# An Improved Version of Residual Power Series Method for Space-Time Fractional Problems

Authors: Mine Aylin Bayrak; Ali Demir; Ebru Ozbilge
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

The task of present research is to establish an enhanced version of residual power series (RPS) technique for the approximate solutions of linear and nonlinear space-time fractional problems with Dirichlet boundary conditions by introducing new parameterλ. The parameter λ allows us to establish the best numerical solutions for space-time fractional differential equations (STFDE). Since each problem has different Dirichlet boundary conditions, the best choice of the parameter λ depends on the problem. This is the major contribution of this research. The illustrated examples also show that the best approximate solutions of various problems are constructed for distinct values of parameter λ. Moreover, the efficiency and reliability of this technique are verified by the numerical examples.


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## Body

## 1. Introduction

The well-established tool fractional differential equations (FDEs) attract growing attention of scientists in a wide range of scientific areas such as biology, physics, and engineering since it is very effective and accurate for modelling many processes in real life [1–11].In FDEs, noninteger derivatives are taken into account to include memory of the system. Fractional mathematical models have results which are very close to the experimental data [12]. Since FDEs are applicable and variable, the progress of them is very rapid in diverse fields of science and engineering. Therefore, studies on solving FDEs produce efficient and reliable techniques to construct numerical and analytical solutions of them [13–20].Diverse methods such as a semianalytical approach, a modified wavelet approach, linearized novel operational matrix-based scheme, hybrid spectral linearized scheme, finite difference/spectral algorithm, innovative operational matrix-based computational scheme, and a Chelyshkov polynomial-based algorithm are utilized to construct the solutions of FDEs [20–22].RPS technique, which can be applied directly to the problem, is one of the most convenient ones to establish approximate solutions with high precision for linear and nonlinear FDEs [21–28]. Moreover, RPS technique allows us to construct the exact solution of the initial value problems whose solutions are polynomial.Various mathematical models including differential equations such as model of vibration equation of large membranes, fractional Black–Scholes option pricing equations, and time–fractional fuzzy vibration equation of large membranes are analyzed by the RPS technique [27, 28].Modified versions of the RPS technique are developed to acquire better approximate solutions of FDEs such as the solutions of STFDE in [29–34].The motivation of this study is to employ a new version of RPS technique to establish better approximate solutions of fractional mathematical models including STFDE in Caputo sense. In the RPS technique, determining a suitable estimation for the initial guess approximations plays an important role. Therefore, in this study, we try to construct best initial guess approximation by introducing a new parameter. The novelty of this method is that new parameterλ is introduced to get the best approximate solution for the fractional mathematical problems including STFDE with Dirichlet boundary conditions. When we are given Dirichlet boundary conditions, we utilize the parameter λ to determine the initial rate of change which enables us to acquire the best approximate solution by the RPS technique.


## 2. Preliminaries

Now, the basic concepts of fractional calculus are presented [1–5].Definition 1.
The Riemann-Liouville (R-L) fractional derivative operatorDα of order α is described as [2, 5]
(1)Dαft=1Γn−αddtn∫0tfxt−xα−n+1dx,n−1<α<n,∂n∂tnft,α=n,where n∈Z+ and α∈R+.Definition 2.
The R-L fractional-order integration operatorJα is described as [2, 5]
(2)Jαft=1Γα∫0tt−xα−1fxdx,x>0,α>0.
Following Podlubny [2], we may have(3)Jαtn=Γn+1Γn+α+1tn+α,Dαtn=Γn+1Γn−α+1tn−α.Definition 3.
Theαth-order derivative of ft in Caputo sense is defined as [2, 5]
(4)CDαft=1Γn−α∫0tfnxt−xα−n+1dx,n−1<α<n,∂nft∂tn,α=n.Definition 4 (see [2, 5]).
(1)
DαJαft=ft(2)
JαDαft=ft−∑k=0nfk0+tk/k!,t>0Definition 5.
k,l-truncated series uklx,t of the RPS method is given as
(5)uklx,t=∑n=0i−1anxn!+∑n=1k∑m=0lfnmxtnα+mΓ1+nα+m,n−1<α≤n,t>0,uklx,t=∑n=0i−1bntn!+λ∑n=1k∑m=0lgnmt−gnm−1txnβ+mΓ1+nβ+m,n−1<β≤n,x>0.


