In this paper the nonlinear reaction diffusion problems with ultraparabolic equations are considered. By using comparison theorem, the existence, uniqueness and asymptotic behavior of solution for the problem are stud...In this paper the nonlinear reaction diffusion problems with ultraparabolic equations are considered. By using comparison theorem, the existence, uniqueness and asymptotic behavior of solution for the problem are studied.展开更多
In this piece of work, using three spatial grid points, we discuss a new two-level implicit cubic spline method of O(k2 + kh2 + h4) for the solution of quasi-linear parabolic equation , 0 0 subject to appropriate init...In this piece of work, using three spatial grid points, we discuss a new two-level implicit cubic spline method of O(k2 + kh2 + h4) for the solution of quasi-linear parabolic equation , 0 0 subject to appropriate initial and Dirichlet boundary conditions, where h > 0, k > 0 are grid sizes in space and time-directions, respectively. The cubic spline approximation produces at each time level a spline function which may be used to obtain the solution at any point in the range of the space variable. The proposed cubic spline method is applicable to parabolic equations having singularity. The stability analysis for diffusion- convection equation shows the unconditionally stable character of the cubic spline method. The numerical tests are performed and comparative results are provided to illustrate the usefulness of the proposed method.展开更多
In this paper,we consider the finite element method for two-dimensional nonlinear modified time-fractional fourth-order diffusion equation.In order to avoid using higher order elements,we introduce an intermediate var...In this paper,we consider the finite element method for two-dimensional nonlinear modified time-fractional fourth-order diffusion equation.In order to avoid using higher order elements,we introduce an intermediate variableσ=∆u and translate the fourth-order derivative of the original problem into a second-order coupled system.We discretize the fractional time derivative terms by using the L1-approximation and discretize the first-order time derivative term by using the second-order backward differentiation formula.In the fully discrete scheme,we implement the finite element method for the spatial approximation.Unconditional stability of the fully discrete scheme is proven and its optimal convergence order is obtained.Numerical experiments are carried out to demonstrate our theoretical analysis.展开更多
A fourth-order degenerate parabolic equation with a viscous term: ?is studied with the initial-boundary conditions ux=wx=0?on {-1,1}×(0,T), u(x,0)=u0(x)?in (-1,1). It can be taken as a thin film equation or a Cah...A fourth-order degenerate parabolic equation with a viscous term: ?is studied with the initial-boundary conditions ux=wx=0?on {-1,1}×(0,T), u(x,0)=u0(x)?in (-1,1). It can be taken as a thin film equation or a Cahn-Hilliard equation with a degenerate mobility. The entropy functional method is introduced to overcome the difficulties that arise from the degenerate mobility m(u)?and the viscosity term. The existence of nonnegative weak solution is obtained.展开更多
A modified version of the Cotte, Lions, Morel and Coil theory for image selective smoothing and edge detection is proposed. Comparing with their model, the most important advantage of this modification is that the con...A modified version of the Cotte, Lions, Morel and Coil theory for image selective smoothing and edge detection is proposed. Comparing with their model, the most important advantage of this modification is that the convolution with Gaussian processes in the filtering process is replaced by solving an initial-boundary value problem for the heat equation, which simplifies the numerical scheme to some extent. Numerical experiments on natural images are presented for this model.展开更多
In this article,non-linear time-fractional diffusion equations are considered to describe oil pollution in the water.The latest technique,fractional reduced differential transform method(FRDTM),is used to ac-quire app...In this article,non-linear time-fractional diffusion equations are considered to describe oil pollution in the water.The latest technique,fractional reduced differential transform method(FRDTM),is used to ac-quire approximate solutions of the time fractional-order diffusion equation and two cases of Allen-Cahn equations.The acquired results are collated with the exact solutions and other results from literature for integer-orderα,which reveal that the proposed method is effective.Hence,FRDTM can be employed to obtain solutions for different types of nonlinear fractional-order IVPs arising in engineering and science.展开更多
In the present work we study the global solvability of the Kolmogorov-Fisher type biological population task with double nonlinear diffusion and qualitative properties of the solution of the task based on the self-sim...In the present work we study the global solvability of the Kolmogorov-Fisher type biological population task with double nonlinear diffusion and qualitative properties of the solution of the task based on the self-similar analysis. In additional, in this paper we consider the model of two competing population with dual nonlinear cross-diffusion.展开更多
An existence and uniqueness result of a renormalized solution for a class of doubly nonlinear parabolic equations with singular coefficient with respect to the unknown and with diffuse measure data is established.A co...An existence and uniqueness result of a renormalized solution for a class of doubly nonlinear parabolic equations with singular coefficient with respect to the unknown and with diffuse measure data is established.A comparison result is also proved for such solutions.展开更多
