To solve the first-order differential equation derived from the problem of a free-falling object and the problem arising from Newton’s law of cooling, the study compares the numerical solutions obtained from Picard’...To solve the first-order differential equation derived from the problem of a free-falling object and the problem arising from Newton’s law of cooling, the study compares the numerical solutions obtained from Picard’s and Taylor’s series methods. We have carried out a descriptive analysis using the MATLAB software. Picard’s and Taylor’s techniques for deriving numerical solutions are both strong mathematical instruments that behave similarly. All first-order differential equations in standard form that have a constant function on the right-hand side share this similarity. As a result, we can conclude that Taylor’s approach is simpler to use, more effective, and more accurate. We will contrast Rung Kutta and Taylor’s methods in more detail in the following section.展开更多
The presented study deals with the investigation of nonlinear Bogoyavlenskii equations with conformable time-derivative which has great importance in plasma physics and non-inspectoral scattering problems.Travelling w...The presented study deals with the investigation of nonlinear Bogoyavlenskii equations with conformable time-derivative which has great importance in plasma physics and non-inspectoral scattering problems.Travelling wave solutions of this nonlinear conformable model are constructed by utilizing two powerful analytical approaches,namely,the modified auxiliary equation method and the Sardar sub-equation method.Many novel soliton solutions are extracted using these methods.Furthermore,3D surface graphs,contour plots and parametric graphs are drawn to show dynamical behavior of some obtained solutions with the aid of symbolic software such as Mathematica.The constructed solutions will help to understand the dynamical framework of nonlinear Bogoyavlenskii equations in the related physical phenomena.展开更多
In this paper,we are concerned with solutions to the fractional Schrodinger-Poisson system■ with prescribed mass ∫_(R^(3))|u|^(2)dx=a^(2),where a> 0 is a prescribed number,μ> 0 is a paremeter,s ∈(0,1),2 <...In this paper,we are concerned with solutions to the fractional Schrodinger-Poisson system■ with prescribed mass ∫_(R^(3))|u|^(2)dx=a^(2),where a> 0 is a prescribed number,μ> 0 is a paremeter,s ∈(0,1),2 <q <2_(s)^(*),and 2_(s)^(*)=6/(3-2s) is the fractional critical Sobolev exponent.In the L2-subcritical case,we show the existence of multiple normalized solutions by using the genus theory and the truncation technique;in the L^(2)-supercritical case,we obtain a couple of normalized solutions by developing a fiber map.Under both cases,to recover the loss of compactness of the energy functional caused by the doubly critical growth,we need to adopt the concentration-compactness principle.Our results complement and improve upon some existing studies on the fractional Schrodinger-Poisson system with a nonlocal critical term.展开更多
This article describes the solution procedure of the fractional Pade-Ⅱ equation and generalized Zakharov equation(GSEs)using the sine-cosine method.Pade-Ⅱ is an important nonlinear wave equation modeling unidirectio...This article describes the solution procedure of the fractional Pade-Ⅱ equation and generalized Zakharov equation(GSEs)using the sine-cosine method.Pade-Ⅱ is an important nonlinear wave equation modeling unidirectional propagation of long-wave in dispersive media and GSEs are used to model the interaction between one-dimensional high,and low-frequency waves.Classes of trigonometric and hyperbolic function solutions in fractional calculus are discussed.Graphical simulations of the numerical solutions are flaunted by MATLAB.展开更多
We study the existence of solutions for Kirchhoff-type equations.Firstly,we use the Sobolev inequality and the weakly lower semi-continuity of the norm to prove that the corresponding function can reach the global min...We study the existence of solutions for Kirchhoff-type equations.Firstly,we use the Sobolev inequality and the weakly lower semi-continuity of the norm to prove that the corresponding function can reach the global minimum.Then,we use the variational method and some analytical techniques to obtain the existence of the positive solution of the equation whenλis small enough.展开更多
