Liquan Mei | Xi'an Jiaotong University (original) (raw)

Papers by Liquan Mei

Radial basis functions are well-known successful tools for interpolation and quasi-interpolation ... more Radial basis functions are well-known successful tools for interpolation and quasi-interpolation of the equal distance or scattered data in high dimensions. Furthermore, their truly mesh-free nature motivated researchers to use them to deal with partial differential equations(PDEs). With more than twenty-year development, radial basis functions have become a powerful and popular method in solving ordinary and partial differential equations now. In this paper, based on the idea of quasi-interpolation and radial basis functions approximation, a fast and accurate numerical method is developed for multi-dimensions Black-Scholes equation for valuation of european options prices on three underlying assets. The advantage of this method is that it does not require solving a resultant full matrix, therefore as indicated in the the numerical computation, this method is effective for option pricing problem.

Physics Letters A, 2012

ABSTRACT In this Letter, the fractional variational iteration method using Heʼs polynomials is im... more ABSTRACT In this Letter, the fractional variational iteration method using Heʼs polynomials is implemented to construct compacton solutions and solitary pattern solutions of nonlinear time-fractional dispersive KdV-type equations involving Jumarieʼs modified Riemann–Liouville derivative. The method yields solutions in the forms of convergent series with easily calculable terms. The obtained results show that the considered method is quite effective, promising and convenient for solving fractional nonlinear dispersive equations. It is found that the time-fractional parameter significantly changes the soliton amplitude of the solitary waves.

Chinese Annals of Mathematics, Series B, 2010

Applied Mathematics and Computation, 2007

Applied Mathematics-A Journal of Chinese Universities, 1999

Applied Numerical Mathematics

Advanced Intelligent Computing Theories and Applications. With Aspects of Artificial Intelligence

In this paper what we study is twofold. Firstly a new weighted support vector machine(WSVM) is in... more In this paper what we study is twofold. Firstly a new weighted support vector machine(WSVM) is introduced, in which different roles of samples are considered. It better processes the singularity in the sample than SVM. Secondly based on grid search and the solution path algorithm, a new algorithm is given to quickly find the optimum parameters. Numerical results show the effectiveness and the stability of the algorithm.

The Astrophysical Journal

We investigate the dependency of the inflow-wind structure of a hot accretion flow on the kinemat... more We investigate the dependency of the inflow-wind structure of a hot accretion flow on the kinematic viscosity coefficient. In this regard, we propose a model for the kinematic viscosity coefficient to mimic the behavior of the magnetorotational instability that would be maximal at the rotation axis. Then, we compare our model with two other prescriptions from numerical simulations of the accretion flow. We solve two-dimensional hydrodynamic equations of hot accretion flows in the presence of thermal conduction. The self-similar approach is also adopted in the radial direction. We calculate the properties of the inflow and the wind such as velocity, density, and angular momentum for three models of the kinematic viscosity prescription. On inspection, we find that in our suggested model the wind is less efficient at extracting the angular momentum outward where the self-similar solutions are applied than it is in two other models. The solutions obtained in this paper might be applicab...

International Journal of Computer Mathematics

From the eigenvalue problem theory, we see that the convergence rate of the biharmonic eigenvalue... more From the eigenvalue problem theory, we see that the convergence rate of the biharmonic eigenvalues obtained by the mixed method in I. Bab ska and J. Osborn, [Eigenvalue Problems, Handbook of Numerical Analysis, Vol. II, North-Holland, Amsterdam, 1991.] is for . In this paper, we give a presentation of the lowest-order virtual element method for the approximation of Kirchhoff plate vibration problem. This discrete scheme is based on a conforming formulation, following the variational formulation of Ciarlet–Raviart method, which allows us to make use of simpler and lower-regularity virtual element space. By using the classical spectral approximation theory in functional analysis, we prove the spectral approximation and optimal convergence order for the eigenvalues. Finally, some numerical experiments are presented, which show that the proposed numerical scheme can achieve the optimal convergence order.

