In this work we devise an algebraic method to uniformly construct rational form solitary wave solutions and Jacobi and Weierstrass doubly periodic wave solutions of physical interest for nonlinear evolution equations....In this work we devise an algebraic method to uniformly construct rational form solitary wave solutions and Jacobi and Weierstrass doubly periodic wave solutions of physical interest for nonlinear evolution equations. With the aid of symbolic computation, we apply the proposed method to solving the (1+1)-dimensional dispersive long wave equation and explicitly construct a series of exact solutions which include the rational form solitary wave solutions and elliptic doubly periodic wave solutions as special cases.展开更多
In this paper, a further extended Jacobi elliptic function rationM expansion method is proposed for constructing new forms of exact solutions to nonlinear partial differential equations by making a more general transf...In this paper, a further extended Jacobi elliptic function rationM expansion method is proposed for constructing new forms of exact solutions to nonlinear partial differential equations by making a more general transformation. For illustration, we apply the method to （2＋1）-dimensionM dispersive long wave equation and successfully obtain many new doubly periodic solutions. When the modulus m→1, these sohitions degenerate as soliton solutions. The method can be also applied to other nonlinear partial differential equations.展开更多
On the basis of Lie group theory,(1 + N)-dimensional time-fractional partial differential equations are studied and the expression of η_α~0 is given. As applications, two special forms of nonlinear time-fractional d...On the basis of Lie group theory,(1 + N)-dimensional time-fractional partial differential equations are studied and the expression of η_α~0 is given. As applications, two special forms of nonlinear time-fractional diffusionconvection equations are investigated by Lie group analysis method. Then the equations are reduced into fractional ordinary differential equations under group transformations. Therefore, the invariant solutions and some exact solutions are obtained.展开更多
The method of the phase plane is emploied to investigate the solitary and periodic traveling waves for a class of nonlinear dispersive partial differential equations.By using the bifurcation theory of dynamical system...The method of the phase plane is emploied to investigate the solitary and periodic traveling waves for a class of nonlinear dispersive partial differential equations.By using the bifurcation theory of dynamical systems to do qualitative analysis,all possible phase portraits in the parametric space for the traveling wave systems are obtained.It can be shown that the existence of a singular straight line in the traveling wave system is the reason why smooth solitary wave solutions converge to solitary cusp wave solution when parameters are varied.The different parameter conditions for the existence of solitary and periodic wave solutions of different kinds are rigorously determined.展开更多
In this paper a finite element model is developed to study cytosolic calcium concen- tration distribution in astrocytes for a two-dimensional steady-state case in presence of excess buffer. The mathematical model of c...In this paper a finite element model is developed to study cytosolic calcium concen- tration distribution in astrocytes for a two-dimensional steady-state case in presence of excess buffer. The mathematical model of calcium diffusion in astrocytes leads to a boundary value problem involving elliptical partial differential equation. The model con- sists of reaction-diffusion phenomena, association and dissociation rates and buffer. A point source of calcium is incorporated in the model. Appropriate boundary conditions have been framed. Finite element method is employed to solve the problem. A MATLAB program has been developed for the entire problem and simulated to compute the numer- ical results. The numerical results have been used to plot calcium concentration profiles in astrocytes. The effect of ECTA, BAPTA and aCa influx on calcium concentration distribution in astrocytes is studied with the help of numerical results.展开更多
We study a second-order parabolic equation with divergence form elliptic operator,having a piecewise constant diffusion coefficient with two points of discontinuity.Such partial differential equations appear in the mo...We study a second-order parabolic equation with divergence form elliptic operator,having a piecewise constant diffusion coefficient with two points of discontinuity.Such partial differential equations appear in the modelization of diffusion phenomena in medium consisting of three kinds of materials.Using probabilistic methods,we present an explicit expression of the fundamental solution under certain conditions.We also derive small-time asymptotic expansion of the PDE’s solutions in the general case.The obtained results are directly usable in applications.展开更多
