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Uncertain Neutron Diffusion
Published in S. Chakraverty, Sukanta Nayak, Neutron Diffusion, 2017
The concept of stochastic differential equation (SDE) was initiated by the great philosopher Einstein in 1905 (Sauer 2012). A mathematical connection between the microscopic random motion of particles and the macroscopic diffusion equation was presented. Later, it was seen that the SDE model plays a prominent role in a range of application areas such as mathematics, physics, chemistry, mechanics, biology, microelectronics, economics and finance. Earlier, SDEs were solved using the Ito integral as an exact method, which is discussed in Malinowski and Michta (2011). But using the exact method proved difficult to study nontrivial problems and hence approximation methods came to be used. In 1982, Rumelin (1982) defined general Runge–Kutta approximations for the solution of SDEs and an explicit form of the correction term has been given. Kloeden and Platen (1992) discussed about the numerical solutions of the SDE. Discrete time strong and weak approximation methods were used by Platen (1999) to investigate the solution of SDEs. Next, Higham (2001) gave a major contribution to solve the approximate solutions of SDEs. Furthermore, Higham and Kloeden (2005) investigated nonlinear SDEs numerically. They presented two implicit methods for the Ito SDEs with Poisson-driven jumps. The first method is a split-step extension of the backward Euler method, and the second method arises from the introduction of a compensated, martingale, form of the Poisson process. Hayes and Allen (2005) solved the stochastic point kinetic reactor problem. They modelled the point stochastic reactor problem into the ordinary time-dependent SDE and studied the stochastic behaviour of the neutron flux.
Nanoscale Fluid Dynamics
Published in Klaus D. Sattler, 21st Century Nanoscience – A Handbook, 2020
Ravi Radhakrishnan, N. Ramakrishnan, David M. Eckmann, Portonovo S. Ayyaswamy
Here, an(y) = ∫ rnwdr. The solution to the Fokker–Planck equation yields the probability distribution of particles which contains the information on Brownian effects. At equilibrium (i.e., when all the time dependence vanishes), the solution can be required to conform to the solutions from equilibrium statistical mechanics. This approach leads to a class of identities for transport coefficients, including the famous Stokes–Einstein diffusivity for particles undergoing Brownian motion to be discussed later in this chapter. Moreover, there is a one-to-one correspondence between the Fokker–Planck equation and a stochastic differential equation (SDE) that encodes for a trajectory of a Brownian particle. The generalized Fokker–Planck equation is written in terms of a generalized order parameter S, given by ∂P(S,t)∂t=DkBT∂∂SP∂F(S)∂S+D∂2P∂S2,
Dynamic System Models and Basic Concepts
Published in Jitendra R. Raol, Girija Gopalratnam, Bhekisipho Twala, Nonlinear Filtering, 2017
Jitendra R. Raol, Girija Gopalratnam, Bhekisipho Twala
where A is the matrix containing the basic dynamic characteristics, B is the control (input) matrix and x0 is the initial condition/value of the state. Stochastic differential equations are used to represent systems which are affected by random disturbances which could be multiplicative or additive. A linear stochastic differential equation is expressed as
Optimal control for a nonlinear stochastic PDE model of cancer growth
Published in Optimization, 2023
Sakine Esmaili, M. R. Eslahchi, Delfim F. M. Torres
In this example, we have solved SOCP for The problem is solved using a combination of euler-maruyama and collocation methods. In this method, the problem on the space domain is discretized using Legendre-Gauss-Lobatto nodes, which results in a SDE. Then, the SDE is solved using the Euler-Maruyama method. In Figure 2, for N=19 and , the effect of stochastic terms on the dynamic and density of alive tumour cells is illustrated. In Figures 3 and 4, for N=13 and , the effects of optimal control variables on the density of tumour cells are illustrated. It is shown that the density of alive cells and proliferative cells decreases under the effects of optimal control. In Figure 5, for N=13 and , sample paths of are shown, where η and are the radius of tumour without control and under the effect of control, respectively. It is illustrated that the optimal control variables result in a relative decrease of tumour radius (Figure 5).
Stability of stochastic differential equations driven by the time-changed Lévy process with impulsive effects
Published in International Journal of Systems Science, 2021
Xiuwei Yin, Wentao Xu, Guangjun Shen
Stochastic differential equations (SDEs) have been applied in various areas, including biology, physics, engineering, economics and finance (see, for example, Sobczyk, 2013 and the references therein). The stability has been one of the most important topics in the study of stochastic differential equations. It has been widely studied in different senses, such as stochastically stable, stochastically asymptotically stable, moment exponentially stable, almost surely stable, mean square polynomial stable (see, for example, Applebaum & Siakalli, 2009; F. Deng et al., 2012; Fei et al., 2020; X. Jin et al., 2020; Li & Mao, 2020; W. Liu et al., 2014; Zhu, 2018; Song et al., 2016, 2018; Tang et al., 2020; Wang, Chen, & Zhuang, 2020; Wang, Chen, Zhuang, & Song, 2020; X. Wu et al., 2019; Yuan, 2005, we can also see Mao, 2007; Mao & Yuan, 2006; Yin & Zhu, 2009 for systematic introduction of stabilities).
ϵ-Nash mean-field games for linear-quadratic systems with random jumps and applications
Published in International Journal of Control, 2021
However, it's worth pointing that out that in the above literatures the dynamics in the stochastic large population systems are driven by Brownian motions. In a financial market, the analysis of price evolution does reveal sudden and rare breaks logically accounted for exogenous events on information. From a probabilistic point of view, such behaviour is naturally modelled by jump-diffusion processes, that is, the individual dynamics evolve by stochastic differential equations (SDEs) driven by both Brownian motions and Poisson random measures. Stochastic processes with random jumps can be widely applied to modelling fluctuations in the financial market, both for risk management and option pricing purposes (see Cadenillas, 2002; Cont & Tankov, 2004; Shi & Wu, 2010, 2011, etc).