We introduce a simple and efficient series solution for a class of nonlinear fractional differential equations of
Caputo's type. The new approach is a modified form of the well-known Taylor series expansion where we overcome the difficulty of computing iterated fractional derivatives, which do not compute in general. The terms of the series are determined sequentially with explicit formula,
where only integer derivatives have to be computed. The efficiency of the new algorithm is illustrated through several examples. Comparison with other series methods such as the Adomian decomposition method and the homotopy perturbation method is made to indicate the efficiency of the new approach. The algorithm can be implemented for a wide class of fractional differential equations with different types of fractional derivatives.
1. Introduction
During the last three decades, fractional calculus caught the attention of many researchers in differential fields of science and engineering. This is, mainly, due to the importance of noninteger order derivatives in modeling certain physical phenomena [1–4]. It turns out that, in some cases, modeling using fractional calculus is more realistic than integer calculus. This is because of the fact that the behavior of many physical phenomena depends not only upon the instantaneous state but also on the previous time history. Fractional derivative, comprising in its definition previous time history about the function, makes it more suitable for modeling systems whose evolution depends upon their current and previous states.
Recently, many researchers got interested in looking at fractional differential equations (FDEs) as new model equations for many physical problems. However, many of these such FDEs do not possess exact analytic solutions. This difficulty prompted many researchers to develop numerical schemes to find approximate solutions. Many numerical methods used to solve integer order differential equations have been adapted to treat FDEs such as the variational iteration (VIM) [5–8], the homotopy analysis method (HAM) [9–14], and the Adomian decomposition method (ADM) [15–20], just to name a few. For a survey of recent development of methods in fractional calculus, the reader is referred to [21]. All these methods can be classified as iterative methods which produce a solution in the form of a series expansion whose terms are generated iteratively. However, for many cases, the iterative process of these methods is not easily implemented. For example, the ADM requires integration at each step to find the next iterate and the ADM requires solving a differential equation. Another approach is the upper-lower iterative method [22]. Quadrature techniques have been implemented to construct different formulations of fractional backward difference methods [23–25]. Also, fractional linear multistep methods presented for special types of the Volterra integral equation [26, 27] have been implemented for several types of fractional differential equations. As a result, a class of higher order backward difference methods have been obtained [28]. For more details one can refer to [29] and the references therein. Convenient and easy presentations to discretize fractional derivative of arbitrary order have been obtained in a form of triangular strip matrices; see [30, 31]. The suggested approach leads to a significant simplification of the solutions of differential equations of fractional order.
In our present work we present a series solution method in the spirit of the Taylor series expansion for a class of nonlinear differential equation of fractional order. The coefficients of the series expansion are also iteratively computed but the iteration process involves only differentiation. Naturally, if the problem is of fractional order, the differentiation is also of fractional order. However, to overcome the use of fractional differentiation, we employ a transformation that allows us to use ordinary differentiation rather than fractional differentiation to recursively compute the coefficient of the series expansion. We see this as an advantage to the abovementioned methods.
In this paper, we consider the initial value problem of fractional order:
(1)Dtαu(x,t)=uxx(x,t)+h(x,t,u),t>0,x∈ℝ,(2)u(x,0)=u0(x),x∈ℝ,
where 0<α<1, h∈C∞(ℝ×ℝ+×ℝ,ℝ), u0(x)∈C∞(ℝ,ℝ), and Dtα is the Caputo partial fractional derivative of order α. For α∈ℝ, n-1<α≤n, n∈ℕ+, the left Caputo fractional derivative is defined by [3]
(3)Dtαu(x,t)=1Γ(n-α)∫0t∂u(x,s)∂s1(t-s)α+1-nds
and satisfies the following properties:
Dtαf(x)=0;
Dtαtr=Γ(r+1)Γ(r-α+1)tr-α,r>n-1,r∈ℝ;
Dtα(∑i=0mcifi(x,t))=∑i=0mciDtαfi(x,t), where c0,c1,…,cm are constants.
The Caputo partial fractional derivative in (3) is related to the Riemann-Liouville partial fractional integral, Itα, of order α, by
(4)Dtαf(x,t)=Itn-α∂nf(x,t)∂tn,
where, for n-1≤α<n,
(5)Itαf(x,t)=1Γ(α)∫0t(t-s)α-1f(x,s)ds.Itα can be considered as the inverse operator of Dtα in the sense
(6)ItαDtαf(x,t)=f(x,t)-∑k=0n-1∂kf(x,0+)∂tktkk!,DtαItαf(x,t)=f(x,t).