In this paper,we mainly focus on the following Choquard equation-{△u-V(x)(I_(a*)|u|^(p))|u|^(p-2)u=λu,x∈R^(N),u∈H^(1)(R^(N))where N≥1,λ∈R will arise as a Lagrange multiplier,0<a<N and N+a/N<p<N+a+2/...In this paper,we mainly focus on the following Choquard equation-{△u-V(x)(I_(a*)|u|^(p))|u|^(p-2)u=λu,x∈R^(N),u∈H^(1)(R^(N))where N≥1,λ∈R will arise as a Lagrange multiplier,0<a<N and N+a/N<p<N+a+2/N Under appropriate hypotheses on V(x),we prove that the above Choquard equation has a normalized ground state solution by utilizing variational methods.展开更多
We study exact solutions to (1 + 1)-dimensional generalized Boussinesq equation with time-space dispersion term by making use of improved sub-equation method, and analyse the dynamical behavior and exact solutions of ...We study exact solutions to (1 + 1)-dimensional generalized Boussinesq equation with time-space dispersion term by making use of improved sub-equation method, and analyse the dynamical behavior and exact solutions of the sub-equation after constructing the nonlinear transformation and constraint conditions. Accordingly, we obtain twenty families of exact solutions such as analytical and singular solitons and singular periodic waves. In addition, we discuss the impact of system parameters on wave propagation.展开更多
In this paper, the evolutionary behavior of N-solitons for a (2 + 1)-dimensional Konopelchenko-Dubrovsky equations is studied by using the Hirota bilinear method and the long wave limit method. Based on the N-soliton ...In this paper, the evolutionary behavior of N-solitons for a (2 + 1)-dimensional Konopelchenko-Dubrovsky equations is studied by using the Hirota bilinear method and the long wave limit method. Based on the N-soliton solution, we first study the evolution from N-soliton to T-order (T=1,2) breather wave solutions via the paired-complexification of parameters, and then we get the N-order rational solutions, M-order (M=1,2) lump solutions, and the hybrid behavior between a variety of different types of solitons combined with the parameter limit technique and the paired-complexification of parameters. Meanwhile, we also provide a large number of three-dimensional figures in order to better show the degeneration of the N-soliton and the interaction behavior between different N-solitons.展开更多
The current study examines the important class of Chebyshev’s differential equations via the application of the efficient Adomian Decomposition Method (ADM) and its modifications. We have proved the effectiveness of ...The current study examines the important class of Chebyshev’s differential equations via the application of the efficient Adomian Decomposition Method (ADM) and its modifications. We have proved the effectiveness of the employed methods by acquiring exact analytical solutions for the governing equations in most cases;while minimal noisy error terms have been observed in a particular method modification. Above all, the presented approaches have rightly affirmed the exactitude of the available literature. More to the point, the application of this methodology could be extended to examine various forms of high-order differential equations, as approximate exact solutions are rapidly attained with less computation stress.展开更多
Multiple objectives to be optimized simultaneously are prevalent in real-life problems. This paper develops a new Pareto Method for bi-objective optimization which yields analytical solutions. The Pareto optimal front...Multiple objectives to be optimized simultaneously are prevalent in real-life problems. This paper develops a new Pareto Method for bi-objective optimization which yields analytical solutions. The Pareto optimal front is obtained in closed-form, enabling the derivation of various solutions in a convenient and efficient way. The advantage of analytical solution is the possibility of deriving accurate, exact and well-understood solutions, which is especially useful for policy analysis. An extension of the method to include multiple objectives is provided with the objectives being classified into two types. Such an extension expands the applicability of the developed techniques.展开更多
In this research work,we constructed the optical soliton solutions of nonlinear complex Kundu-Eckhaus(KE)equation with the help of modified mathematical method.We obtained the solutions in the form of dark solitons,br...In this research work,we constructed the optical soliton solutions of nonlinear complex Kundu-Eckhaus(KE)equation with the help of modified mathematical method.We obtained the solutions in the form of dark solitons,bright solitons and combined dark-bright solitons,travelling wave and periodic wave solutions with general coefficients.In our knowledge earlier reported results of the KE equation with specific coefficients.These obtained solutions are more useful in the development of optical fibers,dynamics of solitons,dynamics of adiabatic parameters,dynamics of fluid,problems of biomedical,industrial phenomena and many other branches.All calculations show that this technique is more powerful,effective,straightforward,and fruitfulness to study analytically other higher-order nonlinear complex PDEs involves in mathematical physics,quantum physics,Geo physics,fluid mechanics,hydrodynamics,mathematical biology,field of engineering and many other physical sciences.展开更多
