This study introduces a novel mathematical model that combines the finite integral transform(FIT)and gradientenhanced physics-informed neural network(g-PINN)to address thermomechanical problems in functionally graded ...This study introduces a novel mathematical model that combines the finite integral transform(FIT)and gradientenhanced physics-informed neural network(g-PINN)to address thermomechanical problems in functionally graded materials with varying properties.The model employs a multilayer heterostructure homogeneous approach within the FIT to linearize and approximate various parameters,such as the thermal conductivity,specific heat,density,stiffness,thermal expansion coefficient,and Poisson’s ratio.The provided FIT and g-PINN techniques are highly proficient in solving the PDEs of energy equations and equations of motion in a spherical domain,particularly when dealing with space-time dependent boundary conditions.The FIT method simplifies the governing partial differential equations into ordinary differential equations for efficient solutions,whereas the g-PINN bypasses linearization,achieving high accuracy with fewer training data(error<3.8%).The approach is applied to a spherical pressure vessel,solving energy and motion equations under complex boundary conditions.Furthermore,extensive parametric studies are conducted herein to demonstrate the impact of different property profiles and radial locations on the transient evolution and dynamic propagation of thermomechanical stresses.However,the accuracy of the presented approach is evaluated by comparing the g-PINN results,which have an error of less than 3.8%.Moreover,this model offers significant potential for optimizing materials in hightemperature reactors and chemical plants,improving safety,extending lifespan,and reducing thermal fatigue under extreme processing conditions.展开更多
In this study,we aimto investigate certain triple integral transformand its application to a class of partial differentialequations.We discuss various properties of the new transformincluding inversion, linearity, exi...In this study,we aimto investigate certain triple integral transformand its application to a class of partial differentialequations.We discuss various properties of the new transformincluding inversion, linearity, existence, scaling andshifting, etc. Then,we derive several results enfolding partial derivatives and establish amulti-convolution theorem.Further, we apply the aforementioned transform to some classical functions and many types of partial differentialequations involving heat equations,wave equations, Laplace equations, and Poisson equations aswell.Moreover,wedraw some figures to illustrate 3-D contour plots for exact solutions of some selected examples involving differentvalues in their variables.展开更多
By using a certain hybrid-type convolution operator,we first introduce a new subclass of normalized analytic functions in the open unit disk.For members of this analytic function class,we then derive several propertie...By using a certain hybrid-type convolution operator,we first introduce a new subclass of normalized analytic functions in the open unit disk.For members of this analytic function class,we then derive several properties and characteristics including(for example)the modified Hadamard products,Holder's inequalities and convolution properties as well as some closure properties under a general family of integral transforms.展开更多
Following exploitation of a coal seam, the final stress field is the sum of in situ stress field and an excavation stress field. Based on this feature, we firstly established a mechanics analytical model of the mining...Following exploitation of a coal seam, the final stress field is the sum of in situ stress field and an excavation stress field. Based on this feature, we firstly established a mechanics analytical model of the mining floor strata. Then the study applied Fourier integral transform to solve a biharmonic equation,obtaining the analytical solution of the stress and displacement of the mining floor. Additionally, this investigation used the Mohr–Coulomb yield criterion to determine the plastic failure depth of the floor strata. The calculation process showed that the plastic failure depth of the floor and floor heave are related to the mining width, burial depth and physical–mechanical properties. The results from an example show that the curve of the plastic failure depth of the mining floor is characterized by a funnel shape and the maximum failure depth generates in the middle of mining floor; and that the maximum and minimum principal stresses change distinctly in the shallow layer and tend to a fixed value with an increase in depth. Based on the displacement results, the maximum floor heave appears in the middle of the stope and its value is 0.107 m. This will provide a basis for floor control. Lastly, we have verified the analytical results using FLAC3 Dto simulate floor excavation and find that there is some deviation between the two results, but their overall tendency is consistent which illustrates that the analysis method can well solve the stress and displacement of the floor.展开更多
The generalized integral transform technique （GITT） is used to find a semianalytical numerical solution for dynamic response of an axially moving Timoshenko beam with clamped-clamped and simply-supported boundary co...The generalized integral transform technique （GITT） is used to find a semianalytical numerical solution for dynamic response of an axially moving Timoshenko beam with clamped-clamped and simply-supported boundary conditions, respectively. The implementation of GITT approach for analyzing the forced vibration equation eliminates the space variable and leads to systems of second-order ordinary differential equations （ODEs） in time. The MATHEMATICA built-in function, NDSolve, is used to numerically solve the resulting transformed ODE system. The good convergence behavior of the suggested eigenfunction expansions is demonstrated for calculating the transverse deflection and the angle of rotation of the beam cross-section. Moreover, parametric studies are performed to analyze the effects of the axially moving speed, the axial tension, and the amplitude of external distributed force on the vibration amplitude of axially moving Timoshenko beams.展开更多
In this paper, the compatibility between the integral type gauge transformation and the additional symmetry of the constrained KP hierarchy is given. And the string-equation constraint in matrix models is also derived.
