A new quadrilateral edge element method is proposed and analyzed for Maxwell equations.This proposed method is based on Duan-Liang quadrilateral element(Math.Comp.73(2004),pp.1–18).When applied to the eigenvalue prob...A new quadrilateral edge element method is proposed and analyzed for Maxwell equations.This proposed method is based on Duan-Liang quadrilateral element(Math.Comp.73(2004),pp.1–18).When applied to the eigenvalue problem,the method is spectral-correct and spurious-free.Stability and error estimates are obtained,including the interpolation error estimates and the error estimates between the finite element solution and the exact solution.The method is suitable for singular solution as well as smooth solution,and consequently,the method is valid for nonconvex domains which may have a number of reentrant corners.Of course,the method is suitable for arbitrary quadrilaterals(under the usual shape-regular condition).展开更多
In this paper,we consider the maximal positive definite solution of the nonlinear matrix equation.By using the idea of Algorithm 2.1 in ZHANG(2013),a new inversion-free method with a stepsize parameter is proposed to ...In this paper,we consider the maximal positive definite solution of the nonlinear matrix equation.By using the idea of Algorithm 2.1 in ZHANG(2013),a new inversion-free method with a stepsize parameter is proposed to obtain the maximal positive definite solution of nonlinear matrix equation X+A^(*)X|^(-α)A=Q with the case 0<α≤1.Based on this method,a new iterative algorithm is developed,and its convergence proof is given.Finally,two numerical examples are provided to show the effectiveness of the proposed method.展开更多
The main purpose of this paper is to use the Chelyshkov-collocation spectral method for solving nonlinear Quadratic integral equations of Volterra type.The method is based on the approximate solutions in terms of Chel...The main purpose of this paper is to use the Chelyshkov-collocation spectral method for solving nonlinear Quadratic integral equations of Volterra type.The method is based on the approximate solutions in terms of Chelyshkov polynomials with unknown coefficients.The Chelyshkov polynomials and their properties are employed to derive the operational matrices of integral and product.The application of these operational matrices for solving the mentioned problem is explained.The error analysis of the proposed method is investigated.Finally,some numerical examples are provided to demonstrate the efficiency of the method.展开更多
The goal of this paper is to investigate the theory of Noether solvability for Volterra singular integral equations(VSIEs)with convolution and Cauchy kernels in a more general function class.To obtain the analytic sol...The goal of this paper is to investigate the theory of Noether solvability for Volterra singular integral equations(VSIEs)with convolution and Cauchy kernels in a more general function class.To obtain the analytic solutions,we transform such equations into boundary value problems with discontinuous coefficients by the properties of Fourier analysis.In view of the analytical Riemann-Hilbert method,the generalized Liouville theorem and Sokhotski-Plemelj formula,we get the uniqueness and existence of solutions for such problems,and study the asymptotic property of solutions at nodes.Therefore,this paper improves the theory of singular integral equations and boundary value problems.展开更多
In this paper,we study an integral system involving m equations■where ui>0 in R^(n),0<α<n,and pi>1(i=1,2,…,m).Based on the optimal integrability intervals,we estimate the decay rates of the positive sol...In this paper,we study an integral system involving m equations■where ui>0 in R^(n),0<α<n,and pi>1(i=1,2,…,m).Based on the optimal integrability intervals,we estimate the decay rates of the positive solutions of the system at infinity.But estimating these rates is difficult because the relation between pi(i=1,2,…,m)is uncertain.To overcome this difficulty,we obtain the asymptotic behavior of all cases by discussing them separately.In addition,we also get the radial symmetry of positive solutions under some integrability condition.展开更多
In this article,we develop the Laplace transform(LT)based Chebyshev spectral collocation method(CSCM)to approximate the time fractional advection-diffusion equation,incorporating the Atangana-Baleanu Caputo(ABC)deriva...In this article,we develop the Laplace transform(LT)based Chebyshev spectral collocation method(CSCM)to approximate the time fractional advection-diffusion equation,incorporating the Atangana-Baleanu Caputo(ABC)derivative.The advection-diffusion equation,which governs the transport of mass,heat,or energy through combined advection and diffusion processes,is central to modeling physical systems with nonlocal behavior.Our numerical scheme employs the LT to transform the time-dependent time-fractional PDEs into a time-independent PDE in LT domain,eliminating the need for classical time-stepping methods that often suffer from stability constraints.For spatial discretization,we employ the CSCM,where the solution is approximated using Lagrange interpolation polynomial based on the Chebyshev collocation nodes,achieving exponential convergence that outperforms the algebraic convergence rates of finite difference and finite element methods.Finally,the solution is reverted to the time domain using contour integration technique.We also establish the existence and uniqueness of the solution for the proposed problem.The performance,efficiency,and accuracy of the proposed method are validated through various fractional advection-diffusion problems.The computed results demonstrate that the proposed method has less computational cost and is highly accurate.展开更多