In this study, we will introduce the modified (G'/G<sup>2</sup>)-expansion method to explore some of the exact traveling wave solutions of some nonlinear partial differential equations namely, Phi-4 eq...In this study, we will introduce the modified (G'/G<sup>2</sup>)-expansion method to explore some of the exact traveling wave solutions of some nonlinear partial differential equations namely, Phi-4 equation, Joseph-Egri (TRLW) equation, and Calogro-Degasperis (CD) equation. As a result, we have obtained solutions for the equations expressed in terms of trigonometric, hyperbolic and rational functions. Moreover, some selected solutions are plotted using some specific values for the parameters.展开更多
This paper concerns the implementation of the orthogonal polynomials using the Galerkin method for solving Volterra integro-differential and Fredholm integro-differential equations. The constructed orthogonal polynomi...This paper concerns the implementation of the orthogonal polynomials using the Galerkin method for solving Volterra integro-differential and Fredholm integro-differential equations. The constructed orthogonal polynomials are used as basis functions in the assumed solution employed. Numerical examples for some selected problems are provided and the results obtained show that the Galerkin method with orthogonal polynomials as basis functions performed creditably well in terms of absolute errors obtained.展开更多
The bending of rectangular plate is divided into the generalized statically determinate bending and the generalized statically indeterminate bending based on the analysis of the completeness of calculating condition a...The bending of rectangular plate is divided into the generalized statically determinate bending and the generalized statically indeterminate bending based on the analysis of the completeness of calculating condition at the corner point. The former can be solved directly by the equilibrium differential equation and the boundary conditions of four edges of the plate. The latter can be solved by using the superposition principle. Making use of the recommended method, the bending of the plate with all kinds of...展开更多
Nanosphere-like Li2FeSiO4/C was synthesized via a solution method using sucrose as carbon sources under a mild condition of time-saving and energy-saving, followed by sintering at high temperatures for crystallization...Nanosphere-like Li2FeSiO4/C was synthesized via a solution method using sucrose as carbon sources under a mild condition of time-saving and energy-saving, followed by sintering at high temperatures for crystallization. The amount of carbon in the composite is less than 10% (mass fraction), and the X-ray diffraction result confirms that the sample is of pure single phase indexed with the orthorhombic Pmn21 space group. The particle size of the Li2FeSiO4/C synthesized at 700 °C for 9 h is very fine and spherical-like with a size of 200 nm. The electrochemical performance of this material, including reversible capacity, cycle number, and charge-discharge characteristics, were tested. The cell of this sample can deliver a discharge capacity of 166 mA-h/g at C/20 rate in the first three cycles. After 30 cycles, the capacity decreases to 158 mA-h/g, and the capacity retention is up to 95%. The results show that this method can prepare nanosphere-like Li2FeSiO4/C composite with good electrochemical performance.展开更多
Cu nanoparticles were prepared by reducing Cu2+ ions with ascorbic acid through aqueous solution reduction method. The effects of solution pH and average size of Cu2O particles on the preparation of Cu nanoparticles ...Cu nanoparticles were prepared by reducing Cu2+ ions with ascorbic acid through aqueous solution reduction method. The effects of solution pH and average size of Cu2O particles on the preparation of Cu nanoparticles were investigated. Cu particles were prepared at pH 3, 5 or 7, with the smallest Cu particles obtained at pH 7. However, Cu particles could not be prepared at pH 9 or 11. The average size of Cu2O particles can affect that of Cu particles. Larger Cu2O particles result in larger Cu particles. In addition, experiments were conducted to explore the reaction process by measuring the X-ray diffraction (XRD) patterns of specimens collected at different time points during the reaction. It was found that Cu(OH)2 was initially formed as a precursor, followed by the formation of Cu2O, which was finally reduced to Cu particles.展开更多
