In regions characterized with great mining depths,complex topography,and intense geological activities,solely relying on lateral pressure coefficients or linear boundary conditions for predicting the in situ stress fi...In regions characterized with great mining depths,complex topography,and intense geological activities,solely relying on lateral pressure coefficients or linear boundary conditions for predicting the in situ stress field of rock bodies can induce substantial deviations and limitations.This study focuses on a typical karst area in Southwest Guizhou,China as its research background.It employs a hybrid approach integrating machine learning,numerical simulations,and field experiments to develop an optimization algorithm for nonlinear prediction of the complex three-dimensional(3D)in situ stress fields.Through collecting and fitting analysis of in situ stress measurement data from the karst region,the distributions of in situ stresses with depth were identified with nonlinear boundary conditions.A prediction model for in situ stress was then established based on artificial neural network(ANN)and genetic algorithm(GA)approach,validated in the typical karst landscape mine,Jinfeng Gold Mine.The results demonstrate that the model's predictions align well with actual measurements,showcasing consistency and regularity.Specifically,the error between the predicted and actual values of the maximum horizontal principal stress was the smallest,with an absolute error 0.01-3 MPa and a relative error of 0.04-15.31%.This model accurately and effectively predicts in situ stresses in complex geological areas.展开更多
Sufficient conditions are obtained for the oscillation of solutions of the systems of quasilinear hyperbolic differential equation with deviating arguments under nonlinear boundary condition.
The purpose of this paper is to investigate the stability and asymptotic behavior of the time-dependent solutions to a linear parabolic equation with nonlinear boundary condition in relation to their corresponding ste...The purpose of this paper is to investigate the stability and asymptotic behavior of the time-dependent solutions to a linear parabolic equation with nonlinear boundary condition in relation to their corresponding steady state solutions. Then, the above results are extended to a semilinear parabolic equation with nonlinear boundary condition by analyzing the corresponding eigenvalue problem and using the method of upper and lower solutions.展开更多
This paper discusses the semidiscrete finite element method for nonlinear hyperbolic equations with nonlinear boundary condition. The superclose property is derived through interpolation instead of the nonlinear H^1 p...This paper discusses the semidiscrete finite element method for nonlinear hyperbolic equations with nonlinear boundary condition. The superclose property is derived through interpolation instead of the nonlinear H^1 projection and integral identity technique. Meanwhile, the global superconvergence is obtained based on the interpolated postprocessing techniques.展开更多
In this paper,we discuss the existence of positive solutions for the second-order singular difference equation boundary value problem -Δ^(2)u(t-1)=λg(t)f(u).t∈[1,T]_(z),u(0)=0,Δu(T)+c(u(T+1))u(T+1)=0,where λ> ...In this paper,we discuss the existence of positive solutions for the second-order singular difference equation boundary value problem -Δ^(2)u(t-1)=λg(t)f(u).t∈[1,T]_(z),u(0)=0,Δu(T)+c(u(T+1))u(T+1)=0,where λ> 0 is a positive parameter,f:(0,∞)→R is continuous,and is allowed to be singular at 0.The existence of positive solutions is established via introducing a new complete continuous operator.展开更多
With prior estimate method, the existence, uniqueness, stability and large time behavior of the solution of second initial-boundary value problem for a fast diffusion equation with nonlinear boundary conditions are in...With prior estimate method, the existence, uniqueness, stability and large time behavior of the solution of second initial-boundary value problem for a fast diffusion equation with nonlinear boundary conditions are investigated. The main results are : 1) there exists only one global weak solution which continuously depends on initial value; 2) when t < T-0, the solution is infinitely continuously differentiable and is a classical solution; 3) the solution converges to zero uniformly as t is large enough.展开更多
