Using a simple and reliable apparatus, the solubilities of adipic acid in water, ethanol, chloroform, n-butanol and acetone are determined by the analytic method. The results are correlated with λh equation, Apelblat...Using a simple and reliable apparatus, the solubilities of adipic acid in water, ethanol, chloroform, n-butanol and acetone are determined by the analytic method. The results are correlated with λh equation, Apelblat equation, and UNIFAC equation. The solubilities calculated by these models are in good agreement with experi-mental data, so that the models can meet the requirements of engineering design.展开更多
Based on the homogenous balance method and with the help of mathematica, the Backlund transformation and the transfer heat equation are derived. Analyzing the heat-transfer equation, the multiple soliton solutions and...Based on the homogenous balance method and with the help of mathematica, the Backlund transformation and the transfer heat equation are derived. Analyzing the heat-transfer equation, the multiple soliton solutions and other exact analytical solution for Whitham-Broer-Kaup equations(WBK) are derived. These solutions contain Fan's, Xie's and Yan's results and other new types of analytical solutions, such as rational function solutions and periodic solutions. The method can also be applied to solve more nonlinear differential equations.展开更多
The mechanism of transport of chemicals in soil is an important research topic of environmental science and engineering, and some models and methods for a variety of solute transport problems have been done. Howeve...The mechanism of transport of chemicals in soil is an important research topic of environmental science and engineering, and some models and methods for a variety of solute transport problems have been done. However. most of previous works are usually for a soil column of infinite dimension. Starting from the one-dimension transient solute transport equation and its boundary and initial condition for a solute transport problem of soil column of finite length, this work has successfully applied a variable transformation to simplify the partial differential equation of solute transport problem. And an analytical serial solution for the simplified equation is then established by the so-called separated variable method and the superposition method. Compared with numerical methods such as finite different method and finite element method, this analytical solution is more accurate and of higher computation efficiency. In addition, the solution procedure presented could be extended for applications such as quality analysis, design of physical experimentation, or parameter estimation and measurement of solute transport problems.展开更多
In the present paper, we construct the analytical exact solutions of some nonlinear evolution equations in mathematical physics; namely the space-time fractional Zakharov–Kuznetsov(ZK) and modified Zakharov–Kuznetso...In the present paper, we construct the analytical exact solutions of some nonlinear evolution equations in mathematical physics; namely the space-time fractional Zakharov–Kuznetsov(ZK) and modified Zakharov–Kuznetsov(m ZK) equations by using fractional sub-equation method. As a result, new types of exact analytical solutions are obtained. The obtained results are shown graphically. Here the fractional derivative is described in the Jumarie's modified Riemann–Liouville sense.展开更多
In this study,by means of homotopy perturbation method(HPM) an approximate solution of the magnetohydrodynamic(MHD) boundary layer flow is obtained.The main feature of the HPM is that it deforms a difficult problem in...In this study,by means of homotopy perturbation method(HPM) an approximate solution of the magnetohydrodynamic(MHD) boundary layer flow is obtained.The main feature of the HPM is that it deforms a difficult problem into a set of problems which are easier to solve.HPM produces analytical expressions for the solution to nonlinear differential equations.The obtained analytic solution is in the form of an infinite power series.In this work,the analytical solution obtained by using only two terms from HPM solution.Comparisons with the exact solution and the solution obtained by the Pade approximants and shooting method show the high accuracy,simplicity and efficiency of this method.展开更多
Both the homotopy analysis method and Galerkin spectral method are applied to find the analytical solutions of the two-dimensional and time-independent Gross-Pitaevskii equation, a nonlinear Schrodinger equation used ...Both the homotopy analysis method and Galerkin spectral method are applied to find the analytical solutions of the two-dimensional and time-independent Gross-Pitaevskii equation, a nonlinear Schrodinger equation used in describing the system of Bose-Einstein condensates trapped in a harmonic potential. The approximate analytical solutions are obtained successfully. Comparisons between the analytical solutions and the numerical solutions have been made. The results indicate that they are agreement very well with each other when the atomic interaction is not too strong.展开更多
The travelling wave solutions of a generalized Camassa-Holm-Degasperis-Procesi equation ut-uxxt + (1 + b)umux = buxuxx + uuxxx are considered where b > 1 and m are positive integers. The qualitative analysis method...The travelling wave solutions of a generalized Camassa-Holm-Degasperis-Procesi equation ut-uxxt + (1 + b)umux = buxuxx + uuxxx are considered where b > 1 and m are positive integers. The qualitative analysis methods of planar autonomous systems yield its phase portraits. Its soliton wave solutions, kink or antikink wave solutions, peakon wave solutions, compacton wave solutions, periodic wave solutions and periodic cusp wave solutions are obtained. Some numerical simulations of these solutions are also given.展开更多
