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Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Population Balance Equations of the Crystallization Process
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作者 Chunlei Ruan Cengceng Dong +2 位作者 Kunfeng Liang Zhijun Liu Xinru Bao 《Computer Modeling in Engineering & Sciences》 SCIE EI 2024年第3期3033-3049,共17页
Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridyna... Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridynamic differential operator(EE–PDDO)was obtained for solving the one-dimensional population balance equation in crystallization.Four different conditions during crystallization were studied:size-independent growth,sizedependent growth in a batch process,nucleation and size-independent growth,and nucleation and size-dependent growth in a continuous process.The high accuracy of the EE–PDDO method was confirmed by comparing it with the numerical results obtained using the second-order upwind and HR-van methods.The method is characterized by non-oscillation and high accuracy,especially in the discontinuous and sharp crystal size distribution.The stability of the EE–PDDO method,choice of weight function in the PDDO method,and optimal time step are also discussed. 展开更多
关键词 Population balance equation CRYsTALLIZATION peridynamic differential operator Euler’s first-order explicit method
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Energy Stable Nodal DG Methods for Maxwell’s Equations of Mixed-Order Form in Nonlinear Optical Media
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作者 Maohui Lyu Vrushali A.Bokil +1 位作者 Yingda Cheng Fengyan Li 《Communications on Applied Mathematics and Computation》 EI 2024年第1期30-63,共34页
In this work,we develop energy stable numerical methods to simulate electromagnetic waves propagating in optical media where the media responses include the linear Lorentz dispersion,the instantaneous nonlinear cubic ... In this work,we develop energy stable numerical methods to simulate electromagnetic waves propagating in optical media where the media responses include the linear Lorentz dispersion,the instantaneous nonlinear cubic Kerr response,and the nonlinear delayed Raman molecular vibrational response.Unlike the first-order PDE-ODE governing equations considered previously in Bokil et al.(J Comput Phys 350:420–452,2017)and Lyu et al.(J Sci Comput 89:1–42,2021),a model of mixed-order form is adopted here that consists of the first-order PDE part for Maxwell’s equations coupled with the second-order ODE part(i.e.,the auxiliary differential equations)modeling the linear and nonlinear dispersion in the material.The main contribution is a new numerical strategy to treat the Kerr and Raman nonlinearities to achieve provable energy stability property within a second-order temporal discretization.A nodal discontinuous Galerkin(DG)method is further applied in space for efficiently handling nonlinear terms at the algebraic level,while preserving the energy stability and achieving high-order accuracy.Indeed with d_(E)as the number of the components of the electric field,only a d_(E)×d_(E)nonlinear algebraic system needs to be solved at each interpolation node,and more importantly,all these small nonlinear systems are completely decoupled over one time step,rendering very high parallel efficiency.We evaluate the proposed schemes by comparing them with the methods in Bokil et al.(2017)and Lyu et al.(2021)(implemented in nodal form)regarding the accuracy,computational efficiency,and energy stability,by a parallel scalability study,and also through the simulations of the soliton-like wave propagation in one dimension,as well as the spatial-soliton propagation and two-beam interactions modeled by the two-dimensional transverse electric(TE)mode of the equations. 展开更多
关键词 Maxwell’s equations Kerr and Raman Discontinuous Galerkin method Energy stability
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Implementation of the Integrated Green’s Function Method for 3D Poisson’s Equation in a Large Aspect Ratio Computational Domain
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作者 Ji Qiang Chad Mitchell +1 位作者 Remi Lehe Arianna Formenti 《Journal of Software Engineering and Applications》 2024年第9期740-749,共10页
The solution of Poisson’s Equation plays an important role in many areas, including modeling high-intensity and high-brightness beams in particle accelerators. For the computational domain with a large aspect ratio, ... The solution of Poisson’s Equation plays an important role in many areas, including modeling high-intensity and high-brightness beams in particle accelerators. For the computational domain with a large aspect ratio, the integrated Green’s function method has been adopted to solve the 3D Poisson equation subject to open boundary conditions. In this paper, we report on the efficient implementation of this method, which can save more than a factor of 50 computing time compared with the direct brute force implementation and its improvement under certain extreme conditions. 展开更多