The preparation of Cu nanoparticles by the aqueous solution reduction method was investigated. The effects of different reaction parameters on the preparation of Cu nanoparticles were studied. The optimum conditions f...The preparation of Cu nanoparticles by the aqueous solution reduction method was investigated. The effects of different reaction parameters on the preparation of Cu nanoparticles were studied. The optimum conditions for preparing well-dispersed nanoparticles were found as follows: 0.4 mol/L NaBH4 was added into solution containing 0.2 mol/L Cu2+, 1.0% gelatin dispersant in mass fraction, and 1.2 mol/L NH3?H2O at pH 12 and 313 K. In addition, a series of experiments were performed to discover the reaction process. NH3?H2O was found to be able to modulate the reaction process. At pH=10, Cu2+ was transformed to Cu(NH3)42+ as precursor after the addition of NH3?H2O, and then Cu(NH3)42+ was reduced by NaBH4 solution. At pH=12, Cu2+ was transformed to Cu(OH)2 as precursor after the addition of NH3?H2O, and Cu(OH)2 was then reduced by NaBH4 solution.展开更多
We study the existence of global-in-time classical solutions for the one-dimensional nonisentropic compressible Euler system for a dusty gas with large initial data.Using the characteristic decomposition method propos...We study the existence of global-in-time classical solutions for the one-dimensional nonisentropic compressible Euler system for a dusty gas with large initial data.Using the characteristic decomposition method proposed by Li et al.(Commun Math Phys 267:1–12,2006),we derive a group of characteristic decompositions for the system.Using these characteristic decompositions,we find a sufficient condition on the initial data to ensure the existence of global-in-time classical solutions.展开更多
In this article the travelling wave solution for a class of nonlinear reaction diffusion problems are considered. Using the homotopic method and the theory of travelling wave transform, the approximate solution for th...In this article the travelling wave solution for a class of nonlinear reaction diffusion problems are considered. Using the homotopic method and the theory of travelling wave transform, the approximate solution for the corresponding problem is obtained.展开更多
The present work describes the application of the method of fundamental solutions (MFS) along with the analog equation method (AEM) and radial basis function (RBF) approximation for solving the 2D isotropic and ...The present work describes the application of the method of fundamental solutions (MFS) along with the analog equation method (AEM) and radial basis function (RBF) approximation for solving the 2D isotropic and anisotropic Helmholtz problems with different wave numbers. The AEM is used to convert the original governing equation into the classical Poisson's equation, and the MFS and RBF approximations are used to derive the homogeneous and particular solutions, respectively. Finally, the satisfaction of the solution consisting of the homogeneous and particular parts to the related governing equation and boundary conditions can produce a system of linear equations, which can be solved with the singular value decomposition (SVD) technique. In the computation, such crucial factors related to the MFS-RBF as the location of the virtual boundary, the differential and integrating strategies, and the variation of shape parameters in multi-quadric (MQ) are fully analyzed to provide useful reference.展开更多
文摘To solve the first-order differential equation derived from the problem of a free-falling object and the problem arising from Newton’s law of cooling, the study compares the numerical solutions obtained from Picard’s and Taylor’s series methods. We have carried out a descriptive analysis using the MATLAB software. Picard’s and Taylor’s techniques for deriving numerical solutions are both strong mathematical instruments that behave similarly. All first-order differential equations in standard form that have a constant function on the right-hand side share this similarity. As a result, we can conclude that Taylor’s approach is simpler to use, more effective, and more accurate. We will contrast Rung Kutta and Taylor’s methods in more detail in the following section.