We consider a system of partial differential equations that describes the interaction of the sterile and fertile species undergoing the sterile insect release method （SIRM）. Unlike in the previous work [M. A. Lewis ...We consider a system of partial differential equations that describes the interaction of the sterile and fertile species undergoing the sterile insect release method （SIRM）. Unlike in the previous work [M. A. Lewis and P. van den Driessche, Waves of extinction from sterile insect release, Math. Biosci. 5 （1992） 221 247] where the habitat is assumed to be the one-dimensional whole space ~, we consider this system in a bounded one- dimensional domain （interval）. Our goal is to derive sufficient conditions for success of the SIRM. We show the existence of the fertile-free steady state and prove its stability. Using the releasing rate as the parameter, and by a saddle-node bifurcation analysis, we obtain conditions for existence of two co-persistence steady states, one stable and the other unstable. Biological implications of our mathematical results are that： （i） when the fertile population is at low level, the SIRM, even with small releasing rate, can successfully eradicate the fertile insects; （ii） when the fertile population is at a higher level, the SIRM can succeed as long as the strength of the sterile releasing is large enough, while the method may also fail if the releasing is not sufficient.展开更多
This paper is devoted to the analysis of the Cauchy problem for a system of PDEs arising in radiative hydrodynamics. This system, which comes from the so-called equilibrium diffusion regime, is a variant of the usual ...This paper is devoted to the analysis of the Cauchy problem for a system of PDEs arising in radiative hydrodynamics. This system, which comes from the so-called equilibrium diffusion regime, is a variant of the usual Euler equations, where the energy and pressure functionals are modified to take into account the effect of radiation and the energy balance containing a nonlinear diffusion term acting on the temperature. The problem is studied in the multi-dimensional framework. The authors identify the existence of a strictly convex entropy and a stability property of the system, and check that the Kawashima-Shizuta condition holds. Then, based on these structure properties, the wellposedness close to a constant state can be proved by using fine energy estimates. The asymptotic decay of the solutions are also investigated.展开更多
文摘In this work we devise an algebraic method to uniformly construct rational form solitary wave solutions and Jacobi and Weierstrass doubly periodic wave solutions of physical interest for nonlinear evolution equations. With the aid of symbolic computation, we apply the proposed method to solving the (1+1)-dimensional dispersive long wave equation and explicitly construct a series of exact solutions which include the rational form solitary wave solutions and elliptic doubly periodic wave solutions as special cases.
基金The project partially supported by the State Key Basic Research Program of China under Grant No. 2004 CB 318000
文摘In this paper, a further extended Jacobi elliptic function rationM expansion method is proposed for constructing new forms of exact solutions to nonlinear partial differential equations by making a more general transformation. For illustration, we apply the method to （2＋1）-dimensionM dispersive long wave equation and successfully obtain many new doubly periodic solutions. When the modulus m→1, these sohitions degenerate as soliton solutions. The method can be also applied to other nonlinear partial differential equations.
基金Supported by the Natural Science Foundation of China under Grant Nos.11371287 and 61663043
文摘On the basis of Lie group theory,(1 + N)-dimensional time-fractional partial differential equations are studied and the expression of η_α~0 is given. As applications, two special forms of nonlinear time-fractional diffusionconvection equations are investigated by Lie group analysis method. Then the equations are reduced into fractional ordinary differential equations under group transformations. Therefore, the invariant solutions and some exact solutions are obtained.
基金National Natural Science Foundation of China(No.19731003,No.19961003)Yunnan Provincial Natural Science Foundation of China(No.1999A0018M,No.2000A0002M)
文摘The method of the phase plane is emploied to investigate the solitary and periodic traveling waves for a class of nonlinear dispersive partial differential equations.By using the bifurcation theory of dynamical systems to do qualitative analysis,all possible phase portraits in the parametric space for the traveling wave systems are obtained.It can be shown that the existence of a singular straight line in the traveling wave system is the reason why smooth solitary wave solutions converge to solitary cusp wave solution when parameters are varied.The different parameter conditions for the existence of solitary and periodic wave solutions of different kinds are rigorously determined.