We present exact solutions for the Klein Gordon equation with a ring-shaped oscillator potential. The energy eigenvalues and the normalized wave functions are obtained for a particle in the presence of non-central osc...We present exact solutions for the Klein Gordon equation with a ring-shaped oscillator potential. The energy eigenvalues and the normalized wave functions are obtained for a particle in the presence of non-central oscillator potential. The angulm＂ functions are expressed in terms of the hypergeometric functions. The radial eigenfunetions have been obtained by using the Laplace integral transform. By means of the Laplace transform method, which is efficient and simple, the radial Klein-Gordon equation is reduced to a first-order differential equation.展开更多
Current research is about the injection of a viscous fluid in the presence of a transverse uniform magnetic field to reduce the sliding drag.There is a slip-on both the slider and the ground in the two cases,for examp...Current research is about the injection of a viscous fluid in the presence of a transverse uniform magnetic field to reduce the sliding drag.There is a slip-on both the slider and the ground in the two cases,for example,a long porous slider and a circular porous slider.By utilizing similarity transformation Navier-Stokes equations are converted into coupled equations which are tackled by Integral Transform Method.Solutions are obtained for different values of Reynolds numbers,velocity slip,and magnetic field.We found that surface slip and Reynolds number has a substantial influence on the lift and drag of long and circular sliders,whereas the magnetic effect is also noticeable.展开更多
In the investigation on fracture mechanics,the potential function was introduced, and the moving differential equation was constructed. By making Laplace and Fourier transformation as well as sine and cosine transform...In the investigation on fracture mechanics,the potential function was introduced, and the moving differential equation was constructed. By making Laplace and Fourier transformation as well as sine and cosine transformation to moving differential equations and various responses, the dual equation which is constructed from boundary conditions lastly was solved. This method of investigating dynamic crack has become a more systematic one that is used widely. Some problems are encountered when the dynamic crack is studied. After the large investigation on the problems, it is discovered that during the process of mathematic derivation, the method is short of precision, and the derived results in this method are accidental and have no credibility.A model for example is taken to explain the problems existing in initial deriving process of the integral_transformation method of dynamic crack.展开更多
This paper serves two purposes. One is to modify Strichartz's results with respect to the asymptotic averages of the Fourier transform of μ on , self-similar measure defined by Hutchinson. Another purpose is to c...This paper serves two purposes. One is to modify Strichartz's results with respect to the asymptotic averages of the Fourier transform of μ on , self-similar measure defined by Hutchinson. Another purpose is to consider a singular integral operator on μ and show that this op- erator is of type (p,p)(1<p<∞).展开更多
This paper deals with the determination of temperature distribution and thermal deflection function of a thin circular plate with the stated conditions. The transient heat conduction equation is solved by using Marchi...This paper deals with the determination of temperature distribution and thermal deflection function of a thin circular plate with the stated conditions. The transient heat conduction equation is solved by using Marchi-Zgrablich transform and Laplace transform simultaneously and the results of temperature distribution and thermal deflection function are obtained in terms of infinite series of Bessel function and it is solved for special case by using Mathcad 2007 software and represented graphically by using Microsoft excel 2007.展开更多
The relation between the normal displacement on the surface of a dynamical elliptical crack and the normal stress over the crack surface was studied. The three dimensional elastodynamic equations and Fourier Laplace...The relation between the normal displacement on the surface of a dynamical elliptical crack and the normal stress over the crack surface was studied. The three dimensional elastodynamic equations and Fourier Laplace transforms are used. Based on the influence function and the inversion of integral transforms, one can find that if the distribution of normal displacement on the surface of a dynamic elliptical crack is a polynomial of degree n in x 1 and x 2 , then the normal pressure acting over the ellipse is also a polynomial P n(x 1,x 2) of the same degree in x 1 and x 2 .展开更多
