Due to their superior properties, the interest in nanostructures is increasing today in engineering. This study presents a new two-noded curved finite element for analyzing the in-plane static behaviors of curved nano...Due to their superior properties, the interest in nanostructures is increasing today in engineering. This study presents a new two-noded curved finite element for analyzing the in-plane static behaviors of curved nanobeams. Opposite to traditional curved finite elements developed by using approximate interpolation functions, the proposed curved finite element is developed by using exact analytical solutions. Although this approach was first introduced for analyzing the mechanical behaviors of macro-scale curved beams by adopting the local theory of elasticity, the exact analytical expressions used in this study were obtained from the solutions of governing equations that were expressed via the differential form of the nonlocal theory of elasticity. Therefore, the effects of shear strain and axial extension included in the analytical formulation are also inherited by the curved finite element developed here. The rigidity matrix and the consistent force vector are developed for a circular finite element. To demonstrate the applicability of the method, static analyses of various curved nanobeams subjected to different boundary conditions and loading scenarios are performed, and the obtained results are compared with the exact analytical ones. The presented study provides an accurate and low computational cost method for researchers to investigate the in-plane static behavior of curved nanobeams.展开更多
Transcritical and supercritical fluids widely exist in aerospace propulsion systems,such as the coolant flow in the regenerative cooling channels of scramjet engines.To numerically simulate the coolant flow,we must ad...Transcritical and supercritical fluids widely exist in aerospace propulsion systems,such as the coolant flow in the regenerative cooling channels of scramjet engines.To numerically simulate the coolant flow,we must address the challenges in solving Riemann problems(RPs)for real fluids under complex flow conditions.In this study,an exact numerical solution for the one-dimensional RP of two-parameter fluids is developed.Due to the comprehensive resolution of fluid thermodynamics,the proposed solution framework is suitable for all forms of the two-parameter equation of state(EoS).The pressure splitting method is introduced to enable parallel calculation of RPs across multiple grid points.Theoretical analysis demonstrates the isentropic nature of weak waves in two-parameter fluids,ensuring that the same mathematical properties as ideal gas could be applied in Newton's iteration.A series of numerical cases validate the effectiveness of the proposed method.A comparative analysis is conducted on the exact Riemann solutions for the real fluid EoS,the ideal gas EoS,and the improved ideal gas EoS under supercritical and transcritical conditions.The results indicate that the improved one produces smaller errors in the calculation of momentum and energy fluxes.展开更多
A polynomial scheme is proposed here to compute exact solutions of nonlinear partial differential equations(NPDEs)based on a series expansions of solutions and a renormalization group(RG)related resummation.The most s...A polynomial scheme is proposed here to compute exact solutions of nonlinear partial differential equations(NPDEs)based on a series expansions of solutions and a renormalization group(RG)related resummation.The most salient feature of the current approach is that only linear algebraic equations need to be solved to implement the resummation for closed-form exact solution and parameter dependence,which does not require any sophisticated analysis like Cole-Hopf transformation or Painlevétest.New exact solutions of typical NPDEs are computed with this novel method,including one-and two-soliton(solitary wave)solutions,periodic solutions of exponential or elliptic function type.Moreover,exact reduced equations may also be conveniently computed for further analysis.展开更多
The investigation centers on a generalized Toda lattice equation featuring four potentials.Initially,the continuous counterpart is first established using the continuous limit approach.Subsequently,leveraging its 3...The investigation centers on a generalized Toda lattice equation featuring four potentials.Initially,the continuous counterpart is first established using the continuous limit approach.Subsequently,leveraging its 3×3 matrix spectrum problem,the discrete generalized(m,3N-m)-fold Darboux transformation(DT)is formulated for this discrete system.Finally,the derived generalized DT is employed to generate precise rational solutions and hybrid rational-exponential solutions,which are thoroughly analyzed and graphically illustrated.These novel findings may offer insights into the lattice dynamical phenomena described by Toda lattice equation.展开更多
