Kerosene-alumina nanofluid flow and heat transfer in the presence of magnetic field are studied. The basic partial differential equations are reduced to ordinary differential equations which are solved semi analytical...Kerosene-alumina nanofluid flow and heat transfer in the presence of magnetic field are studied. The basic partial differential equations are reduced to ordinary differential equations which are solved semi analytically using differential transformation method. Velocity and temperature profiles as well as the skin friction coefficient and the Nusselt number are determined analytically. The influence of pertinent parameters such as magnetic parameter, nanofluid volume fraction, viscosity parameter and Eckert number on the flow and heat transfer characteristics is discussed. Results indicate that skin friction coefficient decreases with increase of magnetic parameter, nanofluid volume fraction and viscosity parameter. Nusselt number increases with increase of magnetic parameter and nanofluid volume fraction while it decreases with increase of Eckert number and viscosity parameter.展开更多
The thermal conduction behavior of the three-dimensional axisymmetric functionally graded circular plate was studied under thermal loads on its top and bottom surfaces. Material properties were taken to be arbitrary d...The thermal conduction behavior of the three-dimensional axisymmetric functionally graded circular plate was studied under thermal loads on its top and bottom surfaces. Material properties were taken to be arbitrary distribution functions of the thickness. A temperature function that satisfies thermal boundary conditions at the edges and the variable separation method were used to reduce equation governing the steady state heat conduction to an ordinary differential equation (ODE) in the thickness coordinate which was solved analytically. Next, resulting variable coefficients ODE due to arbitrary distribution of material properties along thickness coordinate was also solved by the Peano-Baker series. Some numerical examples were given to demonstrate the accuracy, efficiency of the present model, mad to investigate the influence of different distributions of material properties on the temperature field. The numerical results confirm that the influence of different material distributions, gradient indices and thickness of plate to temperature field in plate can not be ignored.展开更多
The steady two-dimensional flow of Powell-Eyring fluid is investigated. The flow is caused by a stretching surface with homogeneous-heterogeneous reactions. The governing nonlinear differential equations are reduced t...The steady two-dimensional flow of Powell-Eyring fluid is investigated. The flow is caused by a stretching surface with homogeneous-heterogeneous reactions. The governing nonlinear differential equations are reduced to the ordinary differential equations by similarity transformations. The analytic solutions are presented in series forms by homotopy analysis method(HAM). Convergence of the obtained series solutions is explicitly discussed. The physical significance of different parameters on the velocity and concentration profiles is discussed through graphical illustrations. It is noticed that the boundary layer thickness increases by increasing the Powell-Eyring fluid material parameter(ε) whereas it decreases by increasing the fluid material parameter(δ). Further, the concentration profile increases when Powell-Eyring fluid material parameters increase. The concentration is also an increasing function of Schmidt number and decreasing function of strength of homogeneous reaction. Also mass transfer rate increases for larger rate of heterogeneous reaction.展开更多
Based on a transformed Painlev~ property and the variable separated ODE method, a function transfor- mation method is proposed to search for exact solutions of some partial differential equations (PDEs) with hyperbo...Based on a transformed Painlev~ property and the variable separated ODE method, a function transfor- mation method is proposed to search for exact solutions of some partial differential equations (PDEs) with hyperbolic or exponential functions. This approach provides a more systematical and convenient handling of the solution process of this kind of nonlinear equations. Its key point is to eradicate the hyperbolic or exponential terms by a transformed Painleve property and reduce the given PDEs to a variable-coefficient the resulting equations by some methods. As an application, are formally derived. ordinary differential equations, then we seek for solutions to exact solutions for the combined sinh-cosh-Gordon equation展开更多
An inverse problem of determining the source term of the non-stationary transport equation was considered.The extra lateral boundary condition was chosen as the observation data.Based on a Carleman estimate on the non...An inverse problem of determining the source term of the non-stationary transport equation was considered.The extra lateral boundary condition was chosen as the observation data.Based on a Carleman estimate on the non-stationary transport equation,a global Lipschitz stability for the inverse problem was proved.展开更多
Basing on the direct method developed by Clarkson and Kruskal, the nearly concentric Korteweg-de Vries (ncKdV) equation can be reduced to three types of (1+1)-dimensional variable coefficients partial differentia...Basing on the direct method developed by Clarkson and Kruskal, the nearly concentric Korteweg-de Vries (ncKdV) equation can be reduced to three types of (1+1)-dimensional variable coefficients partial differential equations (PDEs) and three types of variable coefficients ordinary differential equation. Furthermore, three types of (1+1)-dimensional variable coefficients PDEs are all reduced to constant coefficients PDEs by some transformations.展开更多