关键词 Green’s Function Poisson equation Particle Accelerator
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Derivation of a Revised Tsiolkovsky Rocket Equation That Predicts Combustion Oscillations
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作者 Zaki Harari 《Advances in Aerospace Science and Technology》 2024年第1期10-27,共18页
Our study identifies a subtle deviation from Newton’s third law in the derivation of the ideal rocket equation, also known as the Tsiolkovsky Rocket Equation (TRE). TRE can be derived using a 1D elastic collision mod... Our study identifies a subtle deviation from Newton’s third law in the derivation of the ideal rocket equation, also known as the Tsiolkovsky Rocket Equation (TRE). TRE can be derived using a 1D elastic collision model of the momentum exchange between the differential propellant mass element (dm) and the rocket final mass (m1), in which dm initially travels forward to collide with m1 and rebounds to exit through the exhaust nozzle with a velocity that is known as the effective exhaust velocity ve. We observe that such a model does not explain how dm was able to acquire its initial forward velocity without the support of a reactive mass traveling in the opposite direction. We show instead that the initial kinetic energy of dm is generated from dm itself by a process of self-combustion and expansion. In our ideal rocket with a single particle dm confined inside a hollow tube with one closed end, we show that the process of self-combustion and expansion of dm will result in a pair of differential particles each with a mass dm/2, and each traveling away from one another along the tube axis, from the center of combustion. These two identical particles represent the active and reactive sub-components of dm, co-generated in compliance with Newton’s third law of equal action and reaction. Building on this model, we derive a linear momentum ODE of the system, the solution of which yields what we call the Revised Tsiolkovsky Rocket Equation (RTRE). We show that RTRE has a mathematical form that is similar to TRE, with the exception of the effective exhaust velocity (ve) term. The ve term in TRE is replaced in RTRE by the average of two distinct exhaust velocities that we refer to as fast-jet, vx<sub>1</sub>, and slow-jet, vx<sub>2</sub>. These two velocities correspond, respectively, to the velocities of the detonation pressure wave that is vectored directly towards the exhaust nozzle, and the retonation wave that is initially vectored in the direction of rocket propagation, but subsequently becomes reflected from the thrust surface of the combustion chamber to exit through the exhaust nozzle with a time lag behind the detonation wave. The detonation-retonation phenomenon is supported by experimental evidence in the published literature. Finally, we use a convolution model to simulate the composite exhaust pressure wave, highlighting the frequency spectrum of the pressure perturbations that are generated by the mutual interference between the fast-jet and slow-jet components. Our analysis offers insights into the origin of combustion oscillations in rocket engines, with possible extensions beyond rocket engineering into other fields of combustion engineering. 展开更多
关键词 Tsiolkovsky Rocket equation Ideal Rocket equation Rocket Propulsion Newton’s Third Law Combustion Oscillations Combustion Instability
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High-Order Spatial FDTD Solver of Maxwell’s Equations for Terahertz Radiation Production
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作者 Abdelrahman Mahdy 《Journal of Applied Mathematics and Physics》 2024年第4期1028-1042,共15页
We applied a spatial high-order finite-difference-time-domain (HO-FDTD) scheme to solve 2D Maxwell’s equations in order to develop a fluid model employed to study the production of terahertz radiation by the filament... We applied a spatial high-order finite-difference-time-domain (HO-FDTD) scheme to solve 2D Maxwell’s equations in order to develop a fluid model employed to study the production of terahertz radiation by the filamentation of two femtosecond lasers in air plasma. We examined the performance of the applied scheme, in this context, we implemented the developed model to study selected phenomena in terahertz radiation production, such as the excitation energy and conversion efficiency of the produced THz radiation, in addition to the influence of the pulse chirping on properties of the produced radiation. The obtained numerical results have clarified that the applied HO-FDTD scheme is precisely accurate to solve Maxwell’s equations and sufficiently valid to study the production of terahertz radiation by the filamentation of two femtosecond lasers in air plasma. 展开更多
关键词 The Finite-Difference-Time-Domain Terahertz Radiation Production Filamentation of Femtosecond Laser Maxwell’s equations solution
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On the Approximation of Fractal-Fractional Differential Equations Using Numerical Inverse Laplace Transform Methods
