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首页> 外文期刊>Journal of Computational and Applied Mathematics >Accurate spectral solutions for the parabolic and elliptic partial differential equations by the ultraspherical tau method
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Accurate spectral solutions for the parabolic and elliptic partial differential equations by the ultraspherical tau method

机译：利用超球tau方法求解抛物型和椭圆型偏微分方程的精确谱解

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摘要

We present a double ultraspherical spectral methods that allow the efficient approximate solution for the parabolic partial differential equations in a square subject to the most general inhomogeneous mixed boundary conditions. The differential equations with their boundary and initial conditions are reduced to systems of ordinary differential equations for the time-dependent expansion coefficients. These systems are greatly simplified by using tensor matrix algebra, and are solved by using the step-by-step method. Numerical applications of how to use these methods are described. Numerical results obtained compare favorably with those of the analytical solutions. Accurate double ultraspherical spectral approximations for Poisson's and Helmholtz's equations are also noted. Numerical experiments show that spectral approximation based on Chebyshev polynomials of the first kind is not always better than others based on ultraspherical polynomials. (c) 2004 Elsevier B.V. All rights reserved.
机译：我们提出了一种双超球面光谱方法，该方法可以为抛物线偏微分方程在最一般的不均匀混合边界条件下的有效解。具有边界和初始条件的微分方程被简化为具有时间相关膨胀系数的普通微分方程组。这些系统通过使用张量矩阵代数得到了极大的简化，并通过逐步方法得以解决。描述了如何使用这些方法的数值应用。所获得的数值结果与分析解决方案的数值相比具有优势。还指出了泊松方程和亥姆霍兹方程的精确双超球形光谱近似。数值实验表明，基于第一类切比雪夫多项式的光谱逼近并不总是比基于超球面多项式的光谱逼近更好。 （c）2004 Elsevier B.V.保留所有权利。

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