# An Approximate Solution of the Time-Fractional Fisher Equation with Small Delay by Residual Power Series Method

Authors: Ali Demir; Mine Aylin Bayrak; Ebru Ozbilge
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

An analytical solution of the time-fractional Fisher equation with small delay is established by means of residual the residual power series method (RPSM) where the fractional derivative is taken in the Caputo sense. Taking advantage of small delay, the time-fractional Fisher equation is expanded in powers series of delay termϵ. By using RPSM analytical solution of time-fractional of Fisher equation is constructed. The final results and graphical consequences illustrate that the proposed method in this study is very efficient, effective, and reliable for the solution of the time-fractional Fisher equation with small delay.


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## Body

## 1. Introduction

In last few decades, fractional calculus growing considerable interest is used in bioengineering, thermodynamics, viscoelasticity, control theory, aerodynamics, electromagnetics, signal processing, chemistry, and finance [1–7]. Many numerical methods have been applied and analyzed for differential equations with fractional order derivative of Riemann-Liouville or Caputo sense [6–12]. Delay differential equations (DDEs) can be considered as the generalization of the ordinary differential equations which are appropriate for modelling physical systems with memory. The delay is an important fact of the physical systems such as kinetics, controllers, signal processing, and damping behavior of viscoelastic materials.The RPSM was established as a powerful method for fuzzy differential equations [13]. It has been successfully put into practice in various fields [14–20]. The solution of problems by RPSM is obtained in the form of Maclaurin series. It is an efficient method to find out the coefficients of the series solutions. Construction of multidimensional and multiple solutions for fractional differential equations in the form of power series is an important advantage of RPSM. RPSM is effective and easy to use for solving linear and nonlinear FPDEs without linearization, perturbation, or discretization.In this study, we extend the application of the RPSM in order to establish an approximate solution to time-fractional Fisher equation with small delay(1)Dtαux,t=Dxxux,t+6ux,t-ϵ1-ux,t-ϵ,x∈R,t>0,0<α⩽1,subject to initial condition(2)ux,0=A1x which was originally proposed by Fisher [20] which describes the spatial and temporal propagation of a virile gene in an infinite medium. The time-fractional Fisher equation can be readily solved by many methods [21–25]. Moreover, there are various studies on nonlocal of action of fractional differential equations including space fractional derivative [26–30]. In this article, by making use of the residual power series method (RPSM), we find series solution for the time-fractional Fisher equation with small delay. By taking advantage of the small delay ϵ we expand the term including small delay to Taylor series which reduced the problem to a perturbation problem. Applying RPSM we determine the solution of equations which are the obtained from the coefficients of ϵ. Replacing these solutions in the Taylor series the solution of the original problem is obtained.


## 2. Preliminaries

In this section, the basic definitions and various features for fractional calculus theory are shown [6, 31–35].Definition 1.
The Riemann-Liouville fractional integral of orderα(α≥0) is given as [14](3)Jαfx=1Γα∫0xx-tα-1ftdt,α>0,x>0,(4)J0fx=fx.Definition 2.
The Caputo fractional derivative with orderα is given as [11, 12](5)Dαfx=Jm-αDmfx=∫0xx-tm-α-1dmdtmftdt,m-1<α<m,x>0, where Dm is the classic differential operator with order m.
By the Caputo derivative, we get(6)Dαxβ=0,β<α,(7)Dαxβ=Γβ+1Γβ+1-αxβ-α,β≥α.Definition 3.
Caputo’s time-fractional derivative of orderα of u(x,t) is defined as [11, 12](8)Dtαux,t=1Γm-α∫0tt-ξm-α-1∂mux,ξ∂tmdξ,m-1<α<m∂mux,t∂tm,α=m∈N.Definition 4.
A power series expansion of the form(9)∑m=0∞cmt-t0mα=c0+c1t-t0α+c2t-t02α+⋯,0≤m-1<α≤m,t≥t0 is called fractional power series about t=t0 [25].Definition 5.
A power series expansion of the form(10)∑m=0∞fmxt-t0mα=f0x+f1xt-t0α+f2xt-t02α+⋯,0≤m-1<α≤m,t≥t0 is called multiple fractional power series about t=t0 [11, 12].Theorem 6.
Suppose thatu(x,t) has a multiple fractional power series representation at t=t0 of the form(11)ux,t=∑m=0∞fmxt-t0mα,x∈I,t0≤t≤t0+R.If Dtmαu(x,t), m=0,1,2,…, are continuous on I×(t0,t0+R), then fm(x)=Dtmαu(x,t0)/Γ(mα+1).


