Abstract
Lie symmetry analysis of differential equations provides a powerful and fundamental framework to the exploitation of systematic procedures leading to the integration by quadrature (or at least to lowering the order) of ordinary differential equations, to the determination of invariant solutions of initial and boundary value problems, to the derivation of conservation laws, to the construction of links between different differential equations that turn out to be equivalent. This paper reviews some well known results of Lie group analysis, as well as some recent contributions concerned with the transformation of differential equations to equivalent forms useful to investigate applied problems.
Highlights
IntroductionSymmetry (joined to simplicity) has been, is, and probably will continue to be, an elegant and useful tool in the formulation and exploitation of the laws of nature
Symmetry has been, is, and probably will continue to be, an elegant and useful tool in the formulation and exploitation of the laws of nature
Either classical and well established or recent, concerning the application of Lie symmetries to differential equations, have been presented
Summary
Symmetry (joined to simplicity) has been, is, and probably will continue to be, an elegant and useful tool in the formulation and exploitation of the laws of nature. Symmetry analysis of differential equations was developed and applied by Sophus Lie during the period 1872–1899 [3,4] This theory enables to derive solutions of differential equations in a completely algorithmic way without appealing to special lucky guesses. Lie’s theory is powerful, versatile, and fundamental to the development of systematic procedures leading to the integration by quadrature (or at least to lowering the order) of ordinary differential equations, to the determination of invariant solutions of initial and boundary value problems, to the derivation of conserved quantities, or to the construction of relations between different differential equations that turn out to be equivalent. The key idea of Lie’s theory of symmetry analysis of differential equations relies on the invariance of the latter under a transformation of independent and dependent variables. Any transformation of the independent and dependent variables in turn induces a transformation of the derivatives
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