Transport and diffusion caused by coastal waves have different characteristics from those induced by flows. Through solving the vertical diffusion equation by an analytic method, this paper infers a theoretical formul...Transport and diffusion caused by coastal waves have different characteristics from those induced by flows. Through solving the vertical diffusion equation by an analytic method, this paper infers a theoretical formula of dispersion coefficient under the combined action of current and waves. It divides the general dispersion coefficient into six parts, including coefficients due to tidal current, Stokes drift, wave oscillation and interaction among them. It draws a conclusion that the contribution of dispersive effect induced by coastal waves is mainly produced by Stokes drift, while the contributions to time-averaged dispersion coefficient due to wave orbital motion and interaction between current and waves are very small. The results without tidal current are in agreement with the numerical and experimental results, which proves the correctness of the theoretical derivation. This paper introduces the variation characteristics of both the time-averaged and oscillating dispersion coefficients versus relative water depth, and demonstrates the physical implications of the oscillating mixing coefficient due to waves. We also apply the results to the costal vertical circulation and give its characteristics compared to Stokes drift.展开更多
Creation of fermionic particles by a time-dependent electric field and a space-dependent magnetic field is studied with the Bogoulibov transformation method. Exact analytic solutions of the Dirac equation are obtained...Creation of fermionic particles by a time-dependent electric field and a space-dependent magnetic field is studied with the Bogoulibov transformation method. Exact analytic solutions of the Dirac equation are obtained in terms of the Whittaker functions and the particle creation number density depending on the electric and magnetic fields is determined.展开更多
In this paper, the analytical solutions of Schrodinger equation for Brownian motion in a double well potential are acquired by the homotopy analysis method and the Adomian decomposition method. Double well potential f...In this paper, the analytical solutions of Schrodinger equation for Brownian motion in a double well potential are acquired by the homotopy analysis method and the Adomian decomposition method. Double well potential for Brownian motion is always used to obtain the solutions of Fokker-P1anck equation known as the Klein-Kramers equation, which is suitable for separation and additive Hamiltonians. In essence, we could study the random motion of Brownian particles by solving Schr6dinger equation. The anaiytical results obtained from the two different methods agree with each other well The double well potentiai is affected by two parameters, which are analyzed and discussed in details with the aid of graphical illustrations. According to the final results, the shapes of the double well potential have significant influence on the probability density function.展开更多
The purpose of the paper is to present analytical and numerical solutions of a degenerate parabolic equation with time-fractional derivatives arising in the spatial diffusion of biological populations. The homotopy-pe...The purpose of the paper is to present analytical and numerical solutions of a degenerate parabolic equation with time-fractional derivatives arising in the spatial diffusion of biological populations. The homotopy-perturbation method is employed for solving this class of equations, and the time-fractional derivatives are described in the sense of Caputo. Comparisons are made with those derived by Adomian＇s decomposition method, revealing that the homotopy perturbation method is more accurate and convenient than the Adomian＇s decomposition method. Furthermore, the results reveal that the approximate solution continuously depends on the time-fractional derivative and the proposed method incorporating the Caputo derivatives is a powerful and efficient technique for solving the fractional differential equations without requiring linearization or restrictive assumptions. The basis ideas presented in the paper can be further applied to solve other similar fractional partial differential equations.展开更多
基金Supported by the Natural Science Foundation of Henan Province (0511021700)
文摘Using a simple and reliable apparatus, the solubilities of adipic acid in water, ethanol, chloroform, n-butanol and acetone are determined by the analytic method. The results are correlated with λh equation, Apelblat equation, and UNIFAC equation. The solubilities calculated by these models are in good agreement with experi-mental data, so that the models can meet the requirements of engineering design.
基金Supported by the National Nature Science Foundation of China(10371070)Supported by the Nature Science Foundation of Educational Committee of Liaoning Province(2021401157)
文摘Based on the homogenous balance method and with the help of mathematica, the Backlund transformation and the transfer heat equation are derived. Analyzing the heat-transfer equation, the multiple soliton solutions and other exact analytical solution for Whitham-Broer-Kaup equations(WBK) are derived. These solutions contain Fan's, Xie's and Yan's results and other new types of analytical solutions, such as rational function solutions and periodic solutions. The method can also be applied to solve more nonlinear differential equations.
基金Acknowledgements： The work was supported by the National Natural Science Foundation of China （No. 90502006/D0123）, Hunan provincial Natural Science Foundation of China （No. 06JJ3020） and Scientific Research Fund of Hunan Provincial Education Department （No. 06C500）.
文摘The mechanism of transport of chemicals in soil is an important research topic of environmental science and engineering, and some models and methods for a variety of solute transport problems have been done. However. most of previous works are usually for a soil column of infinite dimension. Starting from the one-dimension transient solute transport equation and its boundary and initial condition for a solute transport problem of soil column of finite length, this work has successfully applied a variable transformation to simplify the partial differential equation of solute transport problem. And an analytical serial solution for the simplified equation is then established by the so-called separated variable method and the superposition method. Compared with numerical methods such as finite different method and finite element method, this analytical solution is more accurate and of higher computation efficiency. In addition, the solution procedure presented could be extended for applications such as quality analysis, design of physical experimentation, or parameter estimation and measurement of solute transport problems.
