Some notes to extend the study on random non-autonomous second order linear differential equations appearing in Mathematical Modeling

The objective of this paper is to complete certain issues from our recent contribution [J. Calatayud, J.-C. Cort\'es, M. Jornet, L. Villafuerte, Random non-autonomous second order linear differential equations: mean square analytic solutions and their statistical properties, Advances in Difference Equations, 2018:392, 1--29 (2018)]. We restate the main theorem therein that deals with the homogeneous case, so that the hypotheses are clearer and also easier to check in applications. Another novelty is that we tackle the non-homogeneous equation with a theorem of existence of mean square analytic solution and a numerical example. We also prove the uniqueness of mean square solution via an habitual Lipschitz condition that extends the classical Picard Theorem to mean square calculus. In this manner, the study on general random non-autonomous second order linear differential equations with analytic data processes is completely resolved. Finally, we relate our exposition based on random power series with polynomial chaos expansions and the random differential transform method, being the latter a reformulation of our random Fr\"obenius method.Comment: 15 pages, 0 figures, 2 table

Numerical solution of random differential models

This paper deals with the construction of a numerical solution of random initial value problems by means of a random improved Euler method. Conditions for the mean square convergence of the proposed method are established. Finally, an illustrative example is included in which the main statistics properties such as the mean and the variance of the stochastic approximation solution process are given. © 2011 Elsevier Ltd.This work has been partially supported by the Spanish M.C.Y.T. grants MTM2009-08587, DPI2010-20891-C02-01, Universidad Politecnica de Valencia grant PAID06-09-2588 and Mexican Conacyt.Cortés López, JC.; Jódar Sánchez, LA.; Villafuerte Altuzar, L.; Company Rossi, R. (2011). Numerical solution of random differential models. Mathematical and Computer Modelling. 54(7):1846-1851. https://doi.org/10.1016/j.mcm.2010.12.037S1846185154

Dissipative vortex solitons in 2D-lattices

We report the existence of stable symmetric vortex-type solutions for two-dimensional nonlinear discrete dissipative systems governed by a cubic-quintic complex Ginzburg-Landau equation. We construct a whole family of vortex solitons with a topological charge S = 1. Surprisingly, the dynamical evolution of unstable solutions of this family does not alter significantly their profile, instead their phase distribution completely changes. They transform into two-charges swirl-vortex solitons. We dynamically excite this novel structure showing its experimental feasibility.Comment: 4 pages, 20 figure

Angular dependence of magnetic properties in Ni nanowire arrays

The angular dependence of the remanence and coercivity of Ni nanowire arrays produced inside the pores of anodic alumina membranes has been studied. By comparing our analytical calculations with our measurements, we conclude that the magnetization reversal in this array is driven by means of the nucleation and propagation of a transverse wall. A simple model based on an adapted Stoner-Wohlfarth model is used to explain the angular dependence of the coercivity

Analysis of the random heat equation via approximate density functions

[EN] In this paper we study the randomized heat equation with homogeneous boundary conditions. The diffusion coefficient is assumed to be a random variable and the initial condition is treated as a stochastic process. The solution of this randomized partial differential equation problem is a stochastic process, which is given by a random series obtained via the classical method of separation of variables. Any stochastic process is determined by its finite-dimensional joint distributions. In this paper, the goal is to obtain approximations to the probability density function of the solution (the first finite-dimensional distributions) under mild conditions. Since the solution is expressed as a random series, we perform approximations to its probability density function. Several illustrative examples are shown.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. (2021). Analysis of the random heat equation via approximate density functions. Romanian Reports in Physics. 73(2):1-10. http://hdl.handle.net/10251/181144S11073

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

[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). $$\mathrm {L}^p$$-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

Analytic solution to the generalized delay diffusion equation with uncertain inputs in the random Lebesgue sense

