The existence of nondecreasing positive solutions for the nonlinear third-order twopoint boundary value problem u′″（t） ＋ q（t）f（t,u（t）,u′（t）） = 0, 0 〈 t 〈 1, u（0） = u″（0） = u′（1） = 0 is studied....The existence of nondecreasing positive solutions for the nonlinear third-order twopoint boundary value problem u′″（t） ＋ q（t）f（t,u（t）,u′（t）） = 0, 0 〈 t 〈 1, u（0） = u″（0） = u′（1） = 0 is studied. The iterative schemes for approximating the solutions are obtained by applying a monotone iterative method.展开更多
We target here to solve numerically a class of nonlinear fractional two-point boundary value problems involving left-and right-sided fractional derivatives.The main ingredient of the proposed method is to recast the p...We target here to solve numerically a class of nonlinear fractional two-point boundary value problems involving left-and right-sided fractional derivatives.The main ingredient of the proposed method is to recast the problem into an equivalent system of weakly singular integral equations.Then,a Legendre-based spectral collocation method is developed for solving the transformed system.Therefore,we can make good use of the advantages of the Gauss quadrature rule.We present the construction and analysis of the collocation method.These results can be indirectly applied to solve fractional optimal control problems by considering the corresponding Euler–Lagrange equations.Two numerical examples are given to confirm the convergence analysis and robustness of the scheme.展开更多
By using the method of upper and lower solution, some conditions of the existence of solutions of nonlinear two-point boundary value problems for 4nth-order nonlinear differential equation are studied.
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 we present a precise integration method based on high order multiple perturbation method and reduction method for solving a class of singular twopoint boundary value problems.Firstly,by employing the met...In this paper we present a precise integration method based on high order multiple perturbation method and reduction method for solving a class of singular twopoint boundary value problems.Firstly,by employing the method of variable coefficient dimensional expanding,the non-homogeneous ordinary differential equations（ODEs） are transformed into homogeneous ODEs.Then the interval is divided evenly,and the transfer matrix in every subinterval is worked out using the high order multiple perturbation method,and a set of algebraic equations is given in the form of matrix by the precise integration relation for each segment,which is worked out by the reduction method.Finally numerical examples are elaboratedd to validate the present method.展开更多
The present paper tackles two-point boundary value problems for fourth-order differential equations as follows:Several existence theorems on multiple positive solutions to the problems are obtained, and some examples ...The present paper tackles two-point boundary value problems for fourth-order differential equations as follows:Several existence theorems on multiple positive solutions to the problems are obtained, and some examples are given to show the validity of these results.展开更多
In this paper, we consider the following problem:The quadratic spline collocation, with uniform mesh and the mid-knot points are taken as the collocation points for this problem is considered. With some assumptions, w...In this paper, we consider the following problem:The quadratic spline collocation, with uniform mesh and the mid-knot points are taken as the collocation points for this problem is considered. With some assumptions, we have proved that the solution of the quadratic spline collocation for the nonlinear problem can be written as a series expansions in integer powers of the mesh-size parameter. This gives us a construction method for using Richardson’s extrapolation. When we have a set of approximate solution with different mesh-size parameter a solution with high accuracy can he obtained by Richardson’s extrapolation.展开更多
By constructing suitable Banach space, an existence theorem is established under a condition of linear growth for the third-order boundary value problem u″′（t）＋f（t,u（t）,u′（t））=0,0〈t〈1,u（0）=u′（0）=u′...By constructing suitable Banach space, an existence theorem is established under a condition of linear growth for the third-order boundary value problem u″′（t）＋f（t,u（t）,u′（t））=0,0〈t〈1,u（0）=u′（0）=u′（1）=0, where the nonlinear term contains first and second derivatives of unknown function. In this theorem the nonlinear term f（t, u, v, w） may be singular at t = 0 and t = 1. The main ingredient is Leray-Schauder nonlinear alternative.展开更多
We consider the nonlinear boundary value problems for elliptic partial differential equations and using a maximum principle for this problem we show uniqueness and continuous dependence on data. We use the strong vers...We consider the nonlinear boundary value problems for elliptic partial differential equations and using a maximum principle for this problem we show uniqueness and continuous dependence on data. We use the strong version of the maximum principle to prove that all solutions of two-point BVP are positives and we also show a numerical example by applying finite difference method for a two-point BVP in one dimension based on discrete version of the maximum principle.展开更多
