This paper presents a design method of H<sub>2</sub> and H<sub>∞</sub>-feedback control loop for nonlinear smooth gene networks that are in control affine form. Formulaic solution methodology ...This paper presents a design method of H<sub>2</sub> and H<sub>∞</sub>-feedback control loop for nonlinear smooth gene networks that are in control affine form. Formulaic solution methodology for solving the nonlinear partial differential equations, namely the Hamilton-Jacobi-Bellman and Hamilton-Jacobi-Isaacs equations through successive Galerkin’s approximation is implemented and the results are compared. Throughout the implementation, there were several caveats that need to be further resolved for practical applications in general cases. Such issues and the clarification of causes are mathematically established and reviewed.展开更多
In this paper, we combine a finite element approach with the natural boundary element method to stduy the weak solvability and Galerkin approximations of a class of semilinear exterior boundary value problems. Our ana...In this paper, we combine a finite element approach with the natural boundary element method to stduy the weak solvability and Galerkin approximations of a class of semilinear exterior boundary value problems. Our analysis is mainly based on the variational formulation with constraints. We discuss the error estimate of the finite element solution and obtain the asymptotic rate of convergence O（h^n) Finally, we also give two numerical examples.展开更多
This paper presents a fully spectral discretization method for solving KdV equations with periodic boundary conditions.Chebyshev pseudospectral approximation in the time direction and Fourier Galerkin approximation in...This paper presents a fully spectral discretization method for solving KdV equations with periodic boundary conditions.Chebyshev pseudospectral approximation in the time direction and Fourier Galerkin approximation in the spatial direction.The expansion coefficients are determined by minimizing an object funictional.Rapid convergence of the method is proved.展开更多
For solving Burgers' equation with periodic boundary conditions, this paper preseats a fully spectral discretisation method: Fourier Galerkin approximation in the spatial direction and Chebyshev pseudospectral app...For solving Burgers' equation with periodic boundary conditions, this paper preseats a fully spectral discretisation method: Fourier Galerkin approximation in the spatial direction and Chebyshev pseudospectral approximation in the time direction. The expansion coefficients are determined by means of minimizing an object functional, and rapid convergence of the method is proved.展开更多
Based on a recent result on linking stochastic differential equations on R^d to （finite-dimensional） Burger-KPZ type nonlinear parabolic partial differential equations, we utilize Galerkin type finite-dimensional ap...Based on a recent result on linking stochastic differential equations on R^d to （finite-dimensional） Burger-KPZ type nonlinear parabolic partial differential equations, we utilize Galerkin type finite-dimensional approximations to characterize the path-independence of the density process of Girsanov transformation for the infinite-dimensionl stochastic evolution equations. Our result provides a link of infinite-dimensional semi-linear stochastic differential equations to infinite-dimensional Burgers-KPZ type nonlinear parabolic partial differential equations. As an application, this characterization result is applied to stochastic heat equation in one space dimension over the unit interval.展开更多
文摘This paper presents a design method of H<sub>2</sub> and H<sub>∞</sub>-feedback control loop for nonlinear smooth gene networks that are in control affine form. Formulaic solution methodology for solving the nonlinear partial differential equations, namely the Hamilton-Jacobi-Bellman and Hamilton-Jacobi-Isaacs equations through successive Galerkin’s approximation is implemented and the results are compared. Throughout the implementation, there were several caveats that need to be further resolved for practical applications in general cases. Such issues and the clarification of causes are mathematically established and reviewed.
文摘In this paper, we combine a finite element approach with the natural boundary element method to stduy the weak solvability and Galerkin approximations of a class of semilinear exterior boundary value problems. Our analysis is mainly based on the variational formulation with constraints. We discuss the error estimate of the finite element solution and obtain the asymptotic rate of convergence O（h^n) Finally, we also give two numerical examples.
文摘This paper presents a fully spectral discretization method for solving KdV equations with periodic boundary conditions.Chebyshev pseudospectral approximation in the time direction and Fourier Galerkin approximation in the spatial direction.The expansion coefficients are determined by minimizing an object funictional.Rapid convergence of the method is proved.
文摘For solving Burgers' equation with periodic boundary conditions, this paper preseats a fully spectral discretisation method: Fourier Galerkin approximation in the spatial direction and Chebyshev pseudospectral approximation in the time direction. The expansion coefficients are determined by means of minimizing an object functional, and rapid convergence of the method is proved.
文摘Based on a recent result on linking stochastic differential equations on R^d to （finite-dimensional） Burger-KPZ type nonlinear parabolic partial differential equations, we utilize Galerkin type finite-dimensional approximations to characterize the path-independence of the density process of Girsanov transformation for the infinite-dimensionl stochastic evolution equations. Our result provides a link of infinite-dimensional semi-linear stochastic differential equations to infinite-dimensional Burgers-KPZ type nonlinear parabolic partial differential equations. As an application, this characterization result is applied to stochastic heat equation in one space dimension over the unit interval.
