# Solution of Space-Time-Fractional Problem by Shehu Variational Iteration Method

Authors: Suleyman Cetinkaya; Ali Demir; Hulya Kodal Sevindir
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this study, we deal with the problem of constructing semianalytical solution of mathematical problems including space-time-fractional linear and nonlinear differential equations. The method, called Shehu Variational Iteration Method (SVIM), applied in this study is a combination of Shehu transform (ST) and variational iteration method (VIM). First, ST is utilized to reduce the time-fractional differential equation with fractional derivative in Liouville-Caputo sense into an integer-order differential equation. Later, VIM is implemented to construct the solution of reduced differential equation. The convergence analysis of this method and illustrated examples confirm that the proposed method is one of best procedures to tackle space-time-fractional differential equations.


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## Body

## 1. Introduction

Last couple of decades, employing fractional differential equations in modelling of processes such as dynamical systems, biology, fluid flow, signal processing, electrical networks, reaction and diffusion procedure, and advection–diffusion–reaction process [1–4] has gained great importance since these models reflect the behaviour of the processes better than integer-order differential equations.Consequently, a great deal of methods such as [3, 4] are established to construct analytical and numerical solutions of fractional differential equations. Moreover, their existence, uniqueness, and stability have been studied by many scientists.One of the significant integral transformations is Shehu transformation proposed by Maitama and Zhao [5]. This linear transformation is a generalization of Laplace transformation. However, the Laplace transformation is obtained by substituting q=1 in Shehu transformation. By this transformation, differential equations are reduced into simpler equations.Various methods such as the homotopy perturbation method (HPM) and VIM are utilized to establish approximate solutions of differential equations of any kind [6, 7]. As a result, it is employed widely to deal with differential equations in various branches of science [8–11]. VIM has been modified by many researchers to improve this method. By modified VIM, the approximate solutions of initial value problems can be established by making use of an initial condition.


## 2. Preliminaries

In this section, preliminaries, notations, and features of the fractional calculus are given [12, 13]. Riemann-Liouville time-fractional integral of a real valued function ux,t is defined as
(1)Itαux,t=1Γα∫0tt−sα−1ux,sds,where α>0 denotes the order of the integral.Theαth-order Liouville-Caputo time-fractional derivative operator of ux,t is defined as
(2)∂αux,t∂tα=Itm−α∂mux,t∂tm=1Γm−α∫0tt−ym−α−1∂mux,y∂ymdy,m−1<α<m,∂mux,t∂tm,α=m.The function(3)Eα,βz=∑k=0∞zkΓαk+β,Reα>0,z,β∈ℂ,is called Mittag-Leffler function depending on two parameters α and β.The following set of functions has Shehu transformation:(4)ft∃P,τ1,τ2>0,ft<Pet/τj,ift∈−1j×0,∞,and it is defined as
(5)Sft=Fp,q=∫0∞e−p/qtftdt,which has the following property:
(6)Stα=∫0∞e−pt/qtαdt=Γα+1qpα+1,Reα>0.The inverse Shehu inverse transform of q/pnα+1 is defined as
(7)S−1qpnα+1=tnαΓnα+1,Reα>0,where n>0 [5].For theαth-order of Liouville-Caputo time-fractional derivative of fx,t, the Shehu transformation has the following form [14]:
(8)S∂αfx,t∂tα=pqαSfx,t−∑k=0n−1pqα−k−1∂kfx,0∂tk,n−1<α≤n,n∈ℕ.


## 3. Main Results

### 3.1. Fractional Shehu Variational Iteration Method

To reveal the fundamental notions of this method, let us take the following space-time-fractional initial value problem in the Liouville-Caputo fractional derivative:(9)DCtαux,t+Ru,DCxβu;x,t+Nu,DCxβu;x,t=gx,t,m−1<α≤m,n−1<β≤n,m,n=1,2,3,⋯,(10)∂m−1ux,t∂tm−1t=0=gm−1x,where N, R, and gx,t denote the nonlinear, linear part of the differential equation, and the source function, respectively.Utilizing Shehu transformation for Equation (9), we have
(11)Sux,t=∑k=0m−1qpk+1∂kux,0∂tk−qpαSRu,DCxβu;x,t+Nu,DCxβu;x,t+qpαSgx,t.Employing the inverse Shehu transformation for Equation (11) leads to
(12)ux,t=kx,t−S−1qpαSRu,DCxβu;x,t+Nu,DCxβu;x,t,where kx,t=S−1q/pαS∑k=0m−1p/qk+1∂kux,0/∂tk+q/pαSgx,t, and so
(13)∂ux,t∂t+∂∂tS−1qpαSRu,DCxβu;x,t+Nu,DCxβu;x,t−∂∂tkx,t=0.The following recurrence relation is established by VIM:(14)um+1x,t=umx,t−∫0t∂umx,τ∂τ+∂∂τS−1qpαSRum,DCxβum;x,τ+Num,DCxβum;x,τ−∂∂τkx,τdτ.Alternately,(15)um+1x,t=kx,t;β−S−1qpαSRum,DCxβum;x,t+Num,DCxβum;x,tis called the m+1th-order of truncated solution.If(16)limm→∞umx,texists, then the analytical solution ux,t=limm→∞umx,t.


