# New Iterative Method for the Solution of Fractional Damped Burger and Fractional Sharma-Tasso-Olver Equations

Authors: Mohammad Jibran Khan; Rashid Nawaz; Samreen Farid; Javed Iqbal
Journal: Complexity
 (2018)

---

## Abstract

The new iterative method has been used to obtain the approximate solutions of time fractional damped Burger and time fractional Sharma-Tasso-Olver equations. Results obtained by the proposed method for different fractional-order derivatives are compared with those obtained by the fractional reduced differential transform method (FRDTM). The 2nd-order approximate solutions by the new iterative method are in good agreement with the exact solution as compared to the 5th-order solution by the FRDTM.


---

## Body

## 1. Introduction

Most of the problems arising in the physical and biological area of science are nonlinear in nature, and it is not always possible to find the exact solution of such problems. These problems become more complicated when they involve fractional derivatives and are modelled through mathematical tools from fractional calculus. Fractional partial differential equations (FPDEs) are tremendous instrument and are widely used to describe many significant phenomena and dynamic processes such as engineering, rheology, acoustic, electrical networks, and viscoelasticity [1–6]. Generally, partial differential equations (PDEs) are hard to tackle, and their fractional-order types are more complicated [7, 8]. Therefore, several analytical and approximate methods can be used for finding their approximate solutions such as Adomian decomposition [9], homotopy analysis [10], tau method [11], residual power series method [12], and optimal homotopy asymptotic method [13]. Though the study of FPDEs has been obstructed due to the absence of proficient and accurate techniques, the derivation of approximate solution of FPDEs remains a hotspot and demands to attempt some dexterous and solid plans which are of interest. Daftardar-Gejji and Jafari proposed an iterative method called the new iterative method (NIM) for finding the approximate solution of differential equations [14]. NIM does not require the need for calculation of tedious Adomian polynomials in nonlinear terms like ADM, the need for determination of a Lagrange multiplier in its algorithm like VIM, and the need for discretization like numerical methods. The proposed method handles linear and nonlinear equations in an easy and straightforward way. Recently, the method has been extended for differential equations of the fractional order [15–17].In the present study, we have implemented NIM for finding the approximate solution of the following fractional-order damped Burger equation.(1)Dtαux,t+ux,tDxux,t−Dx2ux,t+λux,t=0,t>0,0<α≤1.Second, consider the fractional-order Sharma-Tasso-Olver equation of the following form.(2)Dtαux,t+aDxu3x,t+32aDx2u2x,t+aDx3ux,t=0,t>0,0<α≤1,where α is the parameter describing the order of fractional derivatives, ux,t is the function of x and t, and a,λ are constants. The fractional derivatives are described in the Caputo sense. NIM converges rapidly to the exact solution compared to FRDTM, and only at the 2nd iteration does the proposed method yield very encouraging results. The accuracy of the proposed method can further be increased by taking higher-order approximations.


## 2. Definitions

In this section, we have stated some definitions which are relevant to our work.Definition 1.
A functiongy,y>0, is said to be in space Cη,η∈ℜ, if there exists a real number p>η, such that gy=ypg1y, where g1y∈C0,∞. The function gy,y>0, is said to be in space Cηλ if only if gλ∈Cη,λ∈N.Definition 2.
The R-L fractional integral operator of orderα≥0 of a function g∈Cη,η≥−1, is as follows:
(3)Jaαgy=1Γα∫ayy−ηα−1gηdη,α>0,y>a,Ja0gy=gy.Because of certain disadvantages of R-L fractional derivative operator, Caputo proposed modified fractional differential operatorcDα as follow.Definition 3.
Caputo fractional derivative ofgy takes the following form.
(4)cDaαgy=1Γλ−α∫ayy−ηλ−α−1gληdη,where
(5)λ−1<α≤λ,λ∈N,y>a,g∈C−1λ.Definition 4.
Ifλ−1<α≤λ,λ∈N, and g∈Cηλ,η≥−1, then RLDaαJaαgy=gy and JaαcDaαgy=gy−∑k=0λ−1gkay−ak/k!,y>a.The properties of the operatorJaα are shown as follows:
(i)
Jaαgy exists for almost every y∈a,b.
(ii)
JaαJaβgy=Jaα+βgy.
(iii)
JaαJaβgy=JaβJaαgy.
(iv)
Jaαy−aγ=Γγ+1/Γα+γ+1y−aα+γ.In the equations above,g∈Cηλ, α,β>0, η≥−1, and γ≥−1.


