Application of orthogonal expansions for approximate integration of ordinary differential equations

2010 ◽  
Vol 65 (4) ◽  
pp. 172-175 ◽  
Author(s):  
O. B. Arushanyan ◽  
N. I. Volchenskova ◽  
S. F. Zaletkin
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Orthogonal Expansions ◽  
Approximate Integration

scholarly journals Approximate transformations for van der Pol-type equations

2006 ◽  
Vol 2006 ◽  
pp. 1-11 ◽  
Author(s):  
R. K. Gazizov ◽  
C. M. Khalique
Keyword(s):  
Differential Equations ◽  
Small Parameter ◽  
Ordinary Differential Equations ◽  
Transformation Groups ◽  
Van Der Pol ◽  
Approximate Symmetries ◽  
Approximate Integration ◽  
Successive Reduction

Classification of van der Pol-type equations with respect to admitted approximate transformation groups transforming a small parameter is given. It is shown that approximate symmetries transforming the small parameter as well as the usual approximate symmetries can be used for approximate integration (e.g., by method of successive reduction of order) of ordinary differential equations with a small parameter.

scholarly journals To the orthogonal expansion theory of the solution to the Cauchy problem for second-order ordinary differential equations

2018 ◽  
pp. 178-184
Author(s):  
О.Б. Арушанян ◽  
С.Ф. Залеткин
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Second Order ◽  
Orthogonal Expansion ◽  
Orthogonal Expansions ◽  
Canonical Systems ◽  
Theoretical Substantiation ◽  
The Cauchy Problem ◽  
Expansion Theory ◽  
Solvability Theorem

Доказана теорема о разрешимости нелинейной системы уравнений относительно приближенных значений коэффициентов Чебышёва старшей производной, входящей в дифференциальное уравнение. Теорема является теоретическим обоснованием ранее предложенного приближенного метода интегрирования канонических систем обыкновенных дифференциальных уравнений второго порядка на основе ортогональных разложений с использованием многочленов Чебышёва первого рода. A solvability theorem is proved for a nonlinear system of equations with respect to the approximate Chebyshev coefficients of the highest derivative in an ordinary differential equation. This theorem is a theoretical substantiation for the previously proposed approximate method of solving canonical systems of second-order ordinary differential equations using orthogonal expansions on the basis of Chebyshev polynomials of the first kind.

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