Abstract
The main purpose of this paper is to present a new approach to achieving analytical solutions of parameter containing fractional-order differential equations. Using the nonlinear self-adjoint notion, approximate solutions, conservation laws and symmetries of these equations are also obtained via a new formulation of an improved form of the Noether’s theorem. It is indicated that invariant solutions, reduced equations, perturbed or unperturbed symmetries and conservation laws can be obtained by applying a nonlinear self-adjoint notion. The method is applied to the time fractional-order Fokker–Planck equation. We obtained new results in a highly efficient and elegant manner.
Highlights
Fractional partial differential equations are a generalization of classical ordinary calculus with utilizations of integrals and derivatives with an arbitrary order
Noether’s theorem which was introduced by Emmy Noether in 1918 describing general concepts related to symmetry groups and conservation laws is a useful tool in the solutions of perturbed differential equations, see, e.g., [11,12,13]
We present conservation laws of fractional partial differential equations [19,20] with an effective method based on nonlinear self-adjointness
Summary
Fractional partial differential equations are a generalization of classical ordinary calculus with utilizations of integrals and derivatives with an arbitrary order. We are concerned with approximations using a small parameter of the Caputo and Riemann–Liouville type fractional derivative operators. Using this approximation, a fractional-order differential equation may be converted into an integer-order equation [2,3,4,5,6,7]. Noether’s theorem which was introduced by Emmy Noether in 1918 describing general concepts related to symmetry groups and conservation laws is a useful tool in the solutions of perturbed differential equations, see, e.g., [11,12,13]. Because of the importance of perturbed systems to describe the natural phenomena, they generalized the Noether’s theorem to approximated version This generalization helps to find approximate conservation laws of a given system including the related topics [16,17]. B are constants and Dtα is fractional derivative of order α
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