The double constraint equations in the self-dual gravitational theory containing the cosmological term are derived in gravity. Furthermore, in order to deeply study the Lorentzian and Euclidean reality conditions for...The double constraint equations in the self-dual gravitational theory containing the cosmological term are derived in gravity. Furthermore, in order to deeply study the Lorentzian and Euclidean reality conditions for this theory, the relations between constraints are discussed by introducing the double constant conformal transformation and the double complex function method.展开更多
The accurate calculation and measurement of welding temperature field is an important precondition for welding metallurgical analysis and welding process controlling. In this paper, the conformal transformation is fir...The accurate calculation and measurement of welding temperature field is an important precondition for welding metallurgical analysis and welding process controlling. In this paper, the conformal transformation is firstly used to analyze the welding temperature field of featheredged cylinder. The center of the cylinder is chosen as the origin of column coordinate system, and every point may be expressed as complex field vector. The branch isogonality counterchanges the line parallel with the fusion line in half-infinite z-plane to the circle concentric with the fusion line in infinite cylinder. The Laplace equation and Poisson＇s equation still keep validity, so the temperature field equation can be solved. The conformal transformation and equation solution is processed by Matlab program language. It shows that the obtained analytical modeling of temperature field for featheredged cylinder based on conformal transformation is effective and accurate.展开更多
A framework for analytical studies of superconducting systems is presented and illustrated. The formalism, based on the conformal transformation of momentum space, allows one to study the effects of both the dispersio...A framework for analytical studies of superconducting systems is presented and illustrated. The formalism, based on the conformal transformation of momentum space, allows one to study the effects of both the dispersion relation and the structure of the pairing interaction in two-dimensional anisotropic high-T <sub>c</sub> superconductors. In this method, the number of employed degrees of freedom coincides with the dimension of the momentum space, which is different compared to that in the standard Van Hove scenario with a single degree of freedom. A new function, the kernel of the density of states, is defined and its relation to the standard density of states is explained. The versatility of the method is illustrated by analyzing coexistence and competition between spin-singlet and spin-triplet order parameters in superconducting systems with a tight-binding-type dispersion relation and an anisotropic pairing potential. Phase diagrams of stable superconducting states in the coordinates ?· (the ratio of hopping parameters) and n (the carrier concentration) are presented and discussed. Moreover, the role of attractive and repulsive on-site interactions for the stability of the s-wave order parameter is explained.展开更多
The acoustic cloak can manipulate sound waves in surprising ways and enable us to guide the trajectories of waves at will.In contrast to the existing approaches for designing such devices that need exotic material par...The acoustic cloak can manipulate sound waves in surprising ways and enable us to guide the trajectories of waves at will.In contrast to the existing approaches for designing such devices that need exotic material parameters,here we demonstrate how to design a layered and isotropic acoustic cloak based on conformal transformation acoustics.A petal-shaped layered acoustic cloak is designed from the reduction of material parameters of an approximate rectangular cloak.The resultant material parameters span a narrow range,thus strongly facilitating the implementation.The invisibility performance of the cloak for different shapes of acoustic wavefront is verified using a full-wave finite-element analysis.展开更多
An intuitive 2D model of circular electrical impedance tomography （EIT） sensor with small size electrodes is established based on the theory of analytic functions. The validation of the model is proved using the res...An intuitive 2D model of circular electrical impedance tomography （EIT） sensor with small size electrodes is established based on the theory of analytic functions. The validation of the model is proved using the result from the solution of Laplace equation. Suggestions on to electrode optimization and explanation to the ill-condition property of the sensitivity matrix are provided based on the model, which takes electrode distance into account and can be generalized to the sensor with any simple connected region through a conformal transformation. Image reconstruction algorithms based on the model are implemented to show feasibility of the model using experimental data collected from the EIT system developed in Tianjin University. In the simulation with a human chestlike configuration, electrical conductivity distributions are reconstructed using equi-potential backprojection （EBP） and Tikhonov regularization （TR） based on a conformal transformation of the model. The algorithms based on the model are suitable for online image reconstruction and the reconstructed results are aood both in size and position.展开更多