Classification and reduction of the generalized fourth-order nonlinear differential equations arising from theliquid films are considered.It is shown that these equations have solutions on subspaces of the polynomial,...Classification and reduction of the generalized fourth-order nonlinear differential equations arising from theliquid films are considered.It is shown that these equations have solutions on subspaces of the polynomial,exponential ortrigonometric form defined by linear nth-order ordinary differential equations with constant coefficients for n=4,...,9.Several examples of exact solutions are presented.展开更多
Basing on the direct method developed by Clarkson and Kruskal,the nonisospectral BKP equation can bereduced to three types of(1+1)-dimensional variable coefficients partial differential equations(PDEs).Furthermore,ont...Basing on the direct method developed by Clarkson and Kruskal,the nonisospectral BKP equation can bereduced to three types of(1+1)-dimensional variable coefficients partial differential equations(PDEs).Furthermore,onthe basis of the idea of the symmetry group direct method by Lou et al.,three types of reduction PDEs are all reducedto the related constant coefficients PDEs by some transformations.展开更多
This paper is concerned with the order of the solutions of systems of high-order complex algebraic differential equations.By means of Zalcman Lemma,the systems of equations of[1]is extended to more general form.
When one wants to calculate all the three components of magnetization of Heisenberg model under random phase approximation, at least one of the components should be the solution of an ordinary differential equation. I...When one wants to calculate all the three components of magnetization of Heisenberg model under random phase approximation, at least one of the components should be the solution of an ordinary differential equation. In this paper such an equation is established. It is argued that the general expressions of magnetization for any spin quantum number S suggested before are the solution of the ordinary differential equation.展开更多
The symmetries, symmetry reductions, and exact solutions of a coupled nonlinear Schrodinger (CNLS) equation derived from the governing system for atmospheric gravity waves are researched by means of classical Lie gr...The symmetries, symmetry reductions, and exact solutions of a coupled nonlinear Schrodinger (CNLS) equation derived from the governing system for atmospheric gravity waves are researched by means of classical Lie group approach in this paper. Calculation shows the CNLS equation is invariant under some Galilean transformations, scaling transformations, phase shifts, and space-time translations. Some ordinary differential equations are derived from the CNLS equation. Several exact solutions including envelope cnoidal waves, solitary waves and trigonometric function solutions for the CNLS equation are also obtained by making use of symmetries.展开更多
The aim of this paper is to obtain numerical solutions of the one-dimensional,two-dimensional and coupled Burgers' equations through the generalized differential quadrature method(GDQM).The polynomial-based differ...The aim of this paper is to obtain numerical solutions of the one-dimensional,two-dimensional and coupled Burgers' equations through the generalized differential quadrature method(GDQM).The polynomial-based differential quadrature(PDQ) method is employed and the obtained system of ordinary differential equations is solved via the total variation diminishing Runge-Kutta(TVD-RK) method.The numerical solutions are satisfactorily coincident with the exact solutions.The method can compete against the methods applied in the literature.展开更多
Segmentation of three-dimensional(3D) complicated structures is of great importance for many real applications.In this work we combine graph cut minimization method with a variant of the level set idea for 3D segmenta...Segmentation of three-dimensional(3D) complicated structures is of great importance for many real applications.In this work we combine graph cut minimization method with a variant of the level set idea for 3D segmentation based on the Mumford-Shah model.Compared with the traditional approach for solving the Euler-Lagrange equation we do not need to solve any partial differential equations.Instead,the minimum cut on a special designed graph need to be computed.The method is tested on data with complicated structures.It is rather stable with respect to initial value and the algorithm is nearly parameter free.Experiments show that it can solve large problems much faster than traditional approaches.展开更多
Mathematics is very important for the engineering and scientist but to make understand the mathematics is very difficult if without proper tools and suitable measurement. A numerical method is one of the algorithms wh...Mathematics is very important for the engineering and scientist but to make understand the mathematics is very difficult if without proper tools and suitable measurement. A numerical method is one of the algorithms which involved with computer programming. In this paper, Scilab is used to carter the problems related the mathematical models such as Matrices, operation with ODE's and solving the Integration.展开更多
