Linearization of Fourth-order Ordinary Differential Equations by Generalized Sundman Transformations

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Published: Nov 28, 2022
Keywords:
Linearization problem Generalized Sundman transformations Fourth-order ordinary differential equations

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Chanyanit Patcha
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
Warisa Nakpim
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand

Abstract

The linearization problem of fourth-order ordinary differential equations by the generalized Sundman transformation, i.e.,
gif.latex?u&space;=&space;F(x,y)
gif.latex?dt&space;=&space;G(x,y)dx
is considered in the paper. Necessary and sufficient conditions for fourth-order ordinary differential equations to be linearizable into the general form of a linear fourth-order ordinary differential equation are obtained. Here, a complete solution is given for the case gif.latex?F_{x}&space;=&space;0 We also give an example which apply our procedure for a nonlinear fourth-order partial differential equation
gif.latex?u_{tt}&space;=&space;(\kappa&space;\tilde{}u&space;+&space;\gamma&space;\tilde{}u^{2})_{xx}&space;+&space;\nu&space;\tilde{}uu_{xxxx}+\tilde{\mu&space;}u_{xxtt}+a\tilde{}u_{x}u_{xxx}+&space;\tilde{\beta&space;}u^{2}_{xx}) ,
where gif.latex?\tilde{a},\tilde{\beta&space;},\tilde{\gamma&space;},\tilde{\mu&space;},\tilde{\nu&space;} and gif.latex?\tilde{\kappa&space;} are arbitrary constants.

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How to Cite
Patcha, C. ., & Nakpim, W. . (2022). Linearization of Fourth-order Ordinary Differential Equations by Generalized Sundman Transformations. KKU Science Journal, 46(1), 142–153. Retrieved from https://ph01.tci-thaijo.org/index.php/KKUSciJ/article/view/249820
Issue
Vol. 46 No. 1 (2018): January - March 2018
Section
Research Articles
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