In this paper, we consider the initial-boundary value problem of two-dimensional first-order linear hyperbolic equation with variable coefficients. By using the upwind difference method to discretize the spatial deriv...In this paper, we consider the initial-boundary value problem of two-dimensional first-order linear hyperbolic equation with variable coefficients. By using the upwind difference method to discretize the spatial derivative term and the forward and backward Euler method to discretize the time derivative term, the explicit and implicit upwind difference schemes are obtained respectively. It is proved that the explicit upwind scheme is conditionally stable and the implicit upwind scheme is unconditionally stable. Then the convergence of the schemes is derived. Numerical examples verify the results of theoretical analysis.展开更多
In this paper a singularly perturbed linear second order hyperbolic problem with zeroth order reduced equation is discussed. Firstly, an energy inequality of the solution and an estimate of the remainder term of the a...In this paper a singularly perturbed linear second order hyperbolic problem with zeroth order reduced equation is discussed. Firstly, an energy inequality of the solution and an estimate of the remainder term of the asymptotic solution are given. Then an exponentially fitted difference scheme is developed in an equidistant mesh. Finally, uniform convergence in small parameter is proved in the sense of discrete energy norm.展开更多
The conservative form and singular perturbed ordinary differential equation with periodic boundary value problem were studied, and a conservative difference scheme was constructed. Using the method of decomposing the ...The conservative form and singular perturbed ordinary differential equation with periodic boundary value problem were studied, and a conservative difference scheme was constructed. Using the method of decomposing the singular term from its solution and combining an asymptotic expansion of the equation, it is proved that the scheme converges uniformly to the solution of differential equation with order one.展开更多
Much effort has been devoted to researching the common Rosenau equation, but the numerical method of it has not been studied. In this paper, a conservative Crank-Nicolson difference scheme for an initial-boundary valu...Much effort has been devoted to researching the common Rosenau equation, but the numerical method of it has not been studied. In this paper, a conservative Crank-Nicolson difference scheme for an initial-boundary value problem of the generalized Rosenau equation is proposed. Existence and uniqueness of numerical solutions are derived. By method of discrete energy, the second order convergence and stability are discussed. Numerical examples demonstrate the theoretical results.展开更多
In this paper, using nonuniform mesh and exponentially fitted difference method, a uniformly convergent difference scheme for an initial-boundary value problem of linear parabolic differential equation with the nonsmo...In this paper, using nonuniform mesh and exponentially fitted difference method, a uniformly convergent difference scheme for an initial-boundary value problem of linear parabolic differential equation with the nonsmooth boundary layer function with respect to small parameter e is given, and error estimate and numerical result are also given.展开更多
In this paper we discuss, an initial-boundary value problem of hyperbolic type with first derivative with respect to x. The asymptotic solution is constructed and its uniform validity is proved under weader compatibil...In this paper we discuss, an initial-boundary value problem of hyperbolic type with first derivative with respect to x. The asymptotic solution is constructed and its uniform validity is proved under weader compatibility conditions. Then we develop an exponentially fitted difference scheme and establish discrete energy inequality. Finally, we prove that the solution of difference problem uniformly converges to the solution of the original problem.展开更多
In this paper, based on the idea of El-Mistikawy and Werle[1] we construct a difference scheme for a singularly perturbed self-adjoint ordinary differential equation in conservation form. We prove that it is a uniform...In this paper, based on the idea of El-Mistikawy and Werle[1] we construct a difference scheme for a singularly perturbed self-adjoint ordinary differential equation in conservation form. We prove that it is a uniformly convergent second order scheme.展开更多
In this paper, we construct a class of difference schemes with fitted factors for a singular perturbation problem of a self-adjoint ordinary differential equation. Using a different method from [1], by analyzing the t...In this paper, we construct a class of difference schemes with fitted factors for a singular perturbation problem of a self-adjoint ordinary differential equation. Using a different method from [1], by analyzing the truncation errors of schemes, we give the sufficient conditions under which the solution of lite difference scheme converges uniformly to the solution of the differential equation. From this we propose several specific schemes under weaker conditions, and give much higher order of uniform convergence, and applying them to example, obtain the numerical results.展开更多
The conservative form and singular perturbed ordinary differential equation with periodic boundary value problem were studied, and a conservative difference scheme was constructed. Using the method of decomposing the ...The conservative form and singular perturbed ordinary differential equation with periodic boundary value problem were studied, and a conservative difference scheme was constructed. Using the method of decomposing the singular term from its solution and combining an asymptotic expansion of the equation, it is proved that the scheme converges uniformly to the solution of differential equation with order one.展开更多