In this paper, we consider α=p/q rational with gcd(p,q)=1. The paper is organized as follows. In Section 2, we present the series solution method to problem (1) and (2). In Section 3, we present numerical results to illustrate the efficiency of the presented technique. Comparison with other methods such as the Adomian decomposition method (ADM) and the homotopy perturbation method (HPM) will be also presented in Section 3. Finally, we conclude with some remarks in Section 4.
2. Series Method
In this section, we present the series solution method to solve problem (1) and (2) and we give the final result for the ODE version of (1) and (2). Given the order α=p/q, we assume that the solution u(x,t) takes the form
(7)u(x,t)=∑k=0∞ak(x)tk/q,
where u(x,0)=u0(x) and ak(x) are functional coefficients to be determined. Clearly, a0(x)=u0(x). Formal substitution of (7) into (1) gives
(8)Dtα(∑k=0∞ak(x)tk/q)=∑k=0∞ak′′(x)tk/q+h(x,t,∑k=0∞uk(x)tk/q).
Assuming we can interchange the summation and the fractional derivative operator and using property 2 above, we obtain
(9)∑k=1∞ak(x)skt(k-p)/q=∑k=0∞ak′′(x)tkq+h(x,t,∑k=0∞ak(x)tk/q),
where sk=Γ(k/q+1)/Γ(k/q-α+1). Note that if p>1, we will have negative powers of t (for k=1,2,…,p-1) in the sum on the left hand side of (9). To avoid this, we multiply (9) by t(p-1)/q to get
(10)∑k=0∞ak+1(x)sk+1tk/q=∑k=0∞ak′′(x)t(k+p-1)/q+t(p-1)/qh×(x,t,∑k=0∞ak(x)tk/q).
One way of finding the coefficients ak(x), in line with finding the coefficients of a Taylor series, is to recursively apply the operator Dt1/q to (10) and substitute t=0. However, this is not convenient for implementation. To avoid the use of the fractional differentiation, we introduce the change of variable w=t1/q which transforms (10) into
(11)∑k=0∞ak+1(x)sk+1wk=∑k=0∞ak′′(x)wk+p-1+wp-1h×(x,wq,∑k=0∞ak(x)wk).
Now, differentiating ordinarily k times with respect to w and substituting w=0, we find the following recursion relation for k≥0:
(12)ak+1(x)=1sk+1ak-p+1′′(x)+1sk+1k!×[∂k∂wk[wp-1h(x,wq,∑m=0∞am(x)wm)]]w=0,
where we assume that al(x)≡0 for l<0.
We note that if problem (1) and (2) is an ordinary differential equation of fractional order, that is, u≡u(t) and h≡h(t,u), the coefficients ak are real numbers and the recursion relation (12) reduces to
(13)ak+1=1sk+1k![∂k∂wk[wp-1h(wq,∑m=0∞amwm)]]w=0.
Remark 1.
We remark that the present method is different in many ways from the ADM. A main difference between the two methods is that the ADM, in its generation of successive terms, uses fractional integration while the present method uses ordinary differentiation. However, when α=1 (p=q=1), formula (13) will reduce to
(14)ak+1=1(k+1)![∂k∂wk[h(w,∑m=0∞amwm)]]w=0,
which is the well-known Adomian polynomial formula [32].
In Section 3, we present several examples to show the practicality of this approach and make a comparison with other techniques such as the Adomian decomposition method and homotopy perturbation method.
3. Numerical ResultsExample 1.
Consider the fractional initial value problem
(15)Dt1/2y=Γ(32)(y2-t+1),y(0)=0,
with y(t)=t1/2 being the exact solution.
Applying the proposed algorithm, the solution takes the form y=∑k=0∞aktk/2. The zeroth coefficient is readily given by a0=y(0)=0. For k≥0, we have from (13)
(16)ak+1=Γ(3/2)sk+1k![∂k∂wk[(∑m=0∞amwm)2-w2+1]]w=0(17)=Γ(3/2)sk+1k![k!∑m=0kamak-m-2δk-2+δk],
where δj=1 if j=0 and 0 otherwise. With k=0, (17) gives a1=(Γ(3/2)/s1)(a02+1)=1, since s1=Γ(3/2). It can be easily verified that (17) gives ak=0 for k≥2. Hence the solution is
(18)y(t)=∑k=0∞aktk/2=a1t1/2=t1/2,
which is the exact solution.