### 3.2. Convergence Theorem

Now, the convergence of VIM is investigated and required conditions and error estimate [15] are established for Equation (9).The operatorV is introduced as
(17)V=−∫0t∂umx,τ∂τ+∂∂τS−1qpαSRum,DCxβum;x,τ+Num,DCxβum;x,τ−∂∂τkx,τdτ,where vk,k=0,1,2,⋯, denote the components of the solution satisfying
(18)ux,t=limm→∞umx,t=∑k=0∞vk.Theorem 1 [16].
LetV, defined in (5∗), be an operator from a Banach space BS to BS. The series solution ux,t=limm→∞umx,t=∑k=0∞vk as defined in (6∗) converges if 0<p<1 exists such that Vv0+v1+v2+⋯+vk+1≤pVv0+v1+v2+⋯+vk, (i.e., vk+1≤vk), ∀k∈ℕ∪0.Theorem1, obtained from the Banach fixed-point theorem, is utilized to establish a sufficient condition for the convergence of fractional VIM.Theorem 2 [16].
The exact solution of nonlinear problem (9) exists under the condition that the series solution ux,t=∑k=0∞vk defined in (18) converges.Theorem 3 [16].
Suppose that the series solution∑k=0∞vk defined in (18) converges to the solution ux,t. The maximum error Ejx,t for the approximate solution ∑k=0jvk satisfies the following inequality:
(19)Ejx,t≤11−ppj+1v0.The series solution∑k=0∞vk of problem (9) is convergent to an exact solution ux,t, if the conditions
(20)0<χi≤1,∀i∈ℕ∪0, hold where the parameters χi for i∈ℕ∪0 are introduced as
(21)χi=vi+1vi,vi≠0,0,vi=0.Furthermore, the maximum absolute truncation error satisfies the inequality(22)ux,t−∑k=0∞vk≤11−χχj+1v0,where χ=maxχi,i=0,1,2,⋯,j.


## 3.1. Fractional Shehu Variational Iteration Method

To reveal the fundamental notions of this method, let us take the following space-time-fractional initial value problem in the Liouville-Caputo fractional derivative:(9)DCtαux,t+Ru,DCxβu;x,t+Nu,DCxβu;x,t=gx,t,m−1<α≤m,n−1<β≤n,m,n=1,2,3,⋯,(10)∂m−1ux,t∂tm−1t=0=gm−1x,where N, R, and gx,t denote the nonlinear, linear part of the differential equation, and the source function, respectively.Utilizing Shehu transformation for Equation (9), we have
(11)Sux,t=∑k=0m−1qpk+1∂kux,0∂tk−qpαSRu,DCxβu;x,t+Nu,DCxβu;x,t+qpαSgx,t.Employing the inverse Shehu transformation for Equation (11) leads to
(12)ux,t=kx,t−S−1qpαSRu,DCxβu;x,t+Nu,DCxβu;x,t,where kx,t=S−1q/pαS∑k=0m−1p/qk+1∂kux,0/∂tk+q/pαSgx,t, and so
(13)∂ux,t∂t+∂∂tS−1qpαSRu,DCxβu;x,t+Nu,DCxβu;x,t−∂∂tkx,t=0.The following recurrence relation is established by VIM:(14)um+1x,t=umx,t−∫0t∂umx,τ∂τ+∂∂τS−1qpαSRum,DCxβum;x,τ+Num,DCxβum;x,τ−∂∂τkx,τdτ.Alternately,(15)um+1x,t=kx,t;β−S−1qpαSRum,DCxβum;x,t+Num,DCxβum;x,tis called the m+1th-order of truncated solution.If(16)limm→∞umx,texists, then the analytical solution ux,t=limm→∞umx,t.


