# On Effects of a New Method for Fractional Initial Value Problems

Authors: Hülya Kodal Sevindir; Süleyman Çetinkaya; Ali Demir
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

The aim of this study is to analyze nonlinear Liouville-Caputo time-fractional problems by a new technique which is a combination of the iterative and ARA transform methods and is denoted by IAM. First, the ARA transform method and its inverse are utilized to get rid of time fractional derivative. Later, the iterative method is applied to establish the solution of the problem in infinite series form. The main advantages of this method are that it converges to analytic solution of the problem rapidly and implementation of method is easy. Finally, outcomes of the illustrative examples prove the efficiency and accuracy of the method.


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## Body

## 1. Introduction

The mathematical problems including fractional differential equations play a significant role in analysis and modelling of various scientific processes such as damping laws, electrical circuits, fluid mechanics, and relaxation processes since fractional derivative is nonlocal operator [1–5]. As a result, fractional mathematical problems attract a growing attention of numerous scientist from diverse branches of science. The adversity of fractional differential equations is that solving them analytically is hard or impossible. Therefore, numerous numerical methods such as reduced differential transform method (RDTM) [6], Adomian decomposition method (ADM) [7], homotopy perturbation method (HPM) [8], variational iteration method (VIM) [9, 10], homotopy analysis method (HAM) [11], fractional difference method (FDM) [12], and new iterative method (NIM) [13] have been developed to establish numerical solution in series form.ARA transform is a new integral transform method to tackle with any kind of differential equations. The differential equation is reduced into algebraic equation or simpler differential equation by this method. Moreover, the ARA transform method is more applicable than the Laplace transform method since the domain of this method covers the domain of the Laplace transform method [14]. Combination of this transform with other numerical methods produces new and effective methods to construct numerical or analytical solution of differential equations. In this study, the combination of ARA transform method with iterative method is utilized to construct the solution of nonlinear fractional differential problems of biological population. Iterative method is a very common method to obtain the numerical solution of mathematical problems including differential equations [15].The main goal of this research is to establish numerical or analytical solutions of nonlinear Liouville-Caputo time-fractional problems by means of IAM which is a new effective and versatile method. The novelty of this study is that this is the first study in which ARA transform method is applied to fractional differential problem including fractional equation in Liouville-Caputo sense.The rest of the paper is organized as follows: fundamental definitions and properties of fractional calculus and ARA transform, the ARA transform of fundamental functions, and existence of ARA transform for Liouville-Caputo derivative and Riemann-Liouville integral are given in Section2. Implementation and convergence analysis of the method for nonlinear fractional mathematical problems is presented in Section 3. Illustrative examples including Liouville-Caputo time-fractional biological population problem are presented and analyzed in Section 4. Finally, the outcomes of this method are presented in conclusion.