Here,we describe the robust and efficient application of the conventional 3D BEM in solving elasticity problems. We have focused on the precise computation of weakly singular integrals. The conformal Duffy-distance tr...Here,we describe the robust and efficient application of the conventional 3D BEM in solving elasticity problems. We have focused on the precise computation of weakly singular integrals. The conformal Duffy-distance transformation was employed to alleviate near singularities caused from two aspects:(1) the large aspect ratio of elements,i.e.,element shape distortions;and(2)the closeness of element boundaries to field points,i.e.,ill-shaped patches. Then,the rigid body motion method was employed to evaluate strongly singular integrals. Numerical solutions of 3D elastostatic problems demonstrated the high accuracy of the proposed method with coarse meshes and high convergence rates with mesh refinement. Compared with the Duffy transformation and original polar coordinate transformations,the proposed method is insensitive to element shapes.展开更多
The high-order boundary conditions for the problems cf Laplace equation in infinite region have been developed. The improvement in accuracy for numerical solution is achieved by imposing the high-order boundary condit...The high-order boundary conditions for the problems cf Laplace equation in infinite region have been developed. The improvement in accuracy for numerical solution is achieved by imposing the high-order boundary conditions on the exterior boundarv of a reduced finite region in which the numerical method is used. So both the computing efforts and the required storage in computer are reduced. The numerical examples show that the 1st-order boundary condition approaches to the exact boundary condition and it is clearly superior to the traditional boundary condition and the 2nd-order boundary condition.展开更多
Super-resolution imaging is vital for optical applications, such as high capacity information transmission, real-time bio-molecular imaging, and nanolithography. In recent years, technologies and methods of super-reso...Super-resolution imaging is vital for optical applications, such as high capacity information transmission, real-time bio-molecular imaging, and nanolithography. In recent years, technologies and methods of super-resolution imaging have attracted much attention. Different kinds of novel lenses, from the superlens to the super-oscillatory lens, have been designed and fabricated to break through the diffraction limit. However, the effect of the super-resolution imaging in these lenses is not satisfactory due to intrinsic loss, aberration, large sidebands, and so on. Moreover, these lenses also cannot realize multiple super-resolution imaging. In this research, we introduce the solid immersion mechanism to Mikaelian lens(ML) for multiple super-resolution imaging. The effect is robust and valid for broadband frequencies. Based on conformal transformation optics as a bridge linking the solid immersion ML and generalized Maxwell's fish-eye lens(GMFEL), we also discovered the effect of multiple super-resolution imaging in the solid immersion GMFEL.展开更多
In this paper, we have considered a class curves with some geometric properties in a higher dimensional space and obtained the differential equation of such a class curves, which are called the hyperbolas. We have con...In this paper, we have considered a class curves with some geometric properties in a higher dimensional space and obtained the differential equation of such a class curves, which are called the hyperbolas. We have considered also hyperbola-preserving conformal transformation and the relevant physical sense. And therefore obtained other invariant properties under the illustrious concircular transformation.展开更多
