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

2019 ◽  
pp. 13-24
Author(s):  
Gasan Akniyev ◽  
Ramis Gadzhimirzaev
Keyword(s):  
Cauchy Problem ◽  
Fourier Series ◽  
Approximate Calculation ◽  
Fourier Coefficients ◽  
Laguerre Polynomials ◽  
Numerical Realization ◽  
Inner Product ◽  
Sobolev Type ◽  
The Cauchy Problem ◽  
Modified Laguerre Polynomials

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.

scholarly journals On the Uniform Convergence of the Fourier Series by the System of Polynomials Generated by the System of Laguerre Polynomials

2020 ◽  
Vol 20 (4) ◽  
pp. 416-423
Author(s):  
Ramis M. Gadzhimirzaev ◽  
Keyword(s):  
Fourier Series ◽  
Sobolev Space ◽  
Weight Function ◽  
Uniform Convergence ◽  
Laguerre Polynomials ◽  
Inner Product ◽  
Sobolev Type ◽  
Absolutely Continuous ◽  
Differentiable Functions ◽  
Proof Of Convergence

Let w(x) be the Laguerre weight function, 1 ≤ p < ∞, and Lpw be the space of functions f, p-th power of which is integrable with the weight function w(x) on the non-negative axis. For a given positive integer r, let denote by WrLpw the Sobolev space, which consists of r−1 times continuously differentiable functions f, for which the (r−1)-st derivative is absolutely continuous on an arbitrary segment [a, b] of non-negative axis, and the r-th derivative belongs to the space Lpw. In the case when p = 2 we introduce in the space WrL2w an inner product of Sobolev-type, which makes it a Hilbert space. Further, by lαr,n(x), where n = r, r + 1, ..., we denote the polynomials generated by the classical Laguerre polynomials. These polynomials together with functions lαr,n(x) = xn / n! , where n = 0, 1, r − 1, form a complete and orthonormal system in the space WrL2w. In this paper, the problem of uniform convergence on any segment [0,A] of the Fourier series by this system of polynomials to functions from the Sobolev space WrLpw is considered. Earlier, uniform convergence was established for the case p = 2. In this paper, it is proved that uniform convergence of the Fourier series takes place for p > 2 and does not occur for 1 ≤ p < 2. The proof of convergence is based on the fact that WrLpw ⊂ WrL2w for p > 2. The divergence of the Fourier series by the example of the function ecx using the asymptotic behavior of the Laguerre polynomials is established.

scholarly journals Fourier coefficients for Laguerre–Sobolev type orthogonal polynomials

2022 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Alejandro Molano
Keyword(s):  
Orthogonal Polynomials ◽  
Fourier Coefficients ◽  
Finite Interval ◽  
Laguerre Polynomials ◽  
Inner Product ◽  
Sobolev Type ◽  
Content Type ◽  
Constructive Methods ◽  
Reasonable Choice ◽  
Connection Formulas

Purpose In this paper, the authors take the first step in the study of constructive methods by using Sobolev polynomials.Design/methodology/approach To do that, the authors use the connection formulas between Sobolev polynomials and classical Laguerre polynomials, as well as the well-known Fourier coefficients for these latter.Findings Then, the authors compute explicit formulas for the Fourier coefficients of some families of Laguerre–Sobolev type orthogonal polynomials over a finite interval. The authors also describe an oscillatory region in each case as a reasonable choice for approximation purposes.Originality/value In order to take the first step in the study of constructive methods by using Sobolev polynomials, this paper deals with Fourier coefficients for certain families of polynomials orthogonal with respect to the Sobolev type inner product. As far as the authors know, this particular problem has not been addressed in the existing literature.

The Cauchy problem for an equation of Sobolev type with power non-linearity

2005 ◽  
Vol 69 (1) ◽  
pp. 59-111 ◽  
Author(s):  
E I Kaikina ◽  
P I Naumkin ◽  
I A Shishmarev
Keyword(s):  
Cauchy Problem ◽  
Sobolev Type ◽  
The Cauchy Problem

On the critical exponent “instantaneous blow-up” versus “local solubility” in the Cauchy problem for a model equation of Sobolev type

2021 ◽  
Vol 85 (1) ◽  
Author(s):  
Maxim Olegovich Korpusov ◽  
Alexander Anatolyevich Panin ◽  
Andrey Evgenievich Shishkov
Keyword(s):  
Cauchy Problem ◽  
Critical Exponent ◽  
Model Equation ◽  
Blow Up ◽  
Sobolev Type ◽  
The Cauchy Problem
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