﻿ Sobolev orthogonal polynomials generated by modified Laguerre polynomials and the Cauchy problem for ODE systems

# Sobolev orthogonal polynomials generated by modified Laguerre polynomials and the Cauchy problem for ODE systems

### DOI: 10.31029/demr.10.3

We consider the problem of representing a solution of the Cauchy problem for a system of ordinary differential equations (in general, nonlinear) in the form of a Fourier series in polynomials $l_{r, k} (x; b)$ $(k = 0,1,\ldots)$, orthonormal by Sobolev with respect to the scalar product $\langle f,g \rangle=\sum_{\nu=0}^{r-1} f^{(\nu)} (0) g ^ {(\nu)} (0) + \int_ {0} ^ \infty f ^ {(r)} (t) g ^ {(r)} (t) e ^ {-bt} dt$ with $b> 0$, generated by the modified Laguerre \linebreak polynomials $l_k (x; b) = \sqrt {b} L_k (bx)$ by means of the equalities $l_ {r, k} (x; b) = \frac {x ^ k} {k!} \, (k = 0,1 , \ldots, r-1)$, \, $l_ {r, r + k} (x; b) = \frac {1} {(r-1)!} \int_ {0} ^ x (xt) ^ {r-1} {l} _ {k} (t; b) dt \, (k = 0,1, \ldots)$. In the infinite-dimensional Hilbert space of $l_2 ^ m$ $m$ -dimensional sequences $C = (c_0, c_1, \ldots)$ for which the norm $\| C \| = \left (\sum \nolimits_ {j = 0} ^ \infty \sum \nolimits_ {l = 1} ^ {m} (c_j ^ l) ^ 2 \right) ^ \frac12$, the contracting nonlinear operator $A: l_2 ^ m \to l_2 ^ m$ is constructed, the fixed point \linebreak $\hat C = (\hat c_0, \hat c_1, \ldots)$ coincides with the sequence of unknown coefficients of the expansion of the solution of the Cauchy problem in question Fourier series in the system $l_ {1, k} (x; b)$ $(k = 0,1, \ldots)$. The corresponding finite-dimensional analogue $A_N: \mathbb {R} ^ N_m \to \mathbb {R} ^ N_m$ of the operator $A$ is also constructed, which acts in the finite-dimensional space $\mathbb {R} ^ N_m$ of matrices $C$ of dimension $m \times N$, in which the norm $\| C \| _N ^ m = \left (\sum \nolimits_ {j = 0} ^ {N-1} \sum \nolimits_ {l = 1} ^ {m} (c_j ^ l) ^ 2 \right ) ^ \frac12$. The fixed point $\bar C = (\bar c_0, \bar c_1, \ldots, \bar c_ {N-1})$ of the operator $A_N$ is the estimate (approximate value) of the desired point $\hat C_N = (\hat c_0, \hat c_1, \ldots, \hat c_ {N-1})$. An estimate of the error $\| \hat C_N- \bar C_N \| _N ^ m$ is established.

Keywords: Polynomials orthogonal on Sobolev, generated Laguerre polynomials, modified Laguerre polynomials, Cauchy problem for ODE systems.

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