The principle objective of the paper is to study the acoustic radiation problem of the 3D space domain with control boundary. By using the conformal transformation theory, the Green's function for acoustic point s...The principle objective of the paper is to study the acoustic radiation problem of the 3D space domain with control boundary. By using the conformal transformation theory, the Green's function for acoustic point source in the control domain space is obtained. With it, the expression of acoustic radiation function of the control domain is formed. Discussion about the acoustic radiation of pulsating sphere in right-angle space is drawn in the end, including the acoustic radiation directivity effect by the boundary characteristics, acoustic radiation frequency and acoustic source location. Numerical results show that: for the lower frequency radiation, the infection of free surface is significant; for the high frequency radiation, the infection of location is significant on the contrary. The research provides a new method for boundary characteristic problem of the structural-acoustic acoustic.展开更多
The scattering of shear waves (SH waves) by nano-scale arbitrary shape inclusion in infinite plane is studied by complex variable function theory. Firstly, the governing equation and the relationships between stress a...The scattering of shear waves (SH waves) by nano-scale arbitrary shape inclusion in infinite plane is studied by complex variable function theory. Firstly, the governing equation and the relationships between stress and displacement are given by classical elastic theory. Secondly, the arbitrary shape inclusion in the two-dimensional plane is transformed into a unit circle domain by conformal mapping, the incident wave field and the scattered wave field are presented. Next, the stress and displacement boundary conditions are established by considering surface elasticity theory, The infinite algebraic equations for solving the unknown coefficients of the scattered and standing waves are obtained. Finally, the influence of surface effect, non-dimensional wave number, Shear modulus and hole curvature on the dynamic stress concentration factor are analyzed by some examples, the numerical results show that the surface effect weakens the dynamic stress concentration. With the increase of wave number, the dynamic stress concentration factor (DSCF) decreases. Shear modulus and hole curvature have significant effects on DSCF.展开更多
Based on the theory of complex function and the principle of homogenization, harmonic dynamics stress of a radially infinite inhomogeneous medium with a circular cavity is investigated. Due to the symmetry, wave veloc...Based on the theory of complex function and the principle of homogenization, harmonic dynamics stress of a radially infinite inhomogeneous medium with a circular cavity is investigated. Due to the symmetry, wave velocity is assumed to have power-law variation in the radial direction only, and the shear modulus is constant. The Helmholtz equation with a variable coefficient is equivalently transformed into a standard Helmholtz equation with a general conformal transformation method（GCTM）. The displacements and stress fields are proposed. Numerical results show that the wave number and the inhomogeneity parameter of the medium have significant effects on the dynamic stress concentration around the circular cavity. The dynamic stress concentration factor（DSCF） becomes singular when the inhomogeneity parameter of medium is close to zero.展开更多
In this article,several theorems of fractional conformable derivatives and triple Sumudu transform are given and proved.Based on these theorems,a new conformable triple Sumudu decomposition method(CTSDM)is intrduced f...In this article,several theorems of fractional conformable derivatives and triple Sumudu transform are given and proved.Based on these theorems,a new conformable triple Sumudu decomposition method(CTSDM)is intrduced for the solution of singular two-dimensional conformable functional Burger's equation.This method is a combination of the decomposition method(DM)and Conformable triple Sumudu transform.The exact and approximation solutions obtained by using the suggested method in the sense of conformable.Particular examples are given to clarify the possible application of the achieved results and the exact and approximate solution are sketched by using Matlab software.展开更多
This article studies a nonlinear fractional order Lotka-Volterra prey-predator type dynamical system.For the proposed study,we consider the model under the conformable fractional order derivative(CFOD).We investigate ...This article studies a nonlinear fractional order Lotka-Volterra prey-predator type dynamical system.For the proposed study,we consider the model under the conformable fractional order derivative(CFOD).We investigate the mentioned dynamical system for the existence and uniqueness of at least one solution.Indeed,Schauder and Banach fixed point theorems are utilized to prove our claim.Further,an algorithm for the approximate analytical solution to the proposed problem has been established.In this regard,the conformable fractional differential transform(CFDT)technique is used to compute the required results in the form of a series.Using Matlab-16,we simulate the series solution to illustrate our results graphically.Finally,a comparison of our solution to that obtained for the Caputo fractional order derivative via the perturbation method is given.展开更多