G ' G )-expansion method Double non-traveling wave solution (2 + 1)-dimensional Painleve inte... more G ' G )-expansion method Double non-traveling wave solution (2 + 1)-dimensional Painleve integrable Burgers equation (2 + 1)-dimensional breaking soliton equation a b s t r a c t To seek the exact double non-traveling wave solutions of nonlinear partial differential equations, the compound ( G ' G )-expansion method is firstly proposed in this paper. With the aid of symbolic computation, this new method is applied to construct double non- traveling wave solutions of (2 + 1)-dimensional Painleve integrable Burgers equation and (2 + 1)-dimensional breaking soliton equation. As a result, abundant double non-traveling wave solutions including double hyperbolic function solutions, double trigonometric func- tion solutions, double rational solutions, and a series of complexiton solutions of these two equations are obtained via the proposed method. These exact solutions contain arbitrary functions, which may be helpful to explain some complex phenomena. When the parame- ters are ta...

We present a residual-based a posteriori error estimator for the hybrid high-order (HHO) method f... more We present a residual-based a posteriori error estimator for the hybrid high-order (HHO) method for the Stokes model problem. Both the proposed HHO method and error estimator are valid in two and three dimensions and support arbitrary approximation orders on fairly general meshes. The upper bound and lower bound of the error estimator are proved, in which proof, a key ingredient is a novel stabilizer employed in the discrete scheme. By using the given estimator, adaptive algorithm of HHO method is designed to solve model problem. Finally, the expected theoretical results are numerically demonstrated on a variety of meshes for model problem.

In this paper, we give a presentation of virtual element method for the approximation of the vibr... more In this paper, we give a presentation of virtual element method for the approximation of the vibration problem of clamped Kirchhoff plate, which involves the biharmonic eigenvalue problem. Following the theory of Babǔska and Osborn, the error estimates of the discrete scheme for the degree k ≥ 2 of polynomials are standard results. However, when considering the case k = 1, we can not apply the technical framework of classical eigenvalue problem directly. Based on the spectral approximation theory, the theory of mixed virtual element method and mixed finite element method for the Stokes problem, the convergence analysis for eigenvalues and eigenfunctions is analyzed and proved. Finally, some numerical experiments are reported to show that the proposed numerical scheme can achieve the optimal convergence order.

Computers & Mathematics with Applications

Journal of Scientific Computing

Journal of Computational and Applied Mathematics

Numerical Algorithms

This paper presents two second-order and linear finite element schemes for the multi-dimensional ... more This paper presents two second-order and linear finite element schemes for the multi-dimensional nonlinear time-fractional Schrödinger equation. In the first numerical scheme, we adopt the L 2-1 σ formula to approximate the Caputo derivative. However, this scheme requires storing the numerical solution at all previous time steps. In order to overcome this drawback, we develop the F L 2 mathcalFL2\mathcal {F}L2mathcalFL2 -1 σ formula to construct the second numerical scheme, which reduces the computational storage and cost. We prove that both the L 2-1 σ and F L 2 mathcalFL2\mathcal {F}L2mathcalFL2 -1 σ formulas satisfy the three assumptions of the generalized discrete fractional Grönwall inequality. Furthermore, combining with the temporal-spatial error splitting argument, we rigorously prove the unconditional stability and optimal error estimates of these two numerical schemes, which do not require any time-step restrictions dependent on the spatial mesh size. Numerical examples in two and three dimensions are given to illustrate our theoretical results and show that the second scheme based on F L 2 mathcalFL2\mathcal {F}L2mathcalFL2 -1 σ formula can reduce CPU time significantly compared with the first scheme based on L 2-1 σ formula.

Journal of Scientific Computing

In this paper, we construct an efficient Hermite–Galerkin spectral method for the nonlinear react... more In this paper, we construct an efficient Hermite–Galerkin spectral method for the nonlinear reaction–diffusion equations with distributed-order fractional Laplacian in multidimensional unbounded domains. By applying Gauss–Legendre quadrature rule for the distributed integral term, we first approximate the original distributed-order fractional problem by the multi-term fractional-in-space differential equation. Applying Hermite–Galerkin spectral method in space and backward difference method in time, we establish semi-implicit fully discrete scheme. For two- and three-dimensional cases of the original fractional problem, the linear systems are solved by the preconditioned conjugate gradients method. The main advantage of our method is that the original fractional problem is directly solved in the unbounded domains, thus avoiding the errors introduced by the domain truncations. The stability analysis is rigourously established, which shows that our scheme is unconditionally stable under suitable assumption on the nonlinear term. Several numerical examples are presented to validate both stability and accuracy of the numerical method. The numerical results of the fractional Allen–Cahn, Gray–Scott, and Belousov–Zhabotinskii models show that our semi-implicit methods produce good numerical solutions.