Korteweg-de Vries equation is a nonlinear evolutionary partial differential equation that is of third order in space.For the approximation to this equation with the initial and boundary value conditions using the fini...Korteweg-de Vries equation is a nonlinear evolutionary partial differential equation that is of third order in space.For the approximation to this equation with the initial and boundary value conditions using the finite difference method,the difficulty is how to construct matched finite difference schemes at all the inner grid points.In this paper,two finite difference schemes are constructed for the problem.The accuracy is second-order in time and first-order in space.The first scheme is a two-level nonlinear implicit finite difference scheme and the second one is a three-level linearized finite difference scheme.The Browder fixed point theorem is used to prove the existence of the nonlinear implicit finite difference scheme.The con-servation,boundedness,stability,convergence of these schemes are discussed and analyzed by the energy method together with other techniques.The two-level non-linear finite difference scheme is proved to be unconditionally convergent and the three-level linearized one is proved to be conditionally convergent.Some numerical examples illustrate the efficiency of the proposed finite difference schemes.展开更多
Time fractional diffusion equation is usually used to describe the problems involving non-Markovian random walks. This kind of equation is obtained from the standard diffusion equation by replacing the first-order tim...Time fractional diffusion equation is usually used to describe the problems involving non-Markovian random walks. This kind of equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α∈(0, 1). In this paper, an implicit finite difference scheme for solving the time fractional diffusion equation with source term is presented and analyzed, where the fractional derivative is described in the Caputo sense. Stability and convergence of this scheme are rigorously established by a Fourier analysis. And using numerical experiments illustrates the accuracy and effectiveness of the scheme mentioned in this paper.展开更多
In this paper, we investigate a numerical method for the generalized Novikov equation. We propose a conservative finite difference scheme and use Brouwer fixed point theorem to obtain the existence of the solution of ...In this paper, we investigate a numerical method for the generalized Novikov equation. We propose a conservative finite difference scheme and use Brouwer fixed point theorem to obtain the existence of the solution of the corresponding difference equation. We also prove the convergence and stability of the solution by using the discrete energy method. Moreover, we obtain the truncation error of the difference scheme which is .展开更多
The multi-dimensional system of nonlinear partial differential equations is considered. In two-dimensional case, this system describes process of vein formation in higher plants. Variable directions finite difference ...The multi-dimensional system of nonlinear partial differential equations is considered. In two-dimensional case, this system describes process of vein formation in higher plants. Variable directions finite difference scheme is constructed. The stability and convergence of that scheme are studied. Numerical experiments are carried out. The appropriate graphical illustrations and tables are given.展开更多
A conservative difference scheme is presented for the initial-boundary-value problem of a generalized Zakharov equations. On the basis of a prior estimates in L-2 norm, the convergence of the difference solution is pr...A conservative difference scheme is presented for the initial-boundary-value problem of a generalized Zakharov equations. On the basis of a prior estimates in L-2 norm, the convergence of the difference solution is proved in order O(h(2) + r(2)). In the proof, a new skill is used to deal with the term of difference quotient (e(j,k)(n))t. This is necessary, since there is no estimate of E(x, y, t) in L-infinity norm.展开更多
In the poper, the method of separating singularity is applied to study the uniformly difference scheme of a singular perturbation problem for a semilinear ordinary differential equation with mixed boundary value condi...In the poper, the method of separating singularity is applied to study the uniformly difference scheme of a singular perturbation problem for a semilinear ordinary differential equation with mixed boundary value condition. The uniform convergence on small parameter ε of order one for an IVin type difference scheme constructed is proved. At the end of the paper, a numerical example is given. The computing results coincide with the theoretical analysis.展开更多
In this paper,a compact finite difference scheme for the nonlinear fractional integro-differential equation with weak singularity at the initial time is developed,with O(N^(-(2-α))+M^(-4))accuracy order,where N;M den...In this paper,a compact finite difference scheme for the nonlinear fractional integro-differential equation with weak singularity at the initial time is developed,with O(N^(-(2-α))+M^(-4))accuracy order,where N;M denote the numbers of grids in temporal and spatial direction,α ∈(0,1)is the fractional order.To recover the full accuracy based on the regularity requirement of the solution,we adopt the L1 method and the trapezoidal product integration(PI)rule with graded meshes to discretize the Caputo derivative and the Riemann-Liouville integral,respectively,further handle the nonlinear term carefully by the Newton linearized method.Based on the discrete fractional Gr¨onwall inequality and preserved discrete coefficients of Riemann-Liouville fractional integral,the stability and convergence of the proposed scheme are analyzed by the energy method.Theoretical results are also confirmed by a numerical example.展开更多