The Boussinesq equations,pivotal in the analysis of water wave dynamics,effectively model weakly nonlinear and long wave approximations.This study utilizes the complete discriminant system within a polynomial approach...The Boussinesq equations,pivotal in the analysis of water wave dynamics,effectively model weakly nonlinear and long wave approximations.This study utilizes the complete discriminant system within a polynomial approach to derive exact traveling wave solutions for the coupled Boussinesq equation.The solutions are articulated through soliton,trigonometric,rational,and Jacobi elliptic functions.Notably,the introduction of Jacobi elliptic function solutions for this model marks a pioneering advancement.Contour plots of the solutions obtained by assigning values to various parameters are generated and subsequently analyzed.The methodology proposed in this study offers a systematic means to tackle nonlinear partial differential equations in mathematical physics,thereby enhancing comprehension of the physical attributes and dynamics of water waves.展开更多
In order to find closed form solutions of nonintegrable nonlinear ordinary differential equations,numerous tricks have been proposed.The goal of this short review is to explain how a theorem of Eremenko on meromorphic...In order to find closed form solutions of nonintegrable nonlinear ordinary differential equations,numerous tricks have been proposed.The goal of this short review is to explain how a theorem of Eremenko on meromorphic solutions of some nonlinear ODEs together with some classical,19th-century results,can be turned into algorithms(thus avoiding ad hoc assumptions)which provide all(as opposed to some)solutions in a precise class.To illustrate these methods,we present some new such exact solutions,physically relevant.展开更多
The(2+1)-dimensional integrable generalization of the Gardner(2DG)equation is solved via the inverse scattering transform method in this paper.A kind of general solution of the equation is obtained by introducing long...The(2+1)-dimensional integrable generalization of the Gardner(2DG)equation is solved via the inverse scattering transform method in this paper.A kind of general solution of the equation is obtained by introducing long derivatives V_(x),V_(y),V_(t).Two different constraints on the kernel function K are introduced under the reality of the solution u of the 2DG equation.Then,two classes of exact solutions with constant asymptotic values at infinity u|x^(2)+y^(2)→∞→0 are constructed by means of the∂¯-dressing method for the casesσ=1 andσ=i.The rational and multiple pole solutions of the 2DG equation are obtained with the kernel functions of zero-order and higher-order Dirac delta functions,respectively.展开更多
基金supported by Scientific Research Projects Department of Istanbul Technical University.Project Number:MGA-2018-41546.Grant receiver:E.T.
文摘Due to their superior properties, the interest in nanostructures is increasing today in engineering. This study presents a new two-noded curved finite element for analyzing the in-plane static behaviors of curved nanobeams. Opposite to traditional curved finite elements developed by using approximate interpolation functions, the proposed curved finite element is developed by using exact analytical solutions. Although this approach was first introduced for analyzing the mechanical behaviors of macro-scale curved beams by adopting the local theory of elasticity, the exact analytical expressions used in this study were obtained from the solutions of governing equations that were expressed via the differential form of the nonlocal theory of elasticity. Therefore, the effects of shear strain and axial extension included in the analytical formulation are also inherited by the curved finite element developed here. The rigidity matrix and the consistent force vector are developed for a circular finite element. To demonstrate the applicability of the method, static analyses of various curved nanobeams subjected to different boundary conditions and loading scenarios are performed, and the obtained results are compared with the exact analytical ones. The presented study provides an accurate and low computational cost method for researchers to investigate the in-plane static behavior of curved nanobeams.