In this paper, authors describe a Liouville-Green transform to solve a singularly perturbed two-point boundary value problem with right end boundary layer in the interval [0, 1]. They reply Liouville-Green transform i...In this paper, authors describe a Liouville-Green transform to solve a singularly perturbed two-point boundary value problem with right end boundary layer in the interval [0, 1]. They reply Liouville-Green transform into original given problem and finds the numerical solution. Then they implemented this method on two linear examples with right end boundary layer which nicely approximate the exact solution.展开更多
A class of second-order two-point boundary value problem on a measure chain was considered. Under some suitable conditions, by using the Leggett-Williams fixed point theorem in an appropriate cone, the existence of at...A class of second-order two-point boundary value problem on a measure chain was considered. Under some suitable conditions, by using the Leggett-Williams fixed point theorem in an appropriate cone, the existence of at least three positive solutions to this nonlinear problem was obtained.展开更多
This paper describes a numerical solution for a two-point boundary value problem. It includes an algorithm for discretization by mixed finite element method. The discrete scheme allows the utilization a finite element...This paper describes a numerical solution for a two-point boundary value problem. It includes an algorithm for discretization by mixed finite element method. The discrete scheme allows the utilization a finite element method based on piecewise linear approximating functions and we also use the barycentric quadrature rule to compute the stiffness matrix and the L2-norm.展开更多
We consider the nth order nonlinear differential equation on time scales subject to the right focal type two-point boundary conditions We establish a criterion for the existence of at least one positive solution by ut...We consider the nth order nonlinear differential equation on time scales subject to the right focal type two-point boundary conditions We establish a criterion for the existence of at least one positive solution by utilizing Krasnosel’skii fixed point theorem. And then, we establish the existence of at least three positive solutions by utilizing Leggett-Williams fixed point theorem.展开更多
In this paper, we discuss the existence of solution of a nonlinear two-point boundary value problem with a positive parameter Q arising in the study of surfacetension-induced flows of a liquid metal or semiconductor. ...In this paper, we discuss the existence of solution of a nonlinear two-point boundary value problem with a positive parameter Q arising in the study of surfacetension-induced flows of a liquid metal or semiconductor. By applying the Schauder's fixed-point theorem, we prove that the problem admits a solution for 0 ≤ Q ≤ 14.306.It improves the result of 0 ≤ Q ＜ 1 in [2] and 0 ≤ Q ≤ 13.213 in [3].展开更多
Asymmetric tilt boundaries on conventional twin boundaries(TBs)are significant for understanding the role of twins on coordinating plastic deformation in many metallic alloys.However,the formation modes of many asymme...Asymmetric tilt boundaries on conventional twin boundaries(TBs)are significant for understanding the role of twins on coordinating plastic deformation in many metallic alloys.However,the formation modes of many asymmetric tilt boundaries are hard to be accounted for based on traditional theoretical models,and the corresponding solute segregation is complex.Herein,atomic structures of a specific asymmetric boundary on{1012}TBs were reveled using aberration-corrected high-angle annular dark-field scanning transmission electron microscopy(HAADF-STEM),molecular dynamics(MD)and density functional theory(DFT)simulations.Reaction between<a60>M dislocations and the{1012}TB can generate a~61°/25°asymmetric tilt boundary.The segregation of Gd and Zn atoms is closely related to the aggregateddislocations and the interfacial interstices of the asymmetric tilt boundary,which is energetically favorable in reducing the total system energy.展开更多
In this paper,we study the asymptotic behavior of the micropolar fluid flow through a thin domain,assuming zero Dirichlet boundary condition on the top boundary,which is rapidly oscillating,and non-standard boundary c...In this paper,we study the asymptotic behavior of the micropolar fluid flow through a thin domain,assuming zero Dirichlet boundary condition on the top boundary,which is rapidly oscillating,and non-standard boundary conditions on the flat bottom.Assuming“Reynolds roughness regime”,in which the thickness of the domain is very small compared to the wavelength of the roughness(i.e.a very slight roughness),we rigorously derive a generalized Reynolds equation for pressure,clearly showing the roughness-induced effects.Moreover,we give expressions for the average velocity and microrotation.展开更多
