In previous studies, the nonlinear problem of electrohydrodynamic(EHD)ion drag flows in a circular cylindrical conduit has been studied by several authors. However, those studies seldom involve the computation for lar...In previous studies, the nonlinear problem of electrohydrodynamic(EHD)ion drag flows in a circular cylindrical conduit has been studied by several authors. However, those studies seldom involve the computation for large physical parameters such as the electrical Hartmann number and the magnitude parameter for the strength of the nonlinearity due to the existence of strong nonlinearity in these extreme cases. To overcome this faultiness, the newly-developed homotopy Coiflets wavelet method is extended to solve this EHD flow problem with strong nonlinearity. The validity and reliability of the proposed technique are verified. Particularly, the highly accurate homotopy-wavelet solution is obtained for extreme large parameters, which seems to be overlooked before.Discussion about the effects of related physical parameters on the axial velocity field is presented.展开更多
A high-accuracy multiresolution method is proposed to solve mechanics problems subject to complex shapes or irregular domains.To realize this method,we design a new wavelet basis function,by which we construct a fifth...A high-accuracy multiresolution method is proposed to solve mechanics problems subject to complex shapes or irregular domains.To realize this method,we design a new wavelet basis function,by which we construct a fifth-order numerical scheme for the approximation of multi-dimensional functions and their multiple integrals defined in complex domains.In the solution of differential equations,various derivatives of the unknown function are denoted as new functions.Then,the integral relations between these functions are applied in terms of wavelet approximation of multiple integrals.Therefore,the original equation with derivatives of various orders can be converted to a system of algebraic equations with discrete nodal values of the highest-order derivative.During the application of the proposed method,boundary conditions can be automatically included in the integration operations,and relevant matrices can be assured to exhibit perfect sparse patterns.As examples,we consider several second-order mathematics problems defined on regular and irregular domains and the fourth-order bending problems of plates with various shapes.By comparing the solutions obtained by the proposed method with the exact solutions,the new multiresolution method is found to have a convergence rate of fifth order.The solution accuracy of this method with only a few hundreds of nodes can be much higher than that of the finite element method(FEM)with tens of thousands of elements.In addition,because the accuracy order for direct approximation of a function using the proposed basis function is also fifth order,we may conclude that the accuracy of the proposed method is almost independent of the equation order and domain complexity.展开更多
Wavelet methods are a very useful tool in solving integral equations. Both scaling functions and wavelet functions are the key elements of wavelet methods. In this article, we use scaling function interpolation method...Wavelet methods are a very useful tool in solving integral equations. Both scaling functions and wavelet functions are the key elements of wavelet methods. In this article, we use scaling function interpolation method to solve Volterra integral equations of the first kind, and Fredholm-Volterra integral equations. Moreover, we prove convergence theorem for the numerical solution of Volterra integral equations and Freholm-Volterra integral equations. We also present three examples of solving Volterra integral equation and one example of solving Fredholm-Volterra integral equation. Comparisons of the results with other methods are included in the examples.展开更多
A new boundary extension technique based on the Lagrange interpolat- ing polynomial is proposed and used to solve the function approximation defined on an interval by a series of scaling Coiflet functions, where the c...A new boundary extension technique based on the Lagrange interpolat- ing polynomial is proposed and used to solve the function approximation defined on an interval by a series of scaling Coiflet functions, where the coefficients are used as the single-point samplings. The obtained approximation formula can exactly represent any polynomials defined on the interval with the order up to one third of the length of the compact support of the adopted Coiflet function. Based on the Galerkin method, a Coifiet-based solution procedure is established for general two-dimensional p^Laplacian equations, following which the equations can be discretized into a concise matrix form. As examples of applications, the proposed modified wavelet Galerkin method is applied to three typical p-Laplacian equations with strong nonlinearity. The numerical results justify the efficiency and accuracy of the method.展开更多
A high-precision and space-time fully decoupled numerical method is developed for a class of nonlinear initial boundary value problems. It is established based on a proposed Coiflet-based approximation scheme with an ...A high-precision and space-time fully decoupled numerical method is developed for a class of nonlinear initial boundary value problems. It is established based on a proposed Coiflet-based approximation scheme with an adjustable high order for the functions over a bounded interval, which allows the expansion coefficients to be explicitly expressed by the function values at a series of single points. When the solution method is used, the nonlinear initial boundary value problems are first spatially discretized into a series of nonlinear initial value problems by combining the proposed wavelet approximation and the conventional Galerkin method, and a novel high-order step-by-step time integrating approach is then developed for the resulting nonlinear initial value problems with the same function approximation scheme based on the wavelet theory. The solution method is shown to have the N th-order accuracy, as long as the Coiflet with [0, 3 N-1]compact support is adopted, where N can be any positive even number. Typical examples in mechanics are considered to justify the accuracy and efficiency of the method.展开更多
Numerical solutions of singular Fredholm integral equations of the second kind are solved by using Coiflet interpolation method. Error analysis of the method is obtained and examples are presented. It turns out that o...Numerical solutions of singular Fredholm integral equations of the second kind are solved by using Coiflet interpolation method. Error analysis of the method is obtained and examples are presented. It turns out that our method provides better accuracy than other methods.展开更多
In this article, we use scaling function interpolation method to solve linear Fredholm integral equations, and we prove a convergence theorem for the solution of Fredholm integral equations. We present two examples wh...In this article, we use scaling function interpolation method to solve linear Fredholm integral equations, and we prove a convergence theorem for the solution of Fredholm integral equations. We present two examples which have better results than others.展开更多
基金the National Natural Science Foundation of China (No. 11872241)。
文摘In previous studies, the nonlinear problem of electrohydrodynamic(EHD)ion drag flows in a circular cylindrical conduit has been studied by several authors. However, those studies seldom involve the computation for large physical parameters such as the electrical Hartmann number and the magnitude parameter for the strength of the nonlinearity due to the existence of strong nonlinearity in these extreme cases. To overcome this faultiness, the newly-developed homotopy Coiflets wavelet method is extended to solve this EHD flow problem with strong nonlinearity. The validity and reliability of the proposed technique are verified. Particularly, the highly accurate homotopy-wavelet solution is obtained for extreme large parameters, which seems to be overlooked before.Discussion about the effects of related physical parameters on the axial velocity field is presented.