## 2. Preliminary Results

In this section, preliminaries, notations, and features of the fractional calculus are given [12, 16].Definition 1.
Riemann-Liouville time-fractional integral of a real valued functionut is defined as
(1)Itαut=1Γα∫0tt−sα−1usds,where α>0 denotes the order of the integral.Definition 2.
αth order of the Liouville-Caputo fractional derivative of ut is defined as
(2)Dtαut=1Γn−α∫t0tt−sn−α−1unsds,t∈t0,t0+T,where n−1<α<n and unt=dnu/dtn. If α is an integer, then the Liouville-Caputo fractional derivative becomes the integer-order derivative.Definition 3.
αth order of the Liouville-Caputo fractional derivative of ut is defined as
(3)Dtαut=1Γ1−α∫t0tt−s−αu′sds,t∈t0,t0+T,where 0<α<1.Definition 4.
The Mittag-Leffler function with the parametersα and β is given as follows [17].
(4)Eα,βλt−t0α=∑k=0∞λt−t0αkΓαk+β,α,β>0,λ∈R.Ift0=0, α=β=q, then we get
(5)Eq,qλtq=∑k=0∞λtqkΓαk+β,q>0.Moreover, substitutingq=1, in equation (5), we have E1,1λt=eλt. If the reader wants more information, they should refer to [17]. The following functions are used to obtain the solution of the problem discussed in this study.Definition 5.
The ARA integral transform of ordern of the continuous function gt on the interval 0,∞ is defined as [14]
(6)Gngts=Gn,s=s∫0∞tn−1e−stgtdt,s>0.Definition 6.
The inverse of the ARA transform is given by(7)gt=Gn+1−1Gn+1gt=−1n2πi∫c−i∞c+i∞est−1n1sΓn−1∫0ss−xn−1Gn+1,xdx+∑k=0n−1skk!∂kG0∂skds,where Gs=∫0∞e−stgtdt is n−1 times differentiable [14].Now, significant properties which play a vital role for the solution of fractional differential equations are presented as follows:Property 7.
The ARA transform of Mittag-Leffler forn=1 are computed as follows:
(8)Gntβ−1Eα,βλtαs=s∫0∞tn−1e−sttβ−1∑k∞λtαk/Γαk+βdt=∑k∞λkΓαk+βs∫0∞e−sttn+β−2+αkdt=∑k∞λkΓαk+βs∫0∞tn−1e−sttβ−1+αkdt=∑k∞λkΓαk+βΓβ−1+αk+nsβ−1+αk+n−1=1sβ+n−2∑k∞λkΓαk+βΓβ+αk+n−1sαk.
Forn=1,
(9)G1tβ−1Eα,βλtαs=1sβ−1∑k∞λkΓαk+βΓβ+αksαk=1sβ−111−λ/sα=sα−β+1sα−λ.Property 8.
The ARA transform oftpα for p∈ℕ is defined as follows:
(10)Gntpαs=s∫0∞tn−1e−sttpαdt=s∫0∞tpα+n−1e−stdt=Γpα+n1spα+ns∫0∞tpα+n−1e−stΓpα+n1/spα+ndt=Γpα+n1spα+ns=Γpα+nspα+n−1.Property 9 [14].
The ARA transform of convolution is defined as follows:(11)Gnft∗gts=−1n−1s∑j=0n−1cjn−1FjsGn−1−js.
Forn=1, it becomes
(12)G1ft∗gts=sFsGs,where Gs=∫0∞e−stgtdt and Fs=∫0∞e−stftdt.Theorem 10 (the existence of ARA transform for Riemann-Liouville integral).
If the Riemann-Liouville integralItα0RLft of the function ft is piecewise continuous in every finite interval and fulfils
(13)tn−1IR0Ltαft≤Keβt,then ARA transform exists for all s>β.Proof.
By the property of integration the ARA transform of the Riemann-Liouville integral leads to the following(14)s∫0∞tn−1e−stI0RLtαftdt=s∫0γtn−1e−stI0RLtαftdt+s∫γ∞tn−1e−stI0RLtαftdt,piecewise continuity of Itα0RLft implies the existence of first integral on the right side, and the convergence of improper integral on the right side is shown below:
(15)s∫γ∞tn−1e−stI0RLtαftdt≤s∫γ∞e−sttn−1I0RLtαftdt≤s∫γ∞Keβt−stdt=limc⟶∞−sKe−ts−βs−βγc=sKs−βe−γs−β.
As a result, the ARA transform ofItα0RLft exists.☐Theorem 11 (ARA transform of Riemann-Liouville integral forn=1).
If the Riemann-Liouville integralItα0RLft of the function ft is piecewise continuous in every finite interval and fulfils
(16)tn−1I0RLtαft≤Keβt,then ARA transform of it for n=1 is computed as
(17)G1I0RLtαfts=1sαG1fts,for all s>β.Proof.
Forn=1, we have
(18)G1I0RLtαfts=G11Γα∫0tt−xα−1fxdxs.
In terms of convolution, it can be rewritten as follows(19)G11Γαtα−1∗fts,by using convolution property of the ARA transform, we reach the following
(20)1Γαs∫0∞e−sttα−1dt∫0∞e−stftdt=1ΓαsΓαsα∫0∞e−stftdt=1sαs∫0∞e−stftdt=1sαG1fts.☐Theorem 12 (the existence of ARA transform for Liouville-Caputo derivative).
If theαth order of Liouville-Caputo derivative Dtα0Cft of the function ft is piecewise continuous in every finite interval, ft is n times continuously differentiable and fulfils
(21)tn−1Dtα0Cft≤Keβt,then ARA transform exists for all s>β.Proof.
By the property of integration the ARA transform of the Liouville-Caputo derivative leads to the following(22)s∫0∞tn−1e−stDtα0Cftdt=s∫0γtn−1e−stDtα0Cftdt+s∫γ∞tn−1e−stDtα0Cftdt,piecewise continuity of Dtα0Cft implies the existence of first integral on the right side, and the convergence of improper integral on the right side is shown below:
(23)s∫γ∞tn−1e−stDtα0Cftdt≤s∫γ∞e−sttn−1Dtα0Cftdt≤s∫γ∞Keβt−stdt=limc⟶∞−sKe−ts−βs−βγc=sKs−βe−γs−β.
As a result, the ARA transform ofDtα0Cft exists.☐Theorem 13 (ARA transform of Liouville-Caputo derivative forn=1).
If theαth order of Liouville-Caputo derivative Dtα0Cft of the function ft is piecewise continuous in every finite interval and fulfils
(24)tn−1Dtα0Cft≤Keβt,where m−1<α≤m, then ARA transform of it for n=1 is computed as
(25)G1Dtα0Cfts=sαG1fts−∑k=0m−1fk0sα−k,for all s>β.Proof.
The ARA transform of Liouville-Caputo derivative forn=1 can be written as
(26)G1Dtα0Cfts=G1Itm−α0RLfmts.
Takingfmt=gt and using Theorem 11 lead to
(27)G1Itm−α0RLgts=1sm−αG1gts,
Replacinggt by fmt and utilizing integration by parts lead us to the following result
(28)1sm−αG1fmts=sαG1fts−∑k=0m−1fk0sα−k.☐


