# Similarity Solutions to Nonlinear Diffusion/Harry Dym Fractional Equations

Authors: Chao Yue; Guijuan Liu; Kun Li; Hanhui Dong
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

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


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## Body

## 1. 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 [1–9]. 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 fractional 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
(1)∂αu∂tα=un∂nu∂xn,0<α<1,n=2,3and further study nonlinear model of space-fractional diffusion/Harry Dym equation
(2)∂u∂t=uβ∂βu∂xβ,2≤β≤3,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 functionft, the left Riemann-Liouville fractional derivative of order 0<α<1 is given as
(3)dαfdtα=1Γ1−αddt∫0tfτt−ταdτ=1Γ1−αf0tα+∫0tf1τt−ταdτ,where Γ. is the Euler Gamma function
(4)Γα=∫0∞e−zzα−1dz.Similarly, forn−1<β<n, we have
(5)dβfdtβ=1Γn−βdndtn∫0tfτt−τβ+1−ndτ.


## 2. 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 variables t~,x~,u~ in the form [9]
(6)t=λt~,x=λpx~,u=λqu~x~,t~,where p and q are parameters to be determined later. It is easy to verify that the transformed equation reads
(7)λq−α∂αu~∂t~α=λn+1q−npu~n∂nu~∂x~n,which implies that q−α=n+1q−np, that is
(8)p−q=αn.Since(9)tt~=xx~1p,uu~=xx~qp,in order to obtain the travelling-wave similarity solution to time-fractional Harry Dym equation
(10)∂αu∂tα=u3uxxx,0<α<1,we consider the similarity transformation
(11)u=xqpUξ,ξ=ctx−1,where p,q, and c are constants to be determined. We find that
(12)∂αu∂tα=1Γ1−α∂∂t∫0txq/pUct/x−1t−ταdτ.Sety=ct/x−1; then t−τ=x/cξ−y,∂/∂t=x/cd/dξ, and then Equation (12) becomes
(13)∂αu∂tα=1Γ1−αcxddξ∫−1ξxq/pUyx/cαξ−yαxcdy=cαxq−αppdαUξdξα,(14)∂u∂x=xq−ppqpUξ−ξ+1U′ξ,(15)∂2u∂x2=xq−2ppqq−pp2Uξ+2p−qpξ+1U′ξ+ξ+12U′′ξ,(16)∂3u∂x3=xq−3ppqq−pq−2pp3Uξ+3q−p2p−qp2ξ+1U′ξ+3q−6ppξ+12U′′ξ−ξ+13U′′′ξ.Inserting (13) and (16) into Equation (10) leads to
(17)cαxq−αppdαUξdξα=x4q−3ppTξU3ξ,It is easy to find thatq−αp=4q−3p, that is
(18)qp=1−α3.Then, the corresponding ordinary nonlinear fractional Harry Dym equation reads(19)cαdαUξdξα=TξU3ξ.We take special solutions of (19) in the forms:
(20)Uξ=Nξρ,ξ≥00,ξ<0.Substituting (20) into Equation (19) gives
(21)cαNΓ1+ρΓ2−α+ρ1−α+ρξρ−α=qq−pq−2pp3ξ3+3q−p2p−qp2ξ+1ρξ2+3q−6ppξ+12ρρ−1ξ−ξ+13ρρ−1ρ−2N4ξ4ρ−3,which leads to ρ−α=4ρ−3, then
(22)ρ=1−α3.By using (18) and (22), from Equation (21), we get
(23)N=183−2αcαΓ2−1/3αα3−9αΓ3−4/3α13.Hence, we obtain the travelling-wave similarity solutions to Equation (10) as follows:
(24)ux,t=xqpNξρ=Nct−x1−13α=183−2αcαΓ2−1/3αα3−9αΓ3−4/3α13ct−x1−13α.From (23), we get
(25)c=N3α3−9αΓ3−4/3α183−2αΓ2−1/3α1α.Inserting (25) into (24), we finally obtain
(26)ux,t=xqpNξρ=NN3α3−9αΓ3−4/3α183−2αΓ2−1/3α1αt−x1−13α.


## 3. Numerical Solutions of Time-Fractional Diffusion Equation

In order to obtain the numerical solutions of time-fractional diffusion equation(27)∂αu∂tα=u2uxx,0<α<1,we consider the similarity transformation
(28)u=xqpUξ,ξ=x−1pt,where Uξ,ξ,p, and q are constants to be determined. We find that
(29)∂αu∂tα=1Γ1−αx−1pddξ∫0ξxq/pUτx−α/pξ−ταx1pdτ=xq−αpdαUξdξα,(30)∂u∂x=xq−ppqpUξ−1pξU′ξ,(31)∂2u∂x2=xq−2ppqq−pp2Uξ+p−2q+1p2ξU′ξ+1p2ξ2U′′ξ.Substituting (29) and (31) into (27), we have corresponding ordinary nonlinear fractional diffusion equation
(32)dαUξdξα=qq−pp2U3ξ+p−2q+1p2ξU2ξU′ξ+1p2ξ2U2ξU′′ξ,p−q=α2.In what follows, we discuss its numerical solutions. In Ref. [12], suppose that a given function ft has continuous first and second derivatives, then we get
(33)fαt=t1−αΓ2−αf1t+∑n=1∞Γn−1+αΓα−1n!tn∫0tτnf2τdτ+f0tαΓ1−α.Utilizing the integration by parts on the right-hand side gives(34)fαt=1Γ2−αf1t1+∑n=1∞Γn−1+αΓα−1n!t1−α−α−1tαft+∑n=2∞Γn−1+αΓα−1n−1!fttα+Vn~tn−1+α+f0tαΓ1−α,where the moments Vn~ reads
(35)Vn~=1−n∫0tτn−2fτdτ,n=2,3,⋯As an application, we take finite number of terms in sums of (34), that is, we take n=2,3,⋯,N with suitable chosen NN=7. Thus, from following formula for the fractional derivative, we have
(36)fαt=1Γ2−αf1t1+∑n=1NΓn−1+αΓα−1n!t1−α−α−1tαft+∑n=2NΓn−1+αΓα−1n−1!fttα+Vn~tn−1+α+f0tαΓ1−α,where Vn~ is given by (35). Similar to the method proposed in Refs. [13–15], the fractional equation can be replaced by a system of first-order equations of integer order by using (36).In what follows, we consider Equation (32). We utilize the substitution x1=U,x2=U1 and (35) to express the fractional derivative Uα. Then, we have the following system of first-order equations (with ξ=t)
(37)x11=x2,x21=p2x12t2Γ2−αx2t1+∑n=1NΓn−1+αΓα−1n!t1−α−α−1tαx1t+∑n=2NΓn−1+αΓα−1n−1!x1ttα+Vn~tn−1+α+α+1−ptx2t−α2−2αp4t2x1t,with differential equations for variables Vn~,n=2,⋯,7(38)V2~1t=−x1t,V3~1t=−2tx1t,⋯=⋯,V7~1t=−6t5x1t,subject to
(39)x10=x0,x20=v0,V2~0=0,V3~0=0,⋯