Applied Mathematics and Computation

Radial basis functions are well-known successful tools for interpolation and quasi-interpolation ... more Radial basis functions are well-known successful tools for interpolation and quasi-interpolation of the equal distance or scattered data in high dimensions. Furthermore, their truly mesh-free nature motivated researchers to use them to deal with partial differential equations(PDEs). With more than twenty-year development, radial basis functions have become a powerful and popular method in solving ordinary and partial differential equations now. In this paper, based on the idea of quasi-interpolation and radial basis functions approximation, a fast and accurate numerical method is developed for multi-dimensions Black-Scholes equation for valuation of european options prices on three underlying assets. The advantage of this method is that it does not require solving a resultant full matrix, therefore as indicated in the the numerical computation, this method is effective for option pricing problem.

Physics Letters A, 2012

ABSTRACT In this Letter, the fractional variational iteration method using Heʼs polynomials is im... more ABSTRACT In this Letter, the fractional variational iteration method using Heʼs polynomials is implemented to construct compacton solutions and solitary pattern solutions of nonlinear time-fractional dispersive KdV-type equations involving Jumarieʼs modified Riemann–Liouville derivative. The method yields solutions in the forms of convergent series with easily calculable terms. The obtained results show that the considered method is quite effective, promising and convenient for solving fractional nonlinear dispersive equations. It is found that the time-fractional parameter significantly changes the soliton amplitude of the solitary waves.

Chinese Annals of Mathematics, Series B, 2010

Applied Mathematics and Computation, 2007

Applied Mathematics-A Journal of Chinese Universities, 1999

Applied Numerical Mathematics

Advanced Intelligent Computing Theories and Applications. With Aspects of Artificial Intelligence

In this paper what we study is twofold. Firstly a new weighted support vector machine(WSVM) is in... more In this paper what we study is twofold. Firstly a new weighted support vector machine(WSVM) is introduced, in which different roles of samples are considered. It better processes the singularity in the sample than SVM. Secondly based on grid search and the solution path algorithm, a new algorithm is given to quickly find the optimum parameters. Numerical results show the effectiveness and the stability of the algorithm.

The Astrophysical Journal

We investigate the dependency of the inflow-wind structure of a hot accretion flow on the kinemat... more We investigate the dependency of the inflow-wind structure of a hot accretion flow on the kinematic viscosity coefficient. In this regard, we propose a model for the kinematic viscosity coefficient to mimic the behavior of the magnetorotational instability that would be maximal at the rotation axis. Then, we compare our model with two other prescriptions from numerical simulations of the accretion flow. We solve two-dimensional hydrodynamic equations of hot accretion flows in the presence of thermal conduction. The self-similar approach is also adopted in the radial direction. We calculate the properties of the inflow and the wind such as velocity, density, and angular momentum for three models of the kinematic viscosity prescription. On inspection, we find that in our suggested model the wind is less efficient at extracting the angular momentum outward where the self-similar solutions are applied than it is in two other models. The solutions obtained in this paper might be applicab...

International Journal of Computer Mathematics

From the eigenvalue problem theory, we see that the convergence rate of the biharmonic eigenvalue... more From the eigenvalue problem theory, we see that the convergence rate of the biharmonic eigenvalues obtained by the mixed method in I. Bab ska and J. Osborn, [Eigenvalue Problems, Handbook of Numerical Analysis, Vol. II, North-Holland, Amsterdam, 1991.] is for . In this paper, we give a presentation of the lowest-order virtual element method for the approximation of Kirchhoff plate vibration problem. This discrete scheme is based on a conforming formulation, following the variational formulation of Ciarlet–Raviart method, which allows us to make use of simpler and lower-regularity virtual element space. By using the classical spectral approximation theory in functional analysis, we prove the spectral approximation and optimal convergence order for the eigenvalues. Finally, some numerical experiments are presented, which show that the proposed numerical scheme can achieve the optimal convergence order.