## 3. Main Results

In this section, the implementation of IAM for nonlinear fractional partial differential equation with initial conditions is presented. Let us consider the following nonlinear fractional initial value problem:(29)Dtα0Cux,y,t+Rux,y,t+Nux,y,t=gx,y,t,m−1<α≤m,m∈ℕ,(30)∂k∂tkux,y,0=hkx,y,k=0,1,2,⋯,m−1,where Dtα0Cux,y,t, R, N, and gx,y,t represent fractional derivative, the linear equation operator, the general nonlinear differential operator, and the source term, respectively. In order to apply ARA transform to nonlinear fractional initial value problem, ux,y,t must be m times continuously differentiable function with respect to variable t based on Definition 5. In other words, ux,y,t must belong to the Banach space W2m0,T where it is defined as
(31)W2m0,T=u=u.,.,t:u,u1,⋯,um−1∈AC0,T.Without loss of generality takingu=ux,y,t. Making use of the ARA transform on equation (29), we obtain
(32)G1Dtα0Cus+G1Ru+Nus=G1gx,y,ts.Taking the property of the ARA transform into account leads to(33)sαG1us−∑k=0m−1∂k∂tkux,y,0sα−k+G1Ru+Nus=G1gx,y,ts,G1us=1sα∑k=0m−1∂k∂tkux,y,0sα−k−1sαG1Ru+Nus+1sαG1gx,y,ts.Employing inverse ARA transform gives(34)u=G1−11sα∑k=0m−1∂k∂tkux,y,0sα−k+G1gx,y,ts−G1−11sαG1Ru+Nus.Now, the application of the iterative method produces(35)u=∑i=0∞ui.Linearity of the operatorR leads to
(36)R∑i=0∞ui=∑i=0∞Rui,and decomposition of the nonlinear operator N can be written as
(37)N∑i=0∞ui=Nu0+∑i=0∞N∑k=0iuk−N∑k=0i−1uk.Plugging (35), (36), and (37) in (34) allows us to have
(38)∑i=0∞ui=G1−11sα∑k=0m−1∂k∂tkux,y,0sα−k+G1gx,y,ts−G1−11sαG1Rui+Nu0+∑i=1∞N∑k=0iuk−N∑k=0i−1uks.The recurrence relation is obtained as follows:(39)u0=G1−11sα∑k=0m−1∂k∂tkux,y,0sα−k+G1gx,y,ts,(40)u1=−G1−11sαG1Ru0+Nu0s,(41)um+1=G1−11sαG1Rum−N∑k=0muk−N∑k=0m−1uks,m≥1.Finally, the approximate solution ofm-term is constructed as
(42)ux,y,t≅u0+u1+u2+⋯+um,m=1,2,⋯Theorem 14.
In the Banach spaceW2m0,T, the solution of nonlinear fractional differential equation (29) in series form is convergent if the following inequality satisfies for some constant γ, 0<γ<1(43)G1−11sαG1Rum−N∑k=0muk−N∑k=0m−1uks≤γG1−11sαG1Rum−1−N∑k=0m−1uk−N∑k=0m−2uks,∀m∈ℕ,or um+1≤γum.Proof.
(44)Smx,y,t=u1+u2+u3+⋯+um,represent the sequences of partial sums for m≥1. In order to prove that the series solution of nonlinear fractional differential equation (29) is convergent, it is enough to show that Smm≥1 is a Cauchy sequence in a given Banach space B. For this aim, we take
(45)Sm+1−Sm=G1−11sαG1Rum−N∑k=0muk−N∑k=0m−1uks≤γG1−11sαG1Rum−1−N∑k=0m−1uk−N∑k=0m−2uks≤γ2G1−11sαG1Rum−2−N∑k=0m−2uk−N∑k=0m−3uks≤⋯≤γn+1G1−11sα∑k=0m−1∂k∂tkux,y,0sα−k+G1gx,y,ts.
For everyn,m∈ℕ, n≥m, utilizing (43) and triangle inequality successively, we get
(46)Sn−Sm=Sm+1−Sm+Sm+2−Sm+1+⋯+Sn−Sn−1≤γm+11−γn−m1−γG1−11sα∑k=0m−1∂k∂tkux,y,0sα−k+G1gx,y,ts.
The assumption0<γ<1 implies that 1−γn−m≤1 which yields
(47)Sn−Sm≤γm+11−γG1−11sα∑k=0m−1∂k∂tkux,y,0sα−k+G1gx,y,ts.
Boundedness ofG1−11/sα∑k=0m−1∂k/∂tkux,y,0sα−k+G1gx,y,ts implies the following:
(48)limn,m⟶∞Sn−Sm=0.
This result implies that the sequenceSmm≥1 is a Cauchy sequence in Banach space B. As a result, the series solution (35) is convergent.☐