This work presents a numerical methodology for modeling the Winkler supports and nonlinear conditions by proposing new boundary conditions. For the boundary conditions of Winkler support model, the surface tractions a...This work presents a numerical methodology for modeling the Winkler supports and nonlinear conditions by proposing new boundary conditions. For the boundary conditions of Winkler support model, the surface tractions and the displacements normal to the surface of the solid are unknown, but their relationship is known by means of the ballast coefficient, whereas for nonlinear boundary conditions, the displacements normal to the boundary of the solid are zero in the positive direction but are allowed in the negative direction. In those zones, detachments of nodes might appear, leading to a nonlinearity, because the number of nodes that remain fixed or of the detached ones (under tensile tractions) is unknown. The proposed methodology is applied to the 3D elastic receding contact problem using the boundary element method. The surface t r actions and the displacements of the common int erface bet ween the two solids in contac t under the influence of different supports are calculated as well as the boundary zone of the solid where the new boundary conditions are applied. The problem is solved by a double-iterative met hod, so in the final solut ion, t here are no t r act ions or pene trations between the two solids or at the boundary of the solid where the nonlinear boundary conditions are Simula ted. The effectiveness of the proposed method is verified by examples.展开更多
In this paper, the estimate on blow-up rate of the following nonlinear parabolic system is considered：{ut=uxx＋u^l 11v^l 12,vt=vxx＋u^l21v^l22,（x,t）∈（0,1）×（0,T）,ux（0,t）=0,vx（0,t）=0,t∈（0,T）,ux（1,t...In this paper, the estimate on blow-up rate of the following nonlinear parabolic system is considered：{ut=uxx＋u^l 11v^l 12,vt=vxx＋u^l21v^l22,（x,t）∈（0,1）×（0,T）,ux（0,t）=0,vx（0,t）=0,t∈（0,T）,ux（1,t）=（u^p11v^p12）（1,t）,vx（1,t）=（u^p21v^p22）（1,t）,t∈（0,T）,u（x,0）=u0（x）,v（x,0）=v0（x）,x∈（0,1）We will prove that there exist two positive constants such that：c≤max x∈[0,1]u（x,t）（T-t）^r（l1-1）≤C,0〈t〈T,c≤max x∈[0,1] v（x,t）（T-t）^1/（t1-1）≤C,0〈t〈T.where l1=l21α/α2＋l22,r=α1/α2〉1,α1≤α2〈0.展开更多
In this present work,we consider an eigenvalue-type problem involving the p-Laplacian operator under nonlinear boundary conditions,in case g≡0,and we prove the existence of an isolated eigenvalue plus a continuous fa...In this present work,we consider an eigenvalue-type problem involving the p-Laplacian operator under nonlinear boundary conditions,in case g≡0,and we prove the existence of an isolated eigenvalue plus a continuous family of eigenvalues.We also study the maximum principle-type for(P_(λ,g)) with g≥0,g■0.展开更多
The standard finite elements of degree p over the rectangular meshes are applied to solve a kind of nonlinear viscoelastic wave equations with nonlinear boundary conditions, and the superclose property of the continuo...The standard finite elements of degree p over the rectangular meshes are applied to solve a kind of nonlinear viscoelastic wave equations with nonlinear boundary conditions, and the superclose property of the continuous Galerkin approximation is derived without using the nonclassical elliptic projection of the exact solution of the model problem. The global superconvergence of one order higher than the traditional error estimate is also obtained through the postprocessing technique.展开更多
In this paper we consider the periodic solutions of nonlinear parabolic systems with nonlinear boundary conditions.By constructing the Poincare operator,we obtain the existence of W<sub>p</sub><sup>2...In this paper we consider the periodic solutions of nonlinear parabolic systems with nonlinear boundary conditions.By constructing the Poincare operator,we obtain the existence of W<sub>p</sub><sup>2β</sup>-periodic weak solutions under some reasonable assumptions.展开更多
This paper deals with the blow-up of positive solutions of the uniformly pa-rabolic equations ut = Lu + a(x)f(u) subject to nonlinear Neumann boundary conditions . Under suitable assumptions on nonlinear functi-ons f,...This paper deals with the blow-up of positive solutions of the uniformly pa-rabolic equations ut = Lu + a(x)f(u) subject to nonlinear Neumann boundary conditions . Under suitable assumptions on nonlinear functi-ons f, g and initial data U0(x), the blow-up of the solutions in a finite time is proved by the maximum principles. Moreover, the bounds of 'blow-up time' and blow-up rate are obtained.展开更多