The Laplace transformation is a very important integral transform,and it is extensively used in solving ordinary differential equations,partial differential equations,and several types of integro-differential equation...The Laplace transformation is a very important integral transform,and it is extensively used in solving ordinary differential equations,partial differential equations,and several types of integro-differential equations.Our purpose in this study is to introduce the notion of fuzzy double Laplace transform,fuzzy conformable double Laplace transform(FCDLT).We discuss some basic properties of FCDLT.We obtain the solutions of fuzzy partial differential equations(both one-dimensional and two-dimensional cases)through the double Laplace approach.We demonstrate through numerical examples that our proposed method is very successful and convenient for resolving partial differential equations.展开更多
The Cauchy stress equations (1823), the Cosserat couple-stress equations (1909), the Clausius virial equation (1870) and the Maxwell/Weyl equations (1873, 1918) are among the most famous partial differential equations...The Cauchy stress equations (1823), the Cosserat couple-stress equations (1909), the Clausius virial equation (1870) and the Maxwell/Weyl equations (1873, 1918) are among the most famous partial differential equations that can be found today in any textbook dealing with elasticity theory, continuum mechanics, thermodynamics or electromagnetism. Over a manifold of dimension n, their respective numbers are n,n(n−1)/2,1,nwith a total of N=(n+1)(n+2)/2, that is 15 when n=4for space-time. This is also just the number of parameters of the Lie group of conformal transformations with n translations, n(n−1)/2rotations, 1 dilatation and n highly non-linear elations introduced by E. Cartan in 1922. The purpose of this paper is to prove that the form of these equations only depends on the structure of the conformal group for an arbitrary n≥1because they are described as a whole by the (formal) adjoint of the first Spencer operator existing in the Spencer differential sequence. Such a group theoretical implication is obtained by applying totally new differential geometric methods in field theory. In particular, when n=4, the main idea is not to shrink the group from 10 down to 4 or 2 parameters by using the Schwarzschild or Kerr metrics instead of the Minkowski metric, but to enlarge the group from 10 up to 11 or 15 parameters by using the Weyl or conformal group instead of the Poincaré group of space-time. Contrary to the Einstein equations, these equations can be all parametrized by the adjoint of the second Spencer operator through Nn(n−1)/2potentials. These results bring the need to revisit the mathematical foundations of both General Relativity and Gauge Theory according to a clever but rarely quoted paper of H. Poincaré (1901). They strengthen the recent comments we already made about the dual confusions made by Einstein (1915) while following Beltrami (1892), both using the same Einstein operator but ignoring it is self-adjoint in the framework of differential double duality.展开更多
We perform a semi-analytical calculation of the field distributions of a conformal invisible device via mode expansions. For a discrete set of frequencies in the regime of wave optics, the conformal invisible device i...We perform a semi-analytical calculation of the field distributions of a conformal invisible device via mode expansions. For a discrete set of frequencies in the regime of wave optics, the conformal invisible device is perfectly transparent, which stems from the special conformal mapping and the refractive- index profile of the Mikaelian lens.展开更多
We use Weyl transformations between the Minkowski spacetime and dS/AdS spacetime to show that one cannot well define the electrodynamics globally on the ordinary conformal compactification of the Minkowski spacetime(o...We use Weyl transformations between the Minkowski spacetime and dS/AdS spacetime to show that one cannot well define the electrodynamics globally on the ordinary conformal compactification of the Minkowski spacetime(or dS/AdS spacetime),where the electromagnetic field has a sign factor(and thus is discountinuous)at the light cone.This problem is intuitively and clearly shown by the Penrose diagrams,from which one may find the remedy without too much difficulty.We use the Minkowski and dS spacetimes together to cover the compactified space,which in fact leads to the doubled conformal compactification.On this doubled conformal compactification,we obtain the globally well-defined electrodynamics.展开更多