A large-scale view of the magnetospheric cusp is expected to be obtained by the Soft X-ray Imager(SXI)onboard the Solar wind Magnetosphere Ionosphere Link Explorer(SMILE).However,it is challenging to trace the three-d...A large-scale view of the magnetospheric cusp is expected to be obtained by the Soft X-ray Imager(SXI)onboard the Solar wind Magnetosphere Ionosphere Link Explorer(SMILE).However,it is challenging to trace the three-dimensional cusp boundary from a two-dimensional X-ray image because the detected X-ray signals will be integrated along the line of sight.In this work,a global magnetohydrodynamic code was used to simulate the X-ray images and photon count images,assuming an interplanetary magnetic field with a pure Bz component.The assumption of an elliptic cusp boundary at a given altitude was used to trace the equatorward and poleward boundaries of the cusp from a simulated X-ray image.The average discrepancy was less than 0.1 RE.To reduce the influence of instrument effects and cosmic X-ray backgrounds,image denoising was considered before applying the method above to SXI photon count images.The cusp boundaries were reasonably reconstructed from the noisy X-ray image.展开更多
THE mechanical response and deformation mechanisms of pure nickel under nanoindentation were systematically investigated using molecular dynamics(MD)simulations,with a particular focus on the novel interplay between c...THE mechanical response and deformation mechanisms of pure nickel under nanoindentation were systematically investigated using molecular dynamics(MD)simulations,with a particular focus on the novel interplay between crystallographic orientation,grain boundary(GB)proximity,and pore characteristics(size/location).This study compares single-crystal nickel models along[100],[110],and[111]orientations with equiaxed polycrystalline models containing 0,1,and 2.5 nm pores in surface and subsurface configurations.Our results reveal that crystallographic anisotropy manifests as a 24.4%higher elastic modulus and 22.2%greater hardness in[111]-oriented single crystals compared to[100].Pore-GB synergistic effects are found to dominate the deformation behavior:2.5 nm subsurface pores reduce hardness by 25.2%through stress concentration and dislocation annihilation at GBs,whereas surface pores enable mechanical recovery via accelerated dislocation generation post-collapse.Additionally,size-dependent deformation regimes were identified,with 1 nm pores inducing negligible perturbation due to rapid atomic rearrangement,in contrast with persistent softening in 2.5 nm pores.These findings establish atomic-scale design principles for defect engineering in nickel-based aerospace components,demonstrating how crystallographic orientation,pore configuration,and GB interactions collectively govern nanoindentation behavior.展开更多
In this article,we show the existence,uniqueness and stability of bounded solutions to the following quasilinear problems with mean curvature operator(φ'(x′(t)))′=f(t,x),t≥t_(0),lim_(t→∞)x(t)=ψ_(0),lim_(t→...In this article,we show the existence,uniqueness and stability of bounded solutions to the following quasilinear problems with mean curvature operator(φ'(x′(t)))′=f(t,x),t≥t_(0),lim_(t→∞)x(t)=ψ_(0),lim_(t→∞)x′(t)e^(t)=0,where t_(0) and ψ_(0) are real constants,φ(s)=s/√1−s^(2),s∈R with s∈(−1,1),f:[t_(0),∞)×R→R satisfies the Lipschitz or Osgood-type conditions.展开更多
In this paper,we study the nonlinear Riemann boundary value problem with square roots that is represented by a Cauchy-type integral with kernel density in variable exponent Lebesgue spaces.We discuss the odd-order zer...In this paper,we study the nonlinear Riemann boundary value problem with square roots that is represented by a Cauchy-type integral with kernel density in variable exponent Lebesgue spaces.We discuss the odd-order zero-points distribution of the solutions and separate the single valued analytic branch of the solutions with square roots,then convert the problem to a Riemann boundary value problem in variable exponent Lebesgue spaces and discuss the singularity of solutions at individual zeros belonging to curve.We consider two types of cases those where the coefficient is Hölder and those where it is piecewise Hölder.Then we solve the Hilbert boundary value problem with square roots in variable exponent Lebesgue spaces.By discussing the distribution of the odd-order zero-points for solutions and the method of symmetric extension,we convert the Hilbert problem to a Riemann boundary value problem.The equivalence of the transformation is discussed.Finally,we get the solvable conditions and the direct expressions of the solutions in variable exponent Lebesgue spaces.展开更多
基金Supported by the Natural Science Foundation of Zhejiang Province (Y605144)the XNF of Zhejiang University of Media and Communications (XN080012008034)
文摘The existence of nondecreasing positive solutions for the nonlinear third-order twopoint boundary value problem u′″（t） ＋ q（t）f（t,u（t）,u′（t）） = 0, 0 〈 t 〈 1, u（0） = u″（0） = u′（1） = 0 is studied. The iterative schemes for approximating the solutions are obtained by applying a monotone iterative method.
