We study the following modified transitional Korteweg-de Vries equation ut+f(t)upux+uxxx=0, (x,t)∈R+×R+, (p≥2is an even integer) with initial value u(x,0)=g(x)∈H4(R+)and inhomogeneous boundary value u(0,t)=Q(t...We study the following modified transitional Korteweg-de Vries equation ut+f(t)upux+uxxx=0, (x,t)∈R+×R+, (p≥2is an even integer) with initial value u(x,0)=g(x)∈H4(R+)and inhomogeneous boundary value u(0,t)=Q(t)∈C2([ 0,∞ )). Under the conditions either (i) f(t)≤0, f′(t)≥0or (ii) f(t)≤−αwhere α>0, we prove the existence of a unique global classical solution.展开更多
This paper considers the regularity of solutions to mixed boundary value problems in small-angle regions for elliptic equations. By constructing a specific barrier function, we proved that under the assumption of suff...This paper considers the regularity of solutions to mixed boundary value problems in small-angle regions for elliptic equations. By constructing a specific barrier function, we proved that under the assumption of sufficient regularity of boundary conditions and coefficients, as long as the angle is sufficiently small, the regularity of the solution to the mixed boundary value problem of the second-order elliptic equation can reach any order.展开更多
In the poper, the method of separating singularity is applied to study the uniformly difference scheme of a singular perturbation problem for a semilinear ordinary differential equation with mixed boundary value condi...In the poper, the method of separating singularity is applied to study the uniformly difference scheme of a singular perturbation problem for a semilinear ordinary differential equation with mixed boundary value condition. The uniform convergence on small parameter ε of order one for an IVin type difference scheme constructed is proved. At the end of the paper, a numerical example is given. The computing results coincide with the theoretical analysis.展开更多
The singularly perturbed elliptic equation boundary value problem with turning point is considered. Using the method of multiple scales and the comparison theorem, the asymptotic behavior of solution for the boundary ...The singularly perturbed elliptic equation boundary value problem with turning point is considered. Using the method of multiple scales and the comparison theorem, the asymptotic behavior of solution for the boundary value problem is studied.展开更多
In this paper,a compact difference scheme is established for the heat equations with multi-point boundary value conditions.The truncation error of the difference scheme is O(τ2+h^4),where t and h are the temporal ste...In this paper,a compact difference scheme is established for the heat equations with multi-point boundary value conditions.The truncation error of the difference scheme is O(τ2+h^4),where t and h are the temporal step size and the spatial step size.A prior estimate of the difference solution in a weighted norm is obtained.The unique solvability,stability and convergence of the difference scheme are proved by the energy method.The theoretical statements for the solution of the difference scheme are supported by numerical examples.展开更多
In this paper well-conditioning of boundary value problems for systems of second order difference equa-tions is studied.First,a sufficient condition for the existence of a unique bounded solution (for large enough num...In this paper well-conditioning of boundary value problems for systems of second order difference equa-tions is studied.First,a sufficient condition for the existence of a unique bounded solution (for large enough number of steps) of an associated homogeneous system is given.Finally,a sufficient condition for well-condi-tioning,intrinsically related to the problem data is proposed.展开更多
Aim To study a class of boundary value problem of hyperbolic partial functional differential equations with continuous deviating arguments. Methods An averaging technique was used. The multi dimensional problem was...Aim To study a class of boundary value problem of hyperbolic partial functional differential equations with continuous deviating arguments. Methods An averaging technique was used. The multi dimensional problem was reduced to a one dimensional oscillation problem for ordinary differential equations or inequalities. Results and Conclusion The known results of oscillation of solutions for a class of boundary value problem of hyperbolic partial functional differential equations with discrete deviating arguments are generalized, and the oscillatory criteria of solutions for such equation with two kinds of boundary value conditions are obtained.展开更多
A class of singularly perturbed initial boundary value problems for the reaction diffusion equations in a part of domain are considered. Using the operator theory the asymptotic behavior of solution for the problems i...A class of singularly perturbed initial boundary value problems for the reaction diffusion equations in a part of domain are considered. Using the operator theory the asymptotic behavior of solution for the problems is studied.展开更多
This paper studies the prescribed contact angle boundary value problem of a certain type of mean curvature equation.Applying the maximum principle and the moving frame method and based on the location of the maximum p...This paper studies the prescribed contact angle boundary value problem of a certain type of mean curvature equation.Applying the maximum principle and the moving frame method and based on the location of the maximum point,the boundary gradient estimation of the solutions to the equation is obtained.展开更多
In this paper,we study a Dirichlet-type boundary value problem(BVP) of nonlinear fractional differential equation with an order α∈(3,4],where the fractional derivative D~α_(o^+)is the standard Riemann-Liouville fra...In this paper,we study a Dirichlet-type boundary value problem(BVP) of nonlinear fractional differential equation with an order α∈(3,4],where the fractional derivative D~α_(o^+)is the standard Riemann-Liouville fractional derivative.By constructing the Green function and investigating its properties,we obtain some criteria for the existence of one positive solution and two positive solutions for the above BVP.The Krasnosel'skii fixedpoint theorem in cones is used here.We also give an example to illustrate the applicability of our results.展开更多
This paper is concerned with the boundary value problem of a nonlinear fractional differential equation. By means of Schauder fixed-point theorem, an existence result of solution is obtained.
