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
The singularly perturbed nonlinear boundary value problems are considered.Using the stretched variable and the method of boundary layer correction,the formal asymptotic expansion of solution is obtained.And then the u...The singularly perturbed nonlinear boundary value problems are considered.Using the stretched variable and the method of boundary layer correction,the formal asymptotic expansion of solution is obtained.And then the uniform validity of solution is proved by using the differential inequalities.展开更多
In this paper, the existence and uniqueness of solutions for boundary valueproblem x′′′=f(t, x, x′, x″), x(0)=A, x′(0)=B, g(x′(1), x″(1))=0 are studied byusing Volterra type operator and upper and lower soluti...In this paper, the existence and uniqueness of solutions for boundary valueproblem x′′′=f(t, x, x′, x″), x(0)=A, x′(0)=B, g(x′(1), x″(1))=0 are studied byusing Volterra type operator and upper and lower solutions. Our results improve someknown works.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.
This paper is devoted to study the following the singularly perturbed fourth-order ordinary differential equation ∈y（4） =f（t,y＇,y＇＇,y＇＇＇）,0t1,0ε1 with the nonlinear boundary conditions y（0）=y＇（1）=0,p...This paper is devoted to study the following the singularly perturbed fourth-order ordinary differential equation ∈y（4） =f（t,y＇,y＇＇,y＇＇＇）,0t1,0ε1 with the nonlinear boundary conditions y（0）=y＇（1）=0,p（y＇＇（0）,y＇＇＇（0））=0,q（y＇＇（1）,y＇＇＇（1））=0 where f：[0,1]×R3→R is continuous,p,q：R2→R are continuous.Under certain conditions,by introducing an appropriate stretching transformation and constructing boundary layer corrective terms,an asymptotic expansion for the solution of the problem is obtained.And then the uniformly validity of solution is proved by using the differential inequalities.展开更多
In this paper, the authors discuss the global existence and blow-up of the solution to an evolution p-Laplace system with nonlinear sources and nonlinear boundary condition. The authors first establish the local exist...In this paper, the authors discuss the global existence and blow-up of the solution to an evolution p-Laplace system with nonlinear sources and nonlinear boundary condition. The authors first establish the local existence of solutions, then give a necessary and sufficient condition on the global existence of the positive solution.展开更多
A sixth-order accurate wavelet integral collocation method is proposed for solving high-order nonlinear boundary value problems in three dimensions.In order to realize the establishment of this method,an approximate e...A sixth-order accurate wavelet integral collocation method is proposed for solving high-order nonlinear boundary value problems in three dimensions.In order to realize the establishment of this method,an approximate expression of multiple integrals of a continuous function defined in a three-dimensional bounded domain is proposed by combining wavelet expansion and Lagrange boundary extension.Through applying such an integral technique,during the solution of nonlinear partial differential equations,the unknown function and its lower-order partial derivatives can be approximately expressed by its highest-order partial derivative values at nodes.A set of nonlinear algebraic equations with respect to these nodal values of the highest-order partial derivative is obtained using a collocation method.The validation and convergence of the proposed method are examined through several benchmark problems,including the eighth-order two-dimensional and fourth-order three-dimensional boundary value problems and the large deflection bending of von Karman plates.Results demonstrate that the present method has higher accuracy and convergence rate than most existing numerical methods.Most importantly,the convergence rate of the proposed method seems to be independent of the order of the differential equations,because it is always sixth order for second-,fourth-,sixth-,and even eighth-order problems.展开更多
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
We discuss the quenching phenomena for a system of heat equations coupled with nonlinear boundary flux. We determine a critical value for the exponents in the boundary flux, such that only in the super critical case t...We discuss the quenching phenomena for a system of heat equations coupled with nonlinear boundary flux. We determine a critical value for the exponents in the boundary flux, such that only in the super critical case the simultaneous quenching can happen for any solution.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, using the theory of differential inequalities, we study the nonlinear boundary value problem for a class of integro-differential system. Under appropriate assumptions, the existence of solution is prove...In this paper, using the theory of differential inequalities, we study the nonlinear boundary value problem for a class of integro-differential system. Under appropriate assumptions, the existence of solution is proved and the uniformly valid asymptotic expansions for arbitrary n-th order approximation and the estimation of remainder term are obtained simply and conveniently.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
1. Introduction We consider the singular nonlinear boundary value problem where l=v+3/v-1,l+1 is the critical exponent of the embedding of weighted Sobolev space Wt21,2（O, +∞） into Lt2q（O, ∞）, v>2. When v=N-1...1. Introduction We consider the singular nonlinear boundary value problem where l=v+3/v-1,l+1 is the critical exponent of the embedding of weighted Sobolev space Wt21,2（O, +∞） into Lt2q（O, ∞）, v>2. When v=N-1 we can get the radial solutions of problem where 2*=2N/N-2 is the critical exponent of the Sobolev embedding H1（Rn）→LQ（RN）. Kurtz has discussed the existence of κ-node solution of （1.1）, （1.2） for each κ∈N U{0} when the growth rate of |u|l-1u+f（u） is lower then |u|v+3/v-1 i.e.展开更多
This article deals with the degenerate parabolic system with nonlinear boundary flux. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence curve and the critical Fuji...This article deals with the degenerate parabolic system with nonlinear boundary flux. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence curve and the critical Fujita curve for the problem. Especially for the blow-up case, it is rather technical. It comes from the construction of the so-called Zel＇dovich-Kompaneetz-Barenblatt profile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
In this paper nonlinear boundary value problems for discontinuous delayed differen-tial equations are considered.Some existence and boundedness results of solutions are obtained via the method of upper and lower solut...In this paper nonlinear boundary value problems for discontinuous delayed differen-tial equations are considered.Some existence and boundedness results of solutions are obtained via the method of upper and lower solutions,which may be discontinuous.Our analysis can be applied to those phenomena from physics and control theory which have been successfully described by delayed differential equations with discontinuous functions,such as electric,pneu-matic,and hydraulic networks.It is also an important step to precede the design of control signals when finite-time or practical stability control are concerned.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
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
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
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.