基金The Russian Foundation for Basic Research(RFBR)Grant No.19-01-00019.
文摘We target here to solve numerically a class of nonlinear fractional two-point boundary value problems involving left-and right-sided fractional derivatives.The main ingredient of the proposed method is to recast the problem into an equivalent system of weakly singular integral equations.Then,a Legendre-based spectral collocation method is developed for solving the transformed system.Therefore,we can make good use of the advantages of the Gauss quadrature rule.We present the construction and analysis of the collocation method.These results can be indirectly applied to solve fractional optimal control problems by considering the corresponding Euler–Lagrange equations.Two numerical examples are given to confirm the convergence analysis and robustness of the scheme.
文摘By using the method of upper and lower solution, some conditions of the existence of solutions of nonlinear two-point boundary value problems for 4nth-order nonlinear differential equation are studied.
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.
基金supported by the National Natural Science Foundation of China (11132004 and 51078145)the Natural Science Foundation of Guangdong Province (9251064101000016)
文摘In this paper we present a precise integration method based on high order multiple perturbation method and reduction method for solving a class of singular twopoint boundary value problems.Firstly,by employing the method of variable coefficient dimensional expanding,the non-homogeneous ordinary differential equations（ODEs） are transformed into homogeneous ODEs.Then the interval is divided evenly,and the transfer matrix in every subinterval is worked out using the high order multiple perturbation method,and a set of algebraic equations is given in the form of matrix by the precise integration relation for each segment,which is worked out by the reduction method.Finally numerical examples are elaboratedd to validate the present method.
基金The Postdoctoral Science Research Foundation of Zhengzhou University.
文摘The present paper tackles two-point boundary value problems for fourth-order differential equations as follows:Several existence theorems on multiple positive solutions to the problems are obtained, and some examples are given to show the validity of these results.
基金The Project was supported by National Natural Science Foundation of China
文摘In this paper, we consider the following problem:The quadratic spline collocation, with uniform mesh and the mid-knot points are taken as the collocation points for this problem is considered. With some assumptions, we have proved that the solution of the quadratic spline collocation for the nonlinear problem can be written as a series expansions in integer powers of the mesh-size parameter. This gives us a construction method for using Richardson’s extrapolation. When we have a set of approximate solution with different mesh-size parameter a solution with high accuracy can he obtained by Richardson’s extrapolation.
文摘By constructing suitable Banach space, an existence theorem is established under a condition of linear growth for the third-order boundary value problem u″′（t）＋f（t,u（t）,u′（t））=0,0〈t〈1,u（0）=u′（0）=u′（1）=0, where the nonlinear term contains first and second derivatives of unknown function. In this theorem the nonlinear term f（t, u, v, w） may be singular at t = 0 and t = 1. The main ingredient is Leray-Schauder nonlinear alternative.