## 3. New Iterative Method

The basic mathematical theory of NIM is described as follows.Let us consider the following nonlinear equation:(6)vy=fy+ξvy+ℵvy,where fy, y=y1,y,y3,…,yn, is the known function and ξ and N are the linear and nonlinear functions of vy, respectively. According to the basic idea of NIM, the solution of the above equation has the series form.
(7)vy=∑k=0∞vky.The linear operatorξ can be decomposed as
(8)∑k=0∞ξvk=ξ∑k=0∞vk.The decomposition of the nonlinear operatorℵ is as follows:
(9)ℵ∑k=0∞vk=ℵv0+∑k=1∞ℵ∑i=0kvi−ℵ∑i=0k−1vi.Hence, the general equation of (6) takes the following form:
(10)vy=∑k=0∞vky=f+ξ∑k=0∝vk+ℵv0+∑k=1∞ℵ∑i=0kvi−ℵ∑i=0k−1vi,From this, we have(11)v0=f,v1=ξv0+ℵv0,v2=ξv1+ℵv0+v1−ℵv0,vm+1=ξvm+ℵv0+v1+⋯+vm−ℵv0+v1+⋯+vm−1,m=1,2,3…Thek-term series solution of the general equation (6) takes the following form:
(12)v=v0+v1+⋯+vk−1.


## 4. Applications

Example 1
(damped Burger equation).
Consider the damped Burger equation(13)Dtαux,t+ux,tDxux,t−Dx2ux,t+λux,t=0,t>0,0<α≤1,
together with IC(14)ux,0=λx,where λ is a constant. The exact solution of (13) is of the following form:
(15)ux,t=λx2eλt−1.
Using the operatorJα on both sides of (13) using the initial condition and Definition 4 yields
(16)ux,t=λx+Jα−ux,tDxux,t+Dx2ux,t−λux,t,where ξu=JαDx2ux,t−λux,t and ℵu=Jα−ux,tDxux,t.
According to (11), we have
(17)u0x,t=λx,u1x,t=−2λ2tαxΓα+1,u2x,t=2λ3t2αx3−2λtαΓ2α+12/Γα+12Γ3α+1Γ2α+1.
The three-term approximate solution of the above equation is(18)ux,t=λx−2λ2tαxΓα+1+2λ3t2αx3−2λtαΓ2α+12/Γα+12Γ3α+1Γ2α+1.Example 2
(Sharma-Tasso-Olver equation).
One can consider(19)Dtαux,t+aDxu3x,t+32aDx2u2x,t+aDx3ux,t=0,t>0,0<α≤1,
together with IC(20)ux,0=1atanh1ax,where a is a constant. The exact solution of (19) for α=1 is of the following form:
(21)ux,t=1atanh1ax−t.
Using the operatorJα on both sides of (19) using the initial condition and Definition 4 yields
(22)ux,t=1atanh1ax+Jα−aDxu3x,t−32aDx2u2x,t−aDx3ux,t,where ξu=Jα−aDx3ux,t and ℵu=Jα−aDxu3x,t−3/2aDx2u2x,t.
According to (11), we have
(23)u0x,t=1atanh1ax,u1x,t=−tαsech2x/aaΓ1+α,u2x,t=t2αsech2x/aa5/2−3atα−3+2cosh2x/aΓ1+2αsech4x/aΓ1+α2Γ1+3α−2atanhx/aΓ1+2α−6t2αΓ1+3αsech4x/atanhx/aΓ1+α3Γ1+4α.
The three-term approximate solution of the above equation is as follows:(24)ux,t=1atanh1ax−tαsech2x/aaΓ1+α+t2αsech2x/aa5/2−3atα−3+2cosh2x/aΓ1+2αsech4x/aΓ1+α2Γ1+3α−2atanhx/aΓ1+2α−6t2αΓ1+3αsech4x/atanhx/aΓ1+α3Γ1+4α.