基金Supported by BRNS of Bhaba Atomic Research Centre,Mumbai under Department of Atomic Energy,Government of India vide under Grant No.2012/37P/54/BRNS/2382
文摘In the present paper, we construct the analytical exact solutions of some nonlinear evolution equations in mathematical physics; namely the space-time fractional Zakharov–Kuznetsov(ZK) and modified Zakharov–Kuznetsov(m ZK) equations by using fractional sub-equation method. As a result, new types of exact analytical solutions are obtained. The obtained results are shown graphically. Here the fractional derivative is described in the Jumarie's modified Riemann–Liouville sense.
文摘In this study,by means of homotopy perturbation method(HPM) an approximate solution of the magnetohydrodynamic(MHD) boundary layer flow is obtained.The main feature of the HPM is that it deforms a difficult problem into a set of problems which are easier to solve.HPM produces analytical expressions for the solution to nonlinear differential equations.The obtained analytic solution is in the form of an infinite power series.In this work,the analytical solution obtained by using only two terms from HPM solution.Comparisons with the exact solution and the solution obtained by the Pade approximants and shooting method show the high accuracy,simplicity and efficiency of this method.
基金Supported by the National Natural Science Foundation under Grant No. 11047010
文摘Both the homotopy analysis method and Galerkin spectral method are applied to find the analytical solutions of the two-dimensional and time-independent Gross-Pitaevskii equation, a nonlinear Schrodinger equation used in describing the system of Bose-Einstein condensates trapped in a harmonic potential. The approximate analytical solutions are obtained successfully. Comparisons between the analytical solutions and the numerical solutions have been made. The results indicate that they are agreement very well with each other when the atomic interaction is not too strong.
基金supported by China Postdoctoral Science Foundation (Grant Nos. 20100470249, 20100470254)
文摘The travelling wave solutions of a generalized Camassa-Holm-Degasperis-Procesi equation ut-uxxt + (1 + b)umux = buxuxx + uuxxx are considered where b > 1 and m are positive integers. The qualitative analysis methods of planar autonomous systems yield its phase portraits. Its soliton wave solutions, kink or antikink wave solutions, peakon wave solutions, compacton wave solutions, periodic wave solutions and periodic cusp wave solutions are obtained. Some numerical simulations of these solutions are also given.
基金supported by the National Natural Science Foundation of China (Grant Nos. 10672034, 51079024)the Funds for Creative Re-search Groups of China (Grant No. 50921001)
文摘Transport and diffusion caused by coastal waves have different characteristics from those induced by flows. Through solving the vertical diffusion equation by an analytic method, this paper infers a theoretical formula of dispersion coefficient under the combined action of current and waves. It divides the general dispersion coefficient into six parts, including coefficients due to tidal current, Stokes drift, wave oscillation and interaction among them. It draws a conclusion that the contribution of dispersive effect induced by coastal waves is mainly produced by Stokes drift, while the contributions to time-averaged dispersion coefficient due to wave orbital motion and interaction between current and waves are very small. The results without tidal current are in agreement with the numerical and experimental results, which proves the correctness of the theoretical derivation. This paper introduces the variation characteristics of both the time-averaged and oscillating dispersion coefficients versus relative water depth, and demonstrates the physical implications of the oscillating mixing coefficient due to waves. We also apply the results to the costal vertical circulation and give its characteristics compared to Stokes drift.
基金Supported by the Research Fund of Mersin University in TURKEY with project number:2016-1-AP4-1425
文摘Creation of fermionic particles by a time-dependent electric field and a space-dependent magnetic field is studied with the Bogoulibov transformation method. Exact analytic solutions of the Dirac equation are obtained in terms of the Whittaker functions and the particle creation number density depending on the electric and magnetic fields is determined.
基金Supported by National Natural Science Foundation of China under Grant Nos.51276104,51476191
文摘In this paper, the analytical solutions of Schrodinger equation for Brownian motion in a double well potential are acquired by the homotopy analysis method and the Adomian decomposition method. Double well potential for Brownian motion is always used to obtain the solutions of Fokker-P1anck equation known as the Klein-Kramers equation, which is suitable for separation and additive Hamiltonians. In essence, we could study the random motion of Brownian particles by solving Schr6dinger equation. The anaiytical results obtained from the two different methods agree with each other well The double well potentiai is affected by two parameters, which are analyzed and discussed in details with the aid of graphical illustrations. According to the final results, the shapes of the double well potential have significant influence on the probability density function.
文摘The purpose of the paper is to present analytical and numerical solutions of a degenerate parabolic equation with time-fractional derivatives arising in the spatial diffusion of biological populations. The homotopy-perturbation method is employed for solving this class of equations, and the time-fractional derivatives are described in the sense of Caputo. Comparisons are made with those derived by Adomian＇s decomposition method, revealing that the homotopy perturbation method is more accurate and convenient than the Adomian＇s decomposition method. Furthermore, the results reveal that the approximate solution continuously depends on the time-fractional derivative and the proposed method incorporating the Caputo derivatives is a powerful and efficient technique for solving the fractional differential equations without requiring linearization or restrictive assumptions. The basis ideas presented in the paper can be further applied to solve other similar fractional partial differential equations.