First, we compare our results with the Adomian decomposition method (ADM). To apply the ADM, assume that the solution y(t) of (15) and the nonlinear function f(y)=y2can be written in the series form as
(19)y(t)=∑n=0∞yn(t),f(y)=∑n=0∞An,
where An, n=0,1,2,…, are called the Adomian polynomials. These polynomials can be derived by expanding the function f(y) about y0 as follows:
(20)f(y)=f(y0)+f′(y0)y-y01!+f′′(y0)(y-y0)22!+⋯
or
(21)f(y)=f(y0)+f′(y0)∑n=1∞yn1!+f′′(y0)(∑n=1∞yn)22!+⋯
Thus, Ak can be derived as
(22)Ak=1k!dkdβk[f(∑j=0∞βjyj)]β=0,j≥0.
Next, define the fractional differential operator L as L=Dt1/2; then (15) can be written in the form
(23)L(y)=Γ(32)(f(y)-t+1).
And defining the inverse operator as L-1(·)=It1/2, then the solution y(t) of (15) can be written in the form
(24)y(t)=Γ(32)It1/2(f(y))+Γ(32)It1/2((-t+1))=Γ(32)It1/2(∑n=0∞An)+Γ(32)It1/2((-t+1))=∑n=0∞yn(t).
Now balancing the last equality in (24) yields
(25)y0(t)=Γ(32)It1/2(-t+1),yk+1(t)=Γ(32)It1/2(Ak),k=0,1,….
The first few terms generated by ADM are given below:
(26)y0(t)=13(3-2t)t1/2,y1(t)=23t3/2-3245t5/2+64315t7/2,y2(t)=3245t5/2-16641575t7/2+3276859535t9/2-65536654885t11/2⋮
Next, we will compare our results with the homotopy perturbation method (HPM). To apply the HPM, define the homotopy H:[0,∞]×[0,1]→ℝ which satisfies
(27)H(y,p)=(1-p)D1/2y+p(D1/2y-Γ(32)y2-Γ(32)(1-t))=0.
The basic assumption is that the solution of problem (15) can be expressed as a power series in p:(28)y=y0+py1+p2y2+⋯,
where yi(0)=0 for i≥0. The approximate solution of problem (15) can be obtained as
(29)y(t)=limp→1(y0+py1+p2y2+⋯).
The convergence of the last series has been proved in [33].
Substituting (28) into (27) and equating the coefficients of the terms with like powers of p, we have
(30)p0:D1/2y0=0,y0(0)=0,p1:D1/2y1-Γ(32)y02-Γ(32)(1-t)=0,y1(0)=0,p2:D1/2y2-2Γ(32)y0y1=0,y2(0)=0,p3:D1/2y3-Γ(32)(y12+2y0y2)=0,y3(0)=0,⋮
which implies that
(31)y0(t)=0,y1(t)=t-23t3/2,y2(t)=0,y3(t)=23t3/2-3245t5/2+64315t7/2,y4(t)=0,y5(t)=3245t5/2-16641575t7/2+3276859535t9/2-65536654885t11/2.
The approximate solution is
(32)yHPM(t)=t-6475t7/2+3276859535t9/2-65536654885t11/2+⋯.
Figure 1 depicts the exact solution and the approximate solutions yADM(t)=∑k=04yk(t) and yHPM(t)=∑k=04yk(t) obtained by the Adomian decomposition method and the homotopy perturbation method, respectively.
The exact and approximate solutions of Example 1 for 0≤t≤0.5.
Example 2.
Consider the fractional initial value problem
(33)Dt2/3y=12Γ(4/3)t1/3(-3y-1),y(0)=1,
with y(t)=(t-1)2 being the exact solution.
Here α=2/3; hence p=2 and q=3. The solution assumes the form y=∑n=0∞antn/3 with a0=y(0)=1. Then, according to the previous section, we have for k≥0(34)ak+1=12Γ(4/3)sk+1k!×[∂k∂wkw2(-3[∑m=0∞amwm]1/2-1)]w=0.