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作者 Kamran Siraj Ahmad +2 位作者 Kamal Shah Thabet Abdeljawad Bahaaeldin Abdalla 《Computer Modeling in Engineering & Sciences》 SCIE EI 2023年第6期2743-2765,共23页
Laplace transform is one of the powerful tools for solving differential equations in engineering and other science subjects.Using the Laplace transform for solving differential equations,however,sometimes leads to sol... Laplace transform is one of the powerful tools for solving differential equations in engineering and other science subjects.Using the Laplace transform for solving differential equations,however,sometimes leads to solutions in the Laplace domain that are not readily invertible to the real domain by analyticalmeans.Thus,we need numerical inversionmethods to convert the obtained solution fromLaplace domain to a real domain.In this paper,we propose a numerical scheme based on Laplace transform and numerical inverse Laplace transform for the approximate solution of fractal-fractional differential equations with orderα,β.Our proposed numerical scheme is based on three main steps.First,we convert the given fractal-fractional differential equation to fractional-differential equation in Riemann-Liouville sense,and then into Caputo sense.Secondly,we transformthe fractional differential equation in Caputo sense to an equivalent equation in Laplace space.Then the solution of the transformed equation is obtained in Laplace domain.Finally,the solution is converted into the real domain using numerical inversion of Laplace transform.Three inversion methods are evaluated in this paper,and their convergence is also discussed.Three test problems are used to validate the inversion methods.We demonstrate our results with the help of tables and figures.The obtained results show that Euler’s and Talbot’s methods performed better than Stehfest’s method. 展开更多
关键词 Fractal-fractional differential equation power law kernel exponential decay kernel Mittag-Leffler kernel Laplace transform Euler’s method Talbot’s method stehfest’s method
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Tiêu Equation Experimentation of Understanding by Energy Transfer Quantum Mechanics
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作者 Huu S. Tieu Martin F. Loeffler 《Natural Science》 CAS 2023年第3期91-102,共12页
Background: The Tiêu equation has a ground roots approach to the process of Quantum Biology and goes deeper through the incorporation of Quantum Mechanics. The process can be measured in plant, animal, and human ... Background: The Tiêu equation has a ground roots approach to the process of Quantum Biology and goes deeper through the incorporation of Quantum Mechanics. The process can be measured in plant, animal, and human usage through a variety of experimental or testing forms. Animal studies were conducted for which, in the first day of the study all the animals consistently gained dramatic weight, even as a toxic substance was introduced as described in the introduction of the paper to harm animal subjects which induced weight loss through toxicity. Tests can be made by incorporating blood report results. Human patients were also observed to show improvement to their health as administration of the substance was introduced to the biological mechanism and plants were initially exposed to the substance to observe results. This is consistent with the Tiêu equation which provides that wave function is created as the introduction of the substance to the biological mechanism which supports Quantum Mechanics. The Tiêu equation demonstrates that Quantum Mechanics moves a particle by temperature producing energy thru the blood-brain barrier for example. Methods: The methods for the Tiêu equation incorporate animal studies to include the substance administered through laboratory standards using Good Laboratory Practices under Title 40 C.F.R. § 158. Human patients were treated with the substance by medical professionals who are experts in their field and have knowledge to the response of patients. Plant applications were acquired for observation and guidance of ongoing experiments of animals’ representative for the biologics mechanism. Results: The animal studies along with patient blood testing results have been an impressive line that has followed the Tiêu equation to consistently show improvement in the introduction of the innovation to biologic mechanisms. The mechanism responds to the substance by producing energy to the mechanism with efficient effect. For plant observations, plant organisms responded, and were seen as showing improvement thru visual observation. 展开更多
关键词 Tiêu equation Experimentation of Understanding by Energy Transfer Quantum Mechanics Golden sunrise Nutraceutical Huu s. TIȆu schrödinger equation Quantum Mechanics Life Is Quantum Biology
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Solution of Laguerre’s Differential Equations via Modified Adomian Decomposition Method
