A numerical method is proposed to calculate the eigenvalues of the Zakharov–Shabat system based on Chebyshev polynomials. A mapping in the form of tanh(ax) is constructed according to the asymptotic of the potential ...A numerical method is proposed to calculate the eigenvalues of the Zakharov–Shabat system based on Chebyshev polynomials. A mapping in the form of tanh(ax) is constructed according to the asymptotic of the potential function for the Zakharov–Shabat eigenvalue problem. The mapping can distribute Chebyshev nodes very well considering the gradient for the potential function. Using Chebyshev polynomials, tanh(ax) mapping, and Chebyshev nodes, the Zakharov–Shabat eigenvalue problem is transformed into a matrix eigenvalue problem. This method has good convergence for the Satsuma–Yajima potential and the convergence rate is faster than the Fourier collocation method. This method is not only suitable for simple potential functions but also converges quickly for a complex Y-shape potential. It can also be further extended to other linear eigenvalue problems.展开更多
This study presents the Chebyshev polynomials-based Ritz method to examine the thermal buckling and free vibration characteristics of metal foam beams.The analyses include three models for porosity distribution and tw...This study presents the Chebyshev polynomials-based Ritz method to examine the thermal buckling and free vibration characteristics of metal foam beams.The analyses include three models for porosity distribution and two scenarios for thermal distribution.The material properties are assessed under two conditions,i.e.,temperature dependence and temperature independence.The theoretical framework for the beams is based on the higher-order shear deformation theory,which incorporates shear deformations with higher-order polynomials.The governing equations are established from the Lagrange equations,and the beam displacement fields are approximated by the Chebyshev polynomials.Numerical simulations are performed to evaluate the effects of thermal load,slenderness,boundary condition(BC),and porosity distribution on the buckling and vibration behaviors of metal foam beams.The findings highlight the significant influence of temperature-dependent(TD)material properties on metal foam beams'buckling and vibration responses.展开更多
The Chebyshev polynomial approximation is applied to investigate the stochastic period-doubling bifurcation and chaos problems of a stochastic Duffing-van der Pol system with bounded random parameter of exponential pr...The Chebyshev polynomial approximation is applied to investigate the stochastic period-doubling bifurcation and chaos problems of a stochastic Duffing-van der Pol system with bounded random parameter of exponential probability density function subjected to a harmonic excitation. Firstly the stochastic system is reduced into its equivalent deterministic one, and then the responses of stochastic system can be obtained by numerical methods. Nonlinear dynamical behaviour related to stochastic period-doubling bifurcation and chaos in the stochastic system is explored. Numerical simulations show that similar to its counterpart in deterministic nonlinear system of stochastic period-doubling bifurcation and chaos may occur in the stochastic Duffing-van der Pol system even for weak intensity of random parameter. Simply increasing the intensity of the random parameter may result in the period-doubling bifurcation which is absent from the deterministic system.展开更多
The dynamic characteristics of a beam-cable coupled system are investigated using an improved Chebyshev spectral element method in order to observe the effects of adding cables on the beam. The system is modeled as a ...The dynamic characteristics of a beam-cable coupled system are investigated using an improved Chebyshev spectral element method in order to observe the effects of adding cables on the beam. The system is modeled as a double Timoshenko beam system interconnected by discrete springs. Utilizing Chebyshev series expansion and meshing the system according to the locations of its connections, numerical results of the natural frequencies and mode shapes are obtained using only a few elements, and the results are validated by comparing them with the results of a finite-element method. Then the effects of the cable parameters and layout of connections on the natural frequencies and mode shapes of a fixed-pinned beam are studied. The results show that the modes of a beam-cable coupled system can be classified into two types, beam mode and cable mode, according to the dominant deformation. To avoid undesirable vibrations of the cable, its parameters should be controlled in a reasonable range, or the layout of the connections should be optimized.展开更多
A new numerical approach, called the “subdomain Chebyshev spectral method” is presented for calculation of the spatial derivatives in a curved coordinate system, which may be employed for numerical solutions of part...A new numerical approach, called the “subdomain Chebyshev spectral method” is presented for calculation of the spatial derivatives in a curved coordinate system, which may be employed for numerical solutions of partial differential equations defined in a 2D or 3D geological model. The new approach refers to a “strong version” against the “weak version” of the subspace spectral method based on the variational principle or Galerkin’s weighting scheme. We incorporate local nonlinear transformations and global spline interpolations in a curved coordinate system and make the discrete grid exactly matches geometry of the model so that it is achieved to convert the global domain into subdomains and apply Chebyshev points to locally sampling physical quantities and globally computing the spatial derivatives. This new approach not only remains exponential convergence of the standard spectral method in subdomains, but also yields a sparse assembled matrix when applied for the global domain simulations. We conducted 2D and 3D synthetic experiments and compared accuracies of the numerical differentiations with traditional finite difference approaches. The results show that as the points of differentiation vector are larger than five, the subdomain Chebyshev spectral method significantly improve the accuracies of the finite difference approaches.展开更多