In this paper, a singularly perturbed Robin type boundary value problem for second-order ordinary differential equation with discontinuous convection coefficient and source term is considered. A robust-layer-resolving...In this paper, a singularly perturbed Robin type boundary value problem for second-order ordinary differential equation with discontinuous convection coefficient and source term is considered. A robust-layer-resolving numerical method is proposed. An e-uniform global error estimate for the numerical solution and also to the numerical derivative are established. Numerical results are presented, which are in agreement with the theoretical predictions.展开更多
In the theory of Wiman-Valiron on entire function,a theorem on the derivatives of entire functions has important applications to differential equations. We had extended the theorem to the meromorphic functions which h...In the theory of Wiman-Valiron on entire function,a theorem on the derivatives of entire functions has important applications to differential equations. We had extended the theorem to the meromorphic functions which have finite poles. In this paper we give some applications to linner differential equations and algebraic differential equations.展开更多
This paper present an implementation of"modified cubic B-spline differential quadrature method (MCB-DQM)" proposed by Arora & Singh (Applied Mathematics and Computation Vol. 224(1) (2013) 161-177) for numer...This paper present an implementation of"modified cubic B-spline differential quadrature method (MCB-DQM)" proposed by Arora & Singh (Applied Mathematics and Computation Vol. 224(1) (2013) 161-177) for numerical computation of Fokker-Planck equations. The modified cubic B-splines are used as set of basis functions in the differential quadrature to compute the weighting coefficients for the spatial derivatives, which reduces Fokker-Planck equation into system of first-order ordinary differential equations (ODEs), in time. The well known SSP-RK43 scheme is then applied to solve the resulting system of ODEs. The efficiency of proposed method has been confirmed by three examples having their exact solutions. This shows that MCB-DQM results are capable of achieving high accuracy. Advantage of the scheme is that it can be applied very smoothly to solve the linear or nonlinear physical problems, and a very less storage space is required which causes less accumulation of numerical errors.展开更多
Based on the closed connections among the homogeneous balance (HB) method and Clarkson-KruSkal (CK) method, we study the similarity reductions of the generalized variable coefficients 2D KdV equation. In the meant...Based on the closed connections among the homogeneous balance (HB) method and Clarkson-KruSkal (CK) method, we study the similarity reductions of the generalized variable coefficients 2D KdV equation. In the meantime it is shown that this leads to a direct reduction in the form of ordinary differential equation under some integrability conditions between the variable coefficients. Two different cases have been discussed, the search for solutions of those ordinary differential equations yielded many exact travelling and solitonic wave solutions in the form of hyperbolic and trigonometric functions under some constraints between the variable coefficients.展开更多
文摘Kerosene-alumina nanofluid flow and heat transfer in the presence of magnetic field are studied. The basic partial differential equations are reduced to ordinary differential equations which are solved semi analytically using differential transformation method. Velocity and temperature profiles as well as the skin friction coefficient and the Nusselt number are determined analytically. The influence of pertinent parameters such as magnetic parameter, nanofluid volume fraction, viscosity parameter and Eckert number on the flow and heat transfer characteristics is discussed. Results indicate that skin friction coefficient decreases with increase of magnetic parameter, nanofluid volume fraction and viscosity parameter. Nusselt number increases with increase of magnetic parameter and nanofluid volume fraction while it decreases with increase of Eckert number and viscosity parameter.
基金Project(11102136)supported by the National Natural Science Foundation of ChinaProject(2012ZDK04)supported by the Open Project of Guangxi Key Laboratory of Disaster Prevention and Structural Safety,China
文摘The thermal conduction behavior of the three-dimensional axisymmetric functionally graded circular plate was studied under thermal loads on its top and bottom surfaces. Material properties were taken to be arbitrary distribution functions of the thickness. A temperature function that satisfies thermal boundary conditions at the edges and the variable separation method were used to reduce equation governing the steady state heat conduction to an ordinary differential equation (ODE) in the thickness coordinate which was solved analytically. Next, resulting variable coefficients ODE due to arbitrary distribution of material properties along thickness coordinate was also solved by the Peano-Baker series. Some numerical examples were given to demonstrate the accuracy, efficiency of the present model, mad to investigate the influence of different distributions of material properties on the temperature field. The numerical results confirm that the influence of different material distributions, gradient indices and thickness of plate to temperature field in plate can not be ignored.