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作者 Mariam Al-Mazmumy Aishah A. Alsulami 《Journal of Applied Mathematics and Physics》 2023年第1期85-100,共16页
This paper presents a technique for obtaining an exact solution for the well-known Laguerre’s differential equations that arise in the modeling of several phenomena in quantum mechanics and engineering. We utilize an... This paper presents a technique for obtaining an exact solution for the well-known Laguerre’s differential equations that arise in the modeling of several phenomena in quantum mechanics and engineering. We utilize an efficient procedure based on the modified Adomian decomposition method to obtain closed-form solutions of the Laguerre’s and the associated Laguerre’s differential equations. The proposed technique makes sense as the attitudes of the acquired solutions towards the neighboring singular points are correctly taken care of. 展开更多
关键词 Modification Method singular Ordinary Differential equations Laguerre’s equation Associated Laguerre’s equation
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Adomian Modification Methods for the Solution of Chebyshev’s Differential Equations
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作者 Mariam Al Mazmumy Aishah Alsulami +1 位作者 Huda Bakodah Nawal Alzaid 《Applied Mathematics》 2023年第8期512-530,共19页
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
关键词 ADM Modifications Methods Chebyshev’s Differential equations IVPs series solutions
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Verification of the Landau Equation and Hardy’s Inequality
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作者 Salih Yousuf Mohamed Salih 《Applied Mathematics》 2023年第3期208-229,共22页
We prove the L estimate for the isotropic version of the homogeneous landau problem, which was explored by M. Gualdani and N. Guillen. As shown in a region of the smooth potentials range under values of the interactio... We prove the L estimate for the isotropic version of the homogeneous landau problem, which was explored by M. Gualdani and N. Guillen. As shown in a region of the smooth potentials range under values of the interaction exponent (2), a weighted Poincaré inequality is a natural consequence of the traditional weighted Hardy inequality, which in turn implies that the norms of solutions propagate in the L1 space. Now, the L estimate is based on the work of De Giorgi, Nash, and Moser, as well as a few weighted Sobolev inequalities. 展开更多
关键词 Hardy’s Inequality sobolev Inequalities the Landau equation L-Estimate
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Pythagorician Divisors and Applications to Some Diophantine Equations
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作者 François Emmanuel Tanoé Prosper Kouadio Kimou 《Advances in Pure Mathematics》 2023年第2期35-70,共36页
We consider the Pythagoras equation X<sup>2</sup> +Y<sup>2</sup> = Z<sup>2</sup>, and for any solution of the type (a,b = 2<sup>s</sup>b<sub>1 </sub>≠0,c) ... We consider the Pythagoras equation X<sup>2</sup> +Y<sup>2</sup> = Z<sup>2</sup>, and for any solution of the type (a,b = 2<sup>s</sup>b<sub>1 </sub>≠0,c) ∈ N<sup>*3</sup>, s ≥ 2, b<sub>1</sub>odd, (a,b,c) ≡ (±1,0,1)(mod 4), c > a , c > b, and gcd(a,b,c) = 1, we then prove the Pythagorician divisors Theorem, which results in the following: , where (d,d′′) (resp. (e,e<sup>n</sup>)) are unique particular divisors of a and b, such that a = dd′′ (resp. b = ee′′ ), these divisors are called: Pythagorician divisors from a, (resp. from b). Let’s put λ ∈{0,1}, defined by: and S = s -λ (s -1). Then such that . Moreover the map is a bijection. We apply this new tool to obtain a new classification of the primitive, positive and non-trivial solutions of the Pythagoras equations: a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup> via the Pythagorician parameters (d,e,S ). We obtain for (d, e) fixed, the equivalence class of any Pythagorician solution (a,b,c), checking , namely: . We also update the solutions of some Diophantine equations of degree 2, already known, but very important for the resolution of other equations. With this tool of Pythagorean divisors, we have obtained (in another paper) new recurrent methods to solve Fermat’s equation: a<sup>4</sup> + b<sup>4 </sup>= c<sup>4</sup>, other than usual infinite descent method;and to solve congruent numbers problem. We believe that this tool can bring new arguments, for Diophantine resolution, of the general equations of Fermat: a<sup>2p</sup> + b<sup>2p</sup> = c<sup>2p</sup> and a<sup>p</sup> + b<sup>p</sup> = c<sup>p</sup>. MSC2020-Mathematical Sciences Classification System: 11A05-11A51-11D25-11D41-11D72. 展开更多
关键词 Pythagoras equation Pythagorician Triplets Diophantine equations of Degree 2 Factorisation-Gcd-Fermat’s equations
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A Family of Global Attractors for the Generalized Kirchhoff-Beam Equations
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作者 Guoguang Lin Boshi Chen 《Journal of Applied Mathematics and Physics》 2023年第7期1945-1963,共19页