The meshless method is a new numerical technique presented in recent years.It uses the moving least square(MLS)approximation as a shape function.The smoothness of the MLS approximation is determined by that of the bas...The meshless method is a new numerical technique presented in recent years.It uses the moving least square(MLS)approximation as a shape function.The smoothness of the MLS approximation is determined by that of the basic function and of the weight function,and is mainly determined by that of the weight function.Therefore,the weight function greatly affects the accuracy of results obtained.Different kinds of weight functions,such as the spline function, the Gauss function and so on,are proposed recently by many researchers.In the present work,the features of various weight functions are illustrated through solving elasto-static problems using the local boundary integral equation method.The effect of various weight functions on the accuracy, convergence and stability of results obtained is also discussed.Examples show that the weight function proposed by Zhou Weiyuan and Gauss and the quartic spline weight function are better than the others if parameters c and α in Gauss and exponential weight functions are in the range of reasonable values,respectively,and the higher the smoothness of the weight function,the better the features of the solutions.展开更多
The boundary knot method(BKM)is a simple boundary-type meshless method.Due to the use of non-singular general solutions rather than singular fundamental solutions,BKM does not need to consider the artificial boundary....The boundary knot method(BKM)is a simple boundary-type meshless method.Due to the use of non-singular general solutions rather than singular fundamental solutions,BKM does not need to consider the artificial boundary.Therefore,this method has the merits of purely meshless,easy to program,high solution accuracy and so on.In this paper,we investigate the effectiveness of the BKM for solving Helmholtz-type problems under various conditions through a series of novel numerical experiments.The results demonstrate that the BKM is efficient and achieves high computational accuracy for problems with smooth or continuous boundary conditions.However,when applied to discontinuous boundary problems,the method exhibits significant numerical instability,potentially leading to substantial deviations in the computed results.Finally,three potential improvement strategies are proposed to mitigate this limitation.展开更多
In this paper,the convergence of the split-step theta method for stochastic differential equations is analyzed using stochastic C-stability and stochastic B-consistency.The fact that the numerical scheme,which is both...In this paper,the convergence of the split-step theta method for stochastic differential equations is analyzed using stochastic C-stability and stochastic B-consistency.The fact that the numerical scheme,which is both stochastically C-stable and stochastically B-consistent,is convergent has been proved in a previous paper.In order to analyze the convergence of the split-step theta method(θ∈[1/2,1]),the stochastic C-stability and stochastic B-consistency under the condition of global monotonicity have been researched,and the rate of convergence 1/2 has been explored in this paper.It can be seen that the convergence does not require the drift function should satisfy the linear growth condition whenθ=1/2 Furthermore,the rate of the convergence of the split-step scheme for stochastic differential equations with additive noise has been researched and found to be 1.Finally,an example is given to illustrate the convergence with the theoretical results.展开更多
Fatigue analysis of engine turbine blade is an essential issue.Due to various uncertainties during the manufacture and operation,the fatigue damage and life of turbine blade present randomness.In this study,the random...Fatigue analysis of engine turbine blade is an essential issue.Due to various uncertainties during the manufacture and operation,the fatigue damage and life of turbine blade present randomness.In this study,the randomness of structural parameters,working condition and vibration environment are considered for fatigue life predication and reliability assessment.First,the lowcycle fatigue problem is modelled as stochastic static system with random parameters,while the high-cycle fatigue problem is considered as stochastic dynamic system under random excitations.Then,to deal with the two failure modes,the novel Direct Probability Integral Method(DPIM)is proposed,which is efficient and accurate for solving stochastic static and dynamic systems.The probability density functions of accumulated damage and fatigue life of turbine blade for low-cycle and high-cycle fatigue problems are achieved,respectively.Furthermore,the time–frequency hybrid method is advanced to enhance the computational efficiency for governing equation of system.Finally,the results of typical examples demonstrate high accuracy and efficiency of the proposed method by comparison with Monte Carlo simulation and other methods.It is indicated that the DPIM is a unified method for predication of random fatigue life for low-cycle and highcycle fatigue problems.The rotational speed,density,fatigue strength coefficient,and fatigue plasticity index have a high sensitivity to fatigue reliability of engine turbine blade.展开更多