## 4. Illustrative Example

As illustrative examples, nonlinear fractional initial value problems of biological population models are presented in this section to show how IAM is implemented.Example 1.
Consider the following generalized time-fractional biological population model:(49)Dtα0Cu=∂2∂x2u2+∂2∂y2u2+hu,with the initial condition
(50)ux,y,0=xy,where u=ux,y,t,0<α≤1,x,y∈ℝ2,t>0.Employing the ARA transform on equation (49) with the initial condition (50) produces the following:
(51)G1us=xy+1sαG1∂2∂x2u2+∂2∂y2u2+hus.Applying inverse ARA transform to equation (51) leads to
(52)u=xy+G1−11sαG1∂2∂x2u2+∂2∂y2u2+hus.Substituting (35)–(37) into (52) and utilizing (39), the components of the solution are established as follows:
(53)u0=ux,y,0=xy,u1=G1−11sαG1∂2∂x2u02+∂2∂y2u02+hu0s=G1−11sαG1hxys=htαΓα+1xy,u2=G1−11sαG1∂2∂x2u0+u12+∂2∂y2u0+u12+hu0+u1s−G1−11sαG1∂2∂x2u02+∂2∂y2u02+hu0s=G1−11sαG1h+h2tαΓα+1xys−htαΓα+1xy=h2t2αΓ2α+1xy,u3=G1−11sαG1∂2∂x2u0+u1+u22+∂2∂y2u0+u1+u22+hu0+u1+u2s−G1−11sαG1∂2∂x2u0+u12+∂2∂y2u0+u12+hu0+u1s=G1−11sαG1h+h2tαΓα+1+h3t2αΓ2α+1xys−h2t2αΓ2α+1xy−htαΓα+1xy=t3αΓ3α+1xy.Finally, the numerical solution of the problem is constructed as(54)u≅u0+u1+u2+⋯+um,u≅1+htαΓα+1+h2t2αΓ2α+1+h3t3αΓ3α+1+⋯+hmtmαΓmα+1xy.Asm tends to infinity, the numerical solution converges to following analytical solution
(55)ux,y,t=Eα,1htαxy.The exact solution which is computed by takingα=1 can be written as [18]:
(56)ux,y,t=exphtxy.Example 2.
Let us take the following generalized time-fractional biological population model into consideration:(57)Dtα0Cu=∂2∂x2u2+∂2∂y2u2+u,with the initial condition
(58)ux,y,0=sinxsinhy,where u=ux,y,t,0<α≤1,x,y∈ℝ2,t>0.Utilizing the ARA transform on equation (57) with the initial condition (58) produces the following:
(59)G1us=sinxsinhy+1sαG1∂2∂x2u2+∂2∂y2u2+us.Applying inverse ARA transform to equation (59) leads to
(60)u=sinxsinhy+G1−11sαG1∂2∂x2u2+∂2∂y2u2+us.Substituting (35)–(37) into (60) and utilizing (39), the components of the solution are established as follows:
(61)u0=ux,y,0=sinxsinhy,u1=tαΓα+1sinxsinhy,u2=t2αΓ2α+1sinxsinhy,u3=t3αΓ3α+1sinxsinhy,…Finally, the numerical solution of the problem is constructed as(62)u≅u0+u1+u2+⋯+um,u≅1+tαΓα+1+t2αΓ2α+1+t3αΓ3α+1+⋯+tmαΓmα+1sinxsinhy.Asm goes to infinity, the numerical solution converges to following analytical solution
(63)ux,y,t=Eα,1tαsinxsinhy.The exact solution which is computed by takingα=1 can be written as [19]:
(64)ux,y,t=exptsinxsinhy.Example 3.
Consider the following generalized time-fractional biological population model:(65)Dtα0Cu=∂2∂x2u2+∂2∂y2u2+hu1−ru,with the initial condition
(66)ux,y,0=exphr8x+y.where u=ux,y,t,0<α≤1,x,y∈ℝ2,t>0.Employing the ARA transform on equation (65) with the initial condition (66) produces the following:
(67)G1us=exphr8x+y+1sαG1∂2∂x2u2+∂2∂y2u2+hu1−rus.Applying inverse ARA transform to equation (67) leads to
(68)u=exphr8x+y+G1−11sαG1∂2∂x2u2+∂2∂y2u2+hu1−rus.Substituting (35)–(37) into (68) and utilizing (39), the components of the solution are established as follows:
(69)u0=ux,y,0=exphr8x+y,u1=htαΓα+1exphr8x+y,u2=h2t2αΓ2α+1exphr8x+y,u3=h3t3αΓ3α+1exphr8x+y,…Finally, the numerical solution of the problem is constructed as(70)u≅u0+u1+u2+⋯+um,u≅1+htαΓα+1+h2t2αΓ2α+1+h3t3αΓ3α+1+⋯+hmtmαΓmα+1exphr8x+y.Asm tends to infinity, the numerical solution converges to following analytical solution
(71)ux,y,t=Eα,1htαexphr8x+y.The exact solution which is computed by takingα=1 can be written as [20]:
(72)ux,y,t=exphr8x+y+ht.In Table1, absolute errors between the exact solution and the approximate solutions u7 at α=1 for Examples 1, 2, and 3 are presented. Figures 1–6 and outcomes verify that when the order of the fractional derivative α increases to 1, the truncated solutions obtained by IAM continuously get closer to the exact solutions. Notice that the outcomes are compatible with 2D and 3D graphs of exact and approximate solutions u7. Furthermore, we concluded that increasing the number of terms in approximate solutions implies the improvement of accuracy of approximate solutions.Table 1
The absolute errors between the approximate solutionsu7 and the exact solution for Examples 1, 2, and 3 at α=1.
Example1Example2Example3x,y,th=1h=1, r=20.1,0.1,0.12.509104035652854×10−142.50771625687207×10−142.771116669464391×10−130.2,0.2,0.21.298636198576730×10−111.298622320788923×10−117.930767154107343×10−110.3,0.3,0.35.04943808987690×10−105.049211049268365×10−102.27200946945061×10−90.4,0.4,0.46.802539864736446×10−96.801572194348182×10−92.537049281770010×10−80.5,0.5,0.55.127268321025724×10−85.125487789747751×10−81.690687270183844×10−7Figure 1
The 2D graphs of exact and approximate solutionsu7 by IAM for various of α for Example 1 at t=1.5 and h=y=1.Figure 2
3D graphs of exact and approximate solutionsu7 by IAM for various values of α for Example 1 at t=1.5 and h=1.Figure 3
The 2D graphs of exact and approximate solutionsu7 by IAM for various of α for Example 2 at t=1.5 and x=1.Figure 4
3D graphs of exact and approximate solutionsu7 by IAM for various values of α for Example 2 at t=1.5.Figure 5
The 2D graphs of exact and approximate solutionsu7 by IAM for various of α for Example 3 at t=1.5, r=2, and y=h=1.Figure 6
3D graphs of exact and approximate solutionsu7 by IAM for various values of α for Example 3 at t=1.5, r=2, and h=1.


## 5. Conclusion

In this study, nonlinear fractional initial value problem is investigated by a new numerical method IAM which is a combination of the ARA transform and iterative methods. The implementation and convergence analysis of IAM shows how effective and accurate this method is. As an application of this model, Liouville-Caputo time-fractional biological population problems are taken into consideration. The outcomes of these examples also support the effectiveness and accuracy of the method.In the future works, the utilization of ARA transform with other numerical methods is taken into account to analyze substantial nonlinear mathematical problems.



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