In this paper,the boundary value problems of p-Laplacian functional differential equation are studied.By using a fixed point theorem in cones,some criteria for the existence of positive solutions are given.
In this paper, a high accuracy finite volume element method is presented for two-point boundary value problem of second order ordinary differential equation, which differs from the high order generalized difference me...In this paper, a high accuracy finite volume element method is presented for two-point boundary value problem of second order ordinary differential equation, which differs from the high order generalized difference methods. It is proved that the method has optimal order error estimate O(h3) in H1 norm. Finally, two examples show that the method is effective.展开更多
Suffcient conditions for the existence of at least one solution of two-point boundary value problems for second order nonlinear differential equations [φ(x(t))] + kx(t) + g(t,x(t)) = p(t),t ∈(0,π) x(0) = x(π) = 0 ...Suffcient conditions for the existence of at least one solution of two-point boundary value problems for second order nonlinear differential equations [φ(x(t))] + kx(t) + g(t,x(t)) = p(t),t ∈(0,π) x(0) = x(π) = 0 are established,where [φ(x)] =(|x |p-2x) with p > 1.Our result is new even when [φ(x)] = x in above problem,i.e.p = 2.Examples are presented to illustrate the effciency of the theorem in this paper.展开更多
This paper is concerned with the existence and approximation of solutions for a class of first order impulsive functional differential equations with periodic boundary value conditions. A new comparison result is pres...This paper is concerned with the existence and approximation of solutions for a class of first order impulsive functional differential equations with periodic boundary value conditions. A new comparison result is presented and the previous results are extended.展开更多
Studies the existence of solutions of nonlinear two point boundary value problems for nonlinear 4n-th-order differential equationy (4n)=f(t,y,y′,y″,...,y (4n-1))(a)with the boundary conditions g 2i(y (2i)(a),y (2i+1...Studies the existence of solutions of nonlinear two point boundary value problems for nonlinear 4n-th-order differential equationy (4n)=f(t,y,y′,y″,...,y (4n-1))(a)with the boundary conditions g 2i(y (2i)(a),y (2i+1)(a))=0,h 2i(y (2i)(c),y (2i+1)(c))=0,(i=0,1,...,2n-1)(b) where the functions f, g i and h i are continuous with certain monotone properties. For the boundary value problems of nonlinear nth order differential equationy (n)=f(t,y,y′,y″,...,y (n-1))many results have been given at the present time. But the existence of solutions of boundary value problem (a),(b) studied in this paper has not been covered by the above researches. Moreover, the corollary of the important theorem in this paper, i.e. existence of solutions of the boundary value problem.y (4n)=f(t,y,y′,y″,...,y (4n-1)) a 2iy (2i)(a)+a 2i+1y (2i+1)(a)=b 2i,c 2iy (2i)(c)+c 2i+1y (2i+1)(c)=d 2i,(i=0,1,...2n-1)has not been dealt with in previous works.展开更多
In this paper a singular perturbation of boundary value problem for elliptic partial differential equations of higher order is considered by using the differential inequalities. The uniformly valid asymptotic expansio...In this paper a singular perturbation of boundary value problem for elliptic partial differential equations of higher order is considered by using the differential inequalities. The uniformly valid asymptotic expansion in entire region is obtained.展开更多
We study the existence of solutions to the second order three-point boundary value problem：{x″（t）＋f（t,x（t）,x′（t））=0,t≠ti,△x（ti）=Ii（x（ti）,x′（ti））,i=1,2,…,k,△x′（ti）=Ji（x（ti）,x′（t））,i=1...We study the existence of solutions to the second order three-point boundary value problem：{x″（t）＋f（t,x（t）,x′（t））=0,t≠ti,△x（ti）=Ii（x（ti）,x′（ti））,i=1,2,…,k,△x′（ti）=Ji（x（ti）,x′（t））,i=1,2,…,k,x（0）=0=x（1）-αx（η）,where 0〈η〈1,α∈R,and f：[0,1]×R×R→R,Ii：R×R→R,Ji：R×R→R（i=1,2,…,k）are continuous. Our results is new and different from previous results. In particular, we obtain the Green function of the problem, which makes the problem simpler.展开更多
Using the theory of coincidence degree, a class of higher order multi-point boundary value problem for ordinary differential equations are studied. Under the boundary conditions satisfying the resonance case, some new...Using the theory of coincidence degree, a class of higher order multi-point boundary value problem for ordinary differential equations are studied. Under the boundary conditions satisfying the resonance case, some new existence results are obtained by supposing some conditions to the nonlinear term and applying a priori estimates.展开更多