## 3. RPS Technique for Fractional Mathematical Models with Dirichlet Boundary Conditions

We provide an efficient and reliable method for the solution of STFDE with Dirichlet boundary conditions. Let us take the following initial-boundary value problem into consideration:(6)Dtαu=Dxβu+Fu,(7)ux,0=f0x,(8)u0,t=g0t,u1,t=g1t.Thek,0-truncated series uk0x,t is established by taking Equations (6) and (7) into account as follows:
(9)uk0x,t=f00x+∑n=1kfn0xtnαΓ1+nα,t>0,0<α≤1.The use of Equations (1) and (3) leads to the kl-truncated series uklx,t as follows:
(10)uklx,t=g00t+λg01t−g00tx+λ∑n=1k∑m=1lgnmt−gnm−1txnβ+mΓ1+nβ+m,x>0,1<β≤2.An approximation for (6)–(8) is achieved by employing the new RPS technique as follows:
(11)ux,t=12f00x+g00t+λg01t−g00tx+∑n=1∞fn0xtnαΓ1+nα+∑n=1∞gn0txnβΓ1+nβ+λ∑n=1∞gn1t−gn0txnβ+1Γ2+nβ.Notice that we introduce a new parameterλ in this approximate solutions to get a better approximation of the problems (6)–(8). Hence, the k,l-truncated series of uklx,t of ux,t is achieved as follows:
(12)uklx,t=12f00x+g00t+λg01t−g00tx+∑n=1kfn0xtnαΓ1+nα+∑n=1kgn0txnβΓ1+nβ+λ∑n=1k∑m=1lgnmt−gnm−1txnβ+mΓ1+nβ+m.Equation (13) leads to the approximate solution u00x,t to become
(13)u00x,t=12f00x+g00t+λg01t−g00tx.The RPS technique leads to the following definition of thekl residual function Resklx,t as
(14)Resklx,t=Dtαukl−Dxβukl−Fukl.In order to establish the coefficientsgijt,i,j=1,2,3,⋯,k, plunge Equation (12) into Equation (14), and solve the following equation
(15)Dxk−1βDxlReskl0,t=0.To acquireg10t, taking k=1 and l=0 in Equation (14) leads to
(16)Res10x,t=f10x+Dtαg00t+λDtαg01t−Dtαg00tx−Dxβf00x−Dxβf10xtαΓ1+α−g10t−Fu10,where
(17)u10x,t=12f00x+g00t+λg01t−g00tx+f10xtαΓ1+α+g10txβΓ1+β.Employing Equation (15) allows us to have
(18)g10t=f100+Dtαg00t−Dxβf000−Dxβf100tαΓ1+α−Fu100,t.Similarly, we takek=2 and l=0 in Equation (14) to establish the coefficient g20t,
(19)Res20x,t=Dtαu20−Dxβu20−Fu20,where
(20)u20x,t=12f00x+g00t+λg01t−g00tx+f10xtαΓ1+α+f20xt2αΓ1+2α+g10txβΓ1+β+g20tx2βΓ1+2β.Taking Equation (15) into account leads to
(21)g20t=Dxβf100+Dxβf200tαΓ1+α+Dtαg10−DxDxβf000−DxDxβf100tαΓ1+α−DxDxβf200t2αΓ1+2α−DxβFu200,t.As in the previous steps, takingk=1 and l=1 in Equation (14) leads to
(22)Res11x,t=Dtαu11−Dxβu11−Fu11,where
(23)u11x,t=12f00x+g00t+λg01t−g00tx+f10xtαΓ1+α+g10txβΓ1+β+λg11t−g10txβ+1Γ2+β.Using Equation (15) when k=1 and l=1, we get
(24)g11t=g10t+Dtαg01−Dtαg00+1λDxf100−DxDxβf000−DxDxβf100tαΓ1+α−DxFu110,t.To determineg21t, substituting k=2 and l=1 into Equation (14), then
(25)Res21x,t=Dtαu21−Dxβu21−Fu21,where
(26)u21x,t=12f00x+g00t+λg01t−g00tx+f10xtαΓ1+α+f20xt2αΓ1+2α+g10txβΓ1+β+g20tx2βΓ1+2β+λg11t−g10txβ+1Γ2+β+λg21t−g20tx2β+1Γ2+2β.Using Equation (15) when k=2 and l=1, we get
(27)g21t=g20t+Dtαg11−Dtαg10+1λDxβDxf100−DxβDxf200tαΓ1+α−DxβDxDxβf000−DxβDxDxβf100tαΓ1+α−DxβDxDxβf200t2αΓ1+2α−DxβDxFu210,t,and so on.Now, the next step is to obtain the coefficientsfi0x,i=1,2,3,⋯,k, by Equation (9).The equation(28)Dtk−1αResk0x,0=0allows us to establish the required coefficients fijx,i,j=1,2,3,⋯,k. Taking into k=1 in Equation (14) leads to
(29)Res10x,t=Dtαu10−Dxβu10−Fu10,where
(30)u10x,t=f00x+f10xtαΓ1+α,which allows us to determine the coefficients f10x as follows:
(31)f10x=Dxβf00x+Fu10x,0.In a similar form,f20x is acquired by substituting k=2 into Equation (14) then
(32)Res20x,t=Dtαu20−Dxβu20−Fu20,where
(33)u20x,t=f00x+f10xtαΓ1+α+f20xt2αΓ1+2α.Hence, we have the coefficientf20x in the following form:
(34)f20x=Dxβf10x+DtαFu20x,0,and so on. By substituting the coefficients fijx,i,j=1,2,3,⋯,k, into Equations (18), (21), (24), and (27) lead to the coefficients gijt,i,j=1,2,3,⋯,k.As a verification of the new RPS technique, we observe that we obtain as a classical exact solution of the problems (6)–(8) with integer order as β⟶2 and α⟶1.


