We transform the singular integral equations with solutions simultaneously having singularities of higher order at infinite point and at several finite points on the real axis into ones along a closed contour with sol...We transform the singular integral equations with solutions simultaneously having singularities of higher order at infinite point and at several finite points on the real axis into ones along a closed contour with solutions having singularities of higher order, and for the former obtain the extended Neother theorem of complete equation as well as the solutions and the solvable conditions of characteristic equation from the latter. The conclusions drawn by this article contain special cases discussed before.展开更多
Under the appropriate hypotheses subject to the unknown function and the free term, by means of our Lemma, we prove the rationality of order at x = ∞ on two sides for the characteristic singular integral equations wi...Under the appropriate hypotheses subject to the unknown function and the free term, by means of our Lemma, we prove the rationality of order at x = ∞ on two sides for the characteristic singular integral equations with solutions having singularities of higher order on the real axis X. We transform the equations into solving equivalent Riemann boundary value problems with solutions having singularities of higher order and with additional conditions on X. The solutions and the solvable conditions for the former are obtained from the latter.展开更多
In this paper, the difficulties on calculation in solving singular integral equations are overcome when the restriction of curve of integration to be a closed contour is cancelled. When the curve is an open arc and th...In this paper, the difficulties on calculation in solving singular integral equations are overcome when the restriction of curve of integration to be a closed contour is cancelled. When the curve is an open arc and the solutions for singular integral equations possess singularities of higher order, the solution and the solvable condition for characteristic equations as well as the generalized Noether theorem for complete equations are given.展开更多
In this paper, we study on the initial-boundary value problem for nonlinear wave equations of higher-order Kirchhoff type with Strong Dissipation: . At first, we prove the existence and uniqueness of the local solutio...In this paper, we study on the initial-boundary value problem for nonlinear wave equations of higher-order Kirchhoff type with Strong Dissipation: . At first, we prove the existence and uniqueness of the local solution by the Banach contraction mapping principle. Then, by “Concavity” method we establish three blow-up results for certain solutions in the case 1): , in the case 2): and in the case 3): . At last, we consider that the estimation of the upper bounds of the blow-up time is given for deferent initial energy.展开更多
Under some conditions, one seows that the generalized solutions of the first boundary value problem for the equation [GRAPHICS] have the property of finite speed of propagation.
This paper explores the rich structure of peakon and pseudo-peakon solutions for a class of higher-order b-family equations,referred to as the J-th b-family(J-bF)equations.We propose several conjectures concerning the...This paper explores the rich structure of peakon and pseudo-peakon solutions for a class of higher-order b-family equations,referred to as the J-th b-family(J-bF)equations.We propose several conjectures concerning the weak solutions of these equations,including a b-independent pseudo-peakon solution,a b-independent peakon solution,and a b-dependent peakon solution.These conjectures are analytically verified for J≤14 and/or J≤9 using the symbolic computation system MAPLE,which includes a built-in definition of the higher-order derivatives of the sign function.The b-independent pseudo-peakon solution is a 3rd-order pseudo-peakon for general arbitrary constants,with higher-order pseudo-peakons derived under specific parameter constraints.Additionally,we identify both b-independent and b-dependent peakon solutions,highlighting their distinct properties and the nuanced relationship between the parameters b and J.The existence of these solutions underscores the rich dynamical structure of the J-bF equations and generalizes previous results for lower-order equations.Future research directions include higher-order generalizations,rigorous proofs of the conjectures,interactions between different types of peakons and pseudo-peakons,stability analysis,and potential physical applications.These advancements significantly contribute to the understanding of peakon systems and their broader implications in mathematics and physics.展开更多
The basic sets of solutions in classH(orH*)for the characteristic equation and its adjoint equation with Hilbert kernel are given respectively.Thus the expressions of solutions and its solvable conditions are simplifi...The basic sets of solutions in classH(orH*)for the characteristic equation and its adjoint equation with Hilbert kernel are given respectively.Thus the expressions of solutions and its solvable conditions are simplified.On this basis the solutions and the solvable conditions in classH_(1)as well as the generalized Noether theorem for the complete equation are obtained.展开更多
In this paper, we investigate the existence of positive solutions of a class higher order boundary value problems on time scales. The class of boundary value problems educes a four-point （or three-point or two-point...In this paper, we investigate the existence of positive solutions of a class higher order boundary value problems on time scales. The class of boundary value problems educes a four-point （or three-point or two-point） boundary value problems, for which some similar results are established. Our approach relies on the Krasnosel＇skii fixed point theorem. The result of this paper is new and extends previously known results.展开更多
Using the Nevanlinna theory of the value distribution of meromorphic functions, we investigate the existence problem of admissible algebroid solutions of generalized higher order algebraic differential equations.