## 3. RPSM of the Time-Fractional Fisher Equation with Small Delay

Consider the time-fractional Fisher equation (1) with small delay. Since the delay term is very small, we replace u(x,t-ϵ) (1) by the following Taylor series expansion:(12)ux,t-ϵ=ux,t-ϵDtux,t+Oϵ2, where we ignore the higher order terms, which leads to(13)Dtαux,t=Dxxux,t+6ux,t1-ux,t+6ϵuDtu-6ϵ1-uDtu-6ϵ2Dtu2 and assuming that nα=1, we obtain(14)Dtαux,t=Dxxux,t+6ux,t1-ux,t+6ϵuDtnαu-6ϵ1-uDtnαu-6ϵ2Dtnαu2. In this study we obtain the solution of time-fractional Fisher equation with small delay for n=2; hence we get the following equation:(15)Dtαux,t-Dxxux,t-6ux,t1-ux,t-6ϵ2u-1Dt2αu+6ϵ2Dt2αu2=0. Using (12) in (1) we reduce the problem to a perturbation problem for which we use the following series solution:(16)u=u0+ϵu1+ϵ2u2+ϵ3u3+⋯ to obtain the solution of (14). Substituting (15) into (14) leads to the following equation:(17)Dtαu0+ϵu1+ϵ2u2+ϵ3u3+⋯-Dxxu0+ϵu1+ϵ2u2+ϵ3u3+⋯-6u0+ϵu1+ϵ2u2+ϵ3u3+⋯+6u0+ϵu1+ϵ2u2+ϵ3u3+⋯2+12ϵu0+ϵu1+ϵ2u2+ϵ3u3+⋯Dt2αu0+ϵDt2αu1+ϵ2Dt2αu2+ϵ3Dt2αu3+⋯+6ϵDt2αu0+ϵDt2αu1+ϵ2Dt2αu2+ϵ3Dt2αu3+⋯+6ϵ2Dt2αu0+ϵDt2αu1+ϵ2Dt2αu2+ϵ3Dt2αu3+⋯=0, Hence we get the following equations:(18)Dtαu0-Dxxu0-6u0+6u02=0,(19)Dtαu1-Dxxu1-6u1+12u0u1-12u0Dt2αu0+6Dt2αu0=0,(20)Dtαu2-Dxxu2-6u2+6u12+12u0u2-12u0Dt2αu1-12u1Dt2αu0+6Dt2αu1+6Dt2αu02=0,(21)Dtαu3-Dxxu3-6u3+12u0u3+12u1u2-12u0Dt2αu2-12u1Dt2αu1-12u2Dt2αu0+6Dt2αu2+12Dt2αu0Dt2αu1=0 and so on. We apply the RPSM to find out series solution for these equation subject to given initial conditions by replacing its fractional power series expansion with its truncated residual function. From each equation, a repetition formula for the calculation of coefficients is supplied, while coefficients in fractional power series expansion can be calculated by repeatedly fractional differentiation of the truncated residual function [13–20]. The RPSM propose the solutions for (18)-(21) as a fractional power series about the initial point t=0 [13](22)uix,t=∑k=0∞fi,kxtkαΓkα+1,i=0,1,2,3,x∈I,0≤t<R. To obtain the numerical values from this series, let um(x,t) denote the m-th truncated series of u(x,t). That is,(23)ui,mx,t=∑k=0mfi,kxtkαΓkα+1,i=0,1,2,3,x∈I,0≤t<R. By the initial condition, the 0th residual power series approximate solution of u(x,t) can be written as follows:(24)u0x,t=f0x=ux,0=A1x Equation (22) can be written