In recent years there has been a lot of interest in discussing frame depeudences/independences of the cosmological perturbations under the conforlnal transformations. This problem has previously been investigated in L...In recent years there has been a lot of interest in discussing frame depeudences/independences of the cosmological perturbations under the conforlnal transformations. This problem has previously been investigated in Lerlns of the cow^riant approach for a single component universe, and it was found that tile covariant approach is very powerful to pick out the perturbative variables which are both gauge and conformal invariant. In this work, we extend the covariant approach to a universe with multicomponent fluids. We find that similar results can be derived, as e, xpected. In addition, some other interesting perturbations are also identified to be conformal invariant, such as entropy perturbation between two different components.展开更多
Let x ： M^n→S^n＋1 be an immersed hypersurface in the （n ＋1）-dimensional sphere S^n＋1. If, for any points p,q ∈ M^n, there exists a Mobius transformation Ф ： S^n＋l →S^n＋1 such that Ф o x（M^n） = x（M^n）...Let x ： M^n→S^n＋1 be an immersed hypersurface in the （n ＋1）-dimensional sphere S^n＋1. If, for any points p,q ∈ M^n, there exists a Mobius transformation Ф ： S^n＋l →S^n＋1 such that Ф o x（M^n） = x（M^n） and Ф o x（p） = x（q）, then the hypersurface is called a Mobius homogeneous hypersurface. In this paper, the Mobius homogeneous hypersurfaces with three distinct principal curvatures are classified completely up to a Mobius transformation.展开更多
基金辽宁省教育厅高校科研项目，Natural Science Foundation of Liaoning Provence of China
文摘The double constraint equations in the self-dual gravitational theory containing the cosmological term are derived in gravity. Furthermore, in order to deeply study the Lorentzian and Euclidean reality conditions for this theory, the relations between constraints are discussed by introducing the double constant conformal transformation and the double complex function method.
文摘The accurate calculation and measurement of welding temperature field is an important precondition for welding metallurgical analysis and welding process controlling. In this paper, the conformal transformation is firstly used to analyze the welding temperature field of featheredged cylinder. The center of the cylinder is chosen as the origin of column coordinate system, and every point may be expressed as complex field vector. The branch isogonality counterchanges the line parallel with the fusion line in half-infinite z-plane to the circle concentric with the fusion line in infinite cylinder. The Laplace equation and Poisson＇s equation still keep validity, so the temperature field equation can be solved. The conformal transformation and equation solution is processed by Matlab program language. It shows that the obtained analytical modeling of temperature field for featheredged cylinder based on conformal transformation is effective and accurate.
文摘A framework for analytical studies of superconducting systems is presented and illustrated. The formalism, based on the conformal transformation of momentum space, allows one to study the effects of both the dispersion relation and the structure of the pairing interaction in two-dimensional anisotropic high-T <sub>c</sub> superconductors. In this method, the number of employed degrees of freedom coincides with the dimension of the momentum space, which is different compared to that in the standard Van Hove scenario with a single degree of freedom. A new function, the kernel of the density of states, is defined and its relation to the standard density of states is explained. The versatility of the method is illustrated by analyzing coexistence and competition between spin-singlet and spin-triplet order parameters in superconducting systems with a tight-binding-type dispersion relation and an anisotropic pairing potential. Phase diagrams of stable superconducting states in the coordinates ?· (the ratio of hopping parameters) and n (the carrier concentration) are presented and discussed. Moreover, the role of attractive and repulsive on-site interactions for the stability of the s-wave order parameter is explained.
文摘The acoustic cloak can manipulate sound waves in surprising ways and enable us to guide the trajectories of waves at will.In contrast to the existing approaches for designing such devices that need exotic material parameters,here we demonstrate how to design a layered and isotropic acoustic cloak based on conformal transformation acoustics.A petal-shaped layered acoustic cloak is designed from the reduction of material parameters of an approximate rectangular cloak.The resultant material parameters span a narrow range,thus strongly facilitating the implementation.The invisibility performance of the cloak for different shapes of acoustic wavefront is verified using a full-wave finite-element analysis.
基金Supported by National Natural Science Foundation of China (No.60532020,60301008,60472077,50337020), the High Tech-nique Research and Development Program of China (No.2001AA413210).