G ' G )-expansion method Double non-traveling wave solution (2 + 1)-dimensional Painleve inte... more G ' G )-expansion method Double non-traveling wave solution (2 + 1)-dimensional Painleve integrable Burgers equation (2 + 1)-dimensional breaking soliton equation a b s t r a c t To seek the exact double non-traveling wave solutions of nonlinear partial differential equations, the compound ( G ' G )-expansion method is firstly proposed in this paper. With the aid of symbolic computation, this new method is applied to construct double non- traveling wave solutions of (2 + 1)-dimensional Painleve integrable Burgers equation and (2 + 1)-dimensional breaking soliton equation. As a result, abundant double non-traveling wave solutions including double hyperbolic function solutions, double trigonometric func- tion solutions, double rational solutions, and a series of complexiton solutions of these two equations are obtained via the proposed method. These exact solutions contain arbitrary functions, which may be helpful to explain some complex phenomena. When the parame- ters are ta...

We present a residual-based a posteriori error estimator for the hybrid high-order (HHO) method f... more We present a residual-based a posteriori error estimator for the hybrid high-order (HHO) method for the Stokes model problem. Both the proposed HHO method and error estimator are valid in two and three dimensions and support arbitrary approximation orders on fairly general meshes. The upper bound and lower bound of the error estimator are proved, in which proof, a key ingredient is a novel stabilizer employed in the discrete scheme. By using the given estimator, adaptive algorithm of HHO method is designed to solve model problem. Finally, the expected theoretical results are numerically demonstrated on a variety of meshes for model problem.

In this paper, we give a presentation of virtual element method for the approximation of the vibr... more In this paper, we give a presentation of virtual element method for the approximation of the vibration problem of clamped Kirchhoff plate, which involves the biharmonic eigenvalue problem. Following the theory of Babǔska and Osborn, the error estimates of the discrete scheme for the degree k ≥ 2 of polynomials are standard results. However, when considering the case k = 1, we can not apply the technical framework of classical eigenvalue problem directly. Based on the spectral approximation theory, the theory of mixed virtual element method and mixed finite element method for the Stokes problem, the convergence analysis for eigenvalues and eigenfunctions is analyzed and proved. Finally, some numerical experiments are reported to show that the proposed numerical scheme can achieve the optimal convergence order.

Computers & Mathematics with Applications

Journal of Scientific Computing

Journal of Computational and Applied Mathematics

Numerical Algorithms

This paper presents two second-order and linear finite element schemes for the multi-dimensional ... more This paper presents two second-order and linear finite element schemes for the multi-dimensional nonlinear time-fractional Schrödinger equation. In the first numerical scheme, we adopt the L 2-1 σ formula to approximate the Caputo derivative. However, this scheme requires storing the numerical solution at all previous time steps. In order to overcome this drawback, we develop the F L 2 mathcalFL2\mathcal {F}L2mathcalFL2 -1 σ formula to construct the second numerical scheme, which reduces the computational storage and cost. We prove that both the L 2-1 σ and F L 2 mathcalFL2\mathcal {F}L2mathcalFL2 -1 σ formulas satisfy the three assumptions of the generalized discrete fractional Grönwall inequality. Furthermore, combining with the temporal-spatial error splitting argument, we rigorously prove the unconditional stability and optimal error estimates of these two numerical schemes, which do not require any time-step restrictions dependent on the spatial mesh size. Numerical examples in two and three dimensions are given to illustrate our theoretical results and show that the second scheme based on F L 2 mathcalFL2\mathcal {F}L2mathcalFL2 -1 σ formula can reduce CPU time significantly compared with the first scheme based on L 2-1 σ formula.

Journal of Scientific Computing

In this paper, we construct an efficient Hermite–Galerkin spectral method for the nonlinear react... more In this paper, we construct an efficient Hermite–Galerkin spectral method for the nonlinear reaction–diffusion equations with distributed-order fractional Laplacian in multidimensional unbounded domains. By applying Gauss–Legendre quadrature rule for the distributed integral term, we first approximate the original distributed-order fractional problem by the multi-term fractional-in-space differential equation. Applying Hermite–Galerkin spectral method in space and backward difference method in time, we establish semi-implicit fully discrete scheme. For two- and three-dimensional cases of the original fractional problem, the linear systems are solved by the preconditioned conjugate gradients method. The main advantage of our method is that the original fractional problem is directly solved in the unbounded domains, thus avoiding the errors introduced by the domain truncations. The stability analysis is rigourously established, which shows that our scheme is unconditionally stable under suitable assumption on the nonlinear term. Several numerical examples are presented to validate both stability and accuracy of the numerical method. The numerical results of the fractional Allen–Cahn, Gray–Scott, and Belousov–Zhabotinskii models show that our semi-implicit methods produce good numerical solutions.

Applied Mathematics and Computation