基金Project supported by the National Natural Science Foundation of China(No.12525202)。
文摘Transcritical and supercritical fluids widely exist in aerospace propulsion systems,such as the coolant flow in the regenerative cooling channels of scramjet engines.To numerically simulate the coolant flow,we must address the challenges in solving Riemann problems(RPs)for real fluids under complex flow conditions.In this study,an exact numerical solution for the one-dimensional RP of two-parameter fluids is developed.Due to the comprehensive resolution of fluid thermodynamics,the proposed solution framework is suitable for all forms of the two-parameter equation of state(EoS).The pressure splitting method is introduced to enable parallel calculation of RPs across multiple grid points.Theoretical analysis demonstrates the isentropic nature of weak waves in two-parameter fluids,ensuring that the same mathematical properties as ideal gas could be applied in Newton's iteration.A series of numerical cases validate the effectiveness of the proposed method.A comparative analysis is conducted on the exact Riemann solutions for the real fluid EoS,the ideal gas EoS,and the improved ideal gas EoS under supercritical and transcritical conditions.The results indicate that the improved one produces smaller errors in the calculation of momentum and energy fluxes.
基金supported by the National Natural Science Foundation of China under Grant No.12375030the National Natural Science Foundation of China under Grant No.12471084.
文摘A polynomial scheme is proposed here to compute exact solutions of nonlinear partial differential equations(NPDEs)based on a series expansions of solutions and a renormalization group(RG)related resummation.The most salient feature of the current approach is that only linear algebraic equations need to be solved to implement the resummation for closed-form exact solution and parameter dependence,which does not require any sophisticated analysis like Cole-Hopf transformation or Painlevétest.New exact solutions of typical NPDEs are computed with this novel method,including one-and two-soliton(solitary wave)solutions,periodic solutions of exponential or elliptic function type.Moreover,exact reduced equations may also be conveniently computed for further analysis.
基金supported by National Natural Science Foundation of China(Grant No.12071042)Beijing Municipal Natural Science Foundation(Grant No.1242004)。
文摘The investigation centers on a generalized Toda lattice equation featuring four potentials.Initially,the continuous counterpart is first established using the continuous limit approach.Subsequently,leveraging its 3×3 matrix spectrum problem,the discrete generalized(m,3N-m)-fold Darboux transformation(DT)is formulated for this discrete system.Finally,the derived generalized DT is employed to generate precise rational solutions and hybrid rational-exponential solutions,which are thoroughly analyzed and graphically illustrated.These novel findings may offer insights into the lattice dynamical phenomena described by Toda lattice equation.
基金supported by the National Natural Science Foundation of China(Grant No.11925204).
文摘The Boussinesq equations,pivotal in the analysis of water wave dynamics,effectively model weakly nonlinear and long wave approximations.This study utilizes the complete discriminant system within a polynomial approach to derive exact traveling wave solutions for the coupled Boussinesq equation.The solutions are articulated through soliton,trigonometric,rational,and Jacobi elliptic functions.Notably,the introduction of Jacobi elliptic function solutions for this model marks a pioneering advancement.Contour plots of the solutions obtained by assigning values to various parameters are generated and subsequently analyzed.The methodology proposed in this study offers a systematic means to tackle nonlinear partial differential equations in mathematical physics,thereby enhancing comprehension of the physical attributes and dynamics of water waves.
基金partially supported by RGC(No.17307420)supported by NSFC(No.12471077)。
文摘In order to find closed form solutions of nonintegrable nonlinear ordinary differential equations,numerous tricks have been proposed.The goal of this short review is to explain how a theorem of Eremenko on meromorphic solutions of some nonlinear ODEs together with some classical,19th-century results,can be turned into algorithms(thus avoiding ad hoc assumptions)which provide all(as opposed to some)solutions in a precise class.To illustrate these methods,we present some new such exact solutions,physically relevant.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1237125611971475)。
文摘The(2+1)-dimensional integrable generalization of the Gardner(2DG)equation is solved via the inverse scattering transform method in this paper.A kind of general solution of the equation is obtained by introducing long derivatives V_(x),V_(y),V_(t).Two different constraints on the kernel function K are introduced under the reality of the solution u of the 2DG equation.Then,two classes of exact solutions with constant asymptotic values at infinity u|x^(2)+y^(2)→∞→0 are constructed by means of the∂¯-dressing method for the casesσ=1 andσ=i.The rational and multiple pole solutions of the 2DG equation are obtained with the kernel functions of zero-order and higher-order Dirac delta functions,respectively.
