This article presents a numerical solution for the flow of a Newtonian fluid over an impermeable stretching sheet embedded in a porous medium with the power law surface velocity and variable thickness in the presence ...This article presents a numerical solution for the flow of a Newtonian fluid over an impermeable stretching sheet embedded in a porous medium with the power law surface velocity and variable thickness in the presence of thermal radiation. The flow is caused by non-linear stretching of a sheet. Thermal conductivity of the fluid is assumed to vary linearly with temperature. The governing partial differential equations （PDEs） are transformed into a system of coupled non-linear ordinary differential equations （ODEs） with appropriate boundary conditions for various physical parameters. The remaining system of ODEs is solved numerically using a differential transformation method （DTM）. The effects of the porous parameter, the wall thickness parameter, the radiation parameter, the thermal conductivity parameter, and the Prandtl number on the flow and temperature profiles are presented. Moreover, the local skin-friction and the Nusselt numbers are presented. Comparison of the obtained numerical results is made with previously published results in some special cases, with good agreement. The results obtained in this paper confirm the idea that DTM is a powerful mathematical tool and can be applied to a large class of linear and non-linear problems in different fields of science and engineering.展开更多
<Abstract>In this paper,under the Painlevé-integrable condition,the auto-Bcklund transformations in different forms for a variable-coefficient Korteweg-de Vries model with physical interests are obtained ...<Abstract>In this paper,under the Painlevé-integrable condition,the auto-Bcklund transformations in different forms for a variable-coefficient Korteweg-de Vries model with physical interests are obtained through various methods including the Hirota method,truncated Painlevé expansion method,extended variable-coefficient balancing-act method,and Lax pair.Additionally,the compatibility for the truncated Painlevé expansion method and extended variable-coefficient balancing-act method is testified.展开更多
By the application of the extended homogeneous balance method, we derive an auto-Bācklund transformation (BT) for (2＋1)-dimensional variable coefficient generalized KP equations. Based on the BT, in which there are ...By the application of the extended homogeneous balance method, we derive an auto-Bācklund transformation (BT) for (2＋1)-dimensional variable coefficient generalized KP equations. Based on the BT, in which there are two homogeneity equations to be solved, we obtain some exact solutions containing single solitary waves.展开更多
The bilinear form for a nonisospectral and variable-coefficient KdV equation is obtained and some exact soliton solutions are derived through Hirota method and Wronskian technique. We also derive the bilinear transfor...The bilinear form for a nonisospectral and variable-coefficient KdV equation is obtained and some exact soliton solutions are derived through Hirota method and Wronskian technique. We also derive the bilinear transformation from its Lax pairs and find solutions with the help of the obtained bilinear transformation.展开更多
An approximate solution to Richards＇ equation is presented, mathematically describing a sort of unsaturated single phase fluid flow in porous media. The approach is a differential transform method （DTM） with interm...An approximate solution to Richards＇ equation is presented, mathematically describing a sort of unsaturated single phase fluid flow in porous media. The approach is a differential transform method （DTM） with intermediate variables. Two examples are given to demonstrate the accuracy of the presented solution.展开更多
A nonlinear transformation of the Whitham-Broer-Kaup (WBK) model equations in the shallow water small-amplitude regime is derived by using a simplified homogeneous balance method. The WBK model equations are linearize...A nonlinear transformation of the Whitham-Broer-Kaup (WBK) model equations in the shallow water small-amplitude regime is derived by using a simplified homogeneous balance method. The WBK model equations are linearized under the nonlinear transformation. Various exact solutions of the WBK model equations are obtained via the nonlinear transformation with the aid of solutions for the linear equation.展开更多