In this paper, we investigate the positive solutions of strongly coupled nonlinear parabolic systems with nonlinear boundary conditions： {ut-a（u, v）△u=g（u, v）, vt-b（u, v）△v=h（u, v）, δu/δη=d（u, v）, δu...In this paper, we investigate the positive solutions of strongly coupled nonlinear parabolic systems with nonlinear boundary conditions： {ut-a（u, v）△u=g（u, v）, vt-b（u, v）△v=h（u, v）, δu/δη=d（u, v）, δu/δη=f（u, v）.Under appropriate hypotheses on the functions a, b, g, h, d and f, we obtain that the solutions may exist globally or blow up in finite time by utilizing upper and lower solution techniques.展开更多
The authors study the p(x)-Laplacian equations with nonlinear boundary condi- tion. By using the variational method, under appropriate assumptions on the perturbation terms f1(x,u), f2(x,u) and h1(x), h2(x), such that...The authors study the p(x)-Laplacian equations with nonlinear boundary condi- tion. By using the variational method, under appropriate assumptions on the perturbation terms f1(x,u), f2(x,u) and h1(x), h2(x), such that the associated functional satisfies the "mountain pass lemma" and "fountain theorem" respectively, the existence and multiplicity of solutions are obtained. The discussion is based on the theory of variable exponent Lebesgue and Sobolev spaces.展开更多
By employing Mawhin＇s continuation theorem, the existence of solution for a p-Laplacian equation with nonlinear boundary conditions is obtained under simple assumptions.
This paper deals with the blow up properties of solutions to semilinear heat equation u t- Δ u=u p in R N +×(0,T) with the nonlinear boundary condition -ο u ο x 1 = u q for x 1=0,t∈(0,T) ....This paper deals with the blow up properties of solutions to semilinear heat equation u t- Δ u=u p in R N +×(0,T) with the nonlinear boundary condition -ο u ο x 1 = u q for x 1=0,t∈(0,T) .It has been proved that if max( p,q) ≤1,every nonnegative solution is global.When min (p,q) >1 by letting α=1p-1 and β=12(q-1) it follows that if max (α,β)≥N2 ,all nontrivial nonnegative solutions are nonglobal,whereas if max (α,β)<N2 ,there exist both global and nonglobal solutions.Moreover,the exact blow up rates are established.展开更多
The paper deals with the existence of three- solutions for the second- order differential equations with nonlinear boundary value conditions x″=f(t,x,x′) , t∈ [a,b], g1(x(a) ,x′(a) ) =0 , g2 (x(b) ,x′(...The paper deals with the existence of three- solutions for the second- order differential equations with nonlinear boundary value conditions x″=f(t,x,x′) , t∈ [a,b], g1(x(a) ,x′(a) ) =0 , g2 (x(b) ,x′(b) ) =0 , where f :[a,b]× R1× R1→ R1,gi:R1× R1→ R1(i=1 ,2 ) are continuous functions.The methods employed are the coincidence degree theory.As an application,the sufficient conditions under which there are arbitrary odd solutions for the BVP are obtained展开更多
In this paper, the singular perturbation of nonlinear differential equation system with nonlinear boundary conditions is discussed. Under suitable assumptions, with the asymptotic method of Lyusternik-Vishik([1]) and ...In this paper, the singular perturbation of nonlinear differential equation system with nonlinear boundary conditions is discussed. Under suitable assumptions, with the asymptotic method of Lyusternik-Vishik([1]) and fixed point theory, the existence of the solution of the perturbation problem is proved and its uniformly valid asymptotic expansion of higher order is derived.展开更多
This paper presents mechanical quadrature methods （MQMs） for solving nonlinear boundary Helmholtz integral equations. The methods have high accuracy of order O（h3） and low computation complexity. Moreover, the mec...This paper presents mechanical quadrature methods （MQMs） for solving nonlinear boundary Helmholtz integral equations. The methods have high accuracy of order O（h3） and low computation complexity. Moreover, the mechanical quadrature methods are simple without computing any singular integration. A nonlinear system is constructed by discretizing the nonlinear boundary integral equations. The stability and convergence of the system are proved based on an asymptotical compact theory and the Stepleman theorem. Using the h3-Richardson extrapolation algorithms （EAs）, the accuracy to the order of O（h5） is improved. To slove the nonlinear system, the Newton iteration is discussed extensively by using the Ostrowski fixed point theorem. The efficiency of the algorithms is illustrated by numerical examples.展开更多