## 3.2. Convergence Theorem

Now, the convergence of VIM is investigated and required conditions and error estimate [15] are established for Equation (9).The operatorV is introduced as
(17)V=−∫0t∂umx,τ∂τ+∂∂τS−1qpαSRum,DCxβum;x,τ+Num,DCxβum;x,τ−∂∂τkx,τdτ,where vk,k=0,1,2,⋯, denote the components of the solution satisfying
(18)ux,t=limm→∞umx,t=∑k=0∞vk.Theorem 1 [16].
LetV, defined in (5∗), be an operator from a Banach space BS to BS. The series solution ux,t=limm→∞umx,t=∑k=0∞vk as defined in (6∗) converges if 0<p<1 exists such that Vv0+v1+v2+⋯+vk+1≤pVv0+v1+v2+⋯+vk, (i.e., vk+1≤vk), ∀k∈ℕ∪0.Theorem1, obtained from the Banach fixed-point theorem, is utilized to establish a sufficient condition for the convergence of fractional VIM.Theorem 2 [16].
The exact solution of nonlinear problem (9) exists under the condition that the series solution ux,t=∑k=0∞vk defined in (18) converges.Theorem 3 [16].
Suppose that the series solution∑k=0∞vk defined in (18) converges to the solution ux,t. The maximum error Ejx,t for the approximate solution ∑k=0jvk satisfies the following inequality:
(19)Ejx,t≤11−ppj+1v0.The series solution∑k=0∞vk of problem (9) is convergent to an exact solution ux,t, if the conditions
(20)0<χi≤1,∀i∈ℕ∪0, hold where the parameters χi for i∈ℕ∪0 are introduced as
(21)χi=vi+1vi,vi≠0,0,vi=0.Furthermore, the maximum absolute truncation error satisfies the inequality(22)ux,t−∑k=0∞vk≤11−χχj+1v0,where χ=maxχi,i=0,1,2,⋯,j.


