Three-Dimensional Nonlinear Integral Operator with the Modelling of Majorant Function

 تقدم هذه الورقة البحثية طريقة  لايجاد الحل التقريبي لمؤثر فولتيرا التكاملي  الثلاثي الأبعاد غير الخطي في  R3. حيث يتم استخدام مفهوم (Majorant function) وباستخدام طريقة نيوتن المعدلة  لتحويل مؤثر فولتيرا التكاملي  الثلاثي الأبعاد غير الخطي  إلى متتالية  لمؤثر فولتيرا التكاملي  الثلاثي الأبعاد الخطي ومن يتم استخدام طريقة (Gaussian-Legendre)  التربيعية لايجاد الحل التقريبي لمؤثر فولتيرا التكاملي  الثلاثي الأبعاد الخطي من خلال التعامل مع نظام جبري خطي.تم مناقشة وجود ووحدانية الحل للطريقة المستخدمة مع اعطاء أمثلة توضيحية لإظهار دقة وكفاءة الطريقة.In this paper, the process for finding an approximate solution of nonlinear three-dimensional (3D) Volterra type integral operator equation (N3D-VIOE) in R3 is introduced. The modelling of the majorant function (MF) with the modified Newton method (MNM) is employed to convert N3D-VIOE to the linear 3D Volterra type integral operator equation (L3D-VIOE). The method of trapezoidal rule (TR) and collocation points are utilized to determine the approximate solution of L3D-VIOE by dealing with the linear form of the algebraic system. The existence of the approximate solution and its uniqueness are proved, and illustrative examples are provided to show the accuracy and efficiency of the model. Mathematical Subject Classification (2010):  45P05, 45G10, 47H9

An information field theory approach to Bayesian state and parameter estimation in dynamical systems

Dynamical system state estimation and parameter calibration problems are ubiquitous across science and engineering. Bayesian approaches to the problem are the gold standard as they allow for the quantification of uncertainties and enable the seamless fusion of different experimental modalities. When the dynamics are discrete and stochastic, one may employ powerful techniques such as Kalman, particle, or variational filters. Practitioners commonly apply these methods to continuous-time, deterministic dynamical systems after discretizing the dynamics and introducing fictitious transition probabilities. However, approaches based on time-discretization suffer from the curse of dimensionality since the number of random variables grows linearly with the number of time-steps. Furthermore, the introduction of fictitious transition probabilities is an unsatisfactory solution because it increases the number of model parameters and may lead to inference bias. To address these drawbacks, the objective of this paper is to develop a scalable Bayesian approach to state and parameter estimation suitable for continuous-time, deterministic dynamical systems. Our methodology builds upon information field theory. Specifically, we construct a physics-informed prior probability measure on the function space of system responses so that functions that satisfy the physics are more likely. This prior allows us to quantify model form errors. We connect the system's response to observations through a probabilistic model of the measurement process. The joint posterior over the system responses and all parameters is given by Bayes' rule. To approximate the intractable posterior, we develop a stochastic variational inference algorithm. In summary, the developed methodology offers a powerful framework for Bayesian estimation in dynamical systems

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

Effective drifts in dynamical systems with multiplicative noise: A review of recent progress

Noisy dynamical models are employed to describe a wide range of phenomena. Since exact modeling of these phenomena requires access to their microscopic dynamics, whose time scales are typically much shorter than the observable time scales, there is often need to resort to effective mathematical models such as stochastic differential equations (SDEs). In particular, here we consider effective SDEs describing the behavior of systems in the limits when natural time scales become very small. In the presence of multiplicative noise (i.e. noise whose intensity depends upon the system's state), an additional drift term, called noise-induced drift or effective drift, appears. The nature of this noise-induced drift has been recently the subject of a growing number of theoretical and experimental studies. Here, we provide an extensive review of the state of the art in this field. After an introduction, we discuss a minimal model of how multiplicative noise affects the evolution of a system. Next, we consider several case studies with a focus on recent experiments: the Brownian motion of a microscopic particle in thermal equilibrium with a heat bath in the presence of a diffusion gradient; the limiting behavior of a system driven by a colored noise modulated by a multiplicative feedback; and the behavior of an autonomous agent subject to sensorial delay in a noisy environment. This allows us to present the experimental results, as well as mathematical methods and numerical techniques, that can be employed to study a wide range of systems. At the end we give an application-oriented overview of future projects involving noise-induced drifts, including both theory and experiment. © 2016 IOP Publishing Ltd

Mathematical Methods, Modelling and Applications

This volume deals with novel high-quality research results of a wide class of mathematical models with applications in engineering, nature, and social sciences. Analytical and numeric, deterministic and uncertain dimensions are treated. Complex and multidisciplinary models are treated, including novel techniques of obtaining observation data and pattern recognition. Among the examples of treated problems, we encounter problems in engineering, social sciences, physics, biology, and health sciences. The novelty arises with respect to the mathematical treatment of the problem. Mathematical models are built, some of them under a deterministic approach, and other ones taking into account the uncertainty of the data, deriving random models. Several resulting mathematical representations of the models are shown as equations and systems of equations of different types: difference equations, ordinary differential equations, partial differential equations, integral equations, and algebraic equations. Across the chapters of the book, a wide class of approaches can be found to solve the displayed mathematical models, from analytical to numeric techniques, such as finite difference schemes, finite volume methods, iteration schemes, and numerical integration methods

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

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described