Making use of upper and lower solutions and analytical method, the author studies theexistence of positive solution for the singular equation x + f(t, z) = 0 satisfying nonlinear boundary conditions: x (0) = 0, h(x (1...Making use of upper and lower solutions and analytical method, the author studies theexistence of positive solution for the singular equation x + f(t, z) = 0 satisfying nonlinear boundary conditions: x (0) = 0, h(x (1), x’ (1)) = 0, g (z (0), x’(0)) = 0, and x (1) = 0,which extends the result of J. V. Baxley.展开更多
基金financially supported by the National Natural Science Foundation of China(Grant No.52374118)the Science and Technology Support Project of Guizhou Province,China(Project Grant No.Qiankehe Support(2022)General 247).
文摘In regions characterized with great mining depths,complex topography,and intense geological activities,solely relying on lateral pressure coefficients or linear boundary conditions for predicting the in situ stress field of rock bodies can induce substantial deviations and limitations.This study focuses on a typical karst area in Southwest Guizhou,China as its research background.It employs a hybrid approach integrating machine learning,numerical simulations,and field experiments to develop an optimization algorithm for nonlinear prediction of the complex three-dimensional(3D)in situ stress fields.Through collecting and fitting analysis of in situ stress measurement data from the karst region,the distributions of in situ stresses with depth were identified with nonlinear boundary conditions.A prediction model for in situ stress was then established based on artificial neural network(ANN)and genetic algorithm(GA)approach,validated in the typical karst landscape mine,Jinfeng Gold Mine.The results demonstrate that the model's predictions align well with actual measurements,showcasing consistency and regularity.Specifically,the error between the predicted and actual values of the maximum horizontal principal stress was the smallest,with an absolute error 0.01-3 MPa and a relative error of 0.04-15.31%.This model accurately and effectively predicts in situ stresses in complex geological areas.
基金This work is supported in part by NNSF of China(10571126)and in part by Program for New Century Excellent Talents in University.
文摘Sufficient conditions are obtained for the oscillation of solutions of the systems of quasilinear hyperbolic differential equation with deviating arguments under nonlinear boundary condition.
基金The project is supported by National Natural Science Foundation of China (10071026)
文摘The purpose of this paper is to investigate the stability and asymptotic behavior of the time-dependent solutions to a linear parabolic equation with nonlinear boundary condition in relation to their corresponding steady state solutions. Then, the above results are extended to a semilinear parabolic equation with nonlinear boundary condition by analyzing the corresponding eigenvalue problem and using the method of upper and lower solutions.
基金Supported by the National Natural Science Foundation of China (10671184)
文摘This paper discusses the semidiscrete finite element method for nonlinear hyperbolic equations with nonlinear boundary condition. The superclose property is derived through interpolation instead of the nonlinear H^1 projection and integral identity technique. Meanwhile, the global superconvergence is obtained based on the interpolated postprocessing techniques.
基金Supported by the National Natural Science Foundation of China(Grant No.11961060).
文摘In this paper,we discuss the existence of positive solutions for the second-order singular difference equation boundary value problem -Δ^(2)u(t-1)=λg(t)f(u).t∈[1,T]_(z),u(0)=0,Δu(T)+c(u(T+1))u(T+1)=0,where λ> 0 is a positive parameter,f:(0,∞)→R is continuous,and is allowed to be singular at 0.The existence of positive solutions is established via introducing a new complete continuous operator.