## 4. Illustrative Examples

Example 1.
Let us consider following space-time-fractional initial value problem(23)DtαCux,t=Dt2βCux,t,0<α≤1,1<2β≤2,0≤x≤l,t>0,(24)ux,0=Eβ,1xβ.Step 1.
Implementing Shehu transform for (23), we have
(25)Sux,t=qpEβ,1xβ+qpαSDx2βCux,t.Step 2.
Taking the inverse Shehu transform of (25), we get
(26)ux,t=S−1qpEβ,1xβ+S−1qpαSDx2βCux,t,ux,t=Eβ,1xβ+S−1qpαSDx2βCux,t,and so
(27)∂ux,t∂t=∂∂tEβ,1xβ+∂∂tS−1qpαSDx2βCux,t,∂ux,t∂t−∂∂tS−1qpαSDx2βCux,t=0.Step 3.
Employing the variational iteration method, we obtain(28)um+1x,t=umx,t−∫0t∂umx,τ∂τ−∂∂τS−1qpαSDx2βCux,tdτ.Based on the iteration formula (28), we have
(29)u0x,t=Eβ,1xβ,u1x,t=Eβ,1xβ+Eβ,1xβtαΓα+1,u2x,t=Eβ,1xβ1+tαΓα+1+t2αΓ2α+1,u3x,t=Eβ,1xβ1+tαΓα+1+t2αΓ2α+1+t3αΓ3α+1.By using the recurrence relation, we obtained themth approximate solution of (23) as follows:
(30)umx,t=Eβ,1xβ∑k=0mtkαΓkα+1,m=0,1,2,⋯.As a result, the analytical solution of (23) is reached by taking the limit of (30):
(31)ux,t=limm→∞umx,t=Eβ,1xβEα,1tα,where Eα,1tα is the two-parameter Mittag-Leffler function.Notice from Table1 and Figure 1 that the values of the solution for α=β=1 and exact solution are the same which implies that the method implemented in this study is one of the best one for the solution of space-time-fractional differential equations of any order. Moreover, it is clear from Figure 1 that as α and β tend to 1, the corresponding solutions tend to exact solution. Three-dimensional graphs of exact solution and a truncated solution are given in Figure 2.Example 2.
Let us consider the space-time-fractional equation(32)DCtαux,t=DCxβux,t2−ux,tCDxβux,t,0<α,β≤1,0≤x≤l,t>0,with the condition at t=0.
(33)ux,0=3+52Eβ,1xβ.Step 1.
Carrying out Shehu transform of (32), we have
(34)Sux,t=qpux,0+qpαSDCxβux,t2−ux,tCDxβux,t.Step 2.
Enforcing inverse Shehu transform of (34), we obtain
(35)ux,t=ux,0+S−1qpαSDCxβux,t2−ux,tCDxβux,t,and so
(36)∂ux,t∂t−∂∂tS−1qpαSDCxβux,t2−ux,tCDxβux,t=0.Step 3.
Utilizing the variational iteration method, we have(37)um+1x,t=umx,t−∫0t∂umx,τ∂τ−∂∂τS−1qpαSDCxβumx,τ2−umx,τCDxβumx,τdτ.Table 1
Comparison of the exact solution with the truncated solutions by SVIM for variousβ and α for Example 1.
txα=0.5α=0.75α=1β=0.5β=0.75β=1uouruouruouruexact0,2-5-0,0728794142905183-0,05569004754332180,008229747049020030,0082297470490200300,4640496756725130,8653721261758671,221402758160171,221402758160175272,995384456584235,378704685654181,272241875151181,2722418751510,4-5-0,102568289434788-0,07178442614390920,01005183574463360,010051835744633600,6530895165644941,115463969215421,491824697641271,491824697641275384,205469814638303,402959554171221,406416204187221,4064162041870,6-5-0,140869106433447-0,09192742111027440,01227733990306840,012277339903068400,8969647161561011,428467587465761,822118800390511,822118800390515527,674601164535388,539034541079270,426407426153270,4264074261530,8-5-0,190052579642384-0,1170942210868870,01499557682047770,014995576820477701,210133736698361,819536515677072,225540928492472,225540928492475711,908534824601494,908647082167330,299559909649330,2995599096491-5-0,252961550186961-0,1484899528294920,01831563888873420,018315638888734201,610697979184332,307397315397362,718281828459052,718281828459055947,556128411521627,605367523717403,428793492735403,428793492735Figure 1
6th order of truncated solutions for various values ofα and β and exact solution at x=0.3 for Example 1.Figure 2
6th order of truncated solutions forα=β=2/3 and exact solution for Example 1.Based on the iteration formula (37), we have
(38)u0x,t=3+52Eβ,1xβ,u1x,t=3+52Eβ,1xβ−152Eβ,1xβtαΓα+1,u2x,t=3+52Eβ,1xβ−152Eβ,1xβtαΓα+1+452tαΓα+1Eβ,1xβt2αΓ2α+1,u3x,t=3+52Eβ,1xβ−152Eβ,1xβtαΓα+1+452tαΓα+1Eβ,1xβt2αΓ2α+1−1352Eβ,1xβt3αΓ3α+1,u4x,t=3+52Eβ,1xβ−152Eβ,1xβtαΓα+1+452tαΓα+1Eβ,1xβt2αΓ2α+1−1352Eβ,1xβt3αΓ3α+1+4052Eβ,1xβt4αΓ4α+1.By using the recurrence relation, themth approximate solution of (32) is obtained as follows:
(39)umx,t=3+52∑k=0m−3tαkΓkα+1∑l=0mxβlΓlβ+1.Hence, the analytical solution of (32) and (33) is reached by taking the limit of (39):
(40)ux,t=limm→∞umx,t=3+52Eα,1−3tαEβ,1xβ,which is the same as obtained in [17].As in Example1, it is obvious from Table 2 and Figure 3 that the values of the solution for α=β=1 and exact solution are the same and as α and β tend to 1, the corresponding solutions tend to exact solution which indicates that the method employed in this research is a good choice for the solution of space-time-fractional differential equations of any order in Figure 4, 3-dimensional graphs of exact solution and a truncated solution are presented.Table 2
Comparison of the exact solution with the truncated solutions by SVIM for variousβ and α for Example 2.
txα=0.5α=0.75α=1β=0.5β=0.75β=1uouruouruouruexact0,2-52,977231898816412,954688509429833,009244659291213,0092446592912103,144972761825543,704098894948934,372029090235074,37202909023507588,2859013271367194,512854240003206,627171662420206,6271716624200,4-53,019298747863812,985030155645323,005073576590743,0050735765907402,877117869644443,232617614981403,752985529780513,752985529780515-69,290222745321666,2713155938424114,752961233252114,7529612332520,6-53,033463408840222,999288951335173,002784437869613,0027844378696102,786926333445823,011049042370093,413247220553973,413247220553975-122,3489239793886,0053074306665464,331325492773464,33132549277340,8-53,035435945622533,005481963200813,001528131902823,0015281319028202,774366476001952,914815333080623,226794883223533,226794883223535-129,737751709177-20,169981967374036,659345087504236,65934508750421-53,032652691805913,007739660213003,000838656569763,0008386565697602,792088462978492,879732797692613,124467670919663,124467670919665-119,312099237844-29,712329689166521,472640247326621,4726402473266Figure 3
6th order of truncated solutions for various values ofα and β and exact solution at x=0.3 for Example 2.Figure 4
6th order of truncated solutions forα=β=2/3 and exact solution for Example 2.


## 5. Conclusions

In this research, the targeted goal is to construct truncated solutions of linear/nonlinear space-time-fractional initial value problem by employing SVIM, the combination of the Shehu transform and variational iteration method. The main advantage of this method is that its implementation is straightforward and fruitful. Moreover, the illustrated examples reveal that the obtained approximate solutions with high precision converge swiftly to exact analytical solutions.In the future study, this method and its improved modifications are applied to initial value problems including space-time-fractional linear and nonlinear differential equations.



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