## 4. Convergence Analysis

Theorem 6.
Letux,t,Dtkαux,t∈CR1,t0×R1,t0+R2, and Dxkβux,t∈Cx0,R2×x0+R2,R2, where k=0,1,2,⋯,p1+1,j=0,1,2,⋯,n−1, and l=0,1,2,⋯,p2+1,i=0,1,2,⋯,m−1. Moreover, Dtkαux,t can be differentiated n−1 times with respect to ′t′ on t0,t0+R1 and Dxlβux,t can be differentiated m−1 times with respect to ′x′ on x0,x0+R2. Then,
(35)ux,t≅12∑j=0n−1∑i=0p1Fj+iαxt−t0j+iα+λ2∑j=0m−1∑i=0p2Gj+iβtx−x0j+iβ,where Fj+iαx=Dtj+iα/Γj+iα+1Γj+iα+1ux,t0 and Gj+iβt=Dxj+iβ/Γj+iβ+1Γj+iβ+1ux0,t.Moreover, for∃ values ε1 and ε2 such that 0≤ε1≤t and 0≤ε2≤x, the error term has the term as follows:
(36)∥Epx,t∥≤12supt∈0,T∑j=0n−1Dtj+p1+1αΓj+p1+1α+1ux,ε1tj+p1+1α+infλ∈1,2supx∈0,Lλ∑j=0m−1Dxj+p2+1βΓj+p2+1β+1uε2,txj+p2+1β.Proof.
The proof of this theorem follows from the proof 4.3 [27].