A novel three-dimensional beam propagation method (BPM) based on the variable transformed Galerkin's method is introduced for simulating optical field propagation in three-dimensional dielectric structures. The in...A novel three-dimensional beam propagation method (BPM) based on the variable transformed Galerkin's method is introduced for simulating optical field propagation in three-dimensional dielectric structures. The infinite Cartesian x-y plane is mapped into a unit square by a tangent-type function transformation. Consequently, the infinite region problem is converted into the finite region problem. Thus, the boundary truncation is eliminated and the calculation accuracy is promoted. The three-dimensional BPM basic equation is reduced to a set of first-order ordinary differential equations through sinusoidal basis function, which fits arbitrary cladding optical waveguide, then direct solution of the resulting equations by means of the Runge-Kutta method. In addition, the calculation is efficient due to the small matrix derived from the present technique. Both z-invariant and z-variant examples are considered to test both the accuracy and utility of this approach.展开更多
THE range of usage of the quadrature formulae specially designed for the integrals of sometypes can often be extended by the variable transformation. For example, in a quadrature for-mula computing an integral with th...THE range of usage of the quadrature formulae specially designed for the integrals of sometypes can often be extended by the variable transformation. For example, in a quadrature for-mula computing an integral with the weight function g:展开更多
Welding transformer is widely used in industry manufacturing, depleting a large portion of electricity energy.Based on modern computer technology and mathematical programming, optimum design of electro-magnetic device...Welding transformer is widely used in industry manufacturing, depleting a large portion of electricity energy.Based on modern computer technology and mathematical programming, optimum design of electro-magnetic devices leads to highly efficient use of energy and materials. Are welding transformer is optimized here. A mathematical model,considering both productive cost and operating losses, which is called or Economical-through-Life transformer, is established. Mixed penalty function method, mixed dispersing variable method and improved orthogonal method have been applied to carry out the optimization calculations. Result shows that the power factor is quite important in an Economi-cal-through-Life transformer, and that some principles must be followed in the design work. Also discussed are the advantages and disadvantages of the three methods. In the end, the prospect of optimum design of welding transformer is forecast.展开更多
In this paper, we investigate a(3+1)-dimensional generalized variable-coefficient Kadomtsev–Petviashvili equation, which can describe the nonlinear phenomena in fluids or plasmas. Painlev′e analysis is performed for...In this paper, we investigate a(3+1)-dimensional generalized variable-coefficient Kadomtsev–Petviashvili equation, which can describe the nonlinear phenomena in fluids or plasmas. Painlev′e analysis is performed for us to study the integrability, and we find that the equation is not completely integrable. By virtue of the binary Bell polynomials,bilinear form and soliton solutions are obtained, and B¨acklund transformation in the binary-Bell-polynomial form and bilinear form are derived. Soliton collisions are graphically discussed: the solitons keep their original shapes unchanged after the collision except for the phase shifts. Variable coefficients are seen to affect the motion of solitons: when the variable coefficients are chosen as the constants, solitons keep their directions unchanged during the collision; with the variable coefficients as the functions of the temporal coordinate, the one soliton changes its direction.展开更多
In this work,We are looking at the characteristics of micropolar flow in a porous channel that’s being driven by suction or injection.The working of the fluid is described in the flowmodel.We can reduce the governing...In this work,We are looking at the characteristics of micropolar flow in a porous channel that’s being driven by suction or injection.The working of the fluid is described in the flowmodel.We can reduce the governing nonlinear partial differential equations(PDEs)to a model of coupled systems of nonlinear ordinary differential equations using similarity variables(ODEs).In order to obtain the results of a coupled system of nonlinear ODEs,we discuss a method which is known as the differential transform method(DTM).The concern transform is an excellent mathematical tool to obtain the analytical series solution to the nonlinear ODEs.To observe beast agreement between analytical method and numerical method,we compare our result with the Rung-Kutta method of order four(RK4).We also provide simulation plots to the obtained result by using Mathematica.Onthese plots,we discuss the effect of different parameters which arise during the calculation of the flow model equations.展开更多