as(25)umx,t=A1x+∑k=2mAkxtkαΓkα+1,0<α≤1,x∈I,0≤t,k=2,3,… Define the residual function for (18) [35](26)Res0x,t=Dtαu0-Dxxu0-6u0+6u02 and the mth residual function can be expressed as(27)Resmx,t=Dtαum-Dxxum-6um+6um2 From [13–23], by making use of some results such as Res(x,t)=0 for each x∈I and t≥0 and DtrαRes(x,0)=DtrαResm(x,0)=0,r=0,1,2,…,m are used to obtain the solution.Substituting the mth truncated series of u(x,t) into (18), calculating the fractional derivative Dt(m-1)α of Res(x,t),m=1,2,3,…, at t=0 and solving the following obtained algebraic system(28)Dtm-1αResmx,0=0,0<α≤1,m=1,2,3,…, the required coefficients Ak(x),k=2,3,…,m in (24) are determined.In order to determineA2(x), the 1st residual function in (26) can be written as follows:(29)Res0,1x,t=Dtαu0,1-Dxxu0,1-6u0,1+6u0,12 where u0,1(x,t)=A1(x)+A2(x)(tα/Γ1+α). Therefore,(30)Res0,1x,t=A2-A1′′+A2′′tαΓ1+α-6A1+A2tαΓ1+α+6A1+A2tαΓ1+α2. From (27), we deduce that Res1(x,0)=0, which leads to(31)A2x=A1′′+6A1-6A12. Similarly, to obtain A3(x), the 2nd residual function in (26) can be written in the following form:(32)Res0,2x,t=Dtαu0,2-Dxxu0,2-6u0,2+6u0,22, where u0,2(x,t)=A1(x)+A2xtα/Γ1+α+A3(x)(t2α/Γ(1+2α)). Therefore,(33)Res0,2x,t=A2x+A3xtαΓ1+α-A1′′+A2′′tαΓ1+α+A3′′t2αΓ1+2α-6A1+A2tαΓ1+α+A3t2αΓ1+2α+6A1+A2tαΓ1+α+A3t2αΓ1+2α2. The operator Dtα is applied on both sides of (32) as follows:(34)DtαRes0,2x,t=A3-A2′′+A3′′tαΓ1+α-6A2+A3tαΓ1+α+6A2+A3tαΓ1+αA1+A2tαΓ1+α+A3t2αΓ1+2α+6A1+A2tαΓ1+α+A3t2αΓ1+2αA2+A3tαΓ1+α From (27) and (33),(35)A3x=A2′′+6A2-12A1A2. The same manner is repeated as above; the following recurrence results is obtained(36)A4x=A3′′+6A3-12A1A3-12A22 and so on.Thus, we have(37)u0=A1+A1′′+6A1-6A12tαΓ1+α+A2′′+6A2-12A1A2t2αΓ1+2α+A3′′+6A3-12A1A3-12A22t3αΓ1+3α+⋯ Define the residual function for (19)(38)Res1x,t=Dtαu1-Dxxu1-6u1+12u0u1-12u0Dt2αu0+6Dt2αu0. Suppose that(39)u1x=B1x+B2xtαΓ1+α+B3xt2αΓ1+2α+B4xt3αΓ1+3α+⋯ By the initial condition, the 0th residual power series approximate solution of u(x,t) can be written as follows:(40)u1x,0=0=B1x We apply the RPSM to find out Bk(x), k=1,2,3,…,m, in (19). Thus, we have(41)u1=B1+B1′′+6B1-12A1B1+12A1A3-6A3tαΓ1+α+B2′′+6B2-12A2B1-12A1B2+12A2A3+12A1A4-6A4t2αΓ1+2α+B3′′+6B3-12A3B1-12A1B3-24A2B2+12A32+24A2A4t3αΓ1+3α+⋯ Define the residual function for (20)(42)Res2x,t=Dtαu2-Dxxu2-6u2+6u12+12u0u2-12u0Dt2αu1-12u1Dt2αu0+6Dt2αu1+6Dt2αu02. Suppose