The nonlinear dynamic behaviors of viscoelastic axially functionally graded material(AFG)pipes conveying pulsating internal flow are very complex.And the dynamic behavior will induce the failure of the pipes,and resea...The nonlinear dynamic behaviors of viscoelastic axially functionally graded material(AFG)pipes conveying pulsating internal flow are very complex.And the dynamic behavior will induce the failure of the pipes,and research of vibration and stability of pipes becomes a major concern.Considering that the elastic modulus,density,and coefficient of viscoelastic damping of the pipe material vary along the axial direction,the transverse vibration equation of the viscoelastic AFG pipe conveying pulsating fluid is established based on the Euler-Bernoulli beam theory.The generalized integral transform technique(GITT)is used to transform the governing fourth-order partial differential equation into a nonlinear system of fourth-order ordinary differential equations in time.The time domain diagram,phase portraits,Poincarémap and power spectra diagram at different dimensionless pulsation frequencies,are discussed in detail,showing the characteristics of chaotic,periodic,and quasi-periodic motion.The results show that the distributions of the elastic modulus,density,and coefficient of viscoelastic damping have significant effects on the nonlinear dynamic behavior of the viscoelastic AFG pipes.With the increase of the material property coefficient k,the transition between chaotic,periodic,and quasi-periodic motion occurs,especially in the high-frequency region of the flow pulsation.展开更多
This article studies on Cauchy’s function f （z） and its integral, （2πi）J[ f （z）] ≡ ■C f （t）dt/（t z） taken along a closed simple contour C, in regard to their comprehensive properties over the entire z =...This article studies on Cauchy’s function f （z） and its integral, （2πi）J[ f （z）] ≡ ■C f （t）dt/（t z） taken along a closed simple contour C, in regard to their comprehensive properties over the entire z = x ＋ iy plane consisted of the simply connected open domain D ＋ bounded by C and the open domain D outside C. （1） With f （z） assumed to be C n （n ∞-times continuously differentiable） z ∈ D ＋ and in a neighborhood of C, f （z） and its derivatives f （n） （z） are proved uniformly continuous in the closed domain D ＋ = [D ＋ ＋ C]. （2） Cauchy’s integral formulas and their derivatives z ∈ D ＋ （or z ∈ D ） are proved to converge uniformly in D ＋ （or in D = [D ＋C]）, respectively, thereby rendering the integral formulas valid over the entire z-plane. （3） The same claims （as for f （z） and J[ f （z）]） are shown extended to hold for the complement function F（z）, defined to be C n z ∈ D and about C. （4） The uniform convergence theorems for f （z） and F（z） shown for arbitrary contour C are adapted to find special domains in the upper or lower half z-planes and those inside and outside the unit circle ｜z｜ = 1 such that the four general- ized Hilbert-type integral transforms are proved. （5） Further, the singularity distribution of f （z） in D is elucidated by considering the direct problem exemplified with several typ- ical singularities prescribed in D . （6） A comparative study is made between generalized integral formulas and Plemelj’s formulas on their differing basic properties. （7） Physical sig- nificances of these formulas are illustrated with applicationsto nonlinear airfoil theory. （8） Finally, an unsolved inverse problem to determine all the singularities of Cauchy function f （z） in domain D , based on the continuous numerical value of f （z） z ∈ D ＋ = [D ＋ ＋ C], is presented for resolution as a conjecture.展开更多
Starting with governing equations of a saturated soil with anisotropic permeability and based on multiple integral transforms, an analytical layer-element equation is established explicitly in the Laplace-Fourier tran...Starting with governing equations of a saturated soil with anisotropic permeability and based on multiple integral transforms, an analytical layer-element equation is established explicitly in the Laplace-Fourier transformed domain. A global matrix of layered soil can be obtained by assembling a set of analytical layer-elements, which is further solved in the transformed domain by considering boundary conditions. The numerical inversion of LaplaceFourier trans- form is employed to acquire the actual solution. Numerical analysis for 3-D consolidation with anisotropic permeability of a layered soil system is presented, and the influence of anisotropy of permeability on the consolidation behavior is discussed.展开更多
Given the Laplace transform F(s) of a function f(t), we develop a new algorithm to find on approximation to f(t) by the use of the dassical Jacobi polynomials. The main contribution of our work is the development of a...Given the Laplace transform F(s) of a function f(t), we develop a new algorithm to find on approximation to f(t) by the use of the dassical Jacobi polynomials. The main contribution of our work is the development of a new and very effective method to determine the coefficients in the finite series ex-pansion that approximation f(t) in terms of Jacobi polynomials. Some numerical examples are illustrated.展开更多