We raise and partly answer the question: whether there exists a Markov system with respect to which the zeros of the Chebyshev polynomials are dense, but the maximum length of a zero free interval of the nth Chebyshev...We raise and partly answer the question: whether there exists a Markov system with respect to which the zeros of the Chebyshev polynomials are dense, but the maximum length of a zero free interval of the nth Chebyshev polynomial does not tends to zero. We also draw the conclu- tion that a Markov system, under an additional assumption, is dense if and only if the maxi- mum length of a zero free interval of the nth associated Chebyshev polynomial tends to zero.展开更多
Internet of Things(IoT)applications can be found in various industry areas,including critical infrastructure and healthcare,and IoT is one of several technological developments.As a result,tens of billions or possibly...Internet of Things(IoT)applications can be found in various industry areas,including critical infrastructure and healthcare,and IoT is one of several technological developments.As a result,tens of billions or possibly hundreds of billions of devices will be linked together.These smart devices will be able to gather data,process it,and even come to decisions on their own.Security is the most essential thing in these situations.In IoT infrastructure,authenticated key exchange systems are crucial for preserving client and data privacy and guaranteeing the security of data-in-transit(e.g.,via client identification and provision of secure communication).It is still challenging to create secure,authenticated key exchange techniques.The majority of the early authenticated key agreement procedure depended on computationally expensive and resource-intensive pairing,hashing,or modular exponentiation processes.The focus of this paper is to propose an efficient three-party authenticated key exchange procedure(AKEP)using Chebyshev chaotic maps with client anonymity that solves all the problems mentioned above.The proposed three-party AKEP is protected from several attacks.The proposed three-party AKEP can be used in practice for mobile communications and pervasive computing applications,according to statistical experiments and low processing costs.To protect client identification when transferring data over an insecure public network,our three-party AKEP may also offer client anonymity.Finally,the presented procedure offers better security features than the procedures currently available in the literature.展开更多
The current study examines the important class of Chebyshev’s differential equations via the application of the efficient Adomian Decomposition Method (ADM) and its modifications. We have proved the effectiveness of ...The current study examines the important class of Chebyshev’s differential equations via the application of the efficient Adomian Decomposition Method (ADM) and its modifications. We have proved the effectiveness of the employed methods by acquiring exact analytical solutions for the governing equations in most cases;while minimal noisy error terms have been observed in a particular method modification. Above all, the presented approaches have rightly affirmed the exactitude of the available literature. More to the point, the application of this methodology could be extended to examine various forms of high-order differential equations, as approximate exact solutions are rapidly attained with less computation stress.展开更多
基金Project supported by the National Natural Science Foundation of China (Grant Nos.52171251,U2106225,and 52231011)Dalian Science and Technology Innovation Fund (Grant No.2022JJ12GX036)。
文摘A numerical method is proposed to calculate the eigenvalues of the Zakharov–Shabat system based on Chebyshev polynomials. A mapping in the form of tanh(ax) is constructed according to the asymptotic of the potential function for the Zakharov–Shabat eigenvalue problem. The mapping can distribute Chebyshev nodes very well considering the gradient for the potential function. Using Chebyshev polynomials, tanh(ax) mapping, and Chebyshev nodes, the Zakharov–Shabat eigenvalue problem is transformed into a matrix eigenvalue problem. This method has good convergence for the Satsuma–Yajima potential and the convergence rate is faster than the Fourier collocation method. This method is not only suitable for simple potential functions but also converges quickly for a complex Y-shape potential. It can also be further extended to other linear eigenvalue problems.
文摘This study presents the Chebyshev polynomials-based Ritz method to examine the thermal buckling and free vibration characteristics of metal foam beams.The analyses include three models for porosity distribution and two scenarios for thermal distribution.The material properties are assessed under two conditions,i.e.,temperature dependence and temperature independence.The theoretical framework for the beams is based on the higher-order shear deformation theory,which incorporates shear deformations with higher-order polynomials.The governing equations are established from the Lagrange equations,and the beam displacement fields are approximated by the Chebyshev polynomials.Numerical simulations are performed to evaluate the effects of thermal load,slenderness,boundary condition(BC),and porosity distribution on the buckling and vibration behaviors of metal foam beams.The findings highlight the significant influence of temperature-dependent(TD)material properties on metal foam beams'buckling and vibration responses.
基金Project supported by the National Natural Science Foundation of China （Grants Nos 10472091 and 10332030）.