that(43)u2x=C1x+C2xtαΓ1+α+C3xt2αΓ1+2α+C4xt3αΓ1+3α+⋯ By the initial condition, the 0th residual power series approximate solution of u2(x,t) can be written as follows:(44)u2x,0=0=C1x We apply the RPSM to find out Ck(x),k=1,2,3,…,m in (20).Thus, we have(45)u2=C1+C1′′+6C1-6B12-12A1C1+12A1B3+12A3B1-6B3-6A32tαΓ1+α+C2′′+6C2-12B1B2-12A2C1-12A1C2+12A2B3+12A1B4+12A4B1+12A3B2-6B4-12A3A4t2αΓ1+2α+C3′′+6C3-12B3B1-12B22-24A2C2-12A3C1-12A1C3+24A3B3+24A2B4+24A4B2-12A42t3αΓ1+3α+⋯ Define the residual function for (11)(46)Res3x,t=Dtαu3-Dxxu3-6u3+12u0u3+12u1u2-12u0Dt2αu2-12u1Dt2αu1-12u2Dt2αu0+6Dt2αu2+12Dt2αu0Dt2αu1. Suppose that(47)u3x=D1x+D2xtαΓ1+α+D3xt2αΓ1+2α+D4xt3αΓ1+3α+⋯ By the initial condition, the 0th residual power series approximate solution of u2(x,t) can be written as follows:(48)u3x,0=0=D1x We apply the RPSM to find out Dk(x), k=1,2,3,…,m, in (21).Thus, we have(49)u3=D1+D1′′+6D1-12A1D1-12B1C1+12A1C3+12A3C1-6C3-12A3B3tαΓ1+α+D2′′+6D2-12A2D1-12A1D2-12B2C1-12B1C2+12A2C3+12A1C4+12B2B3+12B1B4+12A4C1+12A3C2-6C4-12A4B3-12A3B4t2αΓ1+2α+D3′′+6D3-12A3D1-12A1D3-24A2D2-12B3C1-12B1C3-24B2C2+24A3C3+24A2C412B32+24B2B4+24A4C2-24A4B4t3αΓ1+3α+⋯Forn=3, we obtain u0,u1,u2, and u3 as follows:(50)u0=A1+A1′′+6A1-6A12tαΓ1+α+A2′′+6A2-12A1A2t2αΓ1+2α+A3′′+6A3-12A1A3-12A22t3αΓ1+3α+⋯ and(51)u1=B1+B1′′+6B1-12A1B1+12A1A4-6A4tαΓ1+α+B2′′+6B2-12A2B1-12A1B2+12A2A4+12A1A4-6A4t2αΓ1+2α+B3′′+6B3-12A3B1-12A1B3-24A2B2+12A3A4t3αΓ1+3α+⋯ and(52)u2=C1+C1′′+6C1-6B12-12A1C1+12A1B4+12A4B1-6B4-6A42tαΓ1+α+C2′′+6C2-12B1B2-12A2C1-12A1C2+12A2B4+12A4B2t2αΓ1+2α+C3′′+6C3-12B3B1-12B22-24A2C2-12A3C1-12A1C3+12A3B4+12A4B3t3αΓ1+3α+⋯and(53)u3=D1+D1′′+6D1-12A1D1-12B1C1+12A1C3+12A3C1-6C3-12A3B3tαΓ1+α+D2′′+6D2-12A2D1-12A1D2-12B2C1-12B1C2+12A2C3+12A1C4+12B2B3+12B1B4+12A4C1+12A3C2-6C4-12A4B3-12A3B4t2αΓ1+2α+D3′′+6D3-12A3D1-12A1D3-24A2D2-12B3C1-12B1C3-24B2C2+24A3C3+24A2C412B32+24B2B4+24A4C2-24A4B4t3αΓ1+3α+⋯ For n=4, we have u0,u1,u2, and u3 as follows:(54)u0=A1+A1′′+6A1-6A12tαΓ1+α+A2′′+6A2-12A1A2t2αΓ1+2α+A3′′+6A3-12A1A3-12A22t3αΓ1+3α+⋯ and(55)u1=B1+B1′′+6B1-12A1B1tαΓ1+α+B2′′+6B2-12A2B1-12A1B2t2αΓ1+2α+B3′′+6B3-12A3B1-12A1B3-24A2B2t3αΓ1+3α+⋯ and(56)u2=C1+C1′′+6C1-6B12-12A1C1tαΓ1+α+C2′′+6C2-12B1B2-12A2C1-12A1C2t2αΓ1+2α+C3′′+6C3-12B3B1-12B22-24A2C2-12A3C1-12A1C3t3αΓ1+3α+⋯ and(57)u3=D1+D1′′+6D1-12A1D1-12B1C1tαΓ1+α+D2′′+6D2-12A2D1-12A1D2-12B2C1-12B1C2t2αΓ1+2α+D3′′+6D3-12A3D1-12A1D3-24A2D2-12B3C1-12B1C3-24B2C2t3αΓ1+3α+⋯