This work presents a theoretical study of contact problem. The Fourier integral transform method based on the surface elasticity theory is adopted to derive the fundamental solution for the contact problem with surfac...This work presents a theoretical study of contact problem. The Fourier integral transform method based on the surface elasticity theory is adopted to derive the fundamental solution for the contact problem with surface effects, in which both the surface tension and the surface elasticity are considered. As a special case, the deformation induced by a triangle distribution force is discussed in detail. The results are compared with those of the classical contact problem. At nano-scale, the contributions of the surface tension and the surface elasticity to the stress and displacement are not equal at the contact surface. The surface tension plays a major role to the normal stress, whereas the shear stress is mainly affected by the surface elasticity. In addition, the hardness of material depends strongly on the surface effects. This study is helpful to characterize and measure the mechanical properties of soft materials through nanoindentation.展开更多
The transient response of two coplanar cracks in a piezoelectric ceramic under antiplane mechanical and inplane electric impacting loads is investigated in the present paper. Laplace and Fourier transforms are used to...The transient response of two coplanar cracks in a piezoelectric ceramic under antiplane mechanical and inplane electric impacting loads is investigated in the present paper. Laplace and Fourier transforms are used to reduce the mixed boundary value problems to Cauchy-type singular integral equations in Laplace transform domain, which are solved numerically. The dynamic stress and electric displacement factors are obtained as the functions of time and geometry parameters. The present study shows that the presence of the dynamic electric field will impede or enhance the propagation of the crack in piezoelectric ceramics at different stages of the dynamic electromechanical load. Moreover, the electromechanical response is greatly affected by the ratio of the space of the cracks and the crack length.展开更多
We propose a new two-fold integration transformation in p-q phase space∫∫^∞-∞dpdq/π e^2i（p-x）（q-y）f（p,q）≡G（x,y）,which possesses some well-behaved transformation properties. We apply this transformation t...We propose a new two-fold integration transformation in p-q phase space∫∫^∞-∞dpdq/π e^2i（p-x）（q-y）f（p,q）≡G（x,y）,which possesses some well-behaved transformation properties. We apply this transformation to the Weyl ordering of operators, especially those Q-P ordered and P-Q ordered operators.展开更多
A plane strain analysis based on the generalized Biot＇s equation is utilized to investigate the wave-induced response of a poro-elastic seabed with variable shear modulus. By employing integral transform and Frobenin...A plane strain analysis based on the generalized Biot＇s equation is utilized to investigate the wave-induced response of a poro-elastic seabed with variable shear modulus. By employing integral transform and Frobenins methods, the transient and steady solutions for the wave-inducod pore water pressure, effective stresses and displacements are analytically derived in detail. Verification is available through the reduction to the simple case of homogeneous seabed. The numerical results indicate that the inclusion of variable shear modulus significantly affects the wave-induced seabed response.展开更多
文摘This study introduces a novel mathematical model that combines the finite integral transform(FIT)and gradientenhanced physics-informed neural network(g-PINN)to address thermomechanical problems in functionally graded materials with varying properties.The model employs a multilayer heterostructure homogeneous approach within the FIT to linearize and approximate various parameters,such as the thermal conductivity,specific heat,density,stiffness,thermal expansion coefficient,and Poisson’s ratio.The provided FIT and g-PINN techniques are highly proficient in solving the PDEs of energy equations and equations of motion in a spherical domain,particularly when dealing with space-time dependent boundary conditions.The FIT method simplifies the governing partial differential equations into ordinary differential equations for efficient solutions,whereas the g-PINN bypasses linearization,achieving high accuracy with fewer training data(error<3.8%).The approach is applied to a spherical pressure vessel,solving energy and motion equations under complex boundary conditions.Furthermore,extensive parametric studies are conducted herein to demonstrate the impact of different property profiles and radial locations on the transient evolution and dynamic propagation of thermomechanical stresses.However,the accuracy of the presented approach is evaluated by comparing the g-PINN results,which have an error of less than 3.8%.Moreover,this model offers significant potential for optimizing materials in hightemperature reactors and chemical plants,improving safety,extending lifespan,and reducing thermal fatigue under extreme processing conditions.
