In this paper, we consider a singular nth order three-point boundary value problem with sign changing nonlinearity. By the method of lower solution and topology degree theorem, we investigate the existence of positive...In this paper, we consider a singular nth order three-point boundary value problem with sign changing nonlinearity. By the method of lower solution and topology degree theorem, we investigate the existence of positive solutions to the above problem. Moreover, the associated Green’s function for the above problem is also given. The results of this paper are new and extend the previous known results.展开更多
We study the nonlinear parabolic equations for travelling wave solutions of Burger’s equations. The purpose of the present work is to study various types of Burger’s equations describing waves and those are based on...We study the nonlinear parabolic equations for travelling wave solutions of Burger’s equations. The purpose of the present work is to study various types of Burger’s equations describing waves and those are based on nonlinear equations. We focus on to describe the analytic solution in the special pattern of travelling wave solutions using tan-cot function method. We discuss about inviscid and viscous version of Burger’s equation for fluid flow and investigate the effects of internal friction of a fluid via Reynolds number. By changing the velocity amplitude, the nature of flows with shock wave and disturbance are observed. For numerical solutions, the Crank-Nicolson scheme is introduced to establish the wave solutions.展开更多
This paper gives probabilistic expressions of theminimal and maximal positive solutions of the partial differential equation -1/2△v(x) + γ(x)v(x)α = 0 in D, where D is a regular domain in Rd(d ≥ 3) such that its c...This paper gives probabilistic expressions of theminimal and maximal positive solutions of the partial differential equation -1/2△v(x) + γ(x)v(x)α = 0 in D, where D is a regular domain in Rd(d ≥ 3) such that its complement Dc is compact, γ(x) is a positive bounded integrable function in D, and 1 ＜α≤ 2. As an application, some necessary and sufficient conditions for a compact set to be S-polar are presented.展开更多
In this paper,we consider the positive solutions of fractional three-point boundary value problem of the form D_0~α+u(t) + f(t,u(t),u'(t),…,u^((n-3))(t),u^((n-2))(t)) = 0,u^((i))(0) = 0,0 < i < n - 2,u^(n-...In this paper,we consider the positive solutions of fractional three-point boundary value problem of the form D_0~α+u(t) + f(t,u(t),u'(t),…,u^((n-3))(t),u^((n-2))(t)) = 0,u^((i))(0) = 0,0 < i < n - 2,u^(n-2))(1) -βu^((n-2))(ξ) = 0,where 0 < t < 1,n-l<a≤n,n≥2,ξ,β∈(0,1),βξ^(α-n) < 1.We first transform it into another equivalent boundary value problem.Then,we derive the Green's function for the equivalent boundary value problem and show that it satisfies certain properties.At last,by using some fixed-point theorems,we obtain the existence of positive solution for this problem.Example is given to illustrate the effectiveness of our result.展开更多
In this paper we consider a super-Brownian motion X with branching mechanism k(x)za, where k(x) ＞ 0 is a bounded Holder continuous function on Rd and infx∈Rd k(x) = 0. We prove that if k(x) ≥‖x‖-1(0 ≤ l ＜∞) fo...In this paper we consider a super-Brownian motion X with branching mechanism k(x)za, where k(x) ＞ 0 is a bounded Holder continuous function on Rd and infx∈Rd k(x) = 0. We prove that if k(x) ≥‖x‖-1(0 ≤ l ＜∞) for sufficiently large x, then X has compact support property, and for dimension d = 1, if k(x) ≥ exp(-l‖x‖)(0 ≤ l ＜∞) for sufficiently large x, then X also has compact support property. The maximal order of k(x) for finite time extinction is different between d = 1, d = 2 and d ≥3: it is O(‖x‖-(a+1))in one dimension, O(‖x‖-2(log ‖x‖)-(a+1)) in two dimensions, and O(‖x‖2) in higher dimensions. These growth orders also turn out to be the maximum order for the nonexistence of a positive solution for 1/2△u =k(x)uα.展开更多
基金Project supported by the National Natural Science Foundation of China (10771212)the Natural Science Foundation of Jiangsu Education Office (06KJB110010)
文摘In this paper, we consider a singular nth order three-point boundary value problem with sign changing nonlinearity. By the method of lower solution and topology degree theorem, we investigate the existence of positive solutions to the above problem. Moreover, the associated Green’s function for the above problem is also given. The results of this paper are new and extend the previous known results.
文摘We study the nonlinear parabolic equations for travelling wave solutions of Burger’s equations. The purpose of the present work is to study various types of Burger’s equations describing waves and those are based on nonlinear equations. We focus on to describe the analytic solution in the special pattern of travelling wave solutions using tan-cot function method. We discuss about inviscid and viscous version of Burger’s equation for fluid flow and investigate the effects of internal friction of a fluid via Reynolds number. By changing the velocity amplitude, the nature of flows with shock wave and disturbance are observed. For numerical solutions, the Crank-Nicolson scheme is introduced to establish the wave solutions.
基金supported by the National Natural Science Foundation of China(Grant No.10471003)Foundation for Authors Awarded Excellent Ph.D.Dissertation.
文摘This paper gives probabilistic expressions of theminimal and maximal positive solutions of the partial differential equation -1/2△v(x) + γ(x)v(x)α = 0 in D, where D is a regular domain in Rd(d ≥ 3) such that its complement Dc is compact, γ(x) is a positive bounded integrable function in D, and 1 ＜α≤ 2. As an application, some necessary and sufficient conditions for a compact set to be S-polar are presented.
基金Supported by the National Nature Science Foundation of China(11071001)Supported by the Key Program of Ministry of Education of China(205068)
文摘In this paper,we consider the positive solutions of fractional three-point boundary value problem of the form D_0~α+u(t) + f(t,u(t),u'(t),…,u^((n-3))(t),u^((n-2))(t)) = 0,u^((i))(0) = 0,0 < i < n - 2,u^(n-2))(1) -βu^((n-2))(ξ) = 0,where 0 < t < 1,n-l<a≤n,n≥2,ξ,β∈(0,1),βξ^(α-n) < 1.We first transform it into another equivalent boundary value problem.Then,we derive the Green's function for the equivalent boundary value problem and show that it satisfies certain properties.At last,by using some fixed-point theorems,we obtain the existence of positive solution for this problem.Example is given to illustrate the effectiveness of our result.
基金This work was supported by the N ational Natural Science Foundation of China(Grant Nos.10001020 and 10131040).
文摘In this paper we consider a super-Brownian motion X with branching mechanism k(x)za, where k(x) ＞ 0 is a bounded Holder continuous function on Rd and infx∈Rd k(x) = 0. We prove that if k(x) ≥‖x‖-1(0 ≤ l ＜∞) for sufficiently large x, then X has compact support property, and for dimension d = 1, if k(x) ≥ exp(-l‖x‖)(0 ≤ l ＜∞) for sufficiently large x, then X also has compact support property. The maximal order of k(x) for finite time extinction is different between d = 1, d = 2 and d ≥3: it is O(‖x‖-(a+1))in one dimension, O(‖x‖-2(log ‖x‖)-(a+1)) in two dimensions, and O(‖x‖2) in higher dimensions. These growth orders also turn out to be the maximum order for the nonexistence of a positive solution for 1/2△u =k(x)uα.