In this study,applications of some analytical methods on nonlinear equation of the launching of a rocket with variable mass are investigated.Differential transformation method(DTM),homotopy perturbation method(HPM)and...In this study,applications of some analytical methods on nonlinear equation of the launching of a rocket with variable mass are investigated.Differential transformation method(DTM),homotopy perturbation method(HPM)and least square method(LSM)were applied and their results are compared with numerical solution.An excellent agreement with analytical methods and numerical ones is observed in the results and this reveals that analytical methods are effective and convenient.Also a parametric study is performed here which includes the effect of exhaust velocity(C_(e)),bum rate(BR)of fuel and diameter of cylindrical rocket(d)on the motion of a sample rocket,and contours for showing the sensitivity of these parameters are plotted.The main results indicate that the rocket velocity and altitude are increased with increasing the C_(e) and BR and decreased with increasing the rocket diameter and drag coefficient.展开更多
文摘This article presents a numerical solution for the flow of a Newtonian fluid over an impermeable stretching sheet embedded in a porous medium with the power law surface velocity and variable thickness in the presence of thermal radiation. The flow is caused by non-linear stretching of a sheet. Thermal conductivity of the fluid is assumed to vary linearly with temperature. The governing partial differential equations （PDEs） are transformed into a system of coupled non-linear ordinary differential equations （ODEs） with appropriate boundary conditions for various physical parameters. The remaining system of ODEs is solved numerically using a differential transformation method （DTM）. The effects of the porous parameter, the wall thickness parameter, the radiation parameter, the thermal conductivity parameter, and the Prandtl number on the flow and temperature profiles are presented. Moreover, the local skin-friction and the Nusselt numbers are presented. Comparison of the obtained numerical results is made with previously published results in some special cases, with good agreement. The results obtained in this paper confirm the idea that DTM is a powerful mathematical tool and can be applied to a large class of linear and non-linear problems in different fields of science and engineering.
基金supported by the Key Project of the Ministry of Education under Grant No.106033Specialized Research Fund for the Doctoral Program of Higher Education under Grant No.20060006024+2 种基金Ministry of Education,National Natural Science Foundation of China under Grant Nos.60372095 and 60772023Open Fund of the State Key Laboratory of Software Development Environment under Grant No.SKLSDE-07-001Beijing University of Aeronautics and Astronautics,and National Basic Research Program of China (973 Program) under Grant No.2005CB321901
文摘<Abstract>In this paper,under the Painlevé-integrable condition,the auto-Bcklund transformations in different forms for a variable-coefficient Korteweg-de Vries model with physical interests are obtained through various methods including the Hirota method,truncated Painlevé expansion method,extended variable-coefficient balancing-act method,and Lax pair.Additionally,the compatibility for the truncated Painlevé expansion method and extended variable-coefficient balancing-act method is testified.
文摘By the application of the extended homogeneous balance method, we derive an auto-Bācklund transformation (BT) for (2＋1)-dimensional variable coefficient generalized KP equations. Based on the BT, in which there are two homogeneity equations to be solved, we obtain some exact solutions containing single solitary waves.
文摘The bilinear form for a nonisospectral and variable-coefficient KdV equation is obtained and some exact soliton solutions are derived through Hirota method and Wronskian technique. We also derive the bilinear transformation from its Lax pairs and find solutions with the help of the obtained bilinear transformation.
基金Project supported by the National Basic Research Program of China(973 Program)(No.2011CB013800)
文摘An approximate solution to Richards＇ equation is presented, mathematically describing a sort of unsaturated single phase fluid flow in porous media. The approach is a differential transform method （DTM） with intermediate variables. Two examples are given to demonstrate the accuracy of the presented solution.
文摘A nonlinear transformation of the Whitham-Broer-Kaup (WBK) model equations in the shallow water small-amplitude regime is derived by using a simplified homogeneous balance method. The WBK model equations are linearized under the nonlinear transformation. Various exact solutions of the WBK model equations are obtained via the nonlinear transformation with the aid of solutions for the linear equation.
