A numerical method for solving the Cauchy problem for ODEs using a system of polynomials generated by a system of modified Laguerre polynomials

G. Akniyev, R. Gadzhimirzaev
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引用次数: 0

Abstract

In this paper, we consider a numerical realization of an iterative method for solving the Cauchy problem for ordinary differential equations, based on representing the solution in the form of a Fourier series by the system of polynomials $\{L_{1,n}(x;b)\}_{n=0}^\infty$, orthonormal with respect to the Sobolev-type inner product $$ \langle f,g\rangle=f(0)g(0)+\int_{0}^\infty f'(x)g'(x)\rho(x;b)dx $$ and generated by the system of modified Laguerre polynomials $\{L_{n}(x;b)\}_{n=0}^\infty$, where $b>0$. In the approximate calculation of the Fourier coefficients of the desired solution, the Gauss -- Laguerre quadrature formula is used.
一种利用由修正拉盖尔多项式系统生成的多项式系统求解二阶微分方程柯西问题的数值方法
在本文中,我们考虑了一种求解常微分方程Cauchy问题的迭代方法的数值实现,该方法基于多项式系统$\{L_{1,n}(x;b)\}_{n=0}^\infty$以傅里叶级数的形式表示解,该多项式系统相对于sobolev型内积$$\langle f,g\rangle=f(0)g(0)+\int_{0}^\infty f'(x)g'(x)\rho(x;b)dx$$是正交的,并由修改的Laguerre多项式系统$\{L_{n}(x;b)\}_{n=0}^\infty$生成,其中$b>0$。在近似计算所需解的傅里叶系数时,使用高斯—拉盖尔正交公式。
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