The(2+1)-dimensional integrable generalization of the Gardner(2DG)equation is solved via the inverse scattering transform method in this paper.A kind of general solution of the equation is obtained by introducing long...The(2+1)-dimensional integrable generalization of the Gardner(2DG)equation is solved via the inverse scattering transform method in this paper.A kind of general solution of the equation is obtained by introducing long derivatives V_(x),V_(y),V_(t).Two different constraints on the kernel function K are introduced under the reality of the solution u of the 2DG equation.Then,two classes of exact solutions with constant asymptotic values at infinity u|x^(2)+y^(2)→∞→0 are constructed by means of the∂¯-dressing method for the casesσ=1 andσ=i.The rational and multiple pole solutions of the 2DG equation are obtained with the kernel functions of zero-order and higher-order Dirac delta functions,respectively.展开更多
In this work, we first derive the one-point large deviations principle (LDP) for both the stochastic Cahn–Hilliard equation with small noise and its spatial finite difference method (FDM). Then, we focus on giving th...In this work, we first derive the one-point large deviations principle (LDP) for both the stochastic Cahn–Hilliard equation with small noise and its spatial finite difference method (FDM). Then, we focus on giving the convergence of the one-point large deviations rate function (LDRF) of the spatial FDM, which is about the asymptotical limit of a parametric variational problem. The main idea for proving the convergence of the LDRF of the spatial FDM is via the Γ-convergence of objective functions. This relies on the qualitative analysis of skeleton equations of the original equation and the numerical method. In order to overcome the difficulty that the drift coefficient is not one-sided Lipschitz continuous, we derive the equivalent characterization of the skeleton equation of the spatial FDM and the discrete interpolation inequality to obtain the uniform boundedness of the solution to the underlying skeleton equation. These play important roles in deriving the T-convergence of objective functions.展开更多
In this study,we explore some of the best proximity point results for generalized proximal contractions in the setting of double-controlled metric-type spaces.A non-trivial example is given to elucidate our analysis,a...In this study,we explore some of the best proximity point results for generalized proximal contractions in the setting of double-controlled metric-type spaces.A non-trivial example is given to elucidate our analysis,and some novel results are derived.The discovered results generalize previously known results in the context of a double controlled metric type space environment.This article’s proximity point results are the first of their kind in the realm of controlled metric spaces.To build on the results achieved in this article,we present an application demonstrating the usability of the given results.展开更多
In this paper we propose and analyze a numerical scheme coupling a second-order backward differential formulation(BDF)and the finite element method(FEM)to solve the incompressible resistive magnetohydrodynamic(MHD)equ...In this paper we propose and analyze a numerical scheme coupling a second-order backward differential formulation(BDF)and the finite element method(FEM)to solve the incompressible resistive magnetohydrodynamic(MHD)equations.In the discrete scheme,the pressure variable in the fluid field equation is computed through a Poisson equation,and a linear and decoupled method is adopted to separate both the magnetic and the fluid field functions from the original system.As a result,the original system is divided into several sub-systems for which the numerical solutions can be obtained efficiently.We prove the unique solvability,the unconditional energy stability,and particularly optimal error estimates for the proposed scheme.Numerical results are presented to validate the theory of the scheme.展开更多
This paper determines the exact error order on optimization of adaptive direct methods of approximate solution of the class of Fredholm integral equations of the second kind with kernel belonging to the anisotropic So...This paper determines the exact error order on optimization of adaptive direct methods of approximate solution of the class of Fredholm integral equations of the second kind with kernel belonging to the anisotropic Sobolev classes, and also gives an optimal algorithm.展开更多
