We derive the firstorder approximate symmetries for the Harry Dym equation by the method of approximate transformation groups proposed by Baikov et al. (1989, 1996). Moreover, we investigate the structure of the Lie algebra of symmetries of the perturbed Harry Dym equation. We compute the onedimensional optimal system of subalgebras as well as point out some approximately differential invariants with respect to the generators of Lie algebra and optimal system.
The following nonlinear partial differential equation
In this paper, we analyze the perturbed Harry Dym equation
In this section, we will provide the background definitions and results in approximate symmetry that will be used along this paper. Much of it is stated as in [
The set of transformations (
Let
Equation (
If (
Equations (
Consider the perturbed Harry Dym equation
Let us consider the approximate group generators in the form
At first, we need to determine the auxiliary function
Thus, we derive the following approximate symmetries of the perturbed Harry Dym equation:
Approximate commutators of approximate symmetry of perturbed Harry Dym equation.









 















































































































Equations (
In this section, we determine the structure of the Lie algebra of symmetries of the perturbed Harry Dym equation. The Lie algebra
The radical
Let
Let
Actually, the proposition says that the problem of finding an optimal system of subgroups is equivalent to that of finding an optimal system of subalgebras. For onedimensional subalgebras, this classification problem is essentially the same as the problem of classifying the orbits of the adjoint representation, since each onedimensional subalgebra is determined by a nonzero vector in
Adjoint representation of approximate symmetry of the perturbed Harry Dym equation.
Ad 

























































































































An optimal system of onedimensional approximate Lie algebras of the perturbed Harry Dym equation is provided by
Consider the approximate symmetry algebra
Suppose first that
In this section, we compute some approximately differential invariants of the perturbed Harry Dym equation with respect to the optimal system. Consider the operator
Approximately differential invariants for the perturbed Harry Dym equation.
Operator  Approximate differential invariants 





























In this paper, we investigate the approximate symmetry of the perturbed Harry Dym equation and discuss the structure of its Lie algebra. Moreover, we compute optimal system of onedimensional approximate Lie algebras of the perturbed Harry Dym equation and derive some approximately differential invariants with respect to the generators of Lie algebra and optimal system.