Similarity Solutions to Nonlinear Diffusion/Harry Dym Fractional Equations

By using scalar similarity transformation, nonlinear model of time-fractional di ﬀ usion/Harry Dym equation is transformed to corresponding ordinary fractional di ﬀ erential equations, from which a travelling-wave similarity solution of time-fractional Harry Dym equation is presented. Furthermore, numerical solutions of time-fractional di ﬀ usion equation are discussed. Again, through another similarity transformation, nonlinear model of space-fractional di ﬀ usion/Harry Dym equation is turned into corresponding ordinary di ﬀ erential equations, whose two similarity solutions are also worked out.


Introduction
Nonlinear partial differential equations arise in many fields of engineering, physics, and applied mathematics. During the last few decades, nonlinear fractional partial differential equations have gained much attention due to their applications in many branches of science and engineering such as porous media, fluid flow, fractals, heat conduction, control theory, dynamical processes, and other areas. It is generally known that fractional calculus can propose better results than classical calculus. Many methods have been used to study and analyze fractional differential equations, in which the Lie-group analysis method is an effective tool to investigate symmetries of ordinary and partial differential equations. Later, this method was generalized to study fractional partial differential equations [1][2][3][4][5][6][7][8][9]. Djordjevic and Atanackovic obtained similarity solutions to nonlinear fractional heat conduction equation and Burgers/KdV equations [9]. It is very critical to mention two recent papers [10,11]. First, in Ref. [10], the authors presented and discussed a fractional nonlinear partial differential equation by use of similarity reductions and recovered some interesting results associated with Harry Dym-type equations. In addition, in Ref. [11], the fractional nonlinear space-time wave-diffusion equation was discussed and solved by the similarity method utilizing frac-tional derivatives in the Caputo, Riesz-Feller, and Riesz senses. In this work, we shall treat a nonlinear model of time-fractional diffusion/Harry Dym equation and further study nonlinear model of space-fractional diffusion/Harry Dym equation with2 ≤ β ≤ 3, we can obtain all equations between the diffusion and Harry Dym equation. In the following, we want to study similarity solutions with Equation (1), including a travelling-wave similarity solution and a kind of numerical solutions. Furthermore, two similarity solutions in Equation (2) are also produced.
First of all, we recall several associated notations. For continuous function f ðtÞ, the left Riemann-Liouville fractional derivative of order 0 < α < 1 is given as where Γð:Þ is the Euler Gamma function Similarly, for n − 1 < β < n, we have

Travelling-Wave Similarity Solution of Time-Fractional Harry Dym Equation
Firstly, we will prove that nonlinear model (1) possesses similarity solutions, consider Lie-group scaling transformation by introducing new variablest,x,ũ in the form [9] where p and q are parameters to be determined later. It is easy to verify that the transformed equation reads which implies that q − α = ðn + 1Þq − np, that is Since in order to obtain the travelling-wave similarity solution to time-fractional Harry Dym equation we consider the similarity transformation where p, q, and c are constants to be determined. We find that Set y = ct/x − 1; then t − τ = ðx/cÞðξ − yÞ, ∂/∂t = ðx/cÞðd/ dξÞ, and then Equation (12) becomes Inserting (13) and (16) into Equation (10) leads to It is easy to find that q − αp = 4q − 3p, that is Then, the corresponding ordinary nonlinear fractional Harry Dym equation reads We take special solutions of (19) in the forms: Substituting (20) into Equation (19) gives which leads to ρ − α = 4ρ − 3, then 2 Advances in Mathematical Physics By using (18) and (22), from Equation (21), we get Hence, we obtain the travelling-wave similarity solutions to Equation (10) as follows: From (23), we get Inserting (25) into (24), we finally obtain

Numerical Solutions of Time-Fractional Diffusion Equation
In order to obtain the numerical solutions of time-fractional diffusion equation we consider the similarity transformation where UðξÞ, ξ, p, and q are constants to be determined. We find that Substituting (29) and (31) into (27), we have corresponding ordinary nonlinear fractional diffusion equation In what follows, we discuss its numerical solutions. In Ref. [12], suppose that a given function f ðtÞ has continuous first and second derivatives, then we get Utilizing the integration by parts on the right-hand side gives where the moments f V n reads As an application, we take finite number of terms in sums of (34), that is, we take n = 2, 3, ⋯, N with suitable chosen NðN = 7Þ. Thus, from following formula for the fractional derivative, we have where f V n is given by (35). Similar to the method proposed in Refs. [13][14][15], the fractional equation can be replaced by a system of first-order equations of integer order by using (36).

Advances in Mathematical Physics
In what follows, we consider Equation (32). We utilize the substitution x 1 = U, x 2 = U ð1Þ and (35) to express the fractional derivative U ðαÞ . Then, we have the following system of first-order equations (with ξ = t) with differential equations for variables f V n , n = 2, ⋯, 7 subject to

Two Similarity Solutions of Space-Fractional Diffusion/Harry Dym Equation (2)
Similarity transformation of space-fractional diffusion/Harry Dym Equation (2) is similar to the corresponding discussion of time-fractional diffusion/Harry Dym Equation (1). Take the following transformation: Equation (2) is transformed to which gives that In terms of we have Then, it is easy to find forn − 1 < β < n, by use of definition (5), one can compute that Let we have Set ζ = t −px ; then, we have Substituting the above calculations into Equation (46), we have Inserting the above results into (2) yields which leads to which is equivalent to (42). Thus, we have the following fractional ordinary differential system: Let us consider the solution in this form: