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### Mean square numerical solution of random differential equations: Facts and possibilities

J.C. Cortés, L. Jódar, L. Villafuerte
2007 Computers and Mathematics with Applications
This paper deals with the construction of numerical solutions of random initial value differential problems.  ...  The random Euler method is presented and the conditions for the mean square convergence are established.  ...  This motivates the main goal of this paper that is to introduce the random Euler method for constructing reliable numerical solutions of the problem (1.1) in the sense that approximations be mean square  ...

### Mean-square numerical approximations to random periodic solutions of stochastic differential equations

Qingyi Zhan
This paper is devoted to the possibility of mean-square numerical approximations to random periodic solutions of dissipative stochastic differential equations.  ...  The existence and expression of random periodic solutions are established.  ...  This work is supported by NSFC (Nos. 11021101, 11290142, and 91130003) .  ...

### Introduction: Stochastic differential equations [chapter]

2010 Markov Processes, Feller Semigroups and Evolution Equations
Stationary solutions It is possible to construct stochastic differential equations that have stationary distributions.  ...  The solution is to re-interpret the limit in Definition 4 as a limit in mean square.  ...  However, since ∆W t is of order √ ∆t, we have and the last term on the right is of order ∆t. In fact, and so that the variance of ∆W 2 t is very low for small values of ∆t.  ...

### Linear Partial Differential Equations with Random Forcing

Frederic Y. M. Wan
1972 Studies in applied mathematics (Cambridge)
In any event, the fact that equations (1.9) and (1.10) form a closed system is only a special feature of (1.7) and not true in general for (1.1).  ...  .\$) alone appears in (1.9) and (1.10) so that these become two equations for this sum and (u&. But there does not seem to be any way for us to determine (u:) and (u;) separately.  ...  Acknowledgements The author is grateful to his colleague, Professor Steven Orszag, who read and commented on the manuscript.  ...

### Nonnormality and stochastic differential equations

D. J. Higham, X. Mao
2006 BIT Numerical Mathematics
Here, we show that nonnormality also manifests itself for stochastic differential equations.  ...  Here, we show that nonnormality also manifests itself for stochastic differential equations.  ...  XM thanks the Royal Society and the EPSRC (EP/C548779/1) for financial support.  ...

### Stochastic Differential Equations [chapter]

2005 Financial Derivatives in Theory and Practice
In this paper, I present a brief introduction to methods of stochastic differential equations which may be of relevance to problems of orbital dynamics considered in this workshop.  ...  Included are the Ito calculus, limit theorems for stochastic equations with rapidly varying noise, and the theory of large deviations.  ...  An Ito stochastic differential equation results if b, a are deterministic functions of X, and possibly of time t.  ...

### Burgers Equation Revisited [article]

V. Gurarie
2003 arXiv   pre-print
In particular, the probability that the velocity gradient is negative appears to be p ≈ 0.21 ± 0.01 irrespective of the details of the random force.  ...  This paper studies the 1D pressureless turbulence (the Burgers equation). It shows that reliable numerics in this problem is very easy to produce if one properly discretizes the Burgers equation.  ...  With this restriction, the average velocity vanishes u = 0 and it makes sense to define the root mean square velocity as U rms = u 2 .  ...

### Complex Langevin equations and Schwinger–Dyson equations

Gerald Guralnik, Cengiz Pehlevan
2009 Nuclear Physics B
Stationary distributions of complex Langevin equations are shown to be the complexified path integral solutions of the Schwinger-Dyson equations of the associated quantum field theory.  ...  Specific examples in zero dimensions and on a lattice are given. Relevance to the study of quantum field theory phase space is discussed.  ...  This work is supported in part by funds provided by the US Department of Energy (DoE) under DE-FG02-91ER40688-TaskD.  ...

### Random non-autonomous second order linear differential equations: mean square analytic solutions and their statistical properties

J. Calatayud, J.-C. Cortés, M. Jornet, L. Villafuerte
process solution in the random mean square sense.  ...  In this paper we study random non-autonomous second order linear differential equations by taking advantage of the powerful theory of random difference equations.  ...  By Theorem 3.3, the mean square solution of (24), X(t), is defined and is mean square analytic on (-1, 1). In Tables 12 and 13 we present the numerical experiments.  ...

### Bayesian Numerical Methods for Nonlinear Partial Differential Equations [article]

Junyang Wang, Jon Cockayne, Oksana Chkrebtii, T. J. Sullivan, Chris. J. Oates
2021 arXiv   pre-print
The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied.  ...  However, nonlinear partial differential equations (PDEs) pose substantial challenges from an inferential perspective, most notably the absence of explicit conditioning formula.  ...  JC was supported by Wave 1 of the UKRI Strategic Priorities Fund under the EPSRC Grant EP/T001569/1, particularly the "Digital Twins for Complex Engineering Systems" theme within that grant, and the Alan  ...

### Computational solution of stochastic differential equations

Timothy Sauer
2013 Wiley Interdisciplinary Reviews: Computational Statistics
t, the random variable W t is normally distributed with mean 0 and variance t. (2) For each t 1 < t 2 , the normal random variable W t 2 − W t 1 is independent of the random variable W t 1 , and in fact  ...  Stochastic differential equations (SDEs) provide accessible mathematical models that combine deterministic and probabilistic components of dynamic behavior.  ...  CONCLUSION Numerical methods for the solution of SDEs are essential for the analysis of random phenomena.  ...

### An improved Milstein method for stiff stochastic differential equations

Zhengwei Yin, Siqing Gan
The correction term is derived from an approximation of the difference between the exact solution of stochastic differential equations and the Milstein continuous-time extension.  ...  For a linear scalar test equation, it is shown that the mean-square stability domain of the method is much bigger than that of the Milstein method.  ...  This work is supported by the National Natural Science Foundation of China  ...

### Kernel-based Methods for Stochastic Partial Differential Equations [article]

Qi Ye
2015 arXiv   pre-print
The constructions of these Gaussian random variables provide the kernel-based approximate solutions of the stochastic models.  ...  In the numerical examples of the stochastic Poisson and heat equations, we show that the approximate probability distributions are well-posed for various kinds of kernels such as the compactly supported  ...  Igor Cialenco and my advisor, Prof. Gregory E. Fasshauer, for their guide and assistance of this research topic at Illinois Institute of Technique, Chicago.  ...

### Numerical Procedures for Random Differential Equations

Mohamed Ben Said, Lahcen Azrar, Driss Sarsri
2018 Journal of Applied Mathematics
This methodological approach leads to equations with a reduced number of random variables and thus to a large reduction of CPU time and memory required for the numerical solution.  ...  For random excitations with an arbitrary number of uncertain variables, the IRCM is couplet to the superposition method leading to successive random differential equations with the same main random operator  ...  Acknowledgments This project was funded by the CNRST and the Moroccan Ministry of High Education and Scientific Research with Project PPR2 coordinated by Pr.  ...

### The stochastic Mathieu's equation

F. J Poulin, G. R Flierl
2008 Proceedings of the Royal Society A
The power spectrum of the solutions shows that an increase in stochasticity tends to narrow the width of the subharmonic peak and increase the decay away from this peak.  ...  By numerically integrating the system of equations using a symplectic method, we determine the Lyapunov exponents for a wide range of parameters to quantify how the growth rates vary in parameter space  ...  and NSERC and NSF for financial support during the research of this manuscript.  ...
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