Computational Methods For Nonlocal Partial Differential Equations With Applications In Stochastic Dynamical Systems

Dynamical systems have been extensively considered in the modeling of complex phenomena in biology,chemistry,physics and engineering.Theoretical solutions of the systems are difficult to obtain.Therefore,numerical investigation provides a reasonable way to understand the models.At present,the majority of dynamical systems are characterized by local integer-order equations.However,for the systems with long memory and nonGaussian behaviours with large jumps,the nonlocal fractional-order models are much more reasonable compared with the integer-order models.For example,nonlocal Fokker-Planck equations can be used to describe the gene transcription driven by stable Lévy noise(nonGaussian noise).In this thesis,we consider the computational methods and applications for nonlocal partial differential equations arising in stochastic dynamical systems.This thesis is organized as follows.In Chapter 1,we briefly introduce the development and application of numerical methods to nonlocal partial differential equations.In Chapter 2,we develop several different conserving compact finite difference schemes for solving nonlinear Schr¨odinger equation with wave operator.It is proved that the numerical solutions are bounded and the numerical methods are convergent and stable in maximum norm.Moreover,we apply the Richardson extrapolation to improve the convergence order of the proposed methods in the temporal direction.Finally,several numerical experiments are proposed to illustrate the theoretical results.In Chapter 3,we propose a high-order numerical scheme for solving two-dimensional Riesz space fractional nonlinear reaction-diffusion equations.The numerical scheme is constructed by using a quasi-compact scheme for spatial discretization and Crank-Nicolson scheme for temporal discretization.Furthermore,an alternating direction implicit(ADI)scheme is constructed by introducing small perturbation term.It is proved under some appropriate conditions that the quasi-compact ADI scheme is uniquely solvable and conditionally convergent.A comparison of the proposed method,a extrapolated Crank-Nicolson compact ADI method and a Crank-Nicolson ADI method is given.The numerical results show that the proposed method is comparable.Moreover,we also employ the quasi-compact ADI method to simulate a coupled fractional FitzHugh-Nagume model.In Chapter 4,we adopt the nonlocal equations to characterize the stochastic dynamical behaviors of a genetic regulatory system.We study the most probable trajectories of the concentration evolution for the transcription factor activator in the genetic regulation system,with non-Gaussian stable Lévy noise in the synthesis reaction rate taking into account.We calculate the most probable trajectory by spatially maximizing the probability density of the system path,i.e.,the solution of the associated nonlocal Fokker-Planck equation.We especially examine those most probable trajectories from low concentration state to high concentration state(i.e.,the likely transcription regime)for certain parameters,in order to gain insights into the transcription processes and the tipping time for the transcription likely to occur.Moreover,we have found some peculiar or counter-intuitive phenomena in this gene model system.These findings provide insights for further experimental research,in order to achieve or to avoid specific gene transcriptions,with possible relevance for medical advances.In Chapter 5,we summarize main contributions and discuss some future relevant research issues.