文摘The presented study deals with the investigation of nonlinear Bogoyavlenskii equations with conformable time-derivative which has great importance in plasma physics and non-inspectoral scattering problems.Travelling wave solutions of this nonlinear conformable model are constructed by utilizing two powerful analytical approaches,namely,the modified auxiliary equation method and the Sardar sub-equation method.Many novel soliton solutions are extracted using these methods.Furthermore,3D surface graphs,contour plots and parametric graphs are drawn to show dynamical behavior of some obtained solutions with the aid of symbolic software such as Mathematica.The constructed solutions will help to understand the dynamical framework of nonlinear Bogoyavlenskii equations in the related physical phenomena.
基金supported by the BIT Research and Innovation Promoting Project(2023YCXY046)the NSFC(11771468,11971027,11971061,12171497 and 12271028)+1 种基金the BNSF(1222017)the Fundamental Research Funds for the Central Universities。
文摘In this paper,we are concerned with solutions to the fractional Schrodinger-Poisson system■ with prescribed mass ∫_(R^(3))|u|^(2)dx=a^(2),where a> 0 is a prescribed number,μ> 0 is a paremeter,s ∈(0,1),2 <q <2_(s)^(*),and 2_(s)^(*)=6/(3-2s) is the fractional critical Sobolev exponent.In the L2-subcritical case,we show the existence of multiple normalized solutions by using the genus theory and the truncation technique;in the L^(2)-supercritical case,we obtain a couple of normalized solutions by developing a fiber map.Under both cases,to recover the loss of compactness of the energy functional caused by the doubly critical growth,we need to adopt the concentration-compactness principle.Our results complement and improve upon some existing studies on the fractional Schrodinger-Poisson system with a nonlocal critical term.
文摘This article describes the solution procedure of the fractional Pade-Ⅱ equation and generalized Zakharov equation(GSEs)using the sine-cosine method.Pade-Ⅱ is an important nonlinear wave equation modeling unidirectional propagation of long-wave in dispersive media and GSEs are used to model the interaction between one-dimensional high,and low-frequency waves.Classes of trigonometric and hyperbolic function solutions in fractional calculus are discussed.Graphical simulations of the numerical solutions are flaunted by MATLAB.
文摘We study the existence of solutions for Kirchhoff-type equations.Firstly,we use the Sobolev inequality and the weakly lower semi-continuity of the norm to prove that the corresponding function can reach the global minimum.Then,we use the variational method and some analytical techniques to obtain the existence of the positive solution of the equation whenλis small enough.
基金Supported by National Natural Science Foundation of China(Grant Nos.11671403 and 11671236)Henan Provincial General Natural Science Foundation Project(Grant No.232300420113)National Natural Science Foundation of China Youth Foud of China Youth Foud(Grant No.12101192).
文摘In this paper,we mainly focus on the following Choquard equation-{△u-V(x)(I_(a*)|u|^(p))|u|^(p-2)u=λu,x∈R^(N),u∈H^(1)(R^(N))where N≥1,λ∈R will arise as a Lagrange multiplier,0<a<N and N+a/N<p<N+a+2/N Under appropriate hypotheses on V(x),we prove that the above Choquard equation has a normalized ground state solution by utilizing variational methods.
文摘We study exact solutions to (1 + 1)-dimensional generalized Boussinesq equation with time-space dispersion term by making use of improved sub-equation method, and analyse the dynamical behavior and exact solutions of the sub-equation after constructing the nonlinear transformation and constraint conditions. Accordingly, we obtain twenty families of exact solutions such as analytical and singular solitons and singular periodic waves. In addition, we discuss the impact of system parameters on wave propagation.
文摘In this paper, the evolutionary behavior of N-solitons for a (2 + 1)-dimensional Konopelchenko-Dubrovsky equations is studied by using the Hirota bilinear method and the long wave limit method. Based on the N-soliton solution, we first study the evolution from N-soliton to T-order (T=1,2) breather wave solutions via the paired-complexification of parameters, and then we get the N-order rational solutions, M-order (M=1,2) lump solutions, and the hybrid behavior between a variety of different types of solitons combined with the parameter limit technique and the paired-complexification of parameters. Meanwhile, we also provide a large number of three-dimensional figures in order to better show the degeneration of the N-soliton and the interaction behavior between different N-solitons.