[EN] In this paper, we deal with the randomized generalized diffusion equation with delay:u(t)(t, x) = a(2)u(xx)(t, x) + b(2)u(xx)(t - tau, x),t > tau,0 = 0;u(t,x)=phi(t,x),0 0andl > 0are constant. The coefficientsa(2)andb(2)are nonnegative random variables, and the initial condition phi(t, x)and the solutionu(t, x)are random fields. The separation of variables method develops a formal series solution. We prove that the series satisfies the delay diffusion problem in the random Lebesgue sense rigorously. By truncating the series, the expectation and the variance of the random-field solution can be approximated.Secretaria de Estado de Investigacion, Desarrollo e Innovacion, Grant/Award Number: MTM2017-89664-P; Spanish Ministerio de Economia, Industria y Competitividad (MINECO); Agencia Estatal de Investigacion (AEI); Fondo Europeo de Desarrollo Regional (FEDER UE), Grant/Award Number: MTM2017-89664-PCortés, J.; Jornet, M. (2021). Analytic solution to the generalized delay diffusion equation with uncertain inputs in the random Lebesgue sense. Mathematical Methods in the Applied Sciences. 44(2):2265-2272. https://doi.org/10.1002/mma.6921S22652272442Smith, H. (2011). An Introduction to Delay Differential Equations with Applications to the Life Sciences. Texts in Applied Mathematics. doi:10.1007/978-1-4419-7646-8Driver, R. D. (1977). Ordinary and Delay Differential Equations. Applied Mathematical Sciences. doi:10.1007/978-1-4684-9467-9Kolmanovskii, V., & Myshkis, A. (1999). Introduction to the Theory and Applications of Functional Differential Equations. doi:10.1007/978-94-017-1965-0Wu, J. (1996). Theory and Applications of Partial Functional Differential Equations. Applied Mathematical Sciences. doi:10.1007/978-1-4612-4050-1Diekmann, O., Verduyn Lunel, S. M., van Gils, S. A., & Walther, H.-O. (1995). Delay Equations. Applied Mathematical Sciences. doi:10.1007/978-1-4612-4206-2Hale, J. K. (1977). Theory of Functional Differential Equations. Applied Mathematical Sciences. doi:10.1007/978-1-4612-9892-2Travis, C. C., & Webb, G. F. (1974). Existence and stability for partial functional differential equations. Transactions of the American Mathematical Society, 200, 395-395. doi:10.1090/s0002-9947-1974-0382808-3Bocharov, 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.2014.11.025ErneuxT.Applied Delay Differential Equations Surveys and Tutorials in the Applied Mathematical Sciences Series:Springer New York;2009.Kyrychko, 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:10.1177/1077546309341100Matsumoto, A., & Szidarovszky, F. (2009). Delay Differential Nonlinear Economic Models. Nonlinear Dynamics in Economics, Finance and Social Sciences, 195-214. doi:10.1007/978-3-642-04023-8_11Harding, L., & Neamţu, M. (2016). A Dynamic Model of Unemployment with Migration and Delayed Policy Intervention. Computational Economics, 51(3), 427-462. doi:10.1007/s10614-016-9610-3Xiu, D. (2010). Numerical Methods for Stochastic Computations. doi:10.2307/j.ctv7h0skvLe Maître, O. P., & Knio, O. M. (2010). Spectral Methods for Uncertainty Quantification. Scientific Computation. doi:10.1007/978-90-481-3520-2NeckelT RuppF.Random Differential Equations in Scientific Computing. Walter de Gruyter;2013.Villafuerte, 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.2009.08.061Calatayud, 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/s00009-019-1370-6Martín, J. A., Rodríguez, F., & Company, R. (2004). Analytic solution of mixed problems for thegeneralized diffusion equation with delay. Mathematical and Computer Modelling, 40(3-4), 361-369. doi:10.1016/j.mcm.2003.10.046Martínez-Cervantes, G. (2016). Riemann integrability versus weak continuity. Journal of Mathematical Analysis and Applications, 438(2), 840-855. doi:10.1016/j.jmaa.2016.01.054Cortés, J. C., Sevilla-Peris, P., & Jódar, L. (2005). Analytic-numerical approximating processes of diffusion equation with data uncertainty. Computers & Mathematics with Applications, 49(7-8), 1255-1266. doi:10.1016/j.camwa.2004.05.015Khusainov, 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/s11072-009-0075-3Calatayud, J., Cortés, J. C., & Jornet, M. (2018). Uncertainty quantification for random parabolic equations with nonhomogeneous boundary conditions on a bounded domain via the approximation of the probability density function. Mathematical Methods in the Applied Sciences, 42(17), 5649-5667. doi:10.1002/mma.5333BotevZ RidderA.Variance reductionWiley StatsRef: Statistics Reference Online;2017:1–6.https://doi.org/10.1002/9781118445112.stat0797