This paper presents discontinuous Legendre wavelet Galerkin (DLWG) approach for solving one-dimensional advection-diffusion equation (ADE). Variational formulation of this type equation and corresponding numerical flu...This paper presents discontinuous Legendre wavelet Galerkin (DLWG) approach for solving one-dimensional advection-diffusion equation (ADE). Variational formulation of this type equation and corresponding numerical fluxes are devised by utilizing the advantages of both the Legendre wavelet bases and discontinuous Galerkin (DG) method. The distinctive features of the proposed method are its simple applicability for a variety of boundary conditions and able to effectively approximate the solution of PDEs with less storage space and execution. The results of a numerical experiment are provided to verify the efficiency of the designed new technique.展开更多
Aerosol modelling is very important to study and simulate the behavior of aerosol dynamics in atmospheric environment.In this paper,we consider the general nonlinear aerosol dynamic equations which describe the evolut...Aerosol modelling is very important to study and simulate the behavior of aerosol dynamics in atmospheric environment.In this paper,we consider the general nonlinear aerosol dynamic equations which describe the evolution of the aerosol distribution.Continuous time and discrete time wavelet Galerkin methods are proposed for solving this problem.By using the Schauder’s fixed point theorem and the variational technique,the global existence and uniqueness of solution of continuous time wavelet numerical methods are established for the nonlinear aerosol dynamics with sufficiently smooth initial conditions.Optimal error estimates are obtained for both continuous and discrete time wavelet Galerkin schemes.Numerical examples are given to show the efficiency of the wavelet technique.展开更多
A sampling approximation for a function defined on a bounded interval is proposed by combining the Coiflet-type wavelet expansion and the boundary extension technique. Based on such a wavelet approximation scheme, a G...A sampling approximation for a function defined on a bounded interval is proposed by combining the Coiflet-type wavelet expansion and the boundary extension technique. Based on such a wavelet approximation scheme, a Galerkin procedure is developed for the spatial discretization of the generalized nonlinear Schr6dinger （NLS） equa- tions, and a system of ordinary differential equations for the time dependent unknowns is obtained. Then, the classical fourth-order explicit Runge-Kutta method is used to solve this semi-discretization system. To justify the present method, several widely considered problems are solved as the test examples, and the results demonstrate that the proposed wavelet algorithm has much better accuracy and a faster convergence rate in space than many existing numerical methods.展开更多
In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on [0,1] a...In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on [0,1] are used and are utilized as a basis in Galerkin method to approximate the solution of integral equations. Then, in some examples the mentioned wavelets are compared with each other.展开更多
The aim of this paper is to investigate the numerical solution of the hypersingular integral equation reduced by the harmonic equation. First, we transform the hypersingular integral equation into 2π-periodic hypersi...The aim of this paper is to investigate the numerical solution of the hypersingular integral equation reduced by the harmonic equation. First, we transform the hypersingular integral equation into 2π-periodic hypersingular integral equation with the map x=cot（θ/2）. Second, we initiate the study of the multiscale Galerkin method for the 2π-periodic hypersingular integral equation. The trigonometric wavelets are used as trial functions. Consequently, the 2j＋1 × 2j＋1 stiffness matrix Kj can be partitioned j×j block matrices. Furthermore, these block matrices are zeros except main diagonal block matrices. These main diagonal block matrices are symmetrical and circulant matrices, and hence the solution of the associated linear algebraic system can be solved with the fast Fourier transform and the inverse fast Fourier transform instead of the inverse matrix. Finally, we provide several numerical examples to demonstrate our method has good accuracy even though the exact solutions are multi-peak and almost singular.展开更多
文摘This paper presents discontinuous Legendre wavelet Galerkin (DLWG) approach for solving one-dimensional advection-diffusion equation (ADE). Variational formulation of this type equation and corresponding numerical fluxes are devised by utilizing the advantages of both the Legendre wavelet bases and discontinuous Galerkin (DG) method. The distinctive features of the proposed method are its simple applicability for a variety of boundary conditions and able to effectively approximate the solution of PDEs with less storage space and execution. The results of a numerical experiment are provided to verify the efficiency of the designed new technique.
基金This work was supported partly by National Engineering and Science Research Council of Canada and by Canadian Foundation for Climate and Atmospheric Sciences.
文摘Aerosol modelling is very important to study and simulate the behavior of aerosol dynamics in atmospheric environment.In this paper,we consider the general nonlinear aerosol dynamic equations which describe the evolution of the aerosol distribution.Continuous time and discrete time wavelet Galerkin methods are proposed for solving this problem.By using the Schauder’s fixed point theorem and the variational technique,the global existence and uniqueness of solution of continuous time wavelet numerical methods are established for the nonlinear aerosol dynamics with sufficiently smooth initial conditions.Optimal error estimates are obtained for both continuous and discrete time wavelet Galerkin schemes.Numerical examples are given to show the efficiency of the wavelet technique.
基金supported by the National Natural Science Foundation of China(Nos.11502103 and11421062)the Open Fund of State Key Laboratory of Structural Analysis for Industrial Equipment of China(No.GZ15115)
文摘A sampling approximation for a function defined on a bounded interval is proposed by combining the Coiflet-type wavelet expansion and the boundary extension technique. Based on such a wavelet approximation scheme, a Galerkin procedure is developed for the spatial discretization of the generalized nonlinear Schr6dinger （NLS） equa- tions, and a system of ordinary differential equations for the time dependent unknowns is obtained. Then, the classical fourth-order explicit Runge-Kutta method is used to solve this semi-discretization system. To justify the present method, several widely considered problems are solved as the test examples, and the results demonstrate that the proposed wavelet algorithm has much better accuracy and a faster convergence rate in space than many existing numerical methods.
文摘In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on [0,1] are used and are utilized as a basis in Galerkin method to approximate the solution of integral equations. Then, in some examples the mentioned wavelets are compared with each other.
文摘The aim of this paper is to investigate the numerical solution of the hypersingular integral equation reduced by the harmonic equation. First, we transform the hypersingular integral equation into 2π-periodic hypersingular integral equation with the map x=cot（θ/2）. Second, we initiate the study of the multiscale Galerkin method for the 2π-periodic hypersingular integral equation. The trigonometric wavelets are used as trial functions. Consequently, the 2j＋1 × 2j＋1 stiffness matrix Kj can be partitioned j×j block matrices. Furthermore, these block matrices are zeros except main diagonal block matrices. These main diagonal block matrices are symmetrical and circulant matrices, and hence the solution of the associated linear algebraic system can be solved with the fast Fourier transform and the inverse fast Fourier transform instead of the inverse matrix. Finally, we provide several numerical examples to demonstrate our method has good accuracy even though the exact solutions are multi-peak and almost singular.