基金Project supported by the National Natural Science Foundation of China(No.11925204)the 111 Project(No.B14044)。
文摘A high-accuracy multiresolution method is proposed to solve mechanics problems subject to complex shapes or irregular domains.To realize this method,we design a new wavelet basis function,by which we construct a fifth-order numerical scheme for the approximation of multi-dimensional functions and their multiple integrals defined in complex domains.In the solution of differential equations,various derivatives of the unknown function are denoted as new functions.Then,the integral relations between these functions are applied in terms of wavelet approximation of multiple integrals.Therefore,the original equation with derivatives of various orders can be converted to a system of algebraic equations with discrete nodal values of the highest-order derivative.During the application of the proposed method,boundary conditions can be automatically included in the integration operations,and relevant matrices can be assured to exhibit perfect sparse patterns.As examples,we consider several second-order mathematics problems defined on regular and irregular domains and the fourth-order bending problems of plates with various shapes.By comparing the solutions obtained by the proposed method with the exact solutions,the new multiresolution method is found to have a convergence rate of fifth order.The solution accuracy of this method with only a few hundreds of nodes can be much higher than that of the finite element method(FEM)with tens of thousands of elements.In addition,because the accuracy order for direct approximation of a function using the proposed basis function is also fifth order,we may conclude that the accuracy of the proposed method is almost independent of the equation order and domain complexity.
文摘Wavelet methods are a very useful tool in solving integral equations. Both scaling functions and wavelet functions are the key elements of wavelet methods. In this article, we use scaling function interpolation method to solve Volterra integral equations of the first kind, and Fredholm-Volterra integral equations. Moreover, we prove convergence theorem for the numerical solution of Volterra integral equations and Freholm-Volterra integral equations. We also present three examples of solving Volterra integral equation and one example of solving Fredholm-Volterra integral equation. Comparisons of the results with other methods are included in the examples.
基金supported by the National Natural Science Foundation of China(Nos.11472119 and11421062)
文摘A new boundary extension technique based on the Lagrange interpolat- ing polynomial is proposed and used to solve the function approximation defined on an interval by a series of scaling Coiflet functions, where the coefficients are used as the single-point samplings. The obtained approximation formula can exactly represent any polynomials defined on the interval with the order up to one third of the length of the compact support of the adopted Coiflet function. Based on the Galerkin method, a Coifiet-based solution procedure is established for general two-dimensional p^Laplacian equations, following which the equations can be discretized into a concise matrix form. As examples of applications, the proposed modified wavelet Galerkin method is applied to three typical p-Laplacian equations with strong nonlinearity. The numerical results justify the efficiency and accuracy of the method.
基金Project supported by the National Natural Science Foundation of China(No.11472119)the Fundamental Research Funds for the Central Universities(No.lzujbky-2017-ot11)the 111 Project(No.B14044)
文摘A high-precision and space-time fully decoupled numerical method is developed for a class of nonlinear initial boundary value problems. It is established based on a proposed Coiflet-based approximation scheme with an adjustable high order for the functions over a bounded interval, which allows the expansion coefficients to be explicitly expressed by the function values at a series of single points. When the solution method is used, the nonlinear initial boundary value problems are first spatially discretized into a series of nonlinear initial value problems by combining the proposed wavelet approximation and the conventional Galerkin method, and a novel high-order step-by-step time integrating approach is then developed for the resulting nonlinear initial value problems with the same function approximation scheme based on the wavelet theory. The solution method is shown to have the N th-order accuracy, as long as the Coiflet with [0, 3 N-1]compact support is adopted, where N can be any positive even number. Typical examples in mechanics are considered to justify the accuracy and efficiency of the method.
文摘Numerical solutions of singular Fredholm integral equations of the second kind are solved by using Coiflet interpolation method. Error analysis of the method is obtained and examples are presented. It turns out that our method provides better accuracy than other methods.
文摘In this article, we use scaling function interpolation method to solve linear Fredholm integral equations, and we prove a convergence theorem for the solution of Fredholm integral equations. We present two examples which have better results than others.