文摘An intuitive 2D model of circular electrical impedance tomography （EIT） sensor with small size electrodes is established based on the theory of analytic functions. The validation of the model is proved using the result from the solution of Laplace equation. Suggestions on to electrode optimization and explanation to the ill-condition property of the sensitivity matrix are provided based on the model, which takes electrode distance into account and can be generalized to the sensor with any simple connected region through a conformal transformation. Image reconstruction algorithms based on the model are implemented to show feasibility of the model using experimental data collected from the EIT system developed in Tianjin University. In the simulation with a human chestlike configuration, electrical conductivity distributions are reconstructed using equi-potential backprojection （EBP） and Tikhonov regularization （TR） based on a conformal transformation of the model. The algorithms based on the model are suitable for online image reconstruction and the reconstructed results are aood both in size and position.
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.51879245,41731284&11672360)the Fundamental Research Funds for the Central Universities,China University of Geosciences(Wuhan)(Grant Nos.CUGCJ1821&CUG170645)。
文摘Here,we describe the robust and efficient application of the conventional 3D BEM in solving elasticity problems. We have focused on the precise computation of weakly singular integrals. The conformal Duffy-distance transformation was employed to alleviate near singularities caused from two aspects:(1) the large aspect ratio of elements,i.e.,element shape distortions;and(2)the closeness of element boundaries to field points,i.e.,ill-shaped patches. Then,the rigid body motion method was employed to evaluate strongly singular integrals. Numerical solutions of 3D elastostatic problems demonstrated the high accuracy of the proposed method with coarse meshes and high convergence rates with mesh refinement. Compared with the Duffy transformation and original polar coordinate transformations,the proposed method is insensitive to element shapes.
文摘The high-order boundary conditions for the problems cf Laplace equation in infinite region have been developed. The improvement in accuracy for numerical solution is achieved by imposing the high-order boundary conditions on the exterior boundarv of a reduced finite region in which the numerical method is used. So both the computing efforts and the required storage in computer are reduced. The numerical examples show that the 1st-order boundary condition approaches to the exact boundary condition and it is clearly superior to the traditional boundary condition and the 2nd-order boundary condition.
基金Project supported by the National Natural Science Foundation of China (Grant No. 92050102)the National Key Research and Development Program of China (Grant No. 2020YFA0710100)the Fundamental Research Funds for Central Universities, China (Grant Nos. 20720200074, 20720220134, 202006310051, and 20720220033)。
文摘Super-resolution imaging is vital for optical applications, such as high capacity information transmission, real-time bio-molecular imaging, and nanolithography. In recent years, technologies and methods of super-resolution imaging have attracted much attention. Different kinds of novel lenses, from the superlens to the super-oscillatory lens, have been designed and fabricated to break through the diffraction limit. However, the effect of the super-resolution imaging in these lenses is not satisfactory due to intrinsic loss, aberration, large sidebands, and so on. Moreover, these lenses also cannot realize multiple super-resolution imaging. In this research, we introduce the solid immersion mechanism to Mikaelian lens(ML) for multiple super-resolution imaging. The effect is robust and valid for broadband frequencies. Based on conformal transformation optics as a bridge linking the solid immersion ML and generalized Maxwell's fish-eye lens(GMFEL), we also discovered the effect of multiple super-resolution imaging in the solid immersion GMFEL.
基金Foundation item: Supported by the Natural Science foundation of Henan Education Committee (20021100002)
文摘In this paper, we have considered a class curves with some geometric properties in a higher dimensional space and obtained the differential equation of such a class curves, which are called the hyperbolas. We have considered also hyperbola-preserving conformal transformation and the relevant physical sense. And therefore obtained other invariant properties under the illustrious concircular transformation.
文摘The principle objective of the paper is to study the acoustic radiation problem of the 3D space domain with control boundary. By using the conformal transformation theory, the Green's function for acoustic point source in the control domain space is obtained. With it, the expression of acoustic radiation function of the control domain is formed. Discussion about the acoustic radiation of pulsating sphere in right-angle space is drawn in the end, including the acoustic radiation directivity effect by the boundary characteristics, acoustic radiation frequency and acoustic source location. Numerical results show that: for the lower frequency radiation, the infection of free surface is significant; for the high frequency radiation, the infection of location is significant on the contrary. The research provides a new method for boundary characteristic problem of the structural-acoustic acoustic.