Numerical computation of (34) gives a3=-2, a6=1, and all other coefficients are zero. Thus, our procedure produces the solution
(35)y(t)=a0+a3t+a6t2=1-2t+t2=(t-1)2,
which is the exact solution.
Example 3.
Consider the fractional initial value problem
(36)Dtαy=1-y2,y(0)=0.
For α=p/q, the solution assumes the form y=∑k=0∞aktk/q with a0=y(0)=0. Then according to the previous section, we have for k≥0(37)ak+1=1sk+1k![∂k∂wkwp-1(1-[∑m=0∞amwm]2)]w=0(38)=1sk+1k![∂k∂wkwp-1(1-∑m=0∞Cmwm)]w=0,
where
(39)Cm=∑j=0majam-j.
Simplification of (38) reveals the following recursion, where k≥0:
(40)ak+1={0,ifk<p-1,δk-p+1sk+1-1sk+1Ck-p+1,ifk≥p-1.
Numerical computation of (38) gives a2j=0,j≥0. We consider α=1/2. Since the exact solution, in closed form, is not available, we define the error
(41)En(t)=|Dt1/2yn-(1-yn2)|,
where yn=∑k=0naktk/q. Figure 2 on the left presents the approximate solution y10=∑k=010aktk/2 and on the right the error E10=|Dt1/2y10-(1-y102)|. Table 1 depicts the absolute error En for different values of n at various values of t. From the results presented, it is clear that the series converges and sufficient accuracy is achieved with few terms. However, more terms would be needed for larger values of t which is expected for any initial value problem. We note that, for α=1, we get sk=k and a1=1, a3=-1/3, a5=2/15, a7=-17/315, a9=62/2835,…, and the obtained series solution is
(42)y(t)=t-13t-13t3+215t5-17315t7+622835t9+⋯
which coincides with the Taylor series expansion of the exact solution y(t)=(e2t-1)/(e2t+1).
Error for various t values and different values of n for Example 3.
n
t=0.2
t=0.4
t=0.6
t=0.7
t=1.0
5
0.001798
0.0136945
0.0443229
0.101588
0.193623
10
3.2139×10-6
0.0000989578
0.000725096
0.00295596
0.00874915
15
2.70532×10-10
6.70364×10-8
1.66634×10-6
0.000016172
0.0000938036
20
1.30212×10-13
1.29792×10-10
7.29593×10-9
1.26453×10-7
1.15074×10-6
A plot of y10 and E10, for Example 3, where 0≤t≤1.
Example 4.
Consider the fractional initial value problem
(43)Dtαu=uxx+110u(1-u),u(x,0)=x.
For this example we take α=1/2; hence p=1 and q=2. The solution assumes the form y=∑k=0∞ak(x)tk/2 with a0(x)=u(x,0)=x. Then from (12), we have, for k≥0,
(44)ak+1(x)=1sk+1ak-p+1′′(x)+1sk+1k!×[∂k∂wk[wp-1h(wq,x,∑m=0∞am(x)wm)]]w=0,=1sk+1ak′′(x)+110sk+1k!×[∂k∂wk[∑m=0∞am(x)wm-∑m=0∞Cm(x)wm]]w=0,=1sk+1ak′′(x)+110sk+1(ak(x)-Ck(x)),
where
(45)Ck(x)=∑j=0kaj(x)ak-j(x).
The first few terms of the series solution are
(46)u(x,t)=x+tπ(2621444849845x-2621444849845x2)+tπ(x3-2748779069444704199304805+6871947673623520996524025x-687194767367840332174675x2+13743895347223520996524025x3)+(tπ)3/2(x4-28823037615171174422814637477412005225+41433116571808563216296169626722860875x-3602879701896396838024395795686675375x2+3602879701896396822814637477412005225x3-1801439850948198422814637477412005225x4)+⋯.
Tables 2 and 3 present the error
(47)En(x,t)=|Dt1/2un-unxx-110un(1-un)|
for various values of t and x and n=5,15, where un(x,t)=∑k=0nak(x)tk/2. The presented data indicate the accuracy of the series solutions obtained.