In this paper, we discuss the existence and uniqueness of global solutions, the existence of the family of global attractors and its dimension estimation for generalized Beam-Kirchhoff equation under initial condition... In this paper, we discuss the existence and uniqueness of global solutions, the existence of the family of global attractors and its dimension estimation for generalized Beam-Kirchhoff equation under initial conditions and boundary conditions, using the previous research results for reference. Firstly, the existence of bounded absorption set is proved by using a prior estimation, then the existence and uniqueness of the global solution of the problem is proved by using the classical Galerkin’s method. Finally, Housdorff dimension and fractal dimension of the family of global attractors are estimated by linear variational method and generalized Sobolev-Lieb-Thirring inequality. 展开更多
关键词 Beam-Kirchhoff equation Galerkin’s Method Family of Global Attractors Housdorff Dimension Fractal Dimension
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Long Time Behavior of a Class of Generalized Beam-Kirchhoff Equations
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作者 Guoguang Lin Keshun Peng 《Journal of Applied Mathematics and Physics》 2023年第10期2963-2981,共19页
In this paper, we study the long time behavior of a class of generalized Beam-Kirchhoff equation , and prove the existence and uniqueness of the global solution of this class of equation by Galerkin method by making s... In this paper, we study the long time behavior of a class of generalized Beam-Kirchhoff equation , and prove the existence and uniqueness of the global solution of this class of equation by Galerkin method by making some assumptions about the nonlinear function term . The existence of the family of global attractor and its Hausdorff dimension and Fractal dimension estimation are proved. 展开更多
关键词 Beam-Kirchhoff equation Galerkin’s Method The Family of Global Attractor Dimension Estimation
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Second Approximation of the Generalized Planetary Equation Based upon Golden Metric Tensors
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作者 Nura Yakubu Hayatu Abbba Ibrahim +1 位作者 Musa Hashimu Umar Bappah Alkali 《Journal of High Energy Physics, Gravitation and Cosmology》 CAS 2023年第1期7-26,共20页
In this paper, we consider the Post Einstein Planetary equation of motion. We succeeded in offering a solution using second approximation method, in which we obtained eight exact mathematical solutions that rebel amaz... In this paper, we consider the Post Einstein Planetary equation of motion. We succeeded in offering a solution using second approximation method, in which we obtained eight exact mathematical solutions that rebel amazing theoretical results. To the order of C<sup>-2</sup>, two of these exact solutions are reduced to the approximate solutions from the method of successive approximations. 展开更多
关键词 Golden Matric Tensors Einstein’s Planetary equation
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Complex Maxwell’s Equations
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作者 Mahgoub A. Salih 《Journal of Modern Physics》 2023年第12期1662-1671,共10页
Maxwell’s equations in electromagnetism can be categorized into three dis-tinct groups based on the electromagnetic source when employing quaterni-ons. Each group represents a self-contained system in which Maxwell’... Maxwell’s equations in electromagnetism can be categorized into three dis-tinct groups based on the electromagnetic source when employing quaterni-ons. Each group represents a self-contained system in which Maxwell’s equations are applied and validated concurrently, in contrast to the previous approach that did not account for this. It has been noted that the formulation of these Maxwell equations ultimately results in the formulation of Max-well’s equations utilizing the scalar function. 展开更多
关键词 Maxwell’s equations scalar Function Proca equation Gage Transformation QUATERNION
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Review of Kelvin's Equation and Its Modification in Characterization of Mesoporous Materials 被引量:2
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作者 张颖 阎子峰 胡喜军 《Chinese Journal of Chemical Physics》 SCIE CAS CSCD 北大核心 2006年第2期102-108,共7页
Physical and mathematical models as well as calculation methods of nitrogen bed on porous media have been introduced to evaluate the structural parameters of mesoporous materials. Kelvin's equation is a link between ... Physical and mathematical models as well as calculation methods of nitrogen bed on porous media have been introduced to evaluate the structural parameters of mesoporous materials. Kelvin's equation is a link between the relative adsorbate pressure, the mean pore radius, and pore capillarity on the basis of macroscopic capillary condensation. However, Kelvin's equation has been identified that it underestimates the calculated pore size of a material especially in the boundary of pore size which is between 2 and 4 nm.Various modifications on Kelvin's equation were mentioned in order to develop a new model to improve the accuracy of pore size calculation. The problems on conventional mathematical models were analyzed and discussed. A number of calculation methods on physisorption and pore size, especially fundamental theories of physisorption, basis of models and their deficiencies are reviewed. It can provide guidance on developing a modified Kelvin's equation for pore size calculation. 展开更多