## 5. Numerical Examples

In this section, we provide illustrative examples to prove how effective and reliable the RPS technique proposed in this study is.Example 1.
Consider the STFDE with initial-boundary conditions(37)Dtαu=Dxβu,ux,0=sinx,u0,t=0,u1,t=exp−tsin1,for which
(38)ux,t=exp−tsinxrepresents the exact solution for α=1 and β=2. Based on the RPS technique and above problem, we have the following coefficients:
(39)f00x=sinx,g00t=0,g01t=exp−tsin1,f10x=12i∑k=0∞ik+2xk+2−βΓk+3−β−∑k=0∞ik+2xk+2−βΓk+3−β,g10t=0,g11t=sin1∑k=0∞−1k+1tk+1−αΓk+2−α,f20x=12i∑k=0∞ik+4xk+4−2βΓk+5−2β−∑k=0∞−ik+4xk+4−2βΓk+5−2β,g20t=0,g21t=sin1∑k=0∞−1k+2tk+2−2αΓk+3−2α,f30x=12i∑k=0∞ik+6xk+6−3βΓk+7−3β−∑k=0∞−ik+6xk+6−3βΓk+7−3β,g30t=0,g31t=sin1∑k=0∞−1k+3tk+3−3αΓk+4−3α.In Table1, the approximate solutions are given for various values of λ with α=1 and β=2. Table 2 provides the values of exact solution and an approximate solution u31x,t for the parameter λ=1.22 and various orders of α and β. Based on Table 1, we conclude that approximate solutions have a higher accuracy as adding more components. Moreover, Figure 1 supports our conclusion.Example 2.
Take the initial-boundary value problem including STFDE with source function into consideration(40)Dtαu=Dxβu+3u,ux,0=cosx+π2,u0,t=0,u1,t=exp2tcos1+π2,for which
(41)ux,t=exp2tcosx+π2denotes the exact solution for α=1 and β=2. Based on the RPS technique and above problem, the following coefficients are obtained:
(42)f00x=cosx+π2,g00t=0,g01t=exp2tcos1+π2,f10x=12i∑k=0∞−ik+2xk+2−βΓk+3−β−∑k=0∞ik+2xk+2−βΓk+3−β+3cosx+π2,g10t=0,g11t=−3cos1+π2exp2t+cos1+π2∑k=0∞2k+1tk+1−αΓk+2−α,f20x=12i∑k=0∞ik+2xk+4−2βΓk+5−2β−∑k=0∞−ik+2xk+4−2βΓk+5−2β+62i∑k=0∞−ik+2xk+2−βΓk+3−β−∑k=0∞ik+2xk+2−βΓk+3−β+3cosx+π2+9cosx+π2,g20t=0,g21t=9cos1+π2exp2t−6cos1+π2∑k=0∞2k+1tk+1−αΓk+2−α+cos1+π2∑k=0∞2k+2tk+2−2αΓk+3−2α.⋮Table 1
The approximate values ofu31x,t for various values of λ at α=1 and β=2.
xtExactλ=−1λ=0λ=1λ=2λ=50.30.30.218930.017310.109420.201530.293640.569970.60.162180.012140.080380.148620.216860.421570.90.120150.006110.056670.107220.157770.309420.60.30.418300.033070.209060.385050.561041.089020.60.309880.023200.153580.283960.414340.805480.90.229570.011680.108270.204860.301440.591200.90.30.580300.045870.290030.534180.778331.510800.60.429900.032190.213060.393940.574811.117430.90.318480.016210.150200.284200.418190.82017Table 2
The approximate values ofu31x,t with the parameter λ=1.22 and various orders α and β.
xtExactα=1α=1α=1α=0.9α=0.9α=0.9β=2β=1.9β=1.8β=2β=1.9β=1.80.30.30.218930.221790.226420.230480.216940.222050.226590.60.162180.163630.171290.178310.160140.168060.175240.90.120150.118340.128410.137520.115810.126630.136030.60.30.418300.423770.426830.429440.414890.418530.421750.60.309880.312640.319440.325880.305510.312520.319090.90.229570.226110.235960.245310.220450.231170.241040.90.30.580300.587890.585760.583510.576540.574920.573190.60.429900.433730.435160.436670.422660.424090.425580.90.318480.313680.318070.322670.303820.308690.31383Figure 1
The graphics of approximate solutionu31x,t at t=0.8 for λ=1.22 and various orders α and β.In Table3, the approximate solution is given for various values of λ with α=1 and β=2. Table 4 provides the values of exact solution and the approximation u21x,t with the parameter λ=1.47 and various orders α and β. Based on Table 3, we conclude that the approximate solution has a higher accuracy as adding more components. Moreover, Figure 2 confirms our conclusion.Example 3.
Consider the initial-boundary value problem including fractional Burger equation(43)Dtαu=Dxβu+uux,ux,0=2x,u0,t=0,u1,t=21−2t,for which