文摘The Chebyshev polynomial approximation is applied to investigate the stochastic period-doubling bifurcation and chaos problems of a stochastic Duffing-van der Pol system with bounded random parameter of exponential probability density function subjected to a harmonic excitation. Firstly the stochastic system is reduced into its equivalent deterministic one, and then the responses of stochastic system can be obtained by numerical methods. Nonlinear dynamical behaviour related to stochastic period-doubling bifurcation and chaos in the stochastic system is explored. Numerical simulations show that similar to its counterpart in deterministic nonlinear system of stochastic period-doubling bifurcation and chaos may occur in the stochastic Duffing-van der Pol system even for weak intensity of random parameter. Simply increasing the intensity of the random parameter may result in the period-doubling bifurcation which is absent from the deterministic system.
基金supported by the National Basic Research Program of China (Grant 2013CB733004)
文摘The dynamic characteristics of a beam-cable coupled system are investigated using an improved Chebyshev spectral element method in order to observe the effects of adding cables on the beam. The system is modeled as a double Timoshenko beam system interconnected by discrete springs. Utilizing Chebyshev series expansion and meshing the system according to the locations of its connections, numerical results of the natural frequencies and mode shapes are obtained using only a few elements, and the results are validated by comparing them with the results of a finite-element method. Then the effects of the cable parameters and layout of connections on the natural frequencies and mode shapes of a fixed-pinned beam are studied. The results show that the modes of a beam-cable coupled system can be classified into two types, beam mode and cable mode, according to the dominant deformation. To avoid undesirable vibrations of the cable, its parameters should be controlled in a reasonable range, or the layout of the connections should be optimized.
文摘A new numerical approach, called the “subdomain Chebyshev spectral method” is presented for calculation of the spatial derivatives in a curved coordinate system, which may be employed for numerical solutions of partial differential equations defined in a 2D or 3D geological model. The new approach refers to a “strong version” against the “weak version” of the subspace spectral method based on the variational principle or Galerkin’s weighting scheme. We incorporate local nonlinear transformations and global spline interpolations in a curved coordinate system and make the discrete grid exactly matches geometry of the model so that it is achieved to convert the global domain into subdomains and apply Chebyshev points to locally sampling physical quantities and globally computing the spatial derivatives. This new approach not only remains exponential convergence of the standard spectral method in subdomains, but also yields a sparse assembled matrix when applied for the global domain simulations. We conducted 2D and 3D synthetic experiments and compared accuracies of the numerical differentiations with traditional finite difference approaches. The results show that as the points of differentiation vector are larger than five, the subdomain Chebyshev spectral method significantly improve the accuracies of the finite difference approaches.
文摘We raise and partly answer the question: whether there exists a Markov system with respect to which the zeros of the Chebyshev polynomials are dense, but the maximum length of a zero free interval of the nth Chebyshev polynomial does not tends to zero. We also draw the conclu- tion that a Markov system, under an additional assumption, is dense if and only if the maxi- mum length of a zero free interval of the nth associated Chebyshev polynomial tends to zero.
文摘Internet of Things(IoT)applications can be found in various industry areas,including critical infrastructure and healthcare,and IoT is one of several technological developments.As a result,tens of billions or possibly hundreds of billions of devices will be linked together.These smart devices will be able to gather data,process it,and even come to decisions on their own.Security is the most essential thing in these situations.In IoT infrastructure,authenticated key exchange systems are crucial for preserving client and data privacy and guaranteeing the security of data-in-transit(e.g.,via client identification and provision of secure communication).It is still challenging to create secure,authenticated key exchange techniques.The majority of the early authenticated key agreement procedure depended on computationally expensive and resource-intensive pairing,hashing,or modular exponentiation processes.The focus of this paper is to propose an efficient three-party authenticated key exchange procedure(AKEP)using Chebyshev chaotic maps with client anonymity that solves all the problems mentioned above.The proposed three-party AKEP is protected from several attacks.The proposed three-party AKEP can be used in practice for mobile communications and pervasive computing applications,according to statistical experiments and low processing costs.To protect client identification when transferring data over an insecure public network,our three-party AKEP may also offer client anonymity.Finally,the presented procedure offers better security features than the procedures currently available in the literature.
文摘The current study examines the important class of Chebyshev’s differential equations via the application of the efficient Adomian Decomposition Method (ADM) and its modifications. We have proved the effectiveness of the employed methods by acquiring exact analytical solutions for the governing equations in most cases;while minimal noisy error terms have been observed in a particular method modification. Above all, the presented approaches have rightly affirmed the exactitude of the available literature. More to the point, the application of this methodology could be extended to examine various forms of high-order differential equations, as approximate exact solutions are rapidly attained with less computation stress.
