The(2+1)-dimensional integrable generalization of the Gardner(2DG)equation is solved via the inverse scattering transform method in this paper.A kind of general solution of the equation is obtained by introducing long...The(2+1)-dimensional integrable generalization of the Gardner(2DG)equation is solved via the inverse scattering transform method in this paper.A kind of general solution of the equation is obtained by introducing long derivatives V_(x),V_(y),V_(t).Two different constraints on the kernel function K are introduced under the reality of the solution u of the 2DG equation.Then,two classes of exact solutions with constant asymptotic values at infinity u|x^(2)+y^(2)→∞→0 are constructed by means of the∂¯-dressing method for the casesσ=1 andσ=i.The rational and multiple pole solutions of the 2DG equation are obtained with the kernel functions of zero-order and higher-order Dirac delta functions,respectively.展开更多
This paper aims to investigate the tamed Euler method for the random periodic solution of semilinear SDEs with one-sided Lipschitz coefficient.We introduce a novel approach to analyze mean-square error bounds of the n...This paper aims to investigate the tamed Euler method for the random periodic solution of semilinear SDEs with one-sided Lipschitz coefficient.We introduce a novel approach to analyze mean-square error bounds of the novel schemes,without relying on a priori high-order moment bound of the numerical approximation.The expected order-one mean square convergence is attained for the proposed scheme.Moreover,a numerical example is presented to verify our theoretical analysis.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
The separation of variables is employed to solve Hamiltonian dual form of eigenvalue problem for transverse free vibrations of thin plates, and formulation of the natural mode in closed form is performed. The closed-f...The separation of variables is employed to solve Hamiltonian dual form of eigenvalue problem for transverse free vibrations of thin plates, and formulation of the natural mode in closed form is performed. The closed-form natural mode satisfies the governing equation of the eigenvalue problem of thin plate exactly and is applicable for any types of boundary conditions. With all combinations of simplysupported （S） and clamped （C） boundary conditions applied to the natural mode, the mode shapes are obtained uniquely and two eigenvalue equations are derived with respect to two spatial coordinates, with the aid of which the normal modes and frequencies are solved exactly. It was believed that the exact eigensolutions for cases SSCC, SCCC and CCCC were unable to be obtained, however, they are successfully found in this paper. Comparisons between the present results and the FEM results validate the present exact solutions, which can thus be taken as the benchmark for verifying different approximate approaches.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
Cubic and monoclinic Gd2O3:Eu3+ phosphors in the range of nano-scale and submicron-scale were prepared by a modified solution combustion method.Coexistence of cubic and monoclinic phases was found in the highest lumin...Cubic and monoclinic Gd2O3:Eu3+ phosphors in the range of nano-scale and submicron-scale were prepared by a modified solution combustion method.Coexistence of cubic and monoclinic phases was found in the highest luminescent sample synthesized at 600 oC.In relation to commercial sample,the relative luminescence intensity was 49.8%.The shape of emission spectrum of the sample thus changed and the charge-transfer-state band of excitation spectrum slightly shift toward higher energies.With increasing the anneal...展开更多
Tin(Sn)-lead(Pb)mixed halide perovskites have attracted widespread interest due to their wider re-sponse wavelength and lower toxicity than lead halide perovskites,Among the preparation methods,the two-step method mor...Tin(Sn)-lead(Pb)mixed halide perovskites have attracted widespread interest due to their wider re-sponse wavelength and lower toxicity than lead halide perovskites,Among the preparation methods,the two-step method more easily controls the crystallization rate and is suitable for preparing large-area per-ovskite devices.However,the residual low-conductivity iodide layer in the two-step method can affect carrier transport and device stability,and the different crystallization rates of Sn-and Pb-based per-ovskites may result in poor film quality.Therefore,Sn-Pb mixed perovskites are mainly prepared by a one-step method.Herein,a MAPb_(0.5)Sn_(0.5)I_(3)-based self-powered photodetector without a hole transport layer is fabricated by a two-step method.By adjusting the concentration of the ascorbic acid(AA)addi-tive,the final perovskite film exhibited a pure phase without residues,and the optimal device exhibited a high responsivity(0.276 A W^(-1)),large specific detectivity(2.38×10^(12) Jones),and enhanced stability.This enhancement is mainly attributed to the inhibition of Sn2+oxidation,the control of crystal growth,and the sufficient reaction between organic ammonium salts and bottom halides due to the AA-induced pore structure.展开更多