## 5. Results and Discussion

We have implemented NIM for finding the approximate solutions of the fractional damped Burger equation and fractional Sharma-Tasso-Olver equation. Tables1 and 2 show the numerical results of the 2nd-order NIM which are compared with those of the 5th-order fractional reduced differential transform method (FRDTM) solution [18] for the fractional-order damped Burger equation. Tables 3 and 4 show the comparison of the proposed scheme with the FRDTM for the fractional-order Sharma-Tasso-Olver equation. Figure 1 shows the comparison of 2D plot of the approximate and exact solution by NIM for the classical damped Burger equation. Figure 2 shows the comparison of the approximate solution for different values of α with the exact solution at t=0.01. In Figures 3 and 4, 3D plots of approximate and exact solutions by NIM for the damped Burger equation are given. In Figure 5, the 2D plots of the approximate and exact solution for the classical Sharma-Tasso-Olver equation are given. Figure 6 shows the comparison of the approximate solution for different values of α with the exact solution at t=0.1. The 3D plots of approximate and exact solutions for the Sharma-Tasso-Olver equation are given in Figures 7 and 8. Throughout computations, we take λ=1 and a=4.Table 1
Comparison of numerical results of NIM and FRDTM atα=1 and λ=1.
x
t
5th-order FRDTM [18]
2nd-order NIM
Exact solution
Absolute error
−5
0.002
−4.980060
−4.98006
−4.98006
1.19501 × 10−7
0.004
−4.960239
−4.96024
−4.96024
9.52046 × 10−7
0.006
−4.940535
−4.94054
−4.94054
3.19985 × 10−7
0.008
−4.920949
−4.92096
−4.92095
7.55346 × 10−6
−3
0.002
−2.988036
−2.98804
−2.98804
7.17009 × 10−8
0.004
−2.976143
−2.97614
−2.97614
5.71228 × 10−7
0.006
−2.964321
−2.96432
−2.96432
1.91991 × 10−6
0.008
−2.952569
−2.95257
−2.95257
4.53208 × 10−6
3
0.002
2.988036
2.98804
2.98804
7.17009 × 10−8
0.004
2.976143
2.97614
2.97614
5.71228 × 10−7
0.006
2.964321
2.96432
2.96432
1.91991 × 10−6
0.008
2.952569
2.95257
2.95257
4.53208 × 10−6
5
0.002
4.980060
4.98006
4.98006
1.19501 × 10−7
0.004
4.960239
4.96024
4.96024
9.52046 × 10−7
0.006
4.940535
4.94054
4.94054
3.19985 × 10−6
0.008
4.920949
4.92096
4.92095
7.55346 × 10−6Table 2
Comparison of numerical results of the 2nd-order NIM and the 5th-order FRDTM for different values ofα.
x
t
FRDTM [18] (α=0.5)
FRDTM [18] (α=0.75)
NIM (α=0.5)
NIM (α=0.75)
−5
0.002
−4.211841
−4.876244
−4.55366
−4.89911
0.004
−3.938004
−4.794473
−4.4015
−4.8326
0.006
−3.744961
−4.724456
−4.29706
−4.7758
0.008
−3.592556
−4.661485
−4.21704
−4.72486
−3
0.002
−2.527105
−2.925746
−2.7322
−2.93946
0.004
−2.362803
−2.876684
−2.6409
−2.89956
0.006
−2.246977
−2.834674
−2.57824
−2.86548
0.008
−2.155534
−2.796891
−2.53022
−2.83492
3
0.002
2.527105
2.925746
2.7322
2.93946
0.004
2.362803
2.876684
2.6409
2.89956
0.006
2.246977
2.834674
2.57824
2.86548
0.008
2.155534
2.796891
2.53022
2.83492
5
0.002
4.211841
4.876244
4.55366
4.89911
0.004
3.938004
4.794473
4.4015
4.8326
0.006
3.744961
4.724456
4.29706
4.7758
0.008
3.592556
4.661485
4.21704
4.72486Table 3
Comparison of numerical results of NIM and FRDTM atα=1 and a=4.
x
t
5th-order FRDTM [18]