文摘The current study examines the important class of Chebyshev’s differential equations via the application of the efficient Adomian Decomposition Method (ADM) and its modifications. We have proved the effectiveness of the employed methods by acquiring exact analytical solutions for the governing equations in most cases;while minimal noisy error terms have been observed in a particular method modification. Above all, the presented approaches have rightly affirmed the exactitude of the available literature. More to the point, the application of this methodology could be extended to examine various forms of high-order differential equations, as approximate exact solutions are rapidly attained with less computation stress.
文摘Multiple objectives to be optimized simultaneously are prevalent in real-life problems. This paper develops a new Pareto Method for bi-objective optimization which yields analytical solutions. The Pareto optimal front is obtained in closed-form, enabling the derivation of various solutions in a convenient and efficient way. The advantage of analytical solution is the possibility of deriving accurate, exact and well-understood solutions, which is especially useful for policy analysis. An extension of the method to include multiple objectives is provided with the objectives being classified into two types. Such an extension expands the applicability of the developed techniques.
文摘In this research work,we constructed the optical soliton solutions of nonlinear complex Kundu-Eckhaus(KE)equation with the help of modified mathematical method.We obtained the solutions in the form of dark solitons,bright solitons and combined dark-bright solitons,travelling wave and periodic wave solutions with general coefficients.In our knowledge earlier reported results of the KE equation with specific coefficients.These obtained solutions are more useful in the development of optical fibers,dynamics of solitons,dynamics of adiabatic parameters,dynamics of fluid,problems of biomedical,industrial phenomena and many other branches.All calculations show that this technique is more powerful,effective,straightforward,and fruitfulness to study analytically other higher-order nonlinear complex PDEs involves in mathematical physics,quantum physics,Geo physics,fluid mechanics,hydrodynamics,mathematical biology,field of engineering and many other physical sciences.
文摘In this study, we will introduce the modified (G'/G<sup>2</sup>)-expansion method to explore some of the exact traveling wave solutions of some nonlinear partial differential equations namely, Phi-4 equation, Joseph-Egri (TRLW) equation, and Calogro-Degasperis (CD) equation. As a result, we have obtained solutions for the equations expressed in terms of trigonometric, hyperbolic and rational functions. Moreover, some selected solutions are plotted using some specific values for the parameters.
文摘This paper concerns the implementation of the orthogonal polynomials using the Galerkin method for solving Volterra integro-differential and Fredholm integro-differential equations. The constructed orthogonal polynomials are used as basis functions in the assumed solution employed. Numerical examples for some selected problems are provided and the results obtained show that the Galerkin method with orthogonal polynomials as basis functions performed creditably well in terms of absolute errors obtained.
文摘The bending of rectangular plate is divided into the generalized statically determinate bending and the generalized statically indeterminate bending based on the analysis of the completeness of calculating condition at the corner point. The former can be solved directly by the equilibrium differential equation and the boundary conditions of four edges of the plate. The latter can be solved by using the superposition principle. Making use of the recommended method, the bending of the plate with all kinds of...
基金Project supported by Ministry of Education Key Laboratory of Synthetic and Natural Functional Molecular Chemistry, China Project (2010JK765) supported by the Education Department of Shaanxi Province, China
文摘Nanosphere-like Li2FeSiO4/C was synthesized via a solution method using sucrose as carbon sources under a mild condition of time-saving and energy-saving, followed by sintering at high temperatures for crystallization. The amount of carbon in the composite is less than 10% (mass fraction), and the X-ray diffraction result confirms that the sample is of pure single phase indexed with the orthorhombic Pmn21 space group. The particle size of the Li2FeSiO4/C synthesized at 700 °C for 9 h is very fine and spherical-like with a size of 200 nm. The electrochemical performance of this material, including reversible capacity, cycle number, and charge-discharge characteristics, were tested. The cell of this sample can deliver a discharge capacity of 166 mA-h/g at C/20 rate in the first three cycles. After 30 cycles, the capacity decreases to 158 mA-h/g, and the capacity retention is up to 95%. The results show that this method can prepare nanosphere-like Li2FeSiO4/C composite with good electrochemical performance.