We give general solutions(the explicit solutions) of a class of multi-term impulsive fractional differential equations involving the Riemann-Liouville fractional derivatives. This paper contributes within the domain o...We give general solutions(the explicit solutions) of a class of multi-term impulsive fractional differential equations involving the Riemann-Liouville fractional derivatives. This paper contributes within the domain of impulsive fractional differential equations. The author strongly believes that the article will highly be appreciated by the researchers working in the field of fractional calculus and on fractional differential models.展开更多
We have derived the first Noether theorem and Noether identities in canonical formalism for field theory with higher-order singular Lagrangian,which is a powerful tool toanalyse Dirac constraint for such system. A gau...We have derived the first Noether theorem and Noether identities in canonical formalism for field theory with higher-order singular Lagrangian,which is a powerful tool toanalyse Dirac constraint for such system. A gauge-variant system in canonical variables formalism must has Dirac constraint.For a system with first class constraint (FCC), we have developed an algorithm for construction of the gauge generator of such system. An application to the Podolsky generalized electromagnetic field was given.展开更多
By using the upper and lower solutions method and fixed point theory,we investigate a class of fourth-order singular differential equations with the Sturm-Liouville Boundary conditions.Some sufficient conditions are o...By using the upper and lower solutions method and fixed point theory,we investigate a class of fourth-order singular differential equations with the Sturm-Liouville Boundary conditions.Some sufficient conditions are obtained for the existence of C2[0,1] positive solutions and C3[0,1] positive solutions.展开更多
A higher order asymptotic solution of near-tip field is studied for plane-atrain Mode-Ⅰ quasi-static steady crack growth in the incompressible (v=1/2) elastic perfectly-plastic media.The results show that the near-ti...A higher order asymptotic solution of near-tip field is studied for plane-atrain Mode-Ⅰ quasi-static steady crack growth in the incompressible (v=1/2) elastic perfectly-plastic media.The results show that the near-tip stress and strain are fully continuous,and the strain possesses In (A/r) singularity at the crack tip.The expressions of the stress,strain and velocity in each region are also given.展开更多
Let 0<α,β<n and f,g∈ C([0,∞)×[0,∞))be two nonnegative functions.We study nonnegative classical solutions of the system{(-△)^(α/2)u=f(u,v)in R^(n),(-△)^(β/2)v=g(u,v)in R^(n),and the corresponding eq...Let 0<α,β<n and f,g∈ C([0,∞)×[0,∞))be two nonnegative functions.We study nonnegative classical solutions of the system{(-△)^(α/2)u=f(u,v)in R^(n),(-△)^(β/2)v=g(u,v)in R^(n),and the corresponding equivalent integral system.We classify all such solutions when f(s,t)is nondecreasing in s and increasing in t,g(s,t)is increasing in s and nondecreasing in i,and f(μ^(n-α)s,μ^(n-β)t)/μ^(n-α),g(μ^(n-α)s,μ^(n-β)t)/μ^(n-β)are nonincreasing in μ>0 for all s,t≥0.The main technique we use is the method of moving spheres in integral forms.Since our assumptions are more general than those in the previous literature,some new ideas are introduced to overcome this difficulty.展开更多
It is important to study the propagation and interaction of progressing waves of nonlinear equations in the class of piecewise smooth function. However, there has not been many works on that in multidimensional case. ...It is important to study the propagation and interaction of progressing waves of nonlinear equations in the class of piecewise smooth function. However, there has not been many works on that in multidimensional case. In 1985, J, Rauch & M. Reed have provad the existence and uniqueness of piecewise smooth solution for展开更多
Under the constrained condition induced by the eigenfunction expresson of the potential (u, v)T = (-[A2q, q], [A2p, p])T = f (q, p), the spatial part of the Lax pair of the Kaup-Newell equation is non linearized to be...Under the constrained condition induced by the eigenfunction expresson of the potential (u, v)T = (-[A2q, q], [A2p, p])T = f (q, p), the spatial part of the Lax pair of the Kaup-Newell equation is non linearized to be a completely integrable system (R2N, Adp AND dq, H = H-1) with the Hamiltonian H-1 = -[A3q, p]-1/2[A2p, p][A2q, q]. while the nonlinearization of the time part leads to its N-involutive system {H(m)}. The involutive solution of the compatible fsystem (H-1), (H(m)) is mapped by into the solution of the higher order Kaup-Newell equation.展开更多