文摘In this study,we aimto investigate certain triple integral transformand its application to a class of partial differentialequations.We discuss various properties of the new transformincluding inversion, linearity, existence, scaling andshifting, etc. Then,we derive several results enfolding partial derivatives and establish amulti-convolution theorem.Further, we apply the aforementioned transform to some classical functions and many types of partial differentialequations involving heat equations,wave equations, Laplace equations, and Poisson equations aswell.Moreover,wedraw some figures to illustrate 3-D contour plots for exact solutions of some selected examples involving differentvalues in their variables.
文摘By using a certain hybrid-type convolution operator,we first introduce a new subclass of normalized analytic functions in the open unit disk.For members of this analytic function class,we then derive several properties and characteristics including(for example)the modified Hadamard products,Holder's inequalities and convolution properties as well as some closure properties under a general family of integral transforms.
基金the National Basic Research Program of China(No.2014CB046300)the National Natural Science Foundation of China(No.51174196)
文摘Following exploitation of a coal seam, the final stress field is the sum of in situ stress field and an excavation stress field. Based on this feature, we firstly established a mechanics analytical model of the mining floor strata. Then the study applied Fourier integral transform to solve a biharmonic equation,obtaining the analytical solution of the stress and displacement of the mining floor. Additionally, this investigation used the Mohr–Coulomb yield criterion to determine the plastic failure depth of the floor strata. The calculation process showed that the plastic failure depth of the floor and floor heave are related to the mining width, burial depth and physical–mechanical properties. The results from an example show that the curve of the plastic failure depth of the mining floor is characterized by a funnel shape and the maximum failure depth generates in the middle of mining floor; and that the maximum and minimum principal stresses change distinctly in the shallow layer and tend to a fixed value with an increase in depth. Based on the displacement results, the maximum floor heave appears in the middle of the stope and its value is 0.107 m. This will provide a basis for floor control. Lastly, we have verified the analytical results using FLAC3 Dto simulate floor excavation and find that there is some deviation between the two results, but their overall tendency is consistent which illustrates that the analysis method can well solve the stress and displacement of the floor.
基金Project supported by the Science Foundation of China University of Petroleum in Beijing(No.2462013YJRC003)
文摘The generalized integral transform technique （GITT） is used to find a semianalytical numerical solution for dynamic response of an axially moving Timoshenko beam with clamped-clamped and simply-supported boundary conditions, respectively. The implementation of GITT approach for analyzing the forced vibration equation eliminates the space variable and leads to systems of second-order ordinary differential equations （ODEs） in time. The MATHEMATICA built-in function, NDSolve, is used to numerically solve the resulting transformed ODE system. The good convergence behavior of the suggested eigenfunction expansions is demonstrated for calculating the transverse deflection and the angle of rotation of the beam cross-section. Moreover, parametric studies are performed to analyze the effects of the axially moving speed, the axial tension, and the amplitude of external distributed force on the vibration amplitude of axially moving Timoshenko beams.
基金supported by the Fundamental Research Funds for the Central Universities(2015QNA43)
文摘In this paper, the compatibility between the integral type gauge transformation and the additional symmetry of the constrained KP hierarchy is given. And the string-equation constraint in matrix models is also derived.
文摘We present exact solutions for the Klein Gordon equation with a ring-shaped oscillator potential. The energy eigenvalues and the normalized wave functions are obtained for a particle in the presence of non-central oscillator potential. The angulm＂ functions are expressed in terms of the hypergeometric functions. The radial eigenfunetions have been obtained by using the Laplace integral transform. By means of the Laplace transform method, which is efficient and simple, the radial Klein-Gordon equation is reduced to a first-order differential equation.
文摘Current research is about the injection of a viscous fluid in the presence of a transverse uniform magnetic field to reduce the sliding drag.There is a slip-on both the slider and the ground in the two cases,for example,a long porous slider and a circular porous slider.By utilizing similarity transformation Navier-Stokes equations are converted into coupled equations which are tackled by Integral Transform Method.Solutions are obtained for different values of Reynolds numbers,velocity slip,and magnetic field.We found that surface slip and Reynolds number has a substantial influence on the lift and drag of long and circular sliders,whereas the magnetic effect is also noticeable.