Some doubly-periodic solutions of the Zakharov-Kuznetsov equation are presented. Our approach is to introduce an auxiliary ordinary differential equation and use its Jacobi elliptic function solutions to construct dou...Some doubly-periodic solutions of the Zakharov-Kuznetsov equation are presented. Our approach is to introduce an auxiliary ordinary differential equation and use its Jacobi elliptic function solutions to construct doubly-periodic solutions of the Zakharov-Kuznetsov equation, which has been derived by Gottwald as a two-dimensional model for nonlinear Rossby waves. When the modulus k →1, these solutions reduce to the solitary wave solutions of the equation.展开更多
This paper presents a new and efficient approach for constructing exact solutions to nonlinear differential-difference equations （NLDDEs） and lattice equation. By using this method via symbolic computation system MA...This paper presents a new and efficient approach for constructing exact solutions to nonlinear differential-difference equations （NLDDEs） and lattice equation. By using this method via symbolic computation system MAPLE, we obtained abundant soliton-like and/or period-form solutions to the （2＋1）-dimensional Toda equation. It seems that solitary wave solutions are merely special cases in one family. Furthermore, the method can also be applied to other nonlinear differential-difference equations.展开更多
A 20 wt% Ni/bentonite catalyst was prepared by a solution combustion synthesis (SCS), which exhibited higher activity for the CO_2methanation than that of an impregnation method (IPM), and the catalyst prepared by SCS...A 20 wt% Ni/bentonite catalyst was prepared by a solution combustion synthesis (SCS), which exhibited higher activity for the CO_2methanation than that of an impregnation method (IPM), and the catalyst prepared by SCS showed a CO_2 conversion of 85% and a CH4selectivity of 100% at 300 °C, atmospheric pressure, and 3600 ml·(g cat)-1·h-1, and the catalyst exhibited stable within a 110-h reaction. The results showed higher me- tallic Ni dispersion, smaller Ni particle size, larger specific surface area and lower reduction temperature in the Ni/ bentonite prepared by SCS than that of IPM. And the Ni/bentonite prepared by the SCS moderated the interaction between NiO and bentonite.展开更多
The boundary knot method(BKM) is a boundary-type meshfree method. Only non-singular general solutions are used during the whole solution procedures. The effective condition number(ECN), which depends on the right-hand...The boundary knot method(BKM) is a boundary-type meshfree method. Only non-singular general solutions are used during the whole solution procedures. The effective condition number(ECN), which depends on the right-hand side vector of a linear system, is considered as an alternative criterion to the traditional condition number. In this paper, the effective condition number is used to help determine the position and distribution of the collocation points as well as the quasi-optimal collocation point numbers. During the solution process, we propose an NMN-search algorithm. Numerical examples show that the ECN is reliable to measure the feasibility of the BKM.展开更多
The nonlinear Schrodinger equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas.However,due to the difficulty of solving this equation,in particu...The nonlinear Schrodinger equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas.However,due to the difficulty of solving this equation,in particular in high dimensions,lots of methods are proposed to effectively obtain different kinds of solutions,such as neural networks among others.Recently,a method where some underlying physical laws are embeded into a conventional neural network is proposed to uncover the equation’s dynamical behaviors from spatiotemporal data directly.Compared with traditional neural networks,this method can obtain remarkably accurate solution with extraordinarily less data.Meanwhile,this method also provides a better physical explanation and