On the basis of the Reddy's higher-order theory of composites, this paper introduces a displacement function Phi into it and transforms its three differential equations for symmetric cross-ply composites into only...On the basis of the Reddy's higher-order theory of composites, this paper introduces a displacement function Phi into it and transforms its three differential equations for symmetric cross-ply composites into only one eight-order differential equation generated by the displacement-function. When a proper Phi is chosen, both solutions are obtained, namely, the Navier-type solution of simply supported rectangular laminated plates and the Levy-type solution with the boundary condition where two opposite edges are simply supported and remains are arbitrary. The numerical examples show that the present results coincide well with the existing results in the references, thus validating that the present solving method is reliable. The higher-order theory of Reddy is simpler in calculation but has higher precision than the first-order shear deformation theory because the former has fewer unknows than the latter and requires no shear coefficients.展开更多
This paper investigates a class of 2nth-order singular superlinear problems with Strum-Liouville boundary conditions. We obtain a necessary and sufficient condition for the existence of C 2 n- 2 [0, 1] positive soluti...This paper investigates a class of 2nth-order singular superlinear problems with Strum-Liouville boundary conditions. We obtain a necessary and sufficient condition for the existence of C 2 n- 2 [0, 1] positive solutions, and a sufficient condition, a necessary condition for the existence of C 2 n-1 [0, 1] positive solutions. Relations between the positive solutions and the Green’s functions are depicted. The results are used to judge nonexistence or existence of positive solutions for given boundary value problems.展开更多
This paper uses the Taylor expansion to seek an approximate Korteweg- de Vries equation （KdV） solution to a higher-order traffic flow model with sufficiently large diffusion. It demonstrates the validity of the appr...This paper uses the Taylor expansion to seek an approximate Korteweg- de Vries equation （KdV） solution to a higher-order traffic flow model with sufficiently large diffusion. It demonstrates the validity of the approximate KdV solution considering all the related parameters to ensure the physical boundedness and the stability of the solution. Moreover, when the viscosity coefficient depends on the density and velocity of the flow, the wave speed of the KdV solution is naturally related to either the first or the second characteristic field. The finite element method is extended to solve the model and examine the stability and accuracy of the approximate KdV solution.展开更多
This paper deals with the singular nonlinear third-order periodic boundary value problem u'' + p(3)u = f (t, u), 0 less than or equal to t less than or equal to 2pi, with u((i)) (0) = u((i)) (2pi), i = 0, 1, 2...This paper deals with the singular nonlinear third-order periodic boundary value problem u'' + p(3)u = f (t, u), 0 less than or equal to t less than or equal to 2pi, with u((i)) (0) = u((i)) (2pi), i = 0, 1, 2, where p is an element of (Graphics) and f is singular at t = 0, t = 1 and u = 0. Under suitable weaker conditions than those of [1], it is proved by constructing a special cone in C[0, 2pi] and employing the fixed point index theory that the problem has at least one or at least two positive solutions.展开更多
基金Supported by the NNSF of China (10471107)RFDP of Higher Education of China (20060486001)
文摘We transform the singular integral equations with solutions simultaneously having singularities of higher order at infinite point and at several finite points on the real axis into ones along a closed contour with solutions having singularities of higher order, and for the former obtain the extended Neother theorem of complete equation as well as the solutions and the solvable conditions of characteristic equation from the latter. The conclusions drawn by this article contain special cases discussed before.
基金Supported by the National Natural Science Foundation of China (10471107)the Specialized Research Fund for the Doctoral Program of Higher Education of China (20060486001)
文摘Under the appropriate hypotheses subject to the unknown function and the free term, by means of our Lemma, we prove the rationality of order at x = ∞ on two sides for the characteristic singular integral equations with solutions having singularities of higher order on the real axis X. We transform the equations into solving equivalent Riemann boundary value problems with solutions having singularities of higher order and with additional conditions on X. The solutions and the solvable conditions for the former are obtained from the latter.
基金Foundation item is supported by the NNSF of China(19971064)
文摘In this paper, the difficulties on calculation in solving singular integral equations are overcome when the restriction of curve of integration to be a closed contour is cancelled. When the curve is an open arc and the solutions for singular integral equations possess singularities of higher order, the solution and the solvable condition for characteristic equations as well as the generalized Noether theorem for complete equations are given.
文摘In this paper, we study on the initial-boundary value problem for nonlinear wave equations of higher-order Kirchhoff type with Strong Dissipation: . At first, we prove the existence and uniqueness of the local solution by the Banach contraction mapping principle. Then, by “Concavity” method we establish three blow-up results for certain solutions in the case 1): , in the case 2): and in the case 3): . At last, we consider that the estimation of the upper bounds of the blow-up time is given for deferent initial energy.
