共查询到20条相似文献,搜索用时 31 毫秒
1.
M. M. KHADER 《应用数学和力学(英文版)》2018,39(6):867-876
The proposed method is based on replacement of the unknown function by a truncated series of the shifted Legendre polynomial expansion. An approximate formula of the integer derivative is introduced. Special attention is given to study the convergence analysis and derive an upper bound of the error for the presented approximate formula. The introduced method converts the proposed equation by means of collocation points to a system of algebraic equations with shifted Legendre coefficients. Thus, after solving this system of equations, the shifted Legendre coefficients are obtained. This efficient numerical method is used to solve the system of ordinary differential equations which describe the thin film flow and heat transfer with the effects of the thermal radiation, magnetic field, and slip velocity. 相似文献
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The method of integrated photoelasticity can be very elegantly used to determine the stress distribution in the case of torsionless axisymmetry. The mathematical formulation of such problems often yields Abel's integral equation. One of the ways to solve these equations is by approximating either the known (experimentally determined) or the unknown function by polynomials. The integral equation is thus converted to a system of linear, simultaneous algebraic equations with the coefficients of the approximating polynomial as unknowns. The degree of the approximating polynomial cannot be fixeda priori. The present paper postulates a scheme in which the degree of the approximating polynomial is increased in steps of one, starting from one. Solutions are computed for each degree of the polynomial. It is then possible to pick the solution from the available family of solutions. The computational aspect of the exercise can be very easily taken care of by the algorithms proposed and validated in this paper. The over-determined system of simultaneous equations is solved by the method of singular value decomposition (SVD). The proposed method is validated by first applying it to a test problem. Two cases, which are solved earlier, are then analyzed and the results are compared. 相似文献
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利用滑动最小二乘插值函数作为加权残值法的试函数,分析了该试函数的拟合特性,对试函数中的基函数以及权函数的选取提出了建议;采用最小二乘配点法求出试函数中的系数,进而可得到定解问题的近似解;利用该试函数对薄板的挠曲、中厚板的弯曲两个例子进行了数值计算,并与理论结果或其它数值结果进行对比,结果表明,该试函数适用于多种边值问题,且精度高. 该法简化了选择试函数的过程,尤其适用于工程中的各种数值计算. 相似文献
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复杂梁动力问题的近似分析方法 总被引:22,自引:1,他引:22
本文介绍了在各种复杂条件下,分析梁振动特性的一个近似方法-模态摄动法。这一方法是在等截面均匀梁的模态子空间内实施,将复杂梁的变系数微分方程的求解转化为代数方程组的求解。通过算例,表明这一方法简单实用,且有良好的近似性。 相似文献
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Helmholtz方程的微分容积解法 总被引:1,自引:0,他引:1
用一种新型的数值技术--微分容积法(Differential Cubature Method)求解二维Helmholtz方程的边值问题,几个数值算例表明,该方法稳定收敛,并具有较好的数值精度,本文方法适用于求解具有较小波数的Helmholtz方程。 相似文献
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G. V. Druzhinin 《Journal of Applied Mechanics and Technical Physics》2003,44(6):779-785
A unified method for constructing basis (eigen) functions is proposed to solve problems of mechanics of continuous media, problems of cubature and quadrature, and problems of approximation of hypersurfaces. Numerical-analytical methods are described, which allow obtaining approximate solutions of internal and external boundary-value problems of mechanics of continuous media of a certain class (both linear and nonlinear). The method is based on decomposition of the sought solutions of the considered partial differential equations into series in basis functions. An algorithm is presented for linearization of partial differential equations and reduction of nonlinear boundary-value problems, which are reduced to systems of linear algebraic equations with respect to unknown coefficients without using traditional methods of linearization. 相似文献
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In this paper, an optimization method based on a new class of basis functions, namely generalized polynomials (GPs), is proposed for nonlinear variable-order time fractional diffusion-wave equation. Variable-order time fractional derivative is expressed in the Caputo sense. In the proposed method, solution of the problem under consideration is expanded in terms of GPs with unknown free coefficients and control parameters. In this way, some new operational matrices of the ordinary and fractional derivatives are derived for these basis functions. The residual function and its 2-norm are employed for converting the problem under study to an optimization one and then choosing the unknown free coefficients and control parameters optimally. As a useful result, the necessary conditions of optimality are derived as a system of nonlinear algebraic equations with unknown free coefficients and control parameters. The validity and effectiveness of the method are demonstrated by solving some numerical examples. The results demonstrate that the proposed method is a powerful algorithm with good accuracy for solving such kind of problems. 相似文献
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非线性振动的一个新的渐近解法 总被引:2,自引:0,他引:2
本文在渐近法的基础上,引进谐波平衡思想,得到了一个新的渐近解法。与传统方法相比,应用本文方法求渐近解,不必解微分方程和依靠消除永年项建立补充工程,而是将求解过程转化为一系列的代数运算。因此,本文解法便于手算,更有利于用计算机计算高阶近似。 相似文献
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《Particuology》2022
