Abstract
We obtain the optimal system’s generating operators associated with a generalized Levinson–Smith equation; this one is related to the Liénard equation which is important for physical, mathematical, and engineering points of view. The underlying equation has applications in mechanics and nonlinear dynamics as well. This equation has been widely studied in the qualitative scheme. Here, we treat the equation by using the Lie group method, and we obtain certain operators; using those operators, we characterized all invariants solutions associated with the generalized equation of Levinson Smith considered in this paper. Finally, we classify the Lie algebra associated with the given equation.
Highlights
Lie group symmetry method is a powerful tool employed to study ODEs, PDEs, FPDEs, FODEs, and so on. is theory was introduced in the 19th century by Sophus Lie [1], following the idea of Galois theory in algebra
The Lie group method approach has been applied to solve and analyze different problems in many scientific fields, e.g., in [10], the authors applied the Lie symmetry method to investigate a fourth-order 1 + 2 evolutionary partial differential equation which has been proposed for the image processing noise reduction
Taking into account [22,23,24,25], we present the optimal system associated to the symmetry group of (3), which shows a systematic way to classify the invariant solutions
Summary
Lie group symmetry method is a powerful tool employed to study ODEs, PDEs, FPDEs, FODEs, and so on. is theory was introduced in the 19th century by Sophus Lie [1], following the idea of Galois theory in algebra. Lie group method applied to differential equations has received great interest among researchers in different fields of science such as mathematics, theoretical, and applied physics, due to the physical interpretations of the underlying equations that are studied. As a consequence, this method leads to construct, for example, conservation laws, using the well known Noether’s theorem [2], even more applying Ibragimov’s approach [3]. The Lie group method approach has been applied to solve and analyze different problems in many scientific fields, e.g., in [10], the authors applied the Lie symmetry method to investigate a fourth-order 1 + 2 evolutionary partial differential equation which has been proposed for the image processing noise reduction. Equation (1) can be written as yxx + φ x, y, yxyx c(x, y),
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