文摘In the investigation on fracture mechanics,the potential function was introduced, and the moving differential equation was constructed. By making Laplace and Fourier transformation as well as sine and cosine transformation to moving differential equations and various responses, the dual equation which is constructed from boundary conditions lastly was solved. This method of investigating dynamic crack has become a more systematic one that is used widely. Some problems are encountered when the dynamic crack is studied. After the large investigation on the problems, it is discovered that during the process of mathematic derivation, the method is short of precision, and the derived results in this method are accidental and have no credibility.A model for example is taken to explain the problems existing in initial deriving process of the integral_transformation method of dynamic crack.
文摘This paper serves two purposes. One is to modify Strichartz's results with respect to the asymptotic averages of the Fourier transform of μ on , self-similar measure defined by Hutchinson. Another purpose is to consider a singular integral operator on μ and show that this op- erator is of type (p,p)(1<p<∞).
文摘This paper deals with the determination of temperature distribution and thermal deflection function of a thin circular plate with the stated conditions. The transient heat conduction equation is solved by using Marchi-Zgrablich transform and Laplace transform simultaneously and the results of temperature distribution and thermal deflection function are obtained in terms of infinite series of Bessel function and it is solved for special case by using Mathcad 2007 software and represented graphically by using Microsoft excel 2007.
文摘The relation between the normal displacement on the surface of a dynamical elliptical crack and the normal stress over the crack surface was studied. The three dimensional elastodynamic equations and Fourier Laplace transforms are used. Based on the influence function and the inversion of integral transforms, one can find that if the distribution of normal displacement on the surface of a dynamic elliptical crack is a polynomial of degree n in x 1 and x 2 , then the normal pressure acting over the ellipse is also a polynomial P n(x 1,x 2) of the same degree in x 1 and x 2 .
基金supported by the National Natural Science Foundation of China(52171288,51890914)the Key Research and Development Program of Shandong Province(Major Innovation Project)(2022CXGC020405)+3 种基金the National Ministry of Industry and Information Technology Innovation Special Project-Engineering Demonstration Application of Subsea Oil and Gas Production System-Subject 4“Research on Subsea Christmas Tree and Wellhead Offshore Testing Technology”[MC-201901-S01-04]the Fundamental Research Funds for the Central Universities(20CX02410A)the Development Fund of Shandong Key Laboratory of Oil&Gas Storage and Transportation SafetyCNPq,CAPES and FAPERJ of Brazil。
文摘The nonlinear dynamic behaviors of viscoelastic axially functionally graded material(AFG)pipes conveying pulsating internal flow are very complex.And the dynamic behavior will induce the failure of the pipes,and research of vibration and stability of pipes becomes a major concern.Considering that the elastic modulus,density,and coefficient of viscoelastic damping of the pipe material vary along the axial direction,the transverse vibration equation of the viscoelastic AFG pipe conveying pulsating fluid is established based on the Euler-Bernoulli beam theory.The generalized integral transform technique(GITT)is used to transform the governing fourth-order partial differential equation into a nonlinear system of fourth-order ordinary differential equations in time.The time domain diagram,phase portraits,Poincarémap and power spectra diagram at different dimensionless pulsation frequencies,are discussed in detail,showing the characteristics of chaotic,periodic,and quasi-periodic motion.The results show that the distributions of the elastic modulus,density,and coefficient of viscoelastic damping have significant effects on the nonlinear dynamic behavior of the viscoelastic AFG pipes.With the increase of the material property coefficient k,the transition between chaotic,periodic,and quasi-periodic motion occurs,especially in the high-frequency region of the flow pulsation.