In this paper, two fourth-order accurate compact difference schemes are presented for solving the Helmholtz equation in two space dimensions when the corresponding wave numbers are large. The main idea is to derive an...In this paper, two fourth-order accurate compact difference schemes are presented for solving the Helmholtz equation in two space dimensions when the corresponding wave numbers are large. The main idea is to derive and to study a fourth-order accurate compact difference scheme whose leading truncation term, namely, the O(h^4) term, is independent of the wave number and the solution of the Helmholtz equation. The convergence property of the compact schemes are analyzed and the implementation of solving the resulting linear algebraic system based on a FFT approach is considered. Numerical results are presented, which support our theoretical predictions.展开更多
In this paper, the periodic initial value problem for the following class of nonlinear Schrodinger equation of high order i partial derivative u/partial derivative t + (-1)(m) partial derivative(m)/partial derivative ...In this paper, the periodic initial value problem for the following class of nonlinear Schrodinger equation of high order i partial derivative u/partial derivative t + (-1)(m) partial derivative(m)/partial derivative x(m) (a(x)partial derivative(m)u/partial derivative x(m)) + beta(x)q(\u\(2))u + f(x, t)u = g(x, t) is considered. A leap-frog finite difference scheme is given, and convergence and stability is proved. Finally, it is shown by a numerical example that numerical result is coincident with theoretical result.展开更多
文摘In this paper, we consider the initial-boundary value problem of two-dimensional first-order linear hyperbolic equation with variable coefficients. By using the upwind difference method to discretize the spatial derivative term and the forward and backward Euler method to discretize the time derivative term, the explicit and implicit upwind difference schemes are obtained respectively. It is proved that the explicit upwind scheme is conditionally stable and the implicit upwind scheme is unconditionally stable. Then the convergence of the schemes is derived. Numerical examples verify the results of theoretical analysis.
文摘In this paper a singularly perturbed linear second order hyperbolic problem with zeroth order reduced equation is discussed. Firstly, an energy inequality of the solution and an estimate of the remainder term of the asymptotic solution are given. Then an exponentially fitted difference scheme is developed in an equidistant mesh. Finally, uniform convergence in small parameter is proved in the sense of discrete energy norm.
文摘The conservative form and singular perturbed ordinary differential equation with periodic boundary value problem were studied, and a conservative difference scheme was constructed. Using the method of decomposing the singular term from its solution and combining an asymptotic expansion of the equation, it is proved that the scheme converges uniformly to the solution of differential equation with order one.
基金The National Natural Science Foundation of China (No.40701014)the Scientific Research Fund of Sichuan Provincial Education Department (No.09ZB081)the Research Fund of key Discipline of Xihua University:Applied Mathe-matics (No.XZD0910-09-1)
文摘Much effort has been devoted to researching the common Rosenau equation, but the numerical method of it has not been studied. In this paper, a conservative Crank-Nicolson difference scheme for an initial-boundary value problem of the generalized Rosenau equation is proposed. Existence and uniqueness of numerical solutions are derived. By method of discrete energy, the second order convergence and stability are discussed. Numerical examples demonstrate the theoretical results.
文摘In this paper, using nonuniform mesh and exponentially fitted difference method, a uniformly convergent difference scheme for an initial-boundary value problem of linear parabolic differential equation with the nonsmooth boundary layer function with respect to small parameter e is given, and error estimate and numerical result are also given.
文摘In this paper we discuss, an initial-boundary value problem of hyperbolic type with first derivative with respect to x. The asymptotic solution is constructed and its uniform validity is proved under weader compatibility conditions. Then we develop an exponentially fitted difference scheme and establish discrete energy inequality. Finally, we prove that the solution of difference problem uniformly converges to the solution of the original problem.
文摘In this paper, based on the idea of El-Mistikawy and Werle[1] we construct a difference scheme for a singularly perturbed self-adjoint ordinary differential equation in conservation form. We prove that it is a uniformly convergent second order scheme.
文摘In this paper, we construct a class of difference schemes with fitted factors for a singular perturbation problem of a self-adjoint ordinary differential equation. Using a different method from [1], by analyzing the truncation errors of schemes, we give the sufficient conditions under which the solution of lite difference scheme converges uniformly to the solution of the differential equation. From this we propose several specific schemes under weaker conditions, and give much higher order of uniform convergence, and applying them to example, obtain the numerical results.
文摘The conservative form and singular perturbed ordinary differential equation with periodic boundary value problem were studied, and a conservative difference scheme was constructed. Using the method of decomposing the singular term from its solution and combining an asymptotic expansion of the equation, it is proved that the scheme converges uniformly to the solution of differential equation with order one.
基金The project is supported by National Natural Science Foundation of China grant number No.11671081.