This paper is concerned with the free boundary value problem for multidimensional Navier-Stokes equations with density-dependent viscosity where the flow density vanishes continuously across the free boundary. Local ...This paper is concerned with the free boundary value problem for multidimensional Navier-Stokes equations with density-dependent viscosity where the flow density vanishes continuously across the free boundary. Local （in time） existence of a weak solution is established; in particular, the density is positive and the solution is regular away from the free boundary.展开更多
文摘We study the following modified transitional Korteweg-de Vries equation ut+f(t)upux+uxxx=0, (x,t)∈R+×R+, (p≥2is an even integer) with initial value u(x,0)=g(x)∈H4(R+)and inhomogeneous boundary value u(0,t)=Q(t)∈C2([ 0,∞ )). Under the conditions either (i) f(t)≤0, f′(t)≥0or (ii) f(t)≤−αwhere α>0, we prove the existence of a unique global classical solution.
文摘This paper considers the regularity of solutions to mixed boundary value problems in small-angle regions for elliptic equations. By constructing a specific barrier function, we proved that under the assumption of sufficient regularity of boundary conditions and coefficients, as long as the angle is sufficiently small, the regularity of the solution to the mixed boundary value problem of the second-order elliptic equation can reach any order.
文摘In the poper, the method of separating singularity is applied to study the uniformly difference scheme of a singular perturbation problem for a semilinear ordinary differential equation with mixed boundary value condition. The uniform convergence on small parameter ε of order one for an IVin type difference scheme constructed is proved. At the end of the paper, a numerical example is given. The computing results coincide with the theoretical analysis.
文摘The singularly perturbed elliptic equation boundary value problem with turning point is considered. Using the method of multiple scales and the comparison theorem, the asymptotic behavior of solution for the boundary value problem is studied.
基金The research is supported by the National Natural Science Foundation of China(No.11671081)the Fundamental Research Funds for the Central Universities(No.242017K41044).
文摘In this paper,a compact difference scheme is established for the heat equations with multi-point boundary value conditions.The truncation error of the difference scheme is O(τ2+h^4),where t and h are the temporal step size and the spatial step size.A prior estimate of the difference solution in a weighted norm is obtained.The unique solvability,stability and convergence of the difference scheme are proved by the energy method.The theoretical statements for the solution of the difference scheme are supported by numerical examples.
基金This work has been partially supported by the "Generalitat Valenciana" grant GV1118/93the Spanish D. G. I. C. Y.T. grant PB93-0381
文摘In this paper well-conditioning of boundary value problems for systems of second order difference equa-tions is studied.First,a sufficient condition for the existence of a unique bounded solution (for large enough number of steps) of an associated homogeneous system is given.Finally,a sufficient condition for well-condi-tioning,intrinsically related to the problem data is proposed.
文摘Aim To study a class of boundary value problem of hyperbolic partial functional differential equations with continuous deviating arguments. Methods An averaging technique was used. The multi dimensional problem was reduced to a one dimensional oscillation problem for ordinary differential equations or inequalities. Results and Conclusion The known results of oscillation of solutions for a class of boundary value problem of hyperbolic partial functional differential equations with discrete deviating arguments are generalized, and the oscillatory criteria of solutions for such equation with two kinds of boundary value conditions are obtained.
文摘A class of singularly perturbed initial boundary value problems for the reaction diffusion equations in a part of domain are considered. Using the operator theory the asymptotic behavior of solution for the problems is studied.
基金supported by the National Natural Science Foundation of China (No.12061078)。
文摘This paper studies the prescribed contact angle boundary value problem of a certain type of mean curvature equation.Applying the maximum principle and the moving frame method and based on the location of the maximum point,the boundary gradient estimation of the solutions to the equation is obtained.