generalization.In this paper,based on the above method,we present an improved deep learning method to recover the soliton solutions,breather solution,and rogue wave solutions of the nonlinear Schrodinger equation.In particular,the dynamical behaviors and error analysis about the one-order and two-order rogue waves of nonlinear integrable equations are revealed by the deep neural network with physical constraints for the first time.Moreover,the effects of different numbers of initial points sampled,collocation points sampled,network layers,neurons per hidden layer on the one-order rogue wave dynamics of this equation have been considered with the help of the control variable way under the same initial and boundary conditions.Numerical experiments show that the dynamical behaviors of soliton solutions,breather solution,and rogue wave solutions of the integrable nonlinear Schrodinger equation can be well reconstructed by utilizing this physically-constrained deep learning method.展开更多
In this article, we consider analytical solutions of the time fractional derivative Gardner equation by using the new version of F-expansion method. With this proposed method multiple Jacobi elliptic functions are sit...In this article, we consider analytical solutions of the time fractional derivative Gardner equation by using the new version of F-expansion method. With this proposed method multiple Jacobi elliptic functions are situated in the solution function. As a result, various exact analytical solutions consisting of single and combined Jacobi elliptic functions solutions are obtained.展开更多
The （G＇/G, 1/G）-expansion method for finding exact travelling wave solutions of nonlinear evolution equations, which can be thought of as an extension of the （G＇/G）-expansion method proposed recently, is present...The （G＇/G, 1/G）-expansion method for finding exact travelling wave solutions of nonlinear evolution equations, which can be thought of as an extension of the （G＇/G）-expansion method proposed recently, is presented. By using this method abundant travelling wave so- lutions with arbitrary parameters of the Zakharov equations are successfully obtained. When the parameters are replaced by special values, the well-known solitary wave solutions of the equations are rediscovered from the travelling waves.展开更多
Applying the generalized method, which is a direct and unified algebraic method for constructing multipletravelling wave solutions of nonlinear partial differential equations (PDEs), and implementing in a computer alg...Applying the generalized method, which is a direct and unified algebraic method for constructing multipletravelling wave solutions of nonlinear partial differential equations (PDEs), and implementing in a computer algebraicsystem, we consider the generalized Zakharov-Kuzentsov equation with nonlinear terms of any order. As a result, wecan not only successfully recover the previously known travelling wave solutions found by existing various tanh methodsand other sophisticated methods, but also obtain some new formal solutions. The solutions obtained include kink-shapedsolitons, bell-shaped solitons, singular solitons, and periodic solutions.展开更多
基金Supported by the National Natural Science Foundation of China(Grant Nos.1237125611971475)。
文摘The(2+1)-dimensional integrable generalization of the Gardner(2DG)equation is solved via the inverse scattering transform method in this paper.A kind of general solution of the equation is obtained by introducing long derivatives V_(x),V_(y),V_(t).Two different constraints on the kernel function K are introduced under the reality of the solution u of the 2DG equation.Then,two classes of exact solutions with constant asymptotic values at infinity u|x^(2)+y^(2)→∞→0 are constructed by means of the∂¯-dressing method for the casesσ=1 andσ=i.The rational and multiple pole solutions of the 2DG equation are obtained with the kernel functions of zero-order and higher-order Dirac delta functions,respectively.
基金supported by the National Natural Science Foundation of China(Nos.12471394,12371417)Natural Science Foundation of Changsha(No.kq2502101)。
文摘This paper aims to investigate the tamed Euler method for the random periodic solution of semilinear SDEs with one-sided Lipschitz coefficient.We introduce a novel approach to analyze mean-square error bounds of the novel schemes,without relying on a priori high-order moment bound of the numerical approximation.The expected order-one mean square convergence is attained for the proposed scheme.Moreover,a numerical example is presented to verify our theoretical analysis.