In this article,modified versions of variational iteration algorithms are presented for the numerical simulation of the diffusion of oil pollutions.Three numerical examples are given to demonstrate the applicability a...In this article,modified versions of variational iteration algorithms are presented for the numerical simulation of the diffusion of oil pollutions.Three numerical examples are given to demonstrate the applicability and validity of the proposed algorithms.The obtained results are compared with the existing solutions,which reveal that the proposed methods are very effective and can be used for other nonlinear initial value problems arising in science and engineering.展开更多
This paper deals with backward stochastic differential equations with jumps,whose data(the terminal condition and coefficient) are given functions of jump-diffusion process paths. The author introduces a type of nonli...This paper deals with backward stochastic differential equations with jumps,whose data(the terminal condition and coefficient) are given functions of jump-diffusion process paths. The author introduces a type of nonlinear path-dependent parabolic integrodifferential equations, and then obtains a new type of nonlinear Feynman-Kac formula related to such BSDEs with jumps under some regularity conditions.展开更多
In this paper, let(M~n, g) be an n-dimensional complete Riemannian manifold with the mdimensional Bakry–mery Ricci curvature bounded below. By using the maximum principle, we first prove a Li–Yau type Harnack differ...In this paper, let(M~n, g) be an n-dimensional complete Riemannian manifold with the mdimensional Bakry–mery Ricci curvature bounded below. By using the maximum principle, we first prove a Li–Yau type Harnack differential inequality for positive solutions to the parabolic equation u= LF(u)=ΔF(u)-f·F(u),on compact Riemannian manifolds Mn, where F∈C~2(0, ∞), F>0 and f is a C~2-smooth function defined on M~n. As application, the Harnack differential inequalities for fast diffusion type equation and porous media type equation are derived. On the other hand, we derive a local Hamilton type gradient estimate for positive solutions of the degenerate parabolic equation on complete Riemannian manifolds. As application, related local Hamilton type gradient estimate and Harnack inequality for fast dfiffusion type equation are established. Our results generalize some known results.展开更多
Slow motion for scalar Allen-Cahn type equation is a well-known phenomenon,precise motion law for the dynamics of fronts having been established first using the socalled geometric approach inspired from central manifo...Slow motion for scalar Allen-Cahn type equation is a well-known phenomenon,precise motion law for the dynamics of fronts having been established first using the socalled geometric approach inspired from central manifold theory(see the results of Carr and Pego in 1989). In this paper, the authors present an alternate approach to recover the motion law, and extend it to the case of multiple wells. This method is based on the localized energy identity, and is therefore, at least conceptually, simpler to implement. It also allows to handle collisions and rough initial data.展开更多
In this paper, we present two second-order numerical schemes to solve the fourth order parabolic equation derived from a diffuse interface model with Peng-Robinson Equation of state (EOS) for pure substance. The mas...In this paper, we present two second-order numerical schemes to solve the fourth order parabolic equation derived from a diffuse interface model with Peng-Robinson Equation of state (EOS) for pure substance. The mass conservation, energy decay property, unique solvability and L~ convergence of these two schemes are proved. Numerical results demon- strate the good approximation of the fourth order equation and confirm reliability of these two schemes.展开更多
基金Supported by the NNSF of China(40676016,10471039)the National Key Project for Basics Research(2003CB415101-03 and 2004CB418304)+1 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).
文摘In this paper the nonlinear reaction diffusion problems with ultraparabolic equations are considered. By using comparison theorem, the existence, uniqueness and asymptotic behavior of solution for the problem are studied.
文摘In this piece of work, using three spatial grid points, we discuss a new two-level implicit cubic spline method of O(k2 + kh2 + h4) for the solution of quasi-linear parabolic equation , 0 0 subject to appropriate initial and Dirichlet boundary conditions, where h > 0, k > 0 are grid sizes in space and time-directions, respectively. The cubic spline approximation produces at each time level a spline function which may be used to obtain the solution at any point in the range of the space variable. The proposed cubic spline method is applicable to parabolic equations having singularity. The stability analysis for diffusion- convection equation shows the unconditionally stable character of the cubic spline method. The numerical tests are performed and comparative results are provided to illustrate the usefulness of the proposed method.
文摘In this paper,we consider the finite element method for two-dimensional nonlinear modified time-fractional fourth-order diffusion equation.In order to avoid using higher order elements,we introduce an intermediate variableσ=∆u and translate the fourth-order derivative of the original problem into a second-order coupled system.We discretize the fractional time derivative terms by using the L1-approximation and discretize the first-order time derivative term by using the second-order backward differentiation formula.In the fully discrete scheme,we implement the finite element method for the spatial approximation.Unconditional stability of the fully discrete scheme is proven and its optimal convergence order is obtained.Numerical experiments are carried out to demonstrate our theoretical analysis.
