In this paper,the natural convection of a complex fluid that contains both nanoparticles and gyrotactic microorganisms in a heated square cavity is considered.The Buongiorno model is applied to descirbe the nanofluid ...In this paper,the natural convection of a complex fluid that contains both nanoparticles and gyrotactic microorganisms in a heated square cavity is considered.The Buongiorno model is applied to descirbe the nanofluid behaviours.Both the top and bottom horizontal walls of the cavity are adiabatic,and there is a temperature difference between the left and right vertical walls.The non-dimensional governing equations are obtained when the stream-vorticity formulation of function is used,which are solved by the recently developed robust Coiflet wavelet homotopy analysis method.A rigid verification for the solver is given.Besides,the effects of various physics parameters including the Rayleigh number,the buoyancy ratio parameter,the bioconvection Rayleigh number,the Prandtl number,the Brownian motion parameter,the thermophoresis parameter,the heat generation parameter,the Lewis number,the bioconvection Peclet number and the Schmidt number on this complicated natural convection are examined.It is known that natural convection is closely related to our daily life owing to its wide existence in nature and engineering applications.We believe that our work will make a significant contribution to a better understanding of the natural convection of a complex fluid in a cavity with suspensions of both inorganic nanoparticles and organic microorganisms.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
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
The Coiflet wavelet-homotopy Galerkin method is extended to solve unsteady nonlinear wave equations for the first time.The Korteweg-de Vries(KdV)equation,the Burgers equation and the Korteweg-de Vries-Burgers(KdVB)equ...The Coiflet wavelet-homotopy Galerkin method is extended to solve unsteady nonlinear wave equations for the first time.The Korteweg-de Vries(KdV)equation,the Burgers equation and the Korteweg-de Vries-Burgers(KdVB)equation are examined as illustrative examples.Validity and accuracy of the proposed method are assessed in terms of relative variance and the maximum error norm.Our results are found in good agreement with exact solutions and numerical solutions reported in previous studies.Furthermore,it is found that the solution accuracy is closely related to the resolution level and the convergence-control parameter.It is also found that our proposed method is superior to the traditional homotopy analysis method when dealing with unsteady nonlinear problems.It is expected that this approach can be further used to solve complicated unsteady problems in the fields of science and engineering.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 sixth-order accurate wavelet integral collocation method is proposed for solving high-order nonlinear boundary value problems in three dimensions.In order to realize the establishment of this method,an approximate e...A sixth-order accurate wavelet integral collocation method is proposed for solving high-order nonlinear boundary value problems in three dimensions.In order to realize the establishment of this method,an approximate expression of multiple integrals of a continuous function defined in a three-dimensional bounded domain is proposed by combining wavelet expansion and Lagrange boundary extension.Through applying such an integral technique,during the solution of nonlinear partial differential equations,the unknown function and its lower-order partial derivatives can be approximately expressed by its highest-order partial derivative values at nodes.A set of nonlinear algebraic equations with respect to these nodal values of the highest-order partial derivative is obtained using a collocation method.The validation and convergence of the proposed method are examined through several benchmark problems,including the eighth-order two-dimensional and fourth-order three-dimensional boundary value problems and the large deflection bending of von Karman plates.Results demonstrate that the present method has higher accuracy and convergence rate than most existing numerical methods.Most importantly,the convergence rate of the proposed method seems to be independent of the order of the differential equations,because it is always sixth order for second-,fourth-,sixth-,and even eighth-order problems.展开更多
基金H.Xu is supported by the National Natural Science Foundation of China(Grant No.11872241)This work was partially supported by the Australian Research Council(ARC)through Grants DE150100169,FT160100357 and CE140100003。
文摘In this paper,the natural convection of a complex fluid that contains both nanoparticles and gyrotactic microorganisms in a heated square cavity is considered.The Buongiorno model is applied to descirbe the nanofluid behaviours.Both the top and bottom horizontal walls of the cavity are adiabatic,and there is a temperature difference between the left and right vertical walls.The non-dimensional governing equations are obtained when the stream-vorticity formulation of function is used,which are solved by the recently developed robust Coiflet wavelet homotopy analysis method.A rigid verification for the solver is given.Besides,the effects of various physics parameters including the Rayleigh number,the buoyancy ratio parameter,the bioconvection Rayleigh number,the Prandtl number,the Brownian motion parameter,the thermophoresis parameter,the heat generation parameter,the Lewis number,the bioconvection Peclet number and the Schmidt number on this complicated natural convection are examined.It is known that natural convection is closely related to our daily life owing to its wide existence in nature and engineering applications.We believe that our work will make a significant contribution to a better understanding of the natural convection of a complex fluid in a cavity with suspensions of both inorganic nanoparticles and organic microorganisms.
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.
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.
文摘The Coiflet wavelet-homotopy Galerkin method is extended to solve unsteady nonlinear wave equations for the first time.The Korteweg-de Vries(KdV)equation,the Burgers equation and the Korteweg-de Vries-Burgers(KdVB)equation are examined as illustrative examples.Validity and accuracy of the proposed method are assessed in terms of relative variance and the maximum error norm.Our results are found in good agreement with exact solutions and numerical solutions reported in previous studies.Furthermore,it is found that the solution accuracy is closely related to the resolution level and the convergence-control parameter.It is also found that our proposed method is superior to the traditional homotopy analysis method when dealing with unsteady nonlinear problems.It is expected that this approach can be further used to solve complicated unsteady problems in the fields of science and engineering.
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(Grant Nos.11925204 and 12172154)the 111 Project(Grant No.B14044)the National Key Project of China(Grant No.GJXM92579).
文摘A sixth-order accurate wavelet integral collocation method is proposed for solving high-order nonlinear boundary value problems in three dimensions.In order to realize the establishment of this method,an approximate expression of multiple integrals of a continuous function defined in a three-dimensional bounded domain is proposed by combining wavelet expansion and Lagrange boundary extension.Through applying such an integral technique,during the solution of nonlinear partial differential equations,the unknown function and its lower-order partial derivatives can be approximately expressed by its highest-order partial derivative values at nodes.A set of nonlinear algebraic equations with respect to these nodal values of the highest-order partial derivative is obtained using a collocation method.The validation and convergence of the proposed method are examined through several benchmark problems,including the eighth-order two-dimensional and fourth-order three-dimensional boundary value problems and the large deflection bending of von Karman plates.Results demonstrate that the present method has higher accuracy and convergence rate than most existing numerical methods.Most importantly,the convergence rate of the proposed method seems to be independent of the order of the differential equations,because it is always sixth order for second-,fourth-,sixth-,and even eighth-order problems.