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 (10772014)
文摘The separation of variables is employed to solve Hamiltonian dual form of eigenvalue problem for transverse free vibrations of thin plates, and formulation of the natural mode in closed form is performed. The closed-form natural mode satisfies the governing equation of the eigenvalue problem of thin plate exactly and is applicable for any types of boundary conditions. With all combinations of simplysupported （S） and clamped （C） boundary conditions applied to the natural mode, the mode shapes are obtained uniquely and two eigenvalue equations are derived with respect to two spatial coordinates, with the aid of which the normal modes and frequencies are solved exactly. It was believed that the exact eigensolutions for cases SSCC, SCCC and CCCC were unable to be obtained, however, they are successfully found in this paper. Comparisons between the present results and the FEM results validate the present exact solutions, which can thus be taken as the benchmark for verifying different approximate approaches.
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.
基金supported by the Ministry of Science and Technology of China (2006CB601104)the Foundation of International Joint Research of Beijing (2007N08)+1 种基金Natural Science Foundation of Jiangxi Province (2009GQC0042)Foundation of Jiangxi Educational Committee (GJJ10153)
文摘Cubic and monoclinic Gd2O3:Eu3+ phosphors in the range of nano-scale and submicron-scale were prepared by a modified solution combustion method.Coexistence of cubic and monoclinic phases was found in the highest luminescent sample synthesized at 600 oC.In relation to commercial sample,the relative luminescence intensity was 49.8%.The shape of emission spectrum of the sample thus changed and the charge-transfer-state band of excitation spectrum slightly shift toward higher energies.With increasing the anneal...
基金supported by the National Natural Science Foun-dation of China(Nos.52025028,52332008,52372214,52202273,and U22A20137)the Priority Academic Program Development(PAPD)of Jiangsu Higher Education Institutions.
文摘Tin(Sn)-lead(Pb)mixed halide perovskites have attracted widespread interest due to their wider re-sponse wavelength and lower toxicity than lead halide perovskites,Among the preparation methods,the two-step method more easily controls the crystallization rate and is suitable for preparing large-area per-ovskite devices.However,the residual low-conductivity iodide layer in the two-step method can affect carrier transport and device stability,and the different crystallization rates of Sn-and Pb-based per-ovskites may result in poor film quality.Therefore,Sn-Pb mixed perovskites are mainly prepared by a one-step method.Herein,a MAPb_(0.5)Sn_(0.5)I_(3)-based self-powered photodetector without a hole transport layer is fabricated by a two-step method.By adjusting the concentration of the ascorbic acid(AA)addi-tive,the final perovskite film exhibited a pure phase without residues,and the optimal device exhibited a high responsivity(0.276 A W^(-1)),large specific detectivity(2.38×10^(12) Jones),and enhanced stability.This enhancement is mainly attributed to the inhibition of Sn2+oxidation,the control of crystal growth,and the sufficient reaction between organic ammonium salts and bottom halides due to the AA-induced pore structure.
文摘Some doubly-periodic solutions of the Zakharov-Kuznetsov equation are presented. Our approach is to introduce an auxiliary ordinary differential equation and use its Jacobi elliptic function solutions to construct doubly-periodic solutions of the Zakharov-Kuznetsov equation, which has been derived by Gottwald as a two-dimensional model for nonlinear Rossby waves. When the modulus k →1, these solutions reduce to the solitary wave solutions of the equation.
基金supported by the National Natural Science Foundation of Chinathe Natural Science Foundation of Shandong Province in China (Grant No Y2007G64)
文摘This paper presents a new and efficient approach for constructing exact solutions to nonlinear differential-difference equations （NLDDEs） and lattice equation. By using this method via symbolic computation system MAPLE, we obtained abundant soliton-like and/or period-form solutions to the （2＋1）-dimensional Toda equation. It seems that solitary wave solutions are merely special cases in one family. Furthermore, the method can also be applied to other nonlinear differential-difference equations.
基金Supported by the National Natural Science Foundation of China(21566005)the Natural Science Foundation of Guangxi Province(2016GXNSFFA380015)
文摘A 20 wt% Ni/bentonite catalyst was prepared by a solution combustion synthesis (SCS), which exhibited higher activity for the CO_2methanation than that of an impregnation method (IPM), and the catalyst prepared by SCS showed a CO_2 conversion of 85% and a CH4selectivity of 100% at 300 °C, atmospheric pressure, and 3600 ml·(g cat)-1·h-1, and the catalyst exhibited stable within a 110-h reaction. The results showed higher me- tallic Ni dispersion, smaller Ni particle size, larger specific surface area and lower reduction temperature in the Ni/ bentonite prepared by SCS than that of IPM. And the Ni/bentonite prepared by the SCS moderated the interaction between NiO and bentonite.