Error for various x and t values and n=5.
x
t=0.2
t=0.4
t=0.6
t=0.8
0
8.3726110×10-6
0.0000474301
0.000130878
0.000269023
0.5
2.38184×10-9
3.53911×10-8
1.69571×10-7
5.09316×10-7
1
8.2742×10-6
0.000046751
0.000128695
0.000263925
Error for various x and t values and n=15.
x
t=0.2
t=0.4
t=0.6
t=0.8
0
3.59581×10-11
6.5209×10-9
1.36682×10-7
1.18428×10-6
0.5
1.3542×10-12
2.39819×10-10
4.93086×10-9
4.19941×10-8
1
3.53848×10-11
6.40568×10-9
1.34033×10-7
1.15927×10-6
4. Concluding Remarks
We have presented a new algorithm for obtaining a series solution for a class of fractional differential equations. The algorithm is developed for a class of fractional partial differential equations of the Caputo type. We have applied the new algorithm to different examples. Accurate numerical solutions have been obtained as well as exact solutions for certain problems. The new algorithm is compared with the two well-known methods, the Adomian decomposition method (ADM) and the homotopy perturbation method (HPM), for one example. The exact solution is obtained after one step in the current method and after getting a telescoping sum by the HAM, where an approximate solution is obtained by the ADM. The idea of the new algorithm can be generalized to deal with various types of fractional functional equations.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research is supported by the Individual Research Grant 21S074. The authors would like to express their sincere appreciation to the Research Affairs at the United Arab Emirates University.
HeJ. H.Approximate analytical solution for seepage flow with fractional derivatives in porous mediaHeJ. H.Some applications of nonlinear fractional differential equations and their approximationsMainardiF.Fractional calculus: some basic problems in continum and statistical mechanicsYoungG. O.Definition of physical consistent damping laws with fractional derivativesBaleanuD.MachadoJ. A.CattaniC.BaleanuM. C.YangX. J.Local fractional variational iteration and decomposition methods for wave equation on cantor sets within local fractional operatorsJafariH.KhaliqueC. M.Homotopy perturbation and variational iteration methods for solving fuzzy differential equationsWuG.-C.BaleanuD.Variational iteration method for the Burgers flow with fractional derivatives: new Lagrange multipliersYangX. J.BaleanuD.Fractal heat conduction problem solved by local fractional variation iteration methodDasS.GuptaP. K.Approximate analytical solutions of time-space fractional diffusion equation by Adomian decomposition method and homotopy perturbation methodKademA.BaleanuD.Homotopy perturbation method for the coupled fractional lotka-volterra equationsQinY.-M.ZengD.-Q.Homotopy perturbation method for the q-diffusion equation with a source termSongL.ZhangH.Application of homotopy analysis method to fractional KdV-Burgers-Kuramoto equationSweilamN. H.KhaderM. M.Al-BarR. F.Numerical studies for a multi-order fractional differential equationXuH.Analytical approximations for a population growth model with fractional orderDaftardar-GejjiV.BhalekarS.Solving multi-term linear and non-linear diffusion-wave equations of fractional order by Adomian decomposition methodDuanJ.-S.RachR.BaleanuD.WazwazA.-M.A review of the adomian decomposition method and its applications to fractional differential equationsJafariH.Daftardar-GejjiV.Solving linear and nonlinear fractional diffusion and wave equations by Adomian decompositionYangC.HouJ.An approximate solution of nonlinear fractional differential equation by Laplace transform and Adomian polynomialsYangX. J.BaleabuD.ZhongW.-P.Approximate solutions for diffusion equations on cantor space-timeZengD. Q.QinY.-M.The Laplace-Adomian-Pade technique for the seepage flows with the Riemann-Liouville derivativesBaleanuD.WuG.DuanJ.Some analytical techniques in fractional calculus: realities and challengesAl-RefaiM.Ali HajjiM.Monotone iterative sequences for nonlinear boundary value problems of fractional orderChernJ. T.DiethelmK.An algorithm for the numerical solution of differential equations of fractional orderDiethelmK.LuchkoY.Numerical solution of linear multi-term initial value problems of fractional orderLubichC.On the stability of linear multistep methods for volterra convolution equationsLubichC.Fractional linear multistep methods for Abel-Volterra integral equations of the second kindHairerE.LubichC.SchlichteM.Fast numerical solution of nonlinear Voltera convolution equationsWeilbeerM.PodlubnyI.Matrix approach to discrete fractional calculusPodlubnyI.ChechkinA.SkovranekT.ChenY.Vinagre JaraB. M.Matrix approach to discrete fractional calculus II: partial fractional differential equationsAdomianG.HeJ. H.Coupling method of a homotopy technique and a perturbation technique for non-linear problems