## 4. 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:
(40)t=λt~,x=λpx~,u=λqu~x~,t~.Equation (2) is transformed to
(41)λq−1∂u~∂t~=λq+βq−βpu~β∂βu~∂x~β,which gives that
(42)p−q=1β.In terms of(43)xx~=tt~1p,uu~=tt~q,we have
(44)ξ=t−px,u=tqUξ.Then, it is easy to find(45)ut=tq−1qUξ−pξU′ξ,forn−1<β<n, by use of definition (5), one can compute that
(46)dβux,tdxβ=1Γn−β∂n∂xn∫0xux~,tx−x~β+1−ndx~.Let(47)Ix,t=1Γn−β∫0xux~,tx−x~β+1−ndx~.we have(48)∂nIx,t∂xn=1Γn−β∂n∂xn∫0xtqUt−px~x−x~β+1−ndx~.Setζ=t−px~; then, we have
(49)x−x~=tpξ−ζ,∂n∂xn=t−npdndξn.Substituting the above calculations into Equation (46), we have
(50)dβux,tdxβ=1Γn−βt−npdndξn∫0ξtqUζtptpβ+1−nξ−ζβ+1−ndζ=tq−βpdβUξdξβ.Inserting the above results into (2) yields
(51)tq−1qUξ−pξU′ξ=tq+βq−βpUβξdβUξdξβ,which leads to
(52)p−q=1β,which is equivalent to (42). Thus, we have the following fractional ordinary differential system:
(53)qUξ−pξU′ξ=UβξdβUξdξβ.Let us consider the solution in this form:(54)Uξ=Wξσand then insert (54) into (53); we have
(55)q−pσξσ=WβΓ1+σΓ2+σ−β1−β+σξσ+σβ−β,which gives σ=σ+σβ−β, that is
(56)σ=1.In terms of (52) and (56), from Equation (55), we get
(57)W=Γ3−βββ−21β.Then, we get the following similarity solutions to Equation (2):
(58)ux,t=Γ3−βββ−21βxt−1β.Similar to the discussion of the time-fractional Harry Dym Equation (10), we can obtain the travelling-wave similarity solution to space-fractional diffusion/Harry Dym Equation (2). To obtain this solution a suitable similarity transformation reads
(59)u=tqpUξ,ξ=xt−c¯,where p,q, and c are constants to be determined later. We see that
(60)∂u∂t=tq−ppqpUξ−ξ+c¯U′ξ,(61)∂βu∂xβ=1Γn−β∂n∂xn∫0xtq/pUx¯/t−c¯x−x¯β+1−ndx¯.Setz=x¯/t−c¯, and then x−x¯=tξ−z,∂n/∂xn=t−ndn/dξn; then, Equation (61) becomes
(62)∂βu∂xβ=1Γn−βt−ndndξn∫−1ξtq/pUztβ+1−nξ−zβ+1−ntdz=tq−βppdβUξdξβ.Inserting (60) and (62) into (2) yields
(63)tq−ppqpUξ−ξ+c¯U′ξ=tq+βq−βppUβξdβUξdξβ,which gives rise to
(64)qp=1−1β.Thus, we get the following fractional ordinary differential system:(65)dβUξdξβ=1−1βU1−βξ−ξ+c¯U−βξU′ξ.We take special solutions of (65) in the forms
(66)Uξ=U1ξδ,ξ≥00,ξ<0.Substituting (66) into Equation (65) gives
(67)U1Γ1+δΓ2+δ−β1−β+δξδ−β=U11−β1−1βξ−δξ+c¯ξδ−δβ−1,which implies
(68)δ=1−1β.Then, we have(69)U1=c¯Γ3−β−1/ββ−1Γ2−1/β1β.Thus, we obtain the following travelling-wave similarity solution:(70)ux,t=c¯Γ3−β−1/ββ−1Γ2−1/β1βx−c¯t1−1β.


## 5. Conclusions

In this paper, through similarity transformations, two nonlinear models of time-fractional and space-fractional diffusion/Harry Dym equation are transformed to corresponding ordinary fractional differential equations, from which four similarity solutions are presented, including travelling-wave similarity solutions and numerical solutions. The technology presented in this paper can be applied to other fractional partial differential equations, such as multi-equation systems and (2+1)-dimensional equations, to get more various similarity solutions, which will enrich and supplement the known results.



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