基金Supported by the Natural Science Foundation of Anhui Province(1908085QA09)Higher Education Department of the Ministry of Education(201802358008)
文摘The boundary knot method(BKM) is a boundary-type meshfree method. Only non-singular general solutions are used during the whole solution procedures. The effective condition number(ECN), which depends on the right-hand side vector of a linear system, is considered as an alternative criterion to the traditional condition number. In this paper, the effective condition number is used to help determine the position and distribution of the collocation points as well as the quasi-optimal collocation point numbers. During the solution process, we propose an NMN-search algorithm. Numerical examples show that the ECN is reliable to measure the feasibility of the BKM.
基金supported by the National Natural Science Foundation of China (Grant No. 11675054)the Fund from Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things (Grant No. ZF1213)the Project of Science and Technology Commission of Shanghai Municipality (Grant No. 18dz2271000)。
文摘The nonlinear Schrodinger equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas.However,due to the difficulty of solving this equation,in particular in high dimensions,lots of methods are proposed to effectively obtain different kinds of solutions,such as neural networks among others.Recently,a method where some underlying physical laws are embeded into a conventional neural network is proposed to uncover the equation’s dynamical behaviors from spatiotemporal data directly.Compared with traditional neural networks,this method can obtain remarkably accurate solution with extraordinarily less data.Meanwhile,this method also provides a better physical explanation and generalization.In this paper,based on the above method,we present an improved deep learning method to recover the soliton solutions,breather solution,and rogue wave solutions of the nonlinear Schrodinger equation.In particular,the dynamical behaviors and error analysis about the one-order and two-order rogue waves of nonlinear integrable equations are revealed by the deep neural network with physical constraints for the first time.Moreover,the effects of different numbers of initial points sampled,collocation points sampled,network layers,neurons per hidden layer on the one-order rogue wave dynamics of this equation have been considered with the help of the control variable way under the same initial and boundary conditions.Numerical experiments show that the dynamical behaviors of soliton solutions,breather solution,and rogue wave solutions of the integrable nonlinear Schrodinger equation can be well reconstructed by utilizing this physically-constrained deep learning method.
文摘In this article, we consider analytical solutions of the time fractional derivative Gardner equation by using the new version of F-expansion method. With this proposed method multiple Jacobi elliptic functions are situated in the solution function. As a result, various exact analytical solutions consisting of single and combined Jacobi elliptic functions solutions are obtained.
基金Supported by the International Cooperation and Exchanges Foundation of Henan Province (084300510060)the Youth Science Foundation of Henan University of Science and Technology of China (2008QN026)
文摘The （G＇/G, 1/G）-expansion method for finding exact travelling wave solutions of nonlinear evolution equations, which can be thought of as an extension of the （G＇/G）-expansion method proposed recently, is presented. By using this method abundant travelling wave so- lutions with arbitrary parameters of the Zakharov equations are successfully obtained. When the parameters are replaced by special values, the well-known solitary wave solutions of the equations are rediscovered from the travelling waves.
基金The project supported by National Natural Science Foundation of China under Grant No.10072013the National Key Basic Research Development Program under Grant No.G1998030600
文摘Applying the generalized method, which is a direct and unified algebraic method for constructing multipletravelling wave solutions of nonlinear partial differential equations (PDEs), and implementing in a computer algebraicsystem, we consider the generalized Zakharov-Kuzentsov equation with nonlinear terms of any order. As a result, wecan not only successfully recover the previously known travelling wave solutions found by existing various tanh methodsand other sophisticated methods, but also obtain some new formal solutions. The solutions obtained include kink-shapedsolitons, bell-shaped solitons, singular solitons, and periodic solutions.