We present a fast algorithm based on polynomial interpolation to approximate matrices arising from the discretization of second-kind integral equations where the kernel function is either smooth, non-oscillatory and p...We present a fast algorithm based on polynomial interpolation to approximate matrices arising from the discretization of second-kind integral equations where the kernel function is either smooth, non-oscillatory and possessing only a finite number of singularities or a product of such function with a highly oscillatory coefficient function. Contrast to wavelet-like approximations, ourapproximation matrix is not sparse. However, the approximation can be construced in O(n) operations and requires O(n) storage, where n is the number of quadrature points used in the discretization. Moreover, the matrix-vector multiplication cost is of order O(nlogn). Thus our scheme is well suitable for conjugate gradient type methods. Our numerical results indicate that the algorithm is very accurate and stable for high degree polynomial interpolation.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 elastodynamic problems of piezoelectric hollow cylinders and spheres under radial deformation can be transformed into a second kind Volterra integral equation about a function with respect to time, which greatly s...The elastodynamic problems of piezoelectric hollow cylinders and spheres under radial deformation can be transformed into a second kind Volterra integral equation about a function with respect to time, which greatly simplifies the solving procedure for such elastodynamic problems. Meanwhile, it becomes very important to find a way to solve the second kind Volterra integral equation effectively and quickly. By using an interpolation function to approximate the unknown function, two new recursive formulae were derived, based on which numerical solution can be obtained step by step. The present method can provide accurate numerical results efficiently. It is also very stable for long time calculating.展开更多
In 2D fast multipole method for scattering problems,square quadrature rule is used to discretize the Bessel integral identity for diagonal expansion of 2D Helmholtz kernel,and numerical integration error is introduced...In 2D fast multipole method for scattering problems,square quadrature rule is used to discretize the Bessel integral identity for diagonal expansion of 2D Helmholtz kernel,and numerical integration error is introduced. Taking advantage of the relationship between Euler-Maclaurin formula and trapezoidal quadrature rule,and the relationship between trapezoidal and square quadrature rule,sharp computable bound with analytical form on the error of numerical integration of Bessel integral identity by square quadrature rule is derived in this paper. Numerical experiments are presented at the end to demonstrate the accuracy of the sharp computable bound on the numerical integration error.展开更多
While the approximate solutions of one-dimensional nonlinear Volterra-Fredholm integral equations with smooth kermels are now well understood,no systematic studies of the numerical solutions of their multi-dimensional...While the approximate solutions of one-dimensional nonlinear Volterra-Fredholm integral equations with smooth kermels are now well understood,no systematic studies of the numerical solutions of their multi-dimensional counterparts exist.In this paper,we provide an efficient numerical approach for the multi-dimensional nonlinear Volterra-Fredholm integral equations based on the multi-variate Legendre-collocation approach.Spectral collocation methods for multi-dimensional nonlinear integral equations are known to cause major difficulties from a convergence analysis point of view.Consequently,rigorous error estimates are provided in the weighted Sobolev space showing the exponential decay of the numerical errors.The existence and uniqueness of the numerical solution are established.Numerical experiments are provided to support the theoretical convergence analysis.The results indicate that our spectral collocation method is more flexible with better accuracy than the existing ones.展开更多
基金supported by the National Natural Science Foundation of China(12401482)the second author was supported by the National Natural Science Foundation of China(12371371,12261160361,11971366)supported by the Open Research Fund of Hubei Key Laboratory of Computational Science,Wuhan University.
文摘A new quadrilateral edge element method is proposed and analyzed for Maxwell equations.This proposed method is based on Duan-Liang quadrilateral element(Math.Comp.73(2004),pp.1–18).When applied to the eigenvalue problem,the method is spectral-correct and spurious-free.Stability and error estimates are obtained,including the interpolation error estimates and the error estimates between the finite element solution and the exact solution.The method is suitable for singular solution as well as smooth solution,and consequently,the method is valid for nonconvex domains which may have a number of reentrant corners.Of course,the method is suitable for arbitrary quadrilaterals(under the usual shape-regular condition).
基金Supported in part by Natural Science Foundation of Guangxi(2023GXNSFAA026246)in part by the Central Government's Guide to Local Science and Technology Development Fund(GuikeZY23055044)in part by the National Natural Science Foundation of China(62363003)。
文摘In this paper,we consider the maximal positive definite solution of the nonlinear matrix equation.By using the idea of Algorithm 2.1 in ZHANG(2013),a new inversion-free method with a stepsize parameter is proposed to obtain the maximal positive definite solution of nonlinear matrix equation X+A^(*)X|^(-α)A=Q with the case 0<α≤1.Based on this method,a new iterative algorithm is developed,and its convergence proof is given.Finally,two numerical examples are provided to show the effectiveness of the proposed method.