文摘A fourth-order degenerate parabolic equation with a viscous term: ?is studied with the initial-boundary conditions ux=wx=0?on {-1,1}×(0,T), u(x,0)=u0(x)?in (-1,1). It can be taken as a thin film equation or a Cahn-Hilliard equation with a degenerate mobility. The entropy functional method is introduced to overcome the difficulties that arise from the degenerate mobility m(u)?and the viscosity term. The existence of nonnegative weak solution is obtained.
基金Partially supported by National Natural Science Foundation and the Shanghai Qimingxing grant. # 97QA14040
文摘A modified version of the Cotte, Lions, Morel and Coil theory for image selective smoothing and edge detection is proposed. Comparing with their model, the most important advantage of this modification is that the convolution with Gaussian processes in the filtering process is replaced by solving an initial-boundary value problem for the heat equation, which simplifies the numerical scheme to some extent. Numerical experiments on natural images are presented for this model.
文摘In this article,non-linear time-fractional diffusion equations are considered to describe oil pollution in the water.The latest technique,fractional reduced differential transform method(FRDTM),is used to ac-quire approximate solutions of the time fractional-order diffusion equation and two cases of Allen-Cahn equations.The acquired results are collated with the exact solutions and other results from literature for integer-orderα,which reveal that the proposed method is effective.Hence,FRDTM can be employed to obtain solutions for different types of nonlinear fractional-order IVPs arising in engineering and science.
文摘In the present work we study the global solvability of the Kolmogorov-Fisher type biological population task with double nonlinear diffusion and qualitative properties of the solution of the task based on the self-similar analysis. In additional, in this paper we consider the model of two competing population with dual nonlinear cross-diffusion.
文摘An existence and uniqueness result of a renormalized solution for a class of doubly nonlinear parabolic equations with singular coefficient with respect to the unknown and with diffuse measure data is established.A comparison result is also proved for such solutions.
文摘In this article,modified versions of variational iteration algorithms are presented for the numerical simulation of the diffusion of oil pollutions.Three numerical examples are given to demonstrate the applicability and validity of the proposed algorithms.The obtained results are compared with the existing solutions,which reveal that the proposed methods are very effective and can be used for other nonlinear initial value problems arising in science and engineering.
基金supported by the National Natural Science Foundation of China(Nos.10921101,11471190)the Shandong Provincial Natural Science Foundation of China(No.ZR2014AM002)the Programme of Introducing Talents of Discipline to Universities of China(No.B12023)
文摘This paper deals with backward stochastic differential equations with jumps,whose data(the terminal condition and coefficient) are given functions of jump-diffusion process paths. The author introduces a type of nonlinear path-dependent parabolic integrodifferential equations, and then obtains a new type of nonlinear Feynman-Kac formula related to such BSDEs with jumps under some regularity conditions.
基金Supported by Universities Natural Science Foundation of Anhui Province(Grant No.KJ2016A310)
文摘In this paper, let(M~n, g) be an n-dimensional complete Riemannian manifold with the mdimensional Bakry–mery Ricci curvature bounded below. By using the maximum principle, we first prove a Li–Yau type Harnack differential inequality for positive solutions to the parabolic equation u= LF(u)=ΔF(u)-f·F(u),on compact Riemannian manifolds Mn, where F∈C~2(0, ∞), F>0 and f is a C~2-smooth function defined on M~n. As application, the Harnack differential inequalities for fast diffusion type equation and porous media type equation are derived. On the other hand, we derive a local Hamilton type gradient estimate for positive solutions of the degenerate parabolic equation on complete Riemannian manifolds. As application, related local Hamilton type gradient estimate and Harnack inequality for fast dfiffusion type equation are established. Our results generalize some known results.
文摘Slow motion for scalar Allen-Cahn type equation is a well-known phenomenon,precise motion law for the dynamics of fronts having been established first using the socalled geometric approach inspired from central manifold theory(see the results of Carr and Pego in 1989). In this paper, the authors present an alternate approach to recover the motion law, and extend it to the case of multiple wells. This method is based on the localized energy identity, and is therefore, at least conceptually, simpler to implement. It also allows to handle collisions and rough initial data.
文摘In this paper, we present two second-order numerical schemes to solve the fourth order parabolic equation derived from a diffuse interface model with Peng-Robinson Equation of state (EOS) for pure substance. The mass conservation, energy decay property, unique solvability and L~ convergence of these two schemes are proved. Numerical results demon- strate the good approximation of the fourth order equation and confirm reliability of these two schemes.