文摘The scattering of shear waves (SH waves) by nano-scale arbitrary shape inclusion in infinite plane is studied by complex variable function theory. Firstly, the governing equation and the relationships between stress and displacement are given by classical elastic theory. Secondly, the arbitrary shape inclusion in the two-dimensional plane is transformed into a unit circle domain by conformal mapping, the incident wave field and the scattered wave field are presented. Next, the stress and displacement boundary conditions are established by considering surface elasticity theory, The infinite algebraic equations for solving the unknown coefficients of the scattered and standing waves are obtained. Finally, the influence of surface effect, non-dimensional wave number, Shear modulus and hole curvature on the dynamic stress concentration factor are analyzed by some examples, the numerical results show that the surface effect weakens the dynamic stress concentration. With the increase of wave number, the dynamic stress concentration factor (DSCF) decreases. Shear modulus and hole curvature have significant effects on DSCF.
基金Project supported by the Earthquake Industry Special Science Research Foundation Project(No.201508026-02)the Natural Science Foundation of Heilongjiang Province of China(No.A201310)
文摘Based on the theory of complex function and the principle of homogenization, harmonic dynamics stress of a radially infinite inhomogeneous medium with a circular cavity is investigated. Due to the symmetry, wave velocity is assumed to have power-law variation in the radial direction only, and the shear modulus is constant. The Helmholtz equation with a variable coefficient is equivalently transformed into a standard Helmholtz equation with a general conformal transformation method（GCTM）. The displacements and stress fields are proposed. Numerical results show that the wave number and the inhomogeneity parameter of the medium have significant effects on the dynamic stress concentration around the circular cavity. The dynamic stress concentration factor（DSCF） becomes singular when the inhomogeneity parameter of medium is close to zero.
基金The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding this Research group No(RG-1440-030).
文摘In this article,several theorems of fractional conformable derivatives and triple Sumudu transform are given and proved.Based on these theorems,a new conformable triple Sumudu decomposition method(CTSDM)is intrduced for the solution of singular two-dimensional conformable functional Burger's equation.This method is a combination of the decomposition method(DM)and Conformable triple Sumudu transform.The exact and approximation solutions obtained by using the suggested method in the sense of conformable.Particular examples are given to clarify the possible application of the achieved results and the exact and approximate solution are sketched by using Matlab software.
文摘This article studies a nonlinear fractional order Lotka-Volterra prey-predator type dynamical system.For the proposed study,we consider the model under the conformable fractional order derivative(CFOD).We investigate the mentioned dynamical system for the existence and uniqueness of at least one solution.Indeed,Schauder and Banach fixed point theorems are utilized to prove our claim.Further,an algorithm for the approximate analytical solution to the proposed problem has been established.In this regard,the conformable fractional differential transform(CFDT)technique is used to compute the required results in the form of a series.Using Matlab-16,we simulate the series solution to illustrate our results graphically.Finally,a comparison of our solution to that obtained for the Caputo fractional order derivative via the perturbation method is given.
基金Manar A.Alqudah would like to thank Princess Nourah bint Abdulrahman University Researchers Supporting Project No.(PNURSP2022R14),Princess Nourah bint Abdulrahman University,Riyadh,Saudi Arabia。
文摘The Laplace transformation is a very important integral transform,and it is extensively used in solving ordinary differential equations,partial differential equations,and several types of integro-differential equations.Our purpose in this study is to introduce the notion of fuzzy double Laplace transform,fuzzy conformable double Laplace transform(FCDLT).We discuss some basic properties of FCDLT.We obtain the solutions of fuzzy partial differential equations(both one-dimensional and two-dimensional cases)through the double Laplace approach.We demonstrate through numerical examples that our proposed method is very successful and convenient for resolving partial differential equations.