文摘The main purpose of this paper is to use the Chelyshkov-collocation spectral method for solving nonlinear Quadratic integral equations of Volterra type.The method is based on the approximate solutions in terms of Chelyshkov polynomials with unknown coefficients.The Chelyshkov polynomials and their properties are employed to derive the operational matrices of integral and product.The application of these operational matrices for solving the mentioned problem is explained.The error analysis of the proposed method is investigated.Finally,some numerical examples are provided to demonstrate the efficiency of the method.
基金Supported by National Natural Science Foundation of China(Grant No.11971015).
文摘The goal of this paper is to investigate the theory of Noether solvability for Volterra singular integral equations(VSIEs)with convolution and Cauchy kernels in a more general function class.To obtain the analytic solutions,we transform such equations into boundary value problems with discontinuous coefficients by the properties of Fourier analysis.In view of the analytical Riemann-Hilbert method,the generalized Liouville theorem and Sokhotski-Plemelj formula,we get the uniqueness and existence of solutions for such problems,and study the asymptotic property of solutions at nodes.Therefore,this paper improves the theory of singular integral equations and boundary value problems.
基金supported by the NSFC(11871278)the Postgraduate Research&Practice Innovation Program of Jiangsu Province(KYCX23-1669).
文摘In this paper,we study an integral system involving m equations■where ui>0 in R^(n),0<α<n,and pi>1(i=1,2,…,m).Based on the optimal integrability intervals,we estimate the decay rates of the positive solutions of the system at infinity.But estimating these rates is difficult because the relation between pi(i=1,2,…,m)is uncertain.To overcome this difficulty,we obtain the asymptotic behavior of all cases by discussing them separately.In addition,we also get the radial symmetry of positive solutions under some integrability condition.
基金extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through Large Research Project under grant number RGP2/174/46.
文摘In this article,we develop the Laplace transform(LT)based Chebyshev spectral collocation method(CSCM)to approximate the time fractional advection-diffusion equation,incorporating the Atangana-Baleanu Caputo(ABC)derivative.The advection-diffusion equation,which governs the transport of mass,heat,or energy through combined advection and diffusion processes,is central to modeling physical systems with nonlocal behavior.Our numerical scheme employs the LT to transform the time-dependent time-fractional PDEs into a time-independent PDE in LT domain,eliminating the need for classical time-stepping methods that often suffer from stability constraints.For spatial discretization,we employ the CSCM,where the solution is approximated using Lagrange interpolation polynomial based on the Chebyshev collocation nodes,achieving exponential convergence that outperforms the algebraic convergence rates of finite difference and finite element methods.Finally,the solution is reverted to the time domain using contour integration technique.We also establish the existence and uniqueness of the solution for the proposed problem.The performance,efficiency,and accuracy of the proposed method are validated through various fractional advection-diffusion problems.The computed results demonstrate that the proposed method has less computational cost and is highly accurate.
文摘The meshless method is a new numerical technique presented in recent years.It uses the moving least square(MLS)approximation as a shape function.The smoothness of the MLS approximation is determined by that of the basic function and of the weight function,and is mainly determined by that of the weight function.Therefore,the weight function greatly affects the accuracy of results obtained.Different kinds of weight functions,such as the spline function, the Gauss function and so on,are proposed recently by many researchers.In the present work,the features of various weight functions are illustrated through solving elasto-static problems using the local boundary integral equation method.The effect of various weight functions on the accuracy, convergence and stability of results obtained is also discussed.Examples show that the weight function proposed by Zhou Weiyuan and Gauss and the quartic spline weight function are better than the others if parameters c and α in Gauss and exponential weight functions are in the range of reasonable values,respectively,and the higher the smoothness of the weight function,the better the features of the solutions.