文摘Under some conditions, one seows that the generalized solutions of the first boundary value problem for the equation [GRAPHICS] have the property of finite speed of propagation.
基金supported by the National Natural Science Foundations of China(Grant Nos.12235007,12271324,and 11975131)。
文摘This paper explores the rich structure of peakon and pseudo-peakon solutions for a class of higher-order b-family equations,referred to as the J-th b-family(J-bF)equations.We propose several conjectures concerning the weak solutions of these equations,including a b-independent pseudo-peakon solution,a b-independent peakon solution,and a b-dependent peakon solution.These conjectures are analytically verified for J≤14 and/or J≤9 using the symbolic computation system MAPLE,which includes a built-in definition of the higher-order derivatives of the sign function.The b-independent pseudo-peakon solution is a 3rd-order pseudo-peakon for general arbitrary constants,with higher-order pseudo-peakons derived under specific parameter constraints.Additionally,we identify both b-independent and b-dependent peakon solutions,highlighting their distinct properties and the nuanced relationship between the parameters b and J.The existence of these solutions underscores the rich dynamical structure of the J-bF equations and generalizes previous results for lower-order equations.Future research directions include higher-order generalizations,rigorous proofs of the conjectures,interactions between different types of peakons and pseudo-peakons,stability analysis,and potential physical applications.These advancements significantly contribute to the understanding of peakon systems and their broader implications in mathematics and physics.
基金Supported by the National Natural Science Foundation of China(19971064)Ziqiang Invention Foundation of Wuhan University(201990336)
文摘The basic sets of solutions in classH(orH*)for the characteristic equation and its adjoint equation with Hilbert kernel are given respectively.Thus the expressions of solutions and its solvable conditions are simplified.On this basis the solutions and the solvable conditions in classH_(1)as well as the generalized Noether theorem for the complete equation are obtained.
基金The NSF (11201109) of Chinathe NSF (10040606Q50) of Anhui Province+1 种基金Excellent Talents Foundation (2012SQRL165) of University of Anhui Provincethe NSF (2012kj09) of Heifei Normal University
文摘In this paper, we investigate the existence of positive solutions of a class higher order boundary value problems on time scales. The class of boundary value problems educes a four-point （or three-point or two-point） boundary value problems, for which some similar results are established. Our approach relies on the Krasnosel＇skii fixed point theorem. The result of this paper is new and extends previously known results.
文摘Using the Nevanlinna theory of the value distribution of meromorphic functions, we investigate the existence problem of admissible algebroid solutions of generalized higher order algebraic differential equations.
基金Supported by the Natural Science Foundation of Guangdong Province(Grant No.S2011010001900)the Natural Science Research Project for Colleges and Universities of Guangdong Province(Grant No.2014KTSCX126)+1 种基金the Foundation for High-Level Talents in Guangdong Higher Education(Grant No.201707010425)the Foundations of Guangzhou Science and Technology(Grant No.201804010350).
文摘We give general solutions(the explicit solutions) of a class of multi-term impulsive fractional differential equations involving the Riemann-Liouville fractional derivatives. This paper contributes within the domain of impulsive fractional differential equations. The author strongly believes that the article will highly be appreciated by the researchers working in the field of fractional calculus and on fractional differential models.
文摘We have derived the first Noether theorem and Noether identities in canonical formalism for field theory with higher-order singular Lagrangian,which is a powerful tool toanalyse Dirac constraint for such system. A gauge-variant system in canonical variables formalism must has Dirac constraint.For a system with first class constraint (FCC), we have developed an algorithm for construction of the gauge generator of such system. An application to the Podolsky generalized electromagnetic field was given.
基金Research supported by the National Natural Science Foundation of China(10471075)the Natural Science Foun-dation of Shandong Province of China(Y2006A04)
文摘By using the upper and lower solutions method and fixed point theory,we investigate a class of fourth-order singular differential equations with the Sturm-Liouville Boundary conditions.Some sufficient conditions are obtained for the existence of C2[0,1] positive solutions and C3[0,1] positive solutions.
基金The project supported by National Natural Science Foundation of China
文摘A higher order asymptotic solution of near-tip field is studied for plane-atrain Mode-Ⅰ quasi-static steady crack growth in the incompressible (v=1/2) elastic perfectly-plastic media.The results show that the near-tip stress and strain are fully continuous,and the strain possesses In (A/r) singularity at the crack tip.The expressions of the stress,strain and velocity in each region are also given.