文摘With prior estimate method, the existence, uniqueness, stability and large time behavior of the solution of second initial-boundary value problem for a fast diffusion equation with nonlinear boundary conditions are investigated. The main results are : 1) there exists only one global weak solution which continuously depends on initial value; 2) when t < T-0, the solution is infinitely continuously differentiable and is a classical solution; 3) the solution converges to zero uniformly as t is large enough.
文摘This work presents a numerical methodology for modeling the Winkler supports and nonlinear conditions by proposing new boundary conditions. For the boundary conditions of Winkler support model, the surface tractions and the displacements normal to the surface of the solid are unknown, but their relationship is known by means of the ballast coefficient, whereas for nonlinear boundary conditions, the displacements normal to the boundary of the solid are zero in the positive direction but are allowed in the negative direction. In those zones, detachments of nodes might appear, leading to a nonlinearity, because the number of nodes that remain fixed or of the detached ones (under tensile tractions) is unknown. The proposed methodology is applied to the 3D elastic receding contact problem using the boundary element method. The surface t r actions and the displacements of the common int erface bet ween the two solids in contac t under the influence of different supports are calculated as well as the boundary zone of the solid where the new boundary conditions are applied. The problem is solved by a double-iterative met hod, so in the final solut ion, t here are no t r act ions or pene trations between the two solids or at the boundary of the solid where the nonlinear boundary conditions are Simula ted. The effectiveness of the proposed method is verified by examples.
文摘In this paper, the estimate on blow-up rate of the following nonlinear parabolic system is considered：{ut=uxx＋u^l 11v^l 12,vt=vxx＋u^l21v^l22,（x,t）∈（0,1）×（0,T）,ux（0,t）=0,vx（0,t）=0,t∈（0,T）,ux（1,t）=（u^p11v^p12）（1,t）,vx（1,t）=（u^p21v^p22）（1,t）,t∈（0,T）,u（x,0）=u0（x）,v（x,0）=v0（x）,x∈（0,1）We will prove that there exist two positive constants such that：c≤max x∈[0,1]u（x,t）（T-t）^r（l1-1）≤C,0〈t〈T,c≤max x∈[0,1] v（x,t）（T-t）^1/（t1-1）≤C,0〈t〈T.where l1=l21α/α2＋l22,r=α1/α2〉1,α1≤α2〈0.
文摘In this present work,we consider an eigenvalue-type problem involving the p-Laplacian operator under nonlinear boundary conditions,in case g≡0,and we prove the existence of an isolated eigenvalue plus a continuous family of eigenvalues.We also study the maximum principle-type for(P_(λ,g)) with g≥0,g■0.
基金supported by the National Natural Science Foundation of China under Grant Nos.10671184 and 10971203
文摘The standard finite elements of degree p over the rectangular meshes are applied to solve a kind of nonlinear viscoelastic wave equations with nonlinear boundary conditions, and the superclose property of the continuous Galerkin approximation is derived without using the nonclassical elliptic projection of the exact solution of the model problem. The global superconvergence of one order higher than the traditional error estimate is also obtained through the postprocessing technique.
文摘In this paper we consider the periodic solutions of nonlinear parabolic systems with nonlinear boundary conditions.By constructing the Poincare operator,we obtain the existence of W<sub>p</sub><sup>2β</sup>-periodic weak solutions under some reasonable assumptions.
基金This work is partially supported by Open Project of the Defense Key Laboratory of Science and Technology(OOJS76.4.2JW0810), Talent Teacher Program of Chinese Ministry of Education and the National Science Foundation of China(60174007).
文摘This paper deals with the blow-up of positive solutions of the uniformly pa-rabolic equations ut = Lu + a(x)f(u) subject to nonlinear Neumann boundary conditions . Under suitable assumptions on nonlinear functi-ons f, g and initial data U0(x), the blow-up of the solutions in a finite time is proved by the maximum principles. Moreover, the bounds of 'blow-up time' and blow-up rate are obtained.