## 4. Numerical Results

Consider the following time-fractional Fisher equation with small delay(58)Dtαux,t=Dxxux,t+6ux,t-ϵ1-ux,t-ϵ subject to initial condition(59)ux,0=11+ex2. Then, the exact solution of (1) without delay when α=1 is given by(60)ux,t=11+ex-5t2. It is clear from Figures 1 and 2 and Tables 1-2 that the graph of the solution u(x,t) of time-fractional Fisher equation with small delay with respect to space x in some neighborhood of x=0 solution acts as a critically damped oscillator for α<1/2 whereas it acts as an over damped oscillator for α>1/2. In Figures 3 and 4, as it is seen from the graph of the solution u(x,t) of time-fractional Fisher equation with small delay with respect to time t, the effect of the small time delay is limited for the fractional derivative of order α<1/2; i.e., the solution u(x,t) of time-fractional Fisher equation with small delay keeps close to the solution u(x,t) of Fisher equation without delay. As a result, it is concluded that the solution u(x,t) is stable under small time delay for α<1/2. On the other hand, the effect of the small time delay increases as time increases for α>1/2; i.e., the solution u(x,t) of time-fractional Fisher equation with small delay gets away from the solution u(x,t) of Fisher equation without delay. As a result, it is concluded that the solution u(x,t) is unstable under small time delay for α>1/2.Table 1
The RPS solution for several valuesx and α for t=0.5.
x α = 1 / 4 α = 1 / 3 α = 1 / 2 0.1 -3.95998 -3.72539 -0.046675 0.2 -1.65603 -1.82531 0.52762 0.4 2.76181 1.75026 2.37864 0.6 6.42178 4.62171 3.82190 0.8 8.93800 6.71422 4.70664 1 10.21002 8.02085 5.02524 1.5 9.17997 7.40754 4.10699 2 5.77391 3.99228 2.45076 2.5 2.95270 1.70703 1.21563 3 1.32987 0.70989 0.53796 3.5 0.55378 0.29586 0.22174 4 0.21974 0.12070 0.08746 4.5 0.08465 0.04780 0.03358 5 0.03203 0.01846 0.01268Table 2
The RPS solution for several valuest and α for x=0.5.
t α = 0.5 α = 0.7 α = 0.9 α = 1 0.1 -0.41551 0.75439 1.01269 0.98597 0.2 0.52501 1.11315 1.11369 1.02597 0.4 2.29109 1.76224 1.27825 1.07915 0.6 3.70390 2.24542 1.37108 1.08652 0.8 4.61711 2.51281 1.38106 1.04491 1 5.00312 2.56283 1.31405 0.96068 1.5 4.24695 2.02110 0.94079 0.65201 2 2.61056 1.20375 0.53506 0.36056 2.5 1.32027 0.59918 0.25991 0.17244 3 0.59119 0.26598 0.11381 0.07484 3.5 0.24539 0.10985 0.04664 0.03051 4 0.09719 0.04338 0.01833 0.01195 4.5 0.03740 0.01667 0.00702 0.00457 5 0.01414 0.00629 0.00265 0.00172Figure 1
u solution for t=0.5, ϵ=0.001, and 0≤x≤10.Figure 2
u solution for t=0.5, ϵ=0.001, and 0≤x≤10.Figure 3
u solution for x=0.5, ϵ=0.001, and 0≤t≤50.Figure 4
u solution for x=0.5, ϵ=0.001, and 0≤t≤50.


## 5. Conclusion

Based on the analysis of the solutionu(x,t) of time-fractional Fisher equation with small delay, it is deduced that, at the fractional derivative of order α=1/2, there is a transition of the solution u(x,t) with respect to both variables x and t.



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