文摘Cu nanoparticles were prepared by reducing Cu2+ ions with ascorbic acid through aqueous solution reduction method. The effects of solution pH and average size of Cu2O particles on the preparation of Cu nanoparticles were investigated. Cu particles were prepared at pH 3, 5 or 7, with the smallest Cu particles obtained at pH 7. However, Cu particles could not be prepared at pH 9 or 11. The average size of Cu2O particles can affect that of Cu particles. Larger Cu2O particles result in larger Cu particles. In addition, experiments were conducted to explore the reaction process by measuring the X-ray diffraction (XRD) patterns of specimens collected at different time points during the reaction. It was found that Cu(OH)2 was initially formed as a precursor, followed by the formation of Cu2O, which was finally reduced to Cu particles.
文摘The preparation of Cu nanoparticles by the aqueous solution reduction method was investigated. The effects of different reaction parameters on the preparation of Cu nanoparticles were studied. The optimum conditions for preparing well-dispersed nanoparticles were found as follows: 0.4 mol/L NaBH4 was added into solution containing 0.2 mol/L Cu2+, 1.0% gelatin dispersant in mass fraction, and 1.2 mol/L NH3?H2O at pH 12 and 313 K. In addition, a series of experiments were performed to discover the reaction process. NH3?H2O was found to be able to modulate the reaction process. At pH=10, Cu2+ was transformed to Cu(NH3)42+ as precursor after the addition of NH3?H2O, and then Cu(NH3)42+ was reduced by NaBH4 solution. At pH=12, Cu2+ was transformed to Cu(OH)2 as precursor after the addition of NH3?H2O, and Cu(OH)2 was then reduced by NaBH4 solution.
基金supported by the National Natural Science Foundation of China(12071278).
文摘We study the existence of global-in-time classical solutions for the one-dimensional nonisentropic compressible Euler system for a dusty gas with large initial data.Using the characteristic decomposition method proposed by Li et al.(Commun Math Phys 267:1–12,2006),we derive a group of characteristic decompositions for the system.Using these characteristic decompositions,we find a sufficient condition on the initial data to ensure the existence of global-in-time classical solutions.
基金Supported by the National Natural Sciences Foundation of China(40676016 and 10471039)the National Key Project for Basic Research(2003CB415101-03 and 2004CB418304)+2 种基金the Key Project of the Chinese Academy of Sciences(KZCX3-SW-221)in part by E-Institutes of Shanghai Municipal Education Commission(N.E03004)the Natural Science Foundation of Zeijiang,China(Y606268).
文摘In this article the travelling wave solution for a class of nonlinear reaction diffusion problems are considered. Using the homotopic method and the theory of travelling wave transform, the approximate solution for the corresponding problem is obtained.
文摘The present work describes the application of the method of fundamental solutions (MFS) along with the analog equation method (AEM) and radial basis function (RBF) approximation for solving the 2D isotropic and anisotropic Helmholtz problems with different wave numbers. The AEM is used to convert the original governing equation into the classical Poisson's equation, and the MFS and RBF approximations are used to derive the homogeneous and particular solutions, respectively. Finally, the satisfaction of the solution consisting of the homogeneous and particular parts to the related governing equation and boundary conditions can produce a system of linear equations, which can be solved with the singular value decomposition (SVD) technique. In the computation, such crucial factors related to the MFS-RBF as the location of the virtual boundary, the differential and integrating strategies, and the variation of shape parameters in multi-quadric (MQ) are fully analyzed to provide useful reference.