文摘In this paper, we investigate the positive solutions of strongly coupled nonlinear parabolic systems with nonlinear boundary conditions： {ut-a（u, v）△u=g（u, v）, vt-b（u, v）△v=h（u, v）, δu/δη=d（u, v）, δu/δη=f（u, v）.Under appropriate hypotheses on the functions a, b, g, h, d and f, we obtain that the solutions may exist globally or blow up in finite time by utilizing upper and lower solution techniques.
基金supported by the National Natural Science Foundation of China (No. 10771141)the ZhejiangProvincial Natural Science Foundation of China (No. Y7080008).
文摘The authors study the p(x)-Laplacian equations with nonlinear boundary condi- tion. By using the variational method, under appropriate assumptions on the perturbation terms f1(x,u), f2(x,u) and h1(x), h2(x), such that the associated functional satisfies the "mountain pass lemma" and "fountain theorem" respectively, the existence and multiplicity of solutions are obtained. The discussion is based on the theory of variable exponent Lebesgue and Sobolev spaces.
基金Supported by National Natural Sciences Foundation of China(10371006).
文摘By employing Mawhin＇s continuation theorem, the existence of solution for a p-Laplacian equation with nonlinear boundary conditions is obtained under simple assumptions.
文摘This paper deals with the blow up properties of solutions to semilinear heat equation u t- Δ u=u p in R N +×(0,T) with the nonlinear boundary condition -ο u ο x 1 = u q for x 1=0,t∈(0,T) .It has been proved that if max( p,q) ≤1,every nonnegative solution is global.When min (p,q) >1 by letting α=1p-1 and β=12(q-1) it follows that if max (α,β)≥N2 ,all nontrivial nonnegative solutions are nonglobal,whereas if max (α,β)<N2 ,there exist both global and nonglobal solutions.Moreover,the exact blow up rates are established.
文摘The paper deals with the existence of three- solutions for the second- order differential equations with nonlinear boundary value conditions x″=f(t,x,x′) , t∈ [a,b], g1(x(a) ,x′(a) ) =0 , g2 (x(b) ,x′(b) ) =0 , where f :[a,b]× R1× R1→ R1,gi:R1× R1→ R1(i=1 ,2 ) are continuous functions.The methods employed are the coincidence degree theory.As an application,the sufficient conditions under which there are arbitrary odd solutions for the BVP are obtained
文摘In this paper, the singular perturbation of nonlinear differential equation system with nonlinear boundary conditions is discussed. Under suitable assumptions, with the asymptotic method of Lyusternik-Vishik([1]) and fixed point theory, the existence of the solution of the perturbation problem is proved and its uniformly valid asymptotic expansion of higher order is derived.
基金supported by the National Natural Science Foundation of China(No.10871034)the Natural Science Foundation Project of Chongqing(No.CSTC20-10BB8270)+1 种基金the Air Force Office of Scientific Research(No.FA9550-08-1-0136)the National Science Foundation(No.OCE-0620464)
文摘This paper presents mechanical quadrature methods （MQMs） for solving nonlinear boundary Helmholtz integral equations. The methods have high accuracy of order O（h3） and low computation complexity. Moreover, the mechanical quadrature methods are simple without computing any singular integration. A nonlinear system is constructed by discretizing the nonlinear boundary integral equations. The stability and convergence of the system are proved based on an asymptotical compact theory and the Stepleman theorem. Using the h3-Richardson extrapolation algorithms （EAs）, the accuracy to the order of O（h5） is improved. To slove the nonlinear system, the Newton iteration is discussed extensively by using the Ostrowski fixed point theorem. The efficiency of the algorithms is illustrated by numerical examples.
文摘Making use of upper and lower solutions and analytical method, the author studies theexistence of positive solution for the singular equation x + f(t, z) = 0 satisfying nonlinear boundary conditions: x (0) = 0, h(x (1), x’ (1)) = 0, g (z (0), x’(0)) = 0, and x (1) = 0,which extends the result of J. V. Baxley.