文摘This article studies on Cauchy’s function f （z） and its integral, （2πi）J[ f （z）] ≡ ■C f （t）dt/（t z） taken along a closed simple contour C, in regard to their comprehensive properties over the entire z = x ＋ iy plane consisted of the simply connected open domain D ＋ bounded by C and the open domain D outside C. （1） With f （z） assumed to be C n （n ∞-times continuously differentiable） z ∈ D ＋ and in a neighborhood of C, f （z） and its derivatives f （n） （z） are proved uniformly continuous in the closed domain D ＋ = [D ＋ ＋ C]. （2） Cauchy’s integral formulas and their derivatives z ∈ D ＋ （or z ∈ D ） are proved to converge uniformly in D ＋ （or in D = [D ＋C]）, respectively, thereby rendering the integral formulas valid over the entire z-plane. （3） The same claims （as for f （z） and J[ f （z）]） are shown extended to hold for the complement function F（z）, defined to be C n z ∈ D and about C. （4） The uniform convergence theorems for f （z） and F（z） shown for arbitrary contour C are adapted to find special domains in the upper or lower half z-planes and those inside and outside the unit circle ｜z｜ = 1 such that the four general- ized Hilbert-type integral transforms are proved. （5） Further, the singularity distribution of f （z） in D is elucidated by considering the direct problem exemplified with several typ- ical singularities prescribed in D . （6） A comparative study is made between generalized integral formulas and Plemelj’s formulas on their differing basic properties. （7） Physical sig- nificances of these formulas are illustrated with applicationsto nonlinear airfoil theory. （8） Finally, an unsolved inverse problem to determine all the singularities of Cauchy function f （z） in domain D , based on the continuous numerical value of f （z） z ∈ D ＋ = [D ＋ ＋ C], is presented for resolution as a conjecture.
基金Project supported by the National Natural Science Foundation of China (No. 50578121)
文摘Starting with governing equations of a saturated soil with anisotropic permeability and based on multiple integral transforms, an analytical layer-element equation is established explicitly in the Laplace-Fourier transformed domain. A global matrix of layered soil can be obtained by assembling a set of analytical layer-elements, which is further solved in the transformed domain by considering boundary conditions. The numerical inversion of LaplaceFourier trans- form is employed to acquire the actual solution. Numerical analysis for 3-D consolidation with anisotropic permeability of a layered soil system is presented, and the influence of anisotropy of permeability on the consolidation behavior is discussed.
文摘Given the Laplace transform F(s) of a function f(t), we develop a new algorithm to find on approximation to f(t) by the use of the dassical Jacobi polynomials. The main contribution of our work is the development of a new and very effective method to determine the coefficients in the finite series ex-pansion that approximation f(t) in terms of Jacobi polynomials. Some numerical examples are illustrated.
文摘This work presents a theoretical study of contact problem. The Fourier integral transform method based on the surface elasticity theory is adopted to derive the fundamental solution for the contact problem with surface effects, in which both the surface tension and the surface elasticity are considered. As a special case, the deformation induced by a triangle distribution force is discussed in detail. The results are compared with those of the classical contact problem. At nano-scale, the contributions of the surface tension and the surface elasticity to the stress and displacement are not equal at the contact surface. The surface tension plays a major role to the normal stress, whereas the shear stress is mainly affected by the surface elasticity. In addition, the hardness of material depends strongly on the surface effects. This study is helpful to characterize and measure the mechanical properties of soft materials through nanoindentation.
文摘The transient response of two coplanar cracks in a piezoelectric ceramic under antiplane mechanical and inplane electric impacting loads is investigated in the present paper. Laplace and Fourier transforms are used to reduce the mixed boundary value problems to Cauchy-type singular integral equations in Laplace transform domain, which are solved numerically. The dynamic stress and electric displacement factors are obtained as the functions of time and geometry parameters. The present study shows that the presence of the dynamic electric field will impede or enhance the propagation of the crack in piezoelectric ceramics at different stages of the dynamic electromechanical load. Moreover, the electromechanical response is greatly affected by the ratio of the space of the cracks and the crack length.
基金National Natural Science Foundation of China under Grant Nos.10775097 and 10874174
文摘We propose a new two-fold integration transformation in p-q phase space∫∫^∞-∞dpdq/π e^2i（p-x）（q-y）f（p,q）≡G（x,y）,which possesses some well-behaved transformation properties. We apply this transformation to the Weyl ordering of operators, especially those Q-P ordered and P-Q ordered operators.
基金This project is supported by the National Natural Science Foundation of China (Grant No50479045)
文摘A plane strain analysis based on the generalized Biot＇s equation is utilized to investigate the wave-induced response of a poro-elastic seabed with variable shear modulus. By employing integral transform and Frobenins methods, the transient and steady solutions for the wave-inducod pore water pressure, effective stresses and displacements are analytically derived in detail. Verification is available through the reduction to the simple case of homogeneous seabed. The numerical results indicate that the inclusion of variable shear modulus significantly affects the wave-induced seabed response.
