knowledge, new ideas and trends in this progressive and challenging area. The conference will popularise new successful approaches on resolving hard tasks such as minimisation of transmit power, cooperative and optimal routing

    Limits and dynamics of stochastic neuronal networks with random heterogeneous delays

    Full text link
    Realistic networks display heterogeneous transmission delays. We analyze here the limits of large stochastic multi-populations networks with stochastic coupling and random interconnection delays. We show that depending on the nature of the delays distributions, a quenched or averaged propagation of chaos takes place in these networks, and that the network equations converge towards a delayed McKean-Vlasov equation with distributed delays. Our approach is mostly fitted to neuroscience applications. We instantiate in particular a classical neuronal model, the Wilson and Cowan system, and show that the obtained limit equations have Gaussian solutions whose mean and standard deviation satisfy a closed set of coupled delay differential equations in which the distribution of delays and the noise levels appear as parameters. This allows to uncover precisely the effects of noise, delays and coupling on the dynamics of such heterogeneous networks, in particular their role in the emergence of synchronized oscillations. We show in several examples that not only the averaged delay, but also the dispersion, govern the dynamics of such networks.Comment: Corrected misprint (useless stopping time) in proof of Lemma 1 and clarified a regularity hypothesis (remark 1

    A Survey on Delay-Aware Resource Control for Wireless Systems --- Large Deviation Theory, Stochastic Lyapunov Drift and Distributed Stochastic Learning

    Full text link
    In this tutorial paper, a comprehensive survey is given on several major systematic approaches in dealing with delay-aware control problems, namely the equivalent rate constraint approach, the Lyapunov stability drift approach and the approximate Markov Decision Process (MDP) approach using stochastic learning. These approaches essentially embrace most of the existing literature regarding delay-aware resource control in wireless systems. They have their relative pros and cons in terms of performance, complexity and implementation issues. For each of the approaches, the problem setup, the general solution and the design methodology are discussed. Applications of these approaches to delay-aware resource allocation are illustrated with examples in single-hop wireless networks. Furthermore, recent results regarding delay-aware multi-hop routing designs in general multi-hop networks are elaborated. Finally, the delay performance of the various approaches are compared through simulations using an example of the uplink OFDMA systems.Comment: 58 pages, 8 figures; IEEE Transactions on Information Theory, 201

    Optimization of an avionic VCSEL-based optical link through large signal characterization

    Get PDF
    Optical communication systems have been widely preferred for network communications, especially for Datacoms Local Area Network links. The optical technology is an excellent candidate for on-board systems due to the potential weight saving and EMC immunity. According to the short length of the link and a cost saving, Vertical Cavity Surface Emitting Laser (VCSEL) and multimode fiber are the best solution for gigabit systems. In this context, we propose a modeling of 850nm VCSEL based on the rate equations analysis to predict the optical interconnect performances (jitter, bit error rate). Our aim is to define the operation conditions of VCSEL under large signal modulation in order to maximize the Extinction Ratio (current IOFF below threshold) without affecting link performances. The VCSEL model is developed to provide large signal modulation response. Biasing below threshold causes stochastic turn-on delay. Fluctuations of this delay occur, due to the spontaneous emission. This leads to additional turn-on jitter. These stochastic effects are included in the model by adding the Langevin photon and electron noise sources. The VCSEL behavior under high-speed modulation is studied to observe the transient response and extract the resonance frequency, overshoot and turn-on delay. The associated jitter is evaluated with the standard deviation of the turn-on delay probability density function. Simulations of stochastic and deterministic jitters are realized under different conditions of modulation (OFF current levels). Comparing simulations with measurement results carried out on VCSEL and a short haul gigabit link validates the approach
    • …
    corecore