The particle number density in the Smoluchowski coagulation equation usually cannot be solved as a whole, and it can be decomposed into the following two functions by similarity transformation: one is a function of time (the particle k-th moments), and the other is a function of dimensionless volume (self-preserving size distribution). In this paper, a simple iterative direct numerical simulation (iDNS) is proposed to obtain the similarity solution of the Smoluchowski coagulation equation for Brownian motion from the asymptotic solution of the k-th order moment, which has been solved with the Taylor-series expansion method of moment (TEMOM) in our previous work. The convergence and accuracy of the numerical method are first verified by comparison with previous results about Brownian coagulation in the literature, and then the method is extended to the field of Brownian agglomeration over the entire size range. The results show that the difference between the lognormal function and the self-preserving size distribution is significant. Moreover, the thermodynamic constraint of the algebraic mean volume is also investigated. In short, the asymptotic solution of the TEMOM and the self-preserving size distribution form a one-to-one mapping relationship; thus, a complete method to solve the Smoluchowski coagulation equation asymptotically is established. 相似文献
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提出应用混合摄动$\!$-$\!$-$\!$伽辽金法求解随机桁架结构的几何非线性问题.将含位移项的随机割线弹性模量以及随机响应表示为幂多项式展开,利用高阶摄动方法确定随机结构几何非线性响应的幂多项式展开的各项系数.将随机响应的各阶摄动项假定为伽辽金试函数,运用伽辽金投影对试函数系数进行求解,从而得到随机桁架结构几何非线性响应的显式表达式.同已有的随机伽辽金法相比,本文所给的试函数由摄动解的线性组合而成,在求解非线性问题时,试函数的获取具有自适应性.数值算例结果表明,对于具有不同概率分布的多随机变量问题,本文方法无需对随机变量的概率分布形式进行转换,避免了转换误差,因而比同阶的广义正交多项式方法(generalizedpolynomial chaos, GPC)计算精度高.同时,在结果精度相当时,和GPC方法相比,本文方法得到的试函数系数的非线性方程维度不大,方程的求解工作量小且更易求解.当随机量涨落较大时,混合摄动$\!$-$\!$-$\!$伽辽金法计算所得的结构响应的各阶统计矩比高阶摄动法所得结果更逼近于蒙特卡洛模拟结果,显示了该方法对几何非线性随机问题求解的有效性. 相似文献
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周叮 《应用数学和力学(英文版)》1996,17(12):1189-1192
AN EXACT METHOD OF BENDING OF ELASTIC THIN PLATES WITH ARBITRARY SHAPEZhouDing(周叮)(ReceivedApril10,1995,RevisedApril26.1996.C... 相似文献
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为了提高基于高阶格式的结构动力响应微分求积分析方法的计算效率,发展了一种求解动力方程的快速算法.利用微分求积原理将结构动力方程转化为标准Sylvester方程的形式,通过对系数矩阵进行矩阵分解,进而将动力响应Sylvester方程化为一系列标准线性方程组,采用相关成熟算法求解这些线性方程组后即可获得结构动力时程响应的全部解答.结构动力响应微分求积分析方法为高阶数值方法,一步计算可以获得多个时点处的动力响应.基于本文快速算法,不必直接对矩阵方程进行求解.数值算例表明,本文快速算法能够准确地计算出结构动力响应,具有数值精度高、收敛性好的优点. 相似文献
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In this paper, a new method is presented based on [1]. It can be used to solve the arbitrary nonlinear system of differential equations with variable coefficients. By this method, the general solution for large deformation of nonhomogeneous circular plates resting on an elastic foundation is derived. The convergence of the solution is proved. Finally, it is only necessary to solve a set of nonlinear algebraic equations with three unknowns. The solution obtained by the present method has large convergence range and the computation is simpler and more rapid than other numerical methods.Numerical examples given at the end of this paper indicate that satisfactory results of stress resullants and displacements can be obtained by the present method. The correctness of the theory in this paper is, confirmed. 相似文献
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各向异性平板开孔动应力集中问题的研究 总被引:2,自引:0,他引:2
采用各向异性平板弯曲波动理论及摄动方法,对正交各向异性平板开孔弯曲波的散射及动应力集中问题进行了分析研究,得到了此种平板稳态弯曲波动问题的渐近形式的分析解。同时采用保角映射技术,为求解正交各向异性平板开孔弹性波的散射及动应力集中问题提供了一种统一规范的方法。 相似文献
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The complex function method is used to solve problems of scattering of plane SH-waves on cylindrical canyon topography of
arbitrary shape in anisotropic media. This paper gives the complete function series and general expressions with boundary
condition to approach the solution of steady state scattering of plane SH-waves on two-dimensional canyon topography in anisotropic
media. The problem to be solved can be reduced to a solution of infinite algebraic equation series by using Hermite function
and it's orthogonal conditions. The solution can be obtained directly by using computers. Finally, as an example, computational
results of scattering of plane SH-waves on a semi-cylindrical canyon topography are presented.
The projects sponsored by The Joint Seismological Science Foundation. 相似文献
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An analytical approximate method for strongly nonlinear damped oscillators is proposed. By introducing phase and amplitude
of oscillation as well as a bookkeeping parameter, we rewrite the governing equation into a partial differential equation
with solution being a periodic function of the phase. Based on combination of the Newton’s method with the harmonic balance
method, the partial differential equation is transformed into a set of linear ordinary differential equations in terms of
harmonic coefficients, which can further be converted into systems of linear algebraic equations by using the bookkeeping
parameter expansion. Only a few iterations can provide very accurate approximate analytical solutions even if the nonlinearity
and damping are significant. The method can be applied to general oscillators with odd nonlinearities as well as even ones
even without linear restoring force. Three examples are presented to illustrate the usefulness and effectiveness of the proposed
method. 相似文献
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本文给出了计算复杂渗流场的简化方法.首先由加权余量法得到变系数渗流问题的积分方程.再将区域及边界剖分,对各单元使用形心处的渗透系数而化为正交各向异性问题.最后用正交各向异性问题的基本解,得到对应于边界单元法的线性代数方程组.算例表明方法可靠. 相似文献