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An improvement on the exp-function method when balancing the highest order linear and nonlinear terms. (English) Zbl 1239.35008

Summary: The exp-function method has been widely used to solve different kinds of nonlinear partial differential equations. These nonlinear partial differential equations are transformed first into nonlinear ordinary differential equations and then the ansatz of the exp-function method: \(u=\frac{\sum^p_{n=-c}a_n\exp(n\eta)}{\sum^q_{m=-d}b_n\exp(m\eta)}\) is applied to obtain the solution. However, a part of the solution using this method is to construct the relations between \(c\), \(p\), \(d\) and \(q\) by balancing the highest order linear term with the highest order nonlinear term. It is proved in the present paper that \(c=d\) and \(p=q\) are the only relations that can be obtained by applying this method to any nonlinear ordinary differential equation. Therefore, the additional calculations of balancing the highest order linear term with the highest order nonlinear term are not longer required in the future. Hence, the method becomes more straightforward.

MSC:

35A24 Methods of ordinary differential equations applied to PDEs
35G20 Nonlinear higher-order PDEs
34A34 Nonlinear ordinary differential equations and systems
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