Elastic transformation method for solving the initial value problem of variable coefficient nonlinear ordinary differential equations

Lin Fan; Shunchu Li; Dongfeng Shao; Xueqian Fu; Pan Liu; Qinmin Gui; Lin Fan; Shunchu Li; Dongfeng Shao; Xueqian Fu; Pan Liu; Qinmin Gui

doi:10.3934/math.2022667

AIMS Mathematics

2022, Volume 7, Issue 7: 11972-11991. doi: 10.3934/math.2022667

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Research article

Elastic transformation method for solving the initial value problem of variable coefficient nonlinear ordinary differential equations

1.
Institute of Applied Mathematics, Xihua University, Chengdu, 610039, Sichuan, China
2.
Beijing Dongrunke Petroleum Technology Co., Ltd., Beijing, 100029, China

Received: 03 January 2022 Revised: 08 April 2022 Accepted: 12 April 2022 Published: 20 April 2022
MSC : 34A05, 34A34, 33C45

Aiming at the initial value problems of variable coefficient nonlinear ordinary differential equations, this paper introduces the elastic transformation method into the process of solving the initial value problems of nonlinear ordinary differential equations with variable coefficients. A class of first-order and a class of third-order nonlinear ordinary differential equations with variable coefficients can be transformed into Chebyshev equations through elastic upgrading transformation and elastic reduction transformation respectively. According to the properties of Chebyshev polynomials and the initial conditions, the solutions to the initial value problems of the original first-order and third- order differential equations can be obtained through the elastic inverse transformation, and then the curves of the solutions can be drawn. The introduction of the elastic transformation method not only provides a new idea for solving the initial value problems of nonlinear differential equations, but also expands the solvable classes of ordinary differential equations.
- elastic transformation method,
- initial value problem,
- nonlinear ordinary differential equation,
- variable coefficient,
- Chebyshev equation
Citation: Lin Fan, Shunchu Li, Dongfeng Shao, Xueqian Fu, Pan Liu, Qinmin Gui. Elastic transformation method for solving the initial value problem of variable coefficient nonlinear ordinary differential equations[J]. AIMS Mathematics, 2022, 7(7): 11972-11991. doi: 10.3934/math.2022667

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Abstract

Aiming at the initial value problems of variable coefficient nonlinear ordinary differential equations, this paper introduces the elastic transformation method into the process of solving the initial value problems of nonlinear ordinary differential equations with variable coefficients. A class of first-order and a class of third-order nonlinear ordinary differential equations with variable coefficients can be transformed into Chebyshev equations through elastic upgrading transformation and elastic reduction transformation respectively. According to the properties of Chebyshev polynomials and the initial conditions, the solutions to the initial value problems of the original first-order and third- order differential equations can be obtained through the elastic inverse transformation, and then the curves of the solutions can be drawn. The introduction of the elastic transformation method not only provides a new idea for solving the initial value problems of nonlinear differential equations, but also expands the solvable classes of ordinary differential equations.

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