文摘The steady two-dimensional flow of Powell-Eyring fluid is investigated. The flow is caused by a stretching surface with homogeneous-heterogeneous reactions. The governing nonlinear differential equations are reduced to the ordinary differential equations by similarity transformations. The analytic solutions are presented in series forms by homotopy analysis method(HAM). Convergence of the obtained series solutions is explicitly discussed. The physical significance of different parameters on the velocity and concentration profiles is discussed through graphical illustrations. It is noticed that the boundary layer thickness increases by increasing the Powell-Eyring fluid material parameter(ε) whereas it decreases by increasing the fluid material parameter(δ). Further, the concentration profile increases when Powell-Eyring fluid material parameters increase. The concentration is also an increasing function of Schmidt number and decreasing function of strength of homogeneous reaction. Also mass transfer rate increases for larger rate of heterogeneous reaction.
基金Supported by National Natural Science Foundation of China under Grant No.10926057 Foundation of Zhejiang Educational Committee under Grant No.Y200908784
文摘Based on a transformed Painlev~ property and the variable separated ODE method, a function transfor- mation method is proposed to search for exact solutions of some partial differential equations (PDEs) with hyperbolic or exponential functions. This approach provides a more systematical and convenient handling of the solution process of this kind of nonlinear equations. Its key point is to eradicate the hyperbolic or exponential terms by a transformed Painleve property and reduce the given PDEs to a variable-coefficient the resulting equations by some methods. As an application, are formally derived. ordinary differential equations, then we seek for solutions to exact solutions for the combined sinh-cosh-Gordon equation
文摘An inverse problem of determining the source term of the non-stationary transport equation was considered.The extra lateral boundary condition was chosen as the observation data.Based on a Carleman estimate on the non-stationary transport equation,a global Lipschitz stability for the inverse problem was proved.
基金Supported by K.C. Wong Magna Fund in Ningbo University, NSF of China under Grant Nos. 10747141 and 10735030Zhejiang Provincial Natural Science Foundations of China under Grant No. 605408the Ningbo Natural Science Foundation under Grant Nos. 2007A610049 and 2006A610093
文摘Basing on the direct method developed by Clarkson and Kruskal, the nearly concentric Korteweg-de Vries (ncKdV) equation can be reduced to three types of (1+1)-dimensional variable coefficients partial differential equations (PDEs) and three types of variable coefficients ordinary differential equation. Furthermore, three types of (1+1)-dimensional variable coefficients PDEs are all reduced to constant coefficients PDEs by some transformations.
基金Supported by the National Natural Science Foundation of China under Grant No.10671156the Northwest University Graduate Innovation and Creativity Funds under Grant No.07YZZ15
文摘Classification and reduction of the generalized fourth-order nonlinear differential equations arising from theliquid films are considered.It is shown that these equations have solutions on subspaces of the polynomial,exponential ortrigonometric form defined by linear nth-order ordinary differential equations with constant coefficients for n=4,...,9.Several examples of exact solutions are presented.
基金Supported by National Natural Science Foundation of China under Grant Nos.10747141 and 10735030Zhejiang Provincial Natural Science Foundations under Grant No.605408+2 种基金Ningbo Natural Science Foundation under Grant Nos.2007A610049 and 2008A610017National Basic Research Program of China (973 Program 2007CB814800)K.C.Wong Magna Fund in Ningbo University
文摘Basing on the direct method developed by Clarkson and Kruskal,the nonisospectral BKP equation can bereduced to three types of(1+1)-dimensional variable coefficients partial differential equations(PDEs).Furthermore,onthe basis of the idea of the symmetry group direct method by Lou et al.,three types of reduction PDEs are all reducedto the related constant coefficients PDEs by some transformations.
基金Supported by the Natural Science Foundation of Guangdong Province(04010474) Supported by the Foundation of the Education Department of Anhui Province for Outstanding Young Teachers in University(2011SQRL172)
文摘This paper is concerned with the order of the solutions of systems of high-order complex algebraic differential equations.By means of Zalcman Lemma,the systems of equations of[1]is extended to more general form.