文摘A novel three-dimensional beam propagation method (BPM) based on the variable transformed Galerkin's method is introduced for simulating optical field propagation in three-dimensional dielectric structures. The infinite Cartesian x-y plane is mapped into a unit square by a tangent-type function transformation. Consequently, the infinite region problem is converted into the finite region problem. Thus, the boundary truncation is eliminated and the calculation accuracy is promoted. The three-dimensional BPM basic equation is reduced to a set of first-order ordinary differential equations through sinusoidal basis function, which fits arbitrary cladding optical waveguide, then direct solution of the resulting equations by means of the Runge-Kutta method. In addition, the calculation is efficient due to the small matrix derived from the present technique. Both z-invariant and z-variant examples are considered to test both the accuracy and utility of this approach.
文摘THE range of usage of the quadrature formulae specially designed for the integrals of sometypes can often be extended by the variable transformation. For example, in a quadrature for-mula computing an integral with the weight function g:
文摘Welding transformer is widely used in industry manufacturing, depleting a large portion of electricity energy.Based on modern computer technology and mathematical programming, optimum design of electro-magnetic devices leads to highly efficient use of energy and materials. Are welding transformer is optimized here. A mathematical model,considering both productive cost and operating losses, which is called or Economical-through-Life transformer, is established. Mixed penalty function method, mixed dispersing variable method and improved orthogonal method have been applied to carry out the optimization calculations. Result shows that the power factor is quite important in an Economi-cal-through-Life transformer, and that some principles must be followed in the design work. Also discussed are the advantages and disadvantages of the three methods. In the end, the prospect of optimum design of welding transformer is forecast.
基金Supported by the National Natural Science Foundation of China under Grant No.11272023the Open Fund of State Key Laboratory of Information Photonics and Optical Communications(Beijing University of Posts and Telecommunications)under GrantNo.IPOC2013B008the Fundamental Research Funds for the Central Universities of China under Grant No.2011BUPTYB02
文摘In this paper, we investigate a(3+1)-dimensional generalized variable-coefficient Kadomtsev–Petviashvili equation, which can describe the nonlinear phenomena in fluids or plasmas. Painlev′e analysis is performed for us to study the integrability, and we find that the equation is not completely integrable. By virtue of the binary Bell polynomials,bilinear form and soliton solutions are obtained, and B¨acklund transformation in the binary-Bell-polynomial form and bilinear form are derived. Soliton collisions are graphically discussed: the solitons keep their original shapes unchanged after the collision except for the phase shifts. Variable coefficients are seen to affect the motion of solitons: when the variable coefficients are chosen as the constants, solitons keep their directions unchanged during the collision; with the variable coefficients as the functions of the temporal coordinate, the one soliton changes its direction.
基金Princess Nourah bint Abdulrahman University Researchers Supporting Project No. (PNURSP2023R14)。
文摘In this work,We are looking at the characteristics of micropolar flow in a porous channel that’s being driven by suction or injection.The working of the fluid is described in the flowmodel.We can reduce the governing nonlinear partial differential equations(PDEs)to a model of coupled systems of nonlinear ordinary differential equations using similarity variables(ODEs).In order to obtain the results of a coupled system of nonlinear ODEs,we discuss a method which is known as the differential transform method(DTM).The concern transform is an excellent mathematical tool to obtain the analytical series solution to the nonlinear ODEs.To observe beast agreement between analytical method and numerical method,we compare our result with the Rung-Kutta method of order four(RK4).We also provide simulation plots to the obtained result by using Mathematica.Onthese plots,we discuss the effect of different parameters which arise during the calculation of the flow model equations.
文摘In this study,applications of some analytical methods on nonlinear equation of the launching of a rocket with variable mass are investigated.Differential transformation method(DTM),homotopy perturbation method(HPM)and least square method(LSM)were applied and their results are compared with numerical solution.An excellent agreement with analytical methods and numerical ones is observed in the results and this reveals that analytical methods are effective and convenient.Also a parametric study is performed here which includes the effect of exhaust velocity(C_(e)),bum rate(BR)of fuel and diameter of cylindrical rocket(d)on the motion of a sample rocket,and contours for showing the sensitivity of these parameters are plotted.The main results indicate that the rocket velocity and altitude are increased with increasing the C_(e) and BR and decreased with increasing the rocket diameter and drag coefficient.