基金This research is funded by Vietnam National Foundation for Science and Technology Development(NAFOSTED)under grant number 101.02-2020.22.
文摘Let 0<α,β<n and f,g∈ C([0,∞)×[0,∞))be two nonnegative functions.We study nonnegative classical solutions of the system{(-△)^(α/2)u=f(u,v)in R^(n),(-△)^(β/2)v=g(u,v)in R^(n),and the corresponding equivalent integral system.We classify all such solutions when f(s,t)is nondecreasing in s and increasing in t,g(s,t)is increasing in s and nondecreasing in i,and f(μ^(n-α)s,μ^(n-β)t)/μ^(n-α),g(μ^(n-α)s,μ^(n-β)t)/μ^(n-β)are nonincreasing in μ>0 for all s,t≥0.The main technique we use is the method of moving spheres in integral forms.Since our assumptions are more general than those in the previous literature,some new ideas are introduced to overcome this difficulty.
基金This paper is supported by the National Foundations.
文摘It is important to study the propagation and interaction of progressing waves of nonlinear equations in the class of piecewise smooth function. However, there has not been many works on that in multidimensional case. In 1985, J, Rauch & M. Reed have provad the existence and uniqueness of piecewise smooth solution for
文摘Under the constrained condition induced by the eigenfunction expresson of the potential (u, v)T = (-[A2q, q], [A2p, p])T = f (q, p), the spatial part of the Lax pair of the Kaup-Newell equation is non linearized to be a completely integrable system (R2N, Adp AND dq, H = H-1) with the Hamiltonian H-1 = -[A3q, p]-1/2[A2p, p][A2q, q]. while the nonlinearization of the time part leads to its N-involutive system {H(m)}. The involutive solution of the compatible fsystem (H-1), (H(m)) is mapped by into the solution of the higher order Kaup-Newell equation.
文摘On the basis of the Reddy's higher-order theory of composites, this paper introduces a displacement function Phi into it and transforms its three differential equations for symmetric cross-ply composites into only one eight-order differential equation generated by the displacement-function. When a proper Phi is chosen, both solutions are obtained, namely, the Navier-type solution of simply supported rectangular laminated plates and the Levy-type solution with the boundary condition where two opposite edges are simply supported and remains are arbitrary. The numerical examples show that the present results coincide well with the existing results in the references, thus validating that the present solving method is reliable. The higher-order theory of Reddy is simpler in calculation but has higher precision than the first-order shear deformation theory because the former has fewer unknows than the latter and requires no shear coefficients.
基金Research supported by the National Natural Science Foundation of China (10871116)the Natural Science Foundation of Shandong Province of China (ZR2010AM005)the Doctoral Program Foundation of Education Ministry of China (200804460001)
文摘This paper investigates a class of 2nth-order singular superlinear problems with Strum-Liouville boundary conditions. We obtain a necessary and sufficient condition for the existence of C 2 n- 2 [0, 1] positive solutions, and a sufficient condition, a necessary condition for the existence of C 2 n-1 [0, 1] positive solutions. Relations between the positive solutions and the Green’s functions are depicted. The results are used to judge nonexistence or existence of positive solutions for given boundary value problems.
基金supported by the National Natural Science Foundation of China(Nos.11072141 and11272199)the National Basic Research Program of China(No.2012CB725404)+2 种基金the Shanghai Program for Innovative Research Team in Universitiesthe Research Grants Council of the Hong KongSpecial Administrative Region,China(No.HKU7184/10E)the National Research Foundationof Korea(MEST)(No.NRF-2010-0029446)
文摘This paper uses the Taylor expansion to seek an approximate Korteweg- de Vries equation （KdV） solution to a higher-order traffic flow model with sufficiently large diffusion. It demonstrates the validity of the approximate KdV solution considering all the related parameters to ensure the physical boundedness and the stability of the solution. Moreover, when the viscosity coefficient depends on the density and velocity of the flow, the wave speed of the KdV solution is naturally related to either the first or the second characteristic field. The finite element method is extended to solve the model and examine the stability and accuracy of the approximate KdV solution.
文摘This paper deals with the singular nonlinear third-order periodic boundary value problem u'' + p(3)u = f (t, u), 0 less than or equal to t less than or equal to 2pi, with u((i)) (0) = u((i)) (2pi), i = 0, 1, 2, where p is an element of (Graphics) and f is singular at t = 0, t = 1 and u = 0. Under suitable weaker conditions than those of [1], it is proved by constructing a special cone in C[0, 2pi] and employing the fixed point index theory that the problem has at least one or at least two positive solutions.