基金Supported by the Key Scientific Research Plan of Colleges and Universities in Henan Province(23B140006)。
文摘The boundary knot method(BKM)is a simple boundary-type meshless method.Due to the use of non-singular general solutions rather than singular fundamental solutions,BKM does not need to consider the artificial boundary.Therefore,this method has the merits of purely meshless,easy to program,high solution accuracy and so on.In this paper,we investigate the effectiveness of the BKM for solving Helmholtz-type problems under various conditions through a series of novel numerical experiments.The results demonstrate that the BKM is efficient and achieves high computational accuracy for problems with smooth or continuous boundary conditions.However,when applied to discontinuous boundary problems,the method exhibits significant numerical instability,potentially leading to substantial deviations in the computed results.Finally,three potential improvement strategies are proposed to mitigate this limitation.
基金Supported by the National Natural Science Foundation of China (Grant No. 12301521)the Natural Science Foundation of Shanxi Province (Grant No. 20210302124081)。
文摘In this paper,the convergence of the split-step theta method for stochastic differential equations is analyzed using stochastic C-stability and stochastic B-consistency.The fact that the numerical scheme,which is both stochastically C-stable and stochastically B-consistent,is convergent has been proved in a previous paper.In order to analyze the convergence of the split-step theta method(θ∈[1/2,1]),the stochastic C-stability and stochastic B-consistency under the condition of global monotonicity have been researched,and the rate of convergence 1/2 has been explored in this paper.It can be seen that the convergence does not require the drift function should satisfy the linear growth condition whenθ=1/2 Furthermore,the rate of the convergence of the split-step scheme for stochastic differential equations with additive noise has been researched and found to be 1.Finally,an example is given to illustrate the convergence with the theoretical results.
基金supports of the National Natural Science Foundation of China(Nos.12032008,12102080)the Fundamental Research Funds for the Central Universities,China(No.DUT23RC(3)038)are much appreciated。
文摘Fatigue analysis of engine turbine blade is an essential issue.Due to various uncertainties during the manufacture and operation,the fatigue damage and life of turbine blade present randomness.In this study,the randomness of structural parameters,working condition and vibration environment are considered for fatigue life predication and reliability assessment.First,the lowcycle fatigue problem is modelled as stochastic static system with random parameters,while the high-cycle fatigue problem is considered as stochastic dynamic system under random excitations.Then,to deal with the two failure modes,the novel Direct Probability Integral Method(DPIM)is proposed,which is efficient and accurate for solving stochastic static and dynamic systems.The probability density functions of accumulated damage and fatigue life of turbine blade for low-cycle and high-cycle fatigue problems are achieved,respectively.Furthermore,the time–frequency hybrid method is advanced to enhance the computational efficiency for governing equation of system.Finally,the results of typical examples demonstrate high accuracy and efficiency of the proposed method by comparison with Monte Carlo simulation and other methods.It is indicated that the DPIM is a unified method for predication of random fatigue life for low-cycle and highcycle fatigue problems.The rotational speed,density,fatigue strength coefficient,and fatigue plasticity index have a high sensitivity to fatigue reliability of engine turbine blade.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1237125611971475)。
文摘The(2+1)-dimensional integrable generalization of the Gardner(2DG)equation is solved via the inverse scattering transform method in this paper.A kind of general solution of the equation is obtained by introducing long derivatives V_(x),V_(y),V_(t).Two different constraints on the kernel function K are introduced under the reality of the solution u of the 2DG equation.Then,two classes of exact solutions with constant asymptotic values at infinity u|x^(2)+y^(2)→∞→0 are constructed by means of the∂¯-dressing method for the casesσ=1 andσ=i.The rational and multiple pole solutions of the 2DG equation are obtained with the kernel functions of zero-order and higher-order Dirac delta functions,respectively.
基金supported by the National Natural Science Foundation of China(12201228,12171047)the Fundamental Research Funds for the Central Universities(3034011102)supported by National Key R&D Program of China(2020YFA0713701).
文摘In this work, we first derive the one-point large deviations principle (LDP) for both the stochastic Cahn–Hilliard equation with small noise and its spatial finite difference method (FDM). Then, we focus on giving the convergence of the one-point large deviations rate function (LDRF) of the spatial FDM, which is about the asymptotical limit of a parametric variational problem. The main idea for proving the convergence of the LDRF of the spatial FDM is via the Γ-convergence of objective functions. This relies on the qualitative analysis of skeleton equations of the original equation and the numerical method. In order to overcome the difficulty that the drift coefficient is not one-sided Lipschitz continuous, we derive the equivalent characterization of the skeleton equation of the spatial FDM and the discrete interpolation inequality to obtain the uniform boundedness of the solution to the underlying skeleton equation. These play important roles in deriving the T-convergence of objective functions.