文摘The singularly perturbed nonlinear boundary value problems are considered.Using the stretched variable and the method of boundary layer correction,the formal asymptotic expansion of solution is obtained.And then the uniform validity of solution is proved by using the differential inequalities.
文摘In this paper, the existence and uniqueness of solutions for boundary valueproblem x′′′=f(t, x, x′, x″), x(0)=A, x′(0)=B, g(x′(1), x″(1))=0 are studied byusing Volterra type operator and upper and lower solutions. Our results improve someknown works.
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.
文摘This paper is devoted to study the following the singularly perturbed fourth-order ordinary differential equation ∈y（4） =f（t,y＇,y＇＇,y＇＇＇）,0t1,0ε1 with the nonlinear boundary conditions y（0）=y＇（1）=0,p（y＇＇（0）,y＇＇＇（0））=0,q（y＇＇（1）,y＇＇＇（1））=0 where f：[0,1]×R3→R is continuous,p,q：R2→R are continuous.Under certain conditions,by introducing an appropriate stretching transformation and constructing boundary layer corrective terms,an asymptotic expansion for the solution of the problem is obtained.And then the uniformly validity of solution is proved by using the differential inequalities.
基金supported by a grant from the National High Technology Researchand and Development Program of China (863 Program) (2009AA044501)by NSFC (10776035+2 种基金10771085)by Key Lab of Symbolic Computation and Knowledge Engineering of Ministry of Educationby the 985 program of Jilin University
文摘In this paper, the authors discuss the global existence and blow-up of the solution to an evolution p-Laplace system with nonlinear sources and nonlinear boundary condition. The authors first establish the local existence of solutions, then give a necessary and sufficient condition on the global existence of the positive solution.
基金supported by the National Natural Science Foundation of China(Grant Nos.11925204 and 12172154)the 111 Project(Grant No.B14044)the National Key Project of China(Grant No.GJXM92579).
文摘A sixth-order accurate wavelet integral collocation method is proposed for solving high-order nonlinear boundary value problems in three dimensions.In order to realize the establishment of this method,an approximate expression of multiple integrals of a continuous function defined in a three-dimensional bounded domain is proposed by combining wavelet expansion and Lagrange boundary extension.Through applying such an integral technique,during the solution of nonlinear partial differential equations,the unknown function and its lower-order partial derivatives can be approximately expressed by its highest-order partial derivative values at nodes.A set of nonlinear algebraic equations with respect to these nodal values of the highest-order partial derivative is obtained using a collocation method.The validation and convergence of the proposed method are examined through several benchmark problems,including the eighth-order two-dimensional and fourth-order three-dimensional boundary value problems and the large deflection bending of von Karman plates.Results demonstrate that the present method has higher accuracy and convergence rate than most existing numerical methods.Most importantly,the convergence rate of the proposed method seems to be independent of the order of the differential equations,because it is always sixth order for second-,fourth-,sixth-,and even eighth-order problems.
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partially supported by the NSF of Chinapartially supported by a Specific Foundation for Ph.D. Specialities of Educational Department of China.
文摘We discuss the quenching phenomena for a system of heat equations coupled with nonlinear boundary flux. We determine a critical value for the exponents in the boundary flux, such that only in the super critical case the simultaneous quenching can happen for any solution.
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 (No. 10471039)the Natural Science Foundations of Zhejiang (No Y604127)Supported in part by E-Institutes of Shanghai Municipal Education Commission (No.E03004).
文摘In this paper, using the theory of differential inequalities, we study the nonlinear boundary value problem for a class of integro-differential system. Under appropriate assumptions, the existence of solution is proved and the uniformly valid asymptotic expansions for arbitrary n-th order approximation and the estimation of remainder term are obtained simply and conveniently.
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1. Introduction We consider the singular nonlinear boundary value problem where l=v+3/v-1,l+1 is the critical exponent of the embedding of weighted Sobolev space Wt21,2（O, +∞） into Lt2q（O, ∞）, v>2. When v=N-1 we can get the radial solutions of problem where 2*=2N/N-2 is the critical exponent of the Sobolev embedding H1（Rn）→LQ（RN）. Kurtz has discussed the existence of κ-node solution of （1.1）, （1.2） for each κ∈N U{0} when the growth rate of |u|l-1u+f（u） is lower then |u|v+3/v-1 i.e.
基金supported in part by NSF of China (11071266)in part by NSF project of CQ CSTC (2010BB9218)partially supported by the Educational Science Foundation of Chongqing(KJ101303) China
文摘This article deals with the degenerate parabolic system with nonlinear boundary flux. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence curve and the critical Fujita curve for the problem. Especially for the blow-up case, it is rather technical. It comes from the construction of the so-called Zel＇dovich-Kompaneetz-Barenblatt profile
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.
基金Supported by the National Natural Science Foundation of China(Grants No.70703016 and No.10001024)Research Grant of the Business School of Nanjing University
文摘In this paper nonlinear boundary value problems for discontinuous delayed differen-tial equations are considered.Some existence and boundedness results of solutions are obtained via the method of upper and lower solutions,which may be discontinuous.Our analysis can be applied to those phenomena from physics and control theory which have been successfully described by delayed differential equations with discontinuous functions,such as electric,pneu-matic,and hydraulic networks.It is also an important step to precede the design of control signals when finite-time or practical stability control are concerned.
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
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.