文摘We consider the nonlinear boundary value problems for elliptic partial differential equations and using a maximum principle for this problem we show uniqueness and continuous dependence on data. We use the strong version of the maximum principle to prove that all solutions of two-point BVP are positives and we also show a numerical example by applying finite difference method for a two-point BVP in one dimension based on discrete version of the maximum principle.
文摘In this paper, authors describe a Liouville-Green transform to solve a singularly perturbed two-point boundary value problem with right end boundary layer in the interval [0, 1]. They reply Liouville-Green transform into original given problem and finds the numerical solution. Then they implemented this method on two linear examples with right end boundary layer which nicely approximate the exact solution.
文摘A class of second-order two-point boundary value problem on a measure chain was considered. Under some suitable conditions, by using the Leggett-Williams fixed point theorem in an appropriate cone, the existence of at least three positive solutions to this nonlinear problem was obtained.
文摘This paper describes a numerical solution for a two-point boundary value problem. It includes an algorithm for discretization by mixed finite element method. The discrete scheme allows the utilization a finite element method based on piecewise linear approximating functions and we also use the barycentric quadrature rule to compute the stiffness matrix and the L2-norm.
文摘We consider the nth order nonlinear differential equation on time scales subject to the right focal type two-point boundary conditions We establish a criterion for the existence of at least one positive solution by utilizing Krasnosel’skii fixed point theorem. And then, we establish the existence of at least three positive solutions by utilizing Leggett-Williams fixed point theorem.
基金The work was supported by National Natural Science Foundation(Grant No. 10471129) of China
文摘In this paper, we discuss the existence of solution of a nonlinear two-point boundary value problem with a positive parameter Q arising in the study of surfacetension-induced flows of a liquid metal or semiconductor. By applying the Schauder's fixed-point theorem, we prove that the problem admits a solution for 0 ≤ Q ≤ 14.306.It improves the result of 0 ≤ Q ＜ 1 in [2] and 0 ≤ Q ≤ 13.213 in [3].
基金supported by the Scientific and Technological Developing Scheme of Jilin Province under grants no.YDZJ202301ZYTS538the Chinese Academy of Sciences Youth Innovation Promotion Association under grants number 2023234+3 种基金the National Natural Science Foundation of China under grants number U21A20323the Scientific and Technological Developing Scheme of Jilin Province under grants no.SKL202302038the Major Scientific and Technological Projects of Hebei Province under grants No.23291001Zthe Scientific and Technology Project of Hanjiang District.
文摘Asymmetric tilt boundaries on conventional twin boundaries(TBs)are significant for understanding the role of twins on coordinating plastic deformation in many metallic alloys.However,the formation modes of many asymmetric tilt boundaries are hard to be accounted for based on traditional theoretical models,and the corresponding solute segregation is complex.Herein,atomic structures of a specific asymmetric boundary on{1012}TBs were reveled using aberration-corrected high-angle annular dark-field scanning transmission electron microscopy(HAADF-STEM),molecular dynamics(MD)and density functional theory(DFT)simulations.Reaction between<a60>M dislocations and the{1012}TB can generate a~61°/25°asymmetric tilt boundary.The segregation of Gd and Zn atoms is closely related to the aggregateddislocations and the interfacial interstices of the asymmetric tilt boundary,which is energetically favorable in reducing the total system energy.
文摘In this paper,we study the asymptotic behavior of the micropolar fluid flow through a thin domain,assuming zero Dirichlet boundary condition on the top boundary,which is rapidly oscillating,and non-standard boundary conditions on the flat bottom.Assuming“Reynolds roughness regime”,in which the thickness of the domain is very small compared to the wavelength of the roughness(i.e.a very slight roughness),we rigorously derive a generalized Reynolds equation for pressure,clearly showing the roughness-induced effects.Moreover,we give expressions for the average velocity and microrotation.
基金funded by the National Natural Science Foundation of China(NNSFC)under Grant Numbers 42322408,42188101,and 42441809Additional support was provided by the Climbing Program of the National Space Science Center(NSSC,Grant No.E4PD3005)as well as the Specialized Research Fund for State Key Laboratories of China.