文摘In this paper a finite element model is developed to study cytosolic calcium concen- tration distribution in astrocytes for a two-dimensional steady-state case in presence of excess buffer. The mathematical model of calcium diffusion in astrocytes leads to a boundary value problem involving elliptical partial differential equation. The model con- sists of reaction-diffusion phenomena, association and dissociation rates and buffer. A point source of calcium is incorporated in the model. Appropriate boundary conditions have been framed. Finite element method is employed to solve the problem. A MATLAB program has been developed for the entire problem and simulated to compute the numer- ical results. The numerical results have been used to plot calcium concentration profiles in astrocytes. The effect of ECTA, BAPTA and aCa influx on calcium concentration distribution in astrocytes is studied with the help of numerical results.
基金supported by the National Science Foundation of USA (Grant No. DMS1206276)National Natural Science Foundation of China (Grant No. 1128101)the Research Unit of Tunisia (Grant No. UR11ES53)
文摘We study a second-order parabolic equation with divergence form elliptic operator,having a piecewise constant diffusion coefficient with two points of discontinuity.Such partial differential equations appear in the modelization of diffusion phenomena in medium consisting of three kinds of materials.Using probabilistic methods,we present an explicit expression of the fundamental solution under certain conditions.We also derive small-time asymptotic expansion of the PDE’s solutions in the general case.The obtained results are directly usable in applications.
基金Part of this work was completed when the second author was visiting the Univer- sity of Western Ontario, and he would like to thank the staff in the Department of Applied Mathematics for their help and thank the University for its excellent facilities and support during his stay. The second author was supported by China Scholarship Council, partially sup- ported by NNSF of China （No. 11031002）, by the Heilongjiang Provincial Natural Science Foundation （No. A200806）, and by the Program of Excellent Team and the Science Research Foundation in Harbin Institute of Technology.
文摘We consider a system of partial differential equations that describes the interaction of the sterile and fertile species undergoing the sterile insect release method （SIRM）. Unlike in the previous work [M. A. Lewis and P. van den Driessche, Waves of extinction from sterile insect release, Math. Biosci. 5 （1992） 221 247] where the habitat is assumed to be the one-dimensional whole space ~, we consider this system in a bounded one- dimensional domain （interval）. Our goal is to derive sufficient conditions for success of the SIRM. We show the existence of the fertile-free steady state and prove its stability. Using the releasing rate as the parameter, and by a saddle-node bifurcation analysis, we obtain conditions for existence of two co-persistence steady states, one stable and the other unstable. Biological implications of our mathematical results are that： （i） when the fertile population is at low level, the SIRM, even with small releasing rate, can successfully eradicate the fertile insects; （ii） when the fertile population is at a higher level, the SIRM can succeed as long as the strength of the sterile releasing is large enough, while the method may also fail if the releasing is not sufficient.
基金Project supported by the Fundamental Research Funds for the Central Universities (No. 2009B27514)the National Natural Science Foundation of China (No. 10871059)
文摘This paper is devoted to the analysis of the Cauchy problem for a system of PDEs arising in radiative hydrodynamics. This system, which comes from the so-called equilibrium diffusion regime, is a variant of the usual Euler equations, where the energy and pressure functionals are modified to take into account the effect of radiation and the energy balance containing a nonlinear diffusion term acting on the temperature. The problem is studied in the multi-dimensional framework. The authors identify the existence of a strictly convex entropy and a stability property of the system, and check that the Kawashima-Shizuta condition holds. Then, based on these structure properties, the wellposedness close to a constant state can be proved by using fine energy estimates. The asymptotic decay of the solutions are also investigated.
