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首页> 外文期刊>Applied mathematics and computation >A Taylor-series solution in Cartesian space to GMA-system equations and its application to initial-value problems
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A Taylor-series solution in Cartesian space to GMA-system equations and its application to initial-value problems

机译：笛卡尔空间中GMA系统方程的泰勒级数解及其在初值问题中的应用

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A variable-order, variable-step Taylor-series method in Cartesian space is discussed which makes it possible to solve simultaneous first-order differential equations expressed in GMA-system canonical form with a super high-order accuracy that is comparable to the machine accuracy of the computer. In this method, the stepsize for each differential equation is calculated from the formula derived by setting the ratio of the appropriate two terms of the first few Taylor-series terms at unity, and a minimum value of them is then used as the initial value of the stepsize at each step of the integration and is halved if several criterions are satisfied, Also, when each dependent variable takes a value near zero, the integration is carried out with a very small stepsize. The Taylor-series method constructed here is first applied to the Lotka-Volterra equation to confirm the validity of the calculation algorithm and it is found that the calculated values completely agree with those by the conventional method in which the Taylor-series solution in logarithmic space is used. Second, the differential equation with a solution of cos(t) is solved and the relative error is found to instantaneously increase when the dependent variables take a value near zero and becomes equal to zero or almost zero in the other region. Finally, the differential equation for a two-body problem is solved and it is shown that the accuracies of the calculated values are kept at high levels even after integration over 625 cycles. In conclusion, the Taylor-series method constructed in Cartesian space is considered to have high applicability to simultaneous differential equations and provides high accuracies for the calculated values. (C) 2002 Elsevier Science Inc. All rights reserved. [References: 13]
机译：讨论了笛卡尔空间中的变阶，变步长泰勒级数方法，该方法使得可以求解与机器精度相当的超高阶精度的同时以GMA系统规范形式表示的一阶微分方程。的计算机。在这种方法中，每个微分方程的步长是通过将前几个泰勒级数项中适当的两个项之比设置为1得出的公式计算得出的，然后将它们的最小值用作的初始值。积分的每一步的步长大小，如果满足多个条件，则减半。此外，当每个因变量的值接近零时，积分的步长很小。首先将此处构造的泰勒级数方法应用于Lotka-Volterra方程，以确认计算算法的有效性，并且发现计算值与对数空间中泰勒级数解的常规方法完全一致。用来。其次，求解具有cos（t）解的微分方程，并且当因变量的值接近零并在其他区域变为零或几乎为零时，相对误差立即增加。最后，求解了一个两体问题的微分方程，结果表明，即使经过625个周期的积分，计算值的精度仍保持在较高水平。总之，在笛卡尔空间中构造的泰勒级数方法被认为对联立微分方程具有很高的适用性，并且为计算值提供了很高的精度。 （C）2002 Elsevier Science Inc.保留所有权利。 [参考：13]

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