基金Supported by the Research Fund for the Doctoral Program of High Education of China(20094407110001)Supported by the NSF of Guangdong Province(10151063101000003)
文摘In this paper,we study a Dirichlet-type boundary value problem(BVP) of nonlinear fractional differential equation with an order α∈(3,4],where the fractional derivative D~α_(o^+)is the standard Riemann-Liouville fractional derivative.By constructing the Green function and investigating its properties,we obtain some criteria for the existence of one positive solution and two positive solutions for the above BVP.The Krasnosel'skii fixedpoint theorem in cones is used here.We also give an example to illustrate the applicability of our results.
文摘This paper is concerned with the boundary value problem of a nonlinear fractional differential equation. By means of Schauder fixed-point theorem, an existence result of solution is obtained.
文摘In this paper,the boundary value problems of p-Laplacian functional differential equation are studied.By using a fixed point theorem in cones,some criteria for the existence of positive solutions are given.
基金heprojectissupportedbyNNSFofChina (No .1 9972 0 39) .
文摘In this paper, a high accuracy finite volume element method is presented for two-point boundary value problem of second order ordinary differential equation, which differs from the high order generalized difference methods. It is proved that the method has optimal order error estimate O(h3) in H1 norm. Finally, two examples show that the method is effective.
基金Supported by the Natural Science Foundation of Hunan Province（06JJ50008） Supported by the Natural Science Foundation of Guangdong Province（7004569）
文摘Suffcient conditions for the existence of at least one solution of two-point boundary value problems for second order nonlinear differential equations [φ(x(t))] + kx(t) + g(t,x(t)) = p(t),t ∈(0,π) x(0) = x(π) = 0 are established,where [φ(x)] =(|x |p-2x) with p > 1.Our result is new even when [φ(x)] = x in above problem,i.e.p = 2.Examples are presented to illustrate the effciency of the theorem in this paper.
基金Supported by the National Natural Science Foundation of China (10571050 10871062)Hunan Provincial Innovation Foundation For Postgraduate
文摘This paper is concerned with the existence and approximation of solutions for a class of first order impulsive functional differential equations with periodic boundary value conditions. A new comparison result is presented and the previous results are extended.
文摘Studies the existence of solutions of nonlinear two point boundary value problems for nonlinear 4n-th-order differential equationy (4n)=f(t,y,y′,y″,...,y (4n-1))(a)with the boundary conditions g 2i(y (2i)(a),y (2i+1)(a))=0,h 2i(y (2i)(c),y (2i+1)(c))=0,(i=0,1,...,2n-1)(b) where the functions f, g i and h i are continuous with certain monotone properties. For the boundary value problems of nonlinear nth order differential equationy (n)=f(t,y,y′,y″,...,y (n-1))many results have been given at the present time. But the existence of solutions of boundary value problem (a),(b) studied in this paper has not been covered by the above researches. Moreover, the corollary of the important theorem in this paper, i.e. existence of solutions of the boundary value problem.y (4n)=f(t,y,y′,y″,...,y (4n-1)) a 2iy (2i)(a)+a 2i+1y (2i+1)(a)=b 2i,c 2iy (2i)(c)+c 2i+1y (2i+1)(c)=d 2i,(i=0,1,...2n-1)has not been dealt with in previous works.
文摘In this paper a singular perturbation of boundary value problem for elliptic partial differential equations of higher order is considered by using the differential inequalities. The uniformly valid asymptotic expansion in entire region is obtained.
基金Supported by the National Natural Science Foundation of China(10371006)
文摘We study the existence of solutions to the second order three-point boundary value problem：{x″（t）＋f（t,x（t）,x′（t））=0,t≠ti,△x（ti）=Ii（x（ti）,x′（ti））,i=1,2,…,k,△x′（ti）=Ji（x（ti）,x′（t））,i=1,2,…,k,x（0）=0=x（1）-αx（η）,where 0〈η〈1,α∈R,and f：[0,1]×R×R→R,Ii：R×R→R,Ji：R×R→R（i=1,2,…,k）are continuous. Our results is new and different from previous results. In particular, we obtain the Green function of the problem, which makes the problem simpler.
基金Project supported by the National Natural Science Foundation of China (No.10371006)
文摘Using the theory of coincidence degree, a class of higher order multi-point boundary value problem for ordinary differential equations are studied. Under the boundary conditions satisfying the resonance case, some new existence results are obtained by supposing some conditions to the nonlinear term and applying a priori estimates.
基金partially supported by the NSFC(10871134)the AHRDIHL Project of Beijing Municipality (PHR201006107)
文摘This paper is concerned with the free boundary value problem for multidimensional Navier-Stokes equations with density-dependent viscosity where the flow density vanishes continuously across the free boundary. Local （in time） existence of a weak solution is established; in particular, the density is positive and the solution is regular away from the free boundary.