文摘The Cauchy stress equations (1823), the Cosserat couple-stress equations (1909), the Clausius virial equation (1870) and the Maxwell/Weyl equations (1873, 1918) are among the most famous partial differential equations that can be found today in any textbook dealing with elasticity theory, continuum mechanics, thermodynamics or electromagnetism. Over a manifold of dimension n, their respective numbers are n,n(n−1)/2,1,nwith a total of N=(n+1)(n+2)/2, that is 15 when n=4for space-time. This is also just the number of parameters of the Lie group of conformal transformations with n translations, n(n−1)/2rotations, 1 dilatation and n highly non-linear elations introduced by E. Cartan in 1922. The purpose of this paper is to prove that the form of these equations only depends on the structure of the conformal group for an arbitrary n≥1because they are described as a whole by the (formal) adjoint of the first Spencer operator existing in the Spencer differential sequence. Such a group theoretical implication is obtained by applying totally new differential geometric methods in field theory. In particular, when n=4, the main idea is not to shrink the group from 10 down to 4 or 2 parameters by using the Schwarzschild or Kerr metrics instead of the Minkowski metric, but to enlarge the group from 10 up to 11 or 15 parameters by using the Weyl or conformal group instead of the Poincaré group of space-time. Contrary to the Einstein equations, these equations can be all parametrized by the adjoint of the second Spencer operator through Nn(n−1)/2potentials. These results bring the need to revisit the mathematical foundations of both General Relativity and Gauge Theory according to a clever but rarely quoted paper of H. Poincaré (1901). They strengthen the recent comments we already made about the dual confusions made by Einstein (1915) while following Beltrami (1892), both using the same Einstein operator but ignoring it is self-adjoint in the framework of differential double duality.
基金This work was supported by the National Natural Science Foundation of China for Excellent Young Scientists （Grant No. 61322504）, the Foundation for the Author of National Excellent Doctoral Dissertation of China （Grant No. 201217）, andFundamental Research Funds for the Central Universities （Grant No. 20720170015）.
文摘We perform a semi-analytical calculation of the field distributions of a conformal invisible device via mode expansions. For a discrete set of frequencies in the regime of wave optics, the conformal invisible device is perfectly transparent, which stems from the special conformal mapping and the refractive- index profile of the Mikaelian lens.
基金supported by the National Natural Science Foundation of China(Grant Nos.11075206 and 11175245)
文摘We use Weyl transformations between the Minkowski spacetime and dS/AdS spacetime to show that one cannot well define the electrodynamics globally on the ordinary conformal compactification of the Minkowski spacetime(or dS/AdS spacetime),where the electromagnetic field has a sign factor(and thus is discountinuous)at the light cone.This problem is intuitively and clearly shown by the Penrose diagrams,from which one may find the remedy without too much difficulty.We use the Minkowski and dS spacetimes together to cover the compactified space,which in fact leads to the doubled conformal compactification.On this doubled conformal compactification,we obtain the globally well-defined electrodynamics.
文摘In recent years there has been a lot of interest in discussing frame depeudences/independences of the cosmological perturbations under the conforlnal transformations. This problem has previously been investigated in Lerlns of the cow^riant approach for a single component universe, and it was found that tile covariant approach is very powerful to pick out the perturbative variables which are both gauge and conformal invariant. In this work, we extend the covariant approach to a universe with multicomponent fluids. We find that similar results can be derived, as e, xpected. In addition, some other interesting perturbations are also identified to be conformal invariant, such as entropy perturbation between two different components.
基金supported by the National Natural Science Foundation of China(Nos.11571037,11471021)
文摘Let x ： M^n→S^n＋1 be an immersed hypersurface in the （n ＋1）-dimensional sphere S^n＋1. If, for any points p,q ∈ M^n, there exists a Mobius transformation Ф ： S^n＋l →S^n＋1 such that Ф o x（M^n） = x（M^n） and Ф o x（p） = x（q）, then the hypersurface is called a Mobius homogeneous hypersurface. In this paper, the Mobius homogeneous hypersurfaces with three distinct principal curvatures are classified completely up to a Mobius transformation.