文摘A large-scale view of the magnetospheric cusp is expected to be obtained by the Soft X-ray Imager(SXI)onboard the Solar wind Magnetosphere Ionosphere Link Explorer(SMILE).However,it is challenging to trace the three-dimensional cusp boundary from a two-dimensional X-ray image because the detected X-ray signals will be integrated along the line of sight.In this work,a global magnetohydrodynamic code was used to simulate the X-ray images and photon count images,assuming an interplanetary magnetic field with a pure Bz component.The assumption of an elliptic cusp boundary at a given altitude was used to trace the equatorward and poleward boundaries of the cusp from a simulated X-ray image.The average discrepancy was less than 0.1 RE.To reduce the influence of instrument effects and cosmic X-ray backgrounds,image denoising was considered before applying the method above to SXI photon count images.The cusp boundaries were reasonably reconstructed from the noisy X-ray image.
基金The National Natural Science Foundation of China(Grant No.12462006)Beijing Institute of Structure and Environment Engineering Joint Innovation Fund(No.BQJJ202414).
文摘THE mechanical response and deformation mechanisms of pure nickel under nanoindentation were systematically investigated using molecular dynamics(MD)simulations,with a particular focus on the novel interplay between crystallographic orientation,grain boundary(GB)proximity,and pore characteristics(size/location).This study compares single-crystal nickel models along[100],[110],and[111]orientations with equiaxed polycrystalline models containing 0,1,and 2.5 nm pores in surface and subsurface configurations.Our results reveal that crystallographic anisotropy manifests as a 24.4%higher elastic modulus and 22.2%greater hardness in[111]-oriented single crystals compared to[100].Pore-GB synergistic effects are found to dominate the deformation behavior:2.5 nm subsurface pores reduce hardness by 25.2%through stress concentration and dislocation annihilation at GBs,whereas surface pores enable mechanical recovery via accelerated dislocation generation post-collapse.Additionally,size-dependent deformation regimes were identified,with 1 nm pores inducing negligible perturbation due to rapid atomic rearrangement,in contrast with persistent softening in 2.5 nm pores.These findings establish atomic-scale design principles for defect engineering in nickel-based aerospace components,demonstrating how crystallographic orientation,pore configuration,and GB interactions collectively govern nanoindentation behavior.
基金Supported by the National Natural Science Foundation of China(Grant Nos.12361040,12061064)the National Science Foundation of Gansu Province(Grant No.22JR5RA264)State Scholarship Fund(Grant No.20230862021).
文摘In this article,we show the existence,uniqueness and stability of bounded solutions to the following quasilinear problems with mean curvature operator(φ'(x′(t)))′=f(t,x),t≥t_(0),lim_(t→∞)x(t)=ψ_(0),lim_(t→∞)x′(t)e^(t)=0,where t_(0) and ψ_(0) are real constants,φ(s)=s/√1−s^(2),s∈R with s∈(−1,1),f:[t_(0),∞)×R→R satisfies the Lipschitz or Osgood-type conditions.
基金supported by the National Natural Science Foundation of China(11601525)the Natural Science Foundation of Hunan Province(2024JJ5412),the Changsha Municipal Natural Science Foundation(kq2402193).
文摘In this paper,we study the nonlinear Riemann boundary value problem with square roots that is represented by a Cauchy-type integral with kernel density in variable exponent Lebesgue spaces.We discuss the odd-order zero-points distribution of the solutions and separate the single valued analytic branch of the solutions with square roots,then convert the problem to a Riemann boundary value problem in variable exponent Lebesgue spaces and discuss the singularity of solutions at individual zeros belonging to curve.We consider two types of cases those where the coefficient is Hölder and those where it is piecewise Hölder.Then we solve the Hilbert boundary value problem with square roots in variable exponent Lebesgue spaces.By discussing the distribution of the odd-order zero-points for solutions and the method of symmetric extension,we convert the Hilbert problem to a Riemann boundary value problem.The equivalence of the transformation is discussed.Finally,we get the solvable conditions and the direct expressions of the solutions in variable exponent Lebesgue spaces.