文摘Korteweg-de Vries equation is a nonlinear evolutionary partial differential equation that is of third order in space.For the approximation to this equation with the initial and boundary value conditions using the finite difference method,the difficulty is how to construct matched finite difference schemes at all the inner grid points.In this paper,two finite difference schemes are constructed for the problem.The accuracy is second-order in time and first-order in space.The first scheme is a two-level nonlinear implicit finite difference scheme and the second one is a three-level linearized finite difference scheme.The Browder fixed point theorem is used to prove the existence of the nonlinear implicit finite difference scheme.The con-servation,boundedness,stability,convergence of these schemes are discussed and analyzed by the energy method together with other techniques.The two-level non-linear finite difference scheme is proved to be unconditionally convergent and the three-level linearized one is proved to be conditionally convergent.Some numerical examples illustrate the efficiency of the proposed finite difference schemes.
基金Supported by the Discipline Construction and Teaching Research Fund of LUTcte(20140089)
文摘Time fractional diffusion equation is usually used to describe the problems involving non-Markovian random walks. This kind of equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α∈(0, 1). In this paper, an implicit finite difference scheme for solving the time fractional diffusion equation with source term is presented and analyzed, where the fractional derivative is described in the Caputo sense. Stability and convergence of this scheme are rigorously established by a Fourier analysis. And using numerical experiments illustrates the accuracy and effectiveness of the scheme mentioned in this paper.
文摘In this paper, we investigate a numerical method for the generalized Novikov equation. We propose a conservative finite difference scheme and use Brouwer fixed point theorem to obtain the existence of the solution of the corresponding difference equation. We also prove the convergence and stability of the solution by using the discrete energy method. Moreover, we obtain the truncation error of the difference scheme which is .
文摘The multi-dimensional system of nonlinear partial differential equations is considered. In two-dimensional case, this system describes process of vein formation in higher plants. Variable directions finite difference scheme is constructed. The stability and convergence of that scheme are studied. Numerical experiments are carried out. The appropriate graphical illustrations and tables are given.
文摘A conservative difference scheme is presented for the initial-boundary-value problem of a generalized Zakharov equations. On the basis of a prior estimates in L-2 norm, the convergence of the difference solution is proved in order O(h(2) + r(2)). In the proof, a new skill is used to deal with the term of difference quotient (e(j,k)(n))t. This is necessary, since there is no estimate of E(x, y, t) in L-infinity norm.
文摘In the poper, the method of separating singularity is applied to study the uniformly difference scheme of a singular perturbation problem for a semilinear ordinary differential equation with mixed boundary value condition. The uniform convergence on small parameter ε of order one for an IVin type difference scheme constructed is proved. At the end of the paper, a numerical example is given. The computing results coincide with the theoretical analysis.
基金supported by the National Natural Science Foundation of China(No.11701103,11801095)Young Top-notch Talent Program of Guangdong Province(No.2017GC010379)+2 种基金Natural Science Foundation of Guangdong Province(No.2022A1515012147,2019A1515010876,2017A030310538)the Project of Science and Technology of Guangzhou(No.201904010341,202102020704)the Opening Project of Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University(2021023)。
文摘In this paper,a compact finite difference scheme for the nonlinear fractional integro-differential equation with weak singularity at the initial time is developed,with O(N^(-(2-α))+M^(-4))accuracy order,where N;M denote the numbers of grids in temporal and spatial direction,α ∈(0,1)is the fractional order.To recover the full accuracy based on the regularity requirement of the solution,we adopt the L1 method and the trapezoidal product integration(PI)rule with graded meshes to discretize the Caputo derivative and the Riemann-Liouville integral,respectively,further handle the nonlinear term carefully by the Newton linearized method.Based on the discrete fractional Gr¨onwall inequality and preserved discrete coefficients of Riemann-Liouville fractional integral,the stability and convergence of the proposed scheme are analyzed by the energy method.Theoretical results are also confirmed by a numerical example.
基金supported by Natural Science Foundation of China under grant number 10471047
文摘In this paper, two fourth-order accurate compact difference schemes are presented for solving the Helmholtz equation in two space dimensions when the corresponding wave numbers are large. The main idea is to derive and to study a fourth-order accurate compact difference scheme whose leading truncation term, namely, the O(h^4) term, is independent of the wave number and the solution of the Helmholtz equation. The convergence property of the compact schemes are analyzed and the implementation of solving the resulting linear algebraic system based on a FFT approach is considered. Numerical results are presented, which support our theoretical predictions.
文摘In this paper, the periodic initial value problem for the following class of nonlinear Schrodinger equation of high order i partial derivative u/partial derivative t + (-1)(m) partial derivative(m)/partial derivative x(m) (a(x)partial derivative(m)u/partial derivative x(m)) + beta(x)q(\u\(2))u + f(x, t)u = g(x, t) is considered. A leap-frog finite difference scheme is given, and convergence and stability is proved. Finally, it is shown by a numerical example that numerical result is coincident with theoretical result.