文摘In this study,we explore some of the best proximity point results for generalized proximal contractions in the setting of double-controlled metric-type spaces.A non-trivial example is given to elucidate our analysis,and some novel results are derived.The discovered results generalize previously known results in the context of a double controlled metric type space environment.This article’s proximity point results are the first of their kind in the realm of controlled metric spaces.To build on the results achieved in this article,we present an application demonstrating the usability of the given results.
文摘In this paper we propose and analyze a numerical scheme coupling a second-order backward differential formulation(BDF)and the finite element method(FEM)to solve the incompressible resistive magnetohydrodynamic(MHD)equations.In the discrete scheme,the pressure variable in the fluid field equation is computed through a Poisson equation,and a linear and decoupled method is adopted to separate both the magnetic and the fluid field functions from the original system.As a result,the original system is divided into several sub-systems for which the numerical solutions can be obtained efficiently.We prove the unique solvability,the unconditional energy stability,and particularly optimal error estimates for the proposed scheme.Numerical results are presented to validate the theory of the scheme.
基金Project supported by the Natural Science Foundation of China(10371009)Research Fund for the Doctoral Program Higher Education
文摘This paper determines the exact error order on optimization of adaptive direct methods of approximate solution of the class of Fredholm integral equations of the second kind with kernel belonging to the anisotropic Sobolev classes, and also gives an optimal algorithm.
基金Research supported in part by Hong Kong Research Grant Council grats no.CUHK178/83E
文摘We present a fast algorithm based on polynomial interpolation to approximate matrices arising from the discretization of second-kind integral equations where the kernel function is either smooth, non-oscillatory and possessing only a finite number of singularities or a product of such function with a highly oscillatory coefficient function. Contrast to wavelet-like approximations, ourapproximation matrix is not sparse. However, the approximation can be construced in O(n) operations and requires O(n) storage, where n is the number of quadrature points used in the discretization. Moreover, the matrix-vector multiplication cost is of order O(nlogn). Thus our scheme is well suitable for conjugate gradient type methods. Our numerical results indicate that the algorithm is very accurate and stable for high degree polynomial interpolation.
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 elastodynamic problems of piezoelectric hollow cylinders and spheres under radial deformation can be transformed into a second kind Volterra integral equation about a function with respect to time, which greatly simplifies the solving procedure for such elastodynamic problems. Meanwhile, it becomes very important to find a way to solve the second kind Volterra integral equation effectively and quickly. By using an interpolation function to approximate the unknown function, two new recursive formulae were derived, based on which numerical solution can be obtained step by step. The present method can provide accurate numerical results efficiently. It is also very stable for long time calculating.
基金the National Natural Science Foundation of China (No. 11074170)the Independent Research Program of State Key Laboratory of Machinery System and Vibration (SKLMSV) (No. MSV-MS-2008-05)the Visiting Scholar Program of SKLMSV (No. MSV-2009-06)
文摘In 2D fast multipole method for scattering problems,square quadrature rule is used to discretize the Bessel integral identity for diagonal expansion of 2D Helmholtz kernel,and numerical integration error is introduced. Taking advantage of the relationship between Euler-Maclaurin formula and trapezoidal quadrature rule,and the relationship between trapezoidal and square quadrature rule,sharp computable bound with analytical form on the error of numerical integration of Bessel integral identity by square quadrature rule is derived in this paper. Numerical experiments are presented at the end to demonstrate the accuracy of the sharp computable bound on the numerical integration error.
文摘While the approximate solutions of one-dimensional nonlinear Volterra-Fredholm integral equations with smooth kermels are now well understood,no systematic studies of the numerical solutions of their multi-dimensional counterparts exist.In this paper,we provide an efficient numerical approach for the multi-dimensional nonlinear Volterra-Fredholm integral equations based on the multi-variate Legendre-collocation approach.Spectral collocation methods for multi-dimensional nonlinear integral equations are known to cause major difficulties from a convergence analysis point of view.Consequently,rigorous error estimates are provided in the weighted Sobolev space showing the exponential decay of the numerical errors.The existence and uniqueness of the numerical solution are established.Numerical experiments are provided to support the theoretical convergence analysis.The results indicate that our spectral collocation method is more flexible with better accuracy than the existing ones.