关键词 Kelvin's equation PHYsIsORPTION Adsorption isotherm Hysteresis loop Pore size distribution
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广义概率密度演化方程的Chebyshev拟谱法
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作者 徐亚洲 田锐 《力学学报》 EI CAS CSCD 北大核心 2024年第8期2415-2422,共8页
概率密度演化方法(probability density evolution equation,PDEM)为非线性随机结构的动力响应分析提供了新的途径.通过PDEM获得结构响应概率密度函数(probability density function,PDF)的关键步骤是求解广义概率密度演化方程(generali... 概率密度演化方法(probability density evolution equation,PDEM)为非线性随机结构的动力响应分析提供了新的途径.通过PDEM获得结构响应概率密度函数(probability density function,PDF)的关键步骤是求解广义概率密度演化方程(generalized probability density evolution equation,GDEE).对于GDEE的求解通常采用有限差分法,然而,由于GDEE是初始条件间断的变系数一阶双曲偏微分方程,通过有限差分法求解GDEE可能会面临网格敏感性问题、数值色散和数值耗散现象.文章从全局逼近的角度出发,基于Chebyshev拟谱法为GDEE构造了全局插值格式,解决了数值色散、数值耗散以及网格敏感性问题.考虑GDEE的系数在每个时间步长均为常数,推导了GDEE在每一个时间步长内时域上的序列矩阵指数解.由于序列矩阵指数解形式上是解析的,从而很好地克服了数值稳定性问题.两个数值算例表明,通过Chebyshev拟谱法结合时域的序列矩阵指数解求解GDEE得到的结果与精确解以及Monte Carlo模拟的结果非常吻合,且数值耗散和数值色散现象几乎可以忽略.此外,拟谱法具有高效的收敛性且序列矩阵指数解不受CFL (Courant-Friedrichs-Lewy)条件的限制,因此该方法具有良好的数值稳定性和计算效率. 展开更多
关键词 概率密度演化方法 广义概率密度演化方程 拟谱方法 蒙特卡洛模拟
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Efficient Cell-Centered Multigrid Scheme for the Three-Dimensional Navier-Stokes Equations 被引量:3
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作者 乔渭阳 王占学 蔡元虎 《Chinese Journal of Aeronautics》 SCIE EI CAS CSCD 2002年第4期193-199,共7页
A cell centered scheme for three dimensional Navier Stokes equations, which is based on central difference approximations and Runge Kutta time stepping, is described. By using local time stepping, implicit residual sm... A cell centered scheme for three dimensional Navier Stokes equations, which is based on central difference approximations and Runge Kutta time stepping, is described. By using local time stepping, implicit residual smoothing, a multigrid method, and carefully controlled artificial dissipative terms, good convergence rates are obtained for two and three dimensional flows. The emphases are on the implicit smoothing and artificial dissipative terms with locally variable coefficients which depend on cel... 展开更多
关键词 CFD finite volume method N s equations
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PROPER SPLITTINGS FOR RESTRICTED LINEAR EQUATIONS AND THE GENERALIZED INVERSE A_(T,S)~2 被引量:2
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作者 王国荣 魏益民 《Numerical Mathematics A Journal of Chinese Universities(English Series)》 SCIE 1998年第1期1-13,共13页
This paper presents a proper splitting iterative method for comparing the general restricted linear euqations Ax=b, x ∈T (where, b ∈AT, and T is an arbitrary but fixed subspace of C<sup>m</sup>) and th... This paper presents a proper splitting iterative method for comparing the general restricted linear euqations Ax=b, x ∈T (where, b ∈AT, and T is an arbitrary but fixed subspace of C<sup>m</sup>) and the generalized in A<sub>T,S</sub> For the special case when b ∈AT and dim(T)=dim(AT), this splitting iterative methverse A<sub>T,S</sub> hod converges to A<sub>T,S</sub>b (the unique solution of the general restricted system Ax=bx ∈T). 展开更多
关键词 PROPER sPLITTING REsTRICTED linear equations the generalized inverse A<sup>2sup><sub>Tsub> s.
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Exact invariants and adiabatic invariants of Raitzin's canonical equations of motion for a nonlinear nonholonomic mechanical system 被引量:6
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作者 乔永芬 李仁杰 孙丹姗 《Chinese Physics B》 SCIE EI CAS CSCD 2005年第10期1919-1925,共7页
The exact invariants and the adiabatic invariants of Raitzin's canonical equations of motion for a nonlinear nonholonomic mechanical system are studied. The relations between the invariants and the symmetries of the ... The exact invariants and the adiabatic invariants of Raitzin's canonical equations of motion for a nonlinear nonholonomic mechanical system are studied. The relations between the invariants and the symmetries of the system are established. Based on the concept of higher-order adiabatic invariant of a mechanical system under the action of a small perturbation, the forms of the exact invariants and adiabatic invariants and the conditions for their existence are proved. Finally, the inverse problem of the perturbation to symmetries of the system is studied and an example is also given to illustrate the application of the results. 展开更多
关键词 nonholonomic system Raitzin's canonical equation sYMMETRY PERTURBATION exactinvariant adiabatic invariant
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