基金the State Key Project of Fundamental Research of China under
文摘When one wants to calculate all the three components of magnetization of Heisenberg model under random phase approximation, at least one of the components should be the solution of an ordinary differential equation. In this paper such an equation is established. It is argued that the general expressions of magnetization for any spin quantum number S suggested before are the solution of the ordinary differential equation.
基金supported by the Scientific Research Foundation for the Doctors of University of Electronic Science and Technology of China Zhongshan Institutethe National Natural Science Foundation of China under Grant Nos. 10735030 and 90503006
文摘The symmetries, symmetry reductions, and exact solutions of a coupled nonlinear Schrodinger (CNLS) equation derived from the governing system for atmospheric gravity waves are researched by means of classical Lie group approach in this paper. Calculation shows the CNLS equation is invariant under some Galilean transformations, scaling transformations, phase shifts, and space-time translations. Some ordinary differential equations are derived from the CNLS equation. Several exact solutions including envelope cnoidal waves, solitary waves and trigonometric function solutions for the CNLS equation are also obtained by making use of symmetries.
文摘The aim of this paper is to obtain numerical solutions of the one-dimensional,two-dimensional and coupled Burgers' equations through the generalized differential quadrature method(GDQM).The polynomial-based differential quadrature(PDQ) method is employed and the obtained system of ordinary differential equations is solved via the total variation diminishing Runge-Kutta(TVD-RK) method.The numerical solutions are satisfactorily coincident with the exact solutions.The method can compete against the methods applied in the literature.
基金support from the Centre for Integrated Petroleum Research(CIPR),University of Bergen, Norway,and Singapore MOE Grant T207B2202NRF2007IDMIDM002-010
文摘Segmentation of three-dimensional(3D) complicated structures is of great importance for many real applications.In this work we combine graph cut minimization method with a variant of the level set idea for 3D segmentation based on the Mumford-Shah model.Compared with the traditional approach for solving the Euler-Lagrange equation we do not need to solve any partial differential equations.Instead,the minimum cut on a special designed graph need to be computed.The method is tested on data with complicated structures.It is rather stable with respect to initial value and the algorithm is nearly parameter free.Experiments show that it can solve large problems much faster than traditional approaches.
文摘Mathematics is very important for the engineering and scientist but to make understand the mathematics is very difficult if without proper tools and suitable measurement. A numerical method is one of the algorithms which involved with computer programming. In this paper, Scilab is used to carter the problems related the mathematical models such as Matrices, operation with ODE's and solving the Integration.
基金the Council of Scientific and Industrial Research,New Delhi,India for its financial support.
文摘In this paper, a singularly perturbed Robin type boundary value problem for second-order ordinary differential equation with discontinuous convection coefficient and source term is considered. A robust-layer-resolving numerical method is proposed. An e-uniform global error estimate for the numerical solution and also to the numerical derivative are established. Numerical results are presented, which are in agreement with the theoretical predictions.
文摘In the theory of Wiman-Valiron on entire function,a theorem on the derivatives of entire functions has important applications to differential equations. We had extended the theorem to the meromorphic functions which have finite poles. In this paper we give some applications to linner differential equations and algebraic differential equations.
文摘This paper present an implementation of"modified cubic B-spline differential quadrature method (MCB-DQM)" proposed by Arora & Singh (Applied Mathematics and Computation Vol. 224(1) (2013) 161-177) for numerical computation of Fokker-Planck equations. The modified cubic B-splines are used as set of basis functions in the differential quadrature to compute the weighting coefficients for the spatial derivatives, which reduces Fokker-Planck equation into system of first-order ordinary differential equations (ODEs), in time. The well known SSP-RK43 scheme is then applied to solve the resulting system of ODEs. The efficiency of proposed method has been confirmed by three examples having their exact solutions. This shows that MCB-DQM results are capable of achieving high accuracy. Advantage of the scheme is that it can be applied very smoothly to solve the linear or nonlinear physical problems, and a very less storage space is required which causes less accumulation of numerical errors.
文摘Based on the closed connections among the homogeneous balance (HB) method and Clarkson-KruSkal (CK) method, we study the similarity reductions of the generalized variable coefficients 2D KdV equation. In the meantime it is shown that this leads to a direct reduction in the form of ordinary differential equation under some integrability conditions between the variable coefficients. Two different cases have been discussed, the search for solutions of those ordinary differential equations yielded many exact travelling and solitonic wave solutions in the form of hyperbolic and trigonometric functions under some constraints between the variable coefficients.