2nd-order NIM
Exact solution
Absolute error
−5
0.002
−0.493320
−0.49332
−0.49332
7.13918 × 10−12
0.004
−0.493334
−0.493334
−0.493334
5.70817 × 10−11
0.006
−0.493347
−0.493347
−0.493347
1.92542 × 10−10
0.008
−0.493360
−0.49336
−0.49336
4.56141 × 10−10
−3
0.002
−0.452664
−0.452664
−0.452664
6.65773 × 10−12
0.004
−0.452755
−0.452755
−0.452755
5.333 × 10−11
0.006
−0.452844
−0.452844
−0.452844
1.80218 × 10−10
0.008
−0.452934
−0.452934
−0.452934
4.27726 × 10−10
3
0.002
0.452484
0.452484
0.452484
6.64085 × 10−12
0.004
0.452393
0.452393
0.452393
5.30587 × 10−11
0.006
0.452302
0.452302
0.452302
1.78845 × 10−10
0.008
0.452211
0.452211
0.452211
4.23386 × 10−10
5
0.002
0.493294
0.493294
0.493294
7.14723 × 10−12
0.004
0.493281
0.493281
0.493281
5.721 × 10−11
0.006
0.493267
0.493267
0.493267
1.93192 × 10−10
0.008
0.493254
0.493254
0.493254
4.58194 × 10−10Table 4
Comparison of numerical results of the 2nd-order NIM and the 5th-order FRDTM for different values ofα.
x
t
FRDTM [18] (α=0.5)
FRDTM [18] (α=0.75)
NIM (α=0.5)
NIM (α=0.75)
−5
0.002
−0.493876
−0.49339
−0.49363
−0.493375
0.004
−0.494098
−0.493447
−0.493755
−0.493421
0.006
−0.494262
−0.493496
−0.493849
−0.493461
0.008
−0.494397
−0.49354
−0.493926
−0.493497
−3
0.002
−0.456455
−0.453141
−0.454774
−0.453036
0.004
−0.457972
−0.453523
−0.455639
−0.453348
0.006
−0.459102
−0.453856
−0.456286
−0.45362
0.008
−0.460032
−0.45416
−0.456819
−0.453867
3
0.002
0.448366
0.452001
0.450211
0.452106
0.004
0.446521
0.451607
0.449182
0.451784
0.006
0.445064
0.451259
0.448372
0.4515
0.008
0.443806
0.450937
0.447674
0.451237
5
0.002
0.492686
0.493223
0.492959
0.493238
0.004
0.492412
0.493165
0.492806
0.493191
0.006
0.492194
0.493113
0.492687
0.493149
0.008
0.492007
0.493066
0.492583
0.49311Figure 1
Numerical solution of the classical damped Burger equation with the exact solution att=0.01.Figure 2
Numerical solution of the fractional damped Burger equation with the exact solution for different values ofα at t=0.01.Figure 3
3D plot ofux,t for (DB) equation at α=1.Figure 4
3D plot of the exact solution for the DB equation.Figure 5
Numerical solution of the classical Sharma-Tasso-Olver equation with the exact solution att=0.1.Figure 6
Numerical solution of the fractional Sharma-Tasso-Olver equation with the exact solution for different values ofα at t=0.1.Figure 7
3D plot ofux,t for the STO equation at α=1.Figure 8
3D plot of the exact solution of the STO equation.By forming the numerical values and graphs, it is clear that NIM is a very powerful tool for the solution of fractional partial differential equations. The accuracy of the NIM can further be increased by taking higher-order approximations.


## 6. Conclusion

We have successfully applied NIM to time fractional (DB) and (STO) equations. Results reveal that NIM converges to the desired solution in lesser iteration compared to FRDTM. We can conclude that NIM computationally handles many physical and engineering problems in a simple and straightforward way. The accuracy of this method is also better than that of many methods which are computationally difficult to use.



---

*Source: 3249720-2018-09-05.xml*