(44)ux,t=2x1−2trepresents the exact solution for integer order derivatives. Based on the RPS technique and above problem, the following coefficients are established:
(45)f00x=2x,g00t=0,g01t=21−2t,f10x=4x,g10t=0,f20x=16xg20t=0.Table 3
The approximate values ofu21x,t for various values of λ at α=1 and β=2.
xtExactλ=−1λ=0λ=1λ=2λ=50.30.3-0.53847-0.03648-0.26301-0.48954-0.71608-1.395670.6-0.98116-0.01907-0.43146-0.84385-1.25624-2.493420.9-1.787790.09664-0.65310-1.40284-2.15258-4.401810.60.3-1.02885-0.06983-0.50253-0.93524-1.36794-2.666050.6-1.87468-0.03878-0.82438-1.60998-2.39558-4.752380.9-3.415890.17085-1.24786-2.66657-4.08528-8.341420.90.3-1.42731-0.09709-0.69716-1.29723-1.89730-3.697510.6-2.60074-0.05891-1.14366-2.22840-3.31315-6.567400.9-4.738850.20655-1.73115-3.66885-5.60655-11.41964Table 4
The approximate values ofu21x,t with the parameter λ=1.47 for various orders α and β.
xtExactα=1α=1α=1α=0.9α=0.9α=0.9β=2β=1.9β=1.8β=2β=1.9β=1.80.30.3-0.53847-0.59601-0.60551-0.61355-0.62399-0.63650-0.647180.6-0.98116-1.03768-1.06509-1.08905-1.08716-1.12080-1.150390.9-1.78779-1.75522-1.80729-1.85298-1.81910-1.87986-1.933420.60.3-1.02885-1.13861-1.14310-1.14629-1.18552-1.19207-1.197100.6-1.87468-1.97921-1.99778-2.01391-2.06547-2.08925-2.110300.9-3.41589-3.33337-3.36702-3.39611-3.44429-3.48526-3.521290.90.3-1.42731-1.57926-1.56996-1.56013-1.62855-1.61841-1.607920.6-2.60074-2.73824-2.72758-2.71682-2.83788-2.82787-2.818110.9-4.73885-4.57957-4.55689-4.53486-4.70650-4.68487-4.66435Figure 2
The graphics of approximate solutionu21x,t at t=0.8 and λ=1.47 for various orders α and β.In Table5, the approximate solution is given for various values of λ with α=1 and β=2. Table 6 provides the values of exact solution and the approximation u21x,t with the parameter λ=1.16 and various orders α and β. Based on Table 5, we conclude that the approximate solution has a higher accuracy as adding more components. Moreover, our conclusion is supported by Figure 3.Table 5
The approximate values ofu21x,t for various values of λ at α=1 and β=2.
xtExactλ=−1λ=0λ=1λ=2λ=50.10.10.25000-0.000690.123850.247360.369820.730960.20.33333-0.010450.155680.319960.482380.958550.30.50000-0.054240.195480.441030.682421.381580.41.00000-0.260210.243250.730051.200182.510570.20.10.500000.000450.246830.484870.714581.353700.20.66667-0.019600.309440.623660.923071.732400.31.00000-0.109920.387840.852261.283362.376630.42.00000-0.545680.482031.376402.137443.620550.30.10.750000.005270.368040.702681.009201.759990.21.00000-0.026140.459360.894861.280362.136850.31.50000-0.168470.573961.203891.721322.598620.43.00000-0.881660.711841.855342.548851.929360.40.11.000000.015610.486610.890951.228611.841610.21.33333-0.028770.603521.117291.512551.987200.32.00000-0.231340.750721.466121.914841.661030.44.00000-1.293430.928212.083192.17149-3.96359Table 6
The approximate values ofu21x,t with the parameter λ=1.16 for various orders α and β.
xtExactα=1α=1α=1α=0.9α=0.9α=0.9β=2β=1.9β=1.8β=2β=1.9β=1.80.10.10.250000.267020.266180.264990.276670.275790.274540.20.333330.346070.344680.342720.365100.363650.361600.30.500000.479930.477350.473710.508790.506140.502410.41.000000.806390.798900.788330.845290.837810.827260.20.10.500000.522180.516960.510100.540890.535410.528220.20.666670.672560.663950.652630.709710.700690.688840.31.000000.923480.907470.886430.980270.963860.942290.42.000001.507121.460651.399561.585121.538731.477760.30.10.750000.753610.738820.720180.780220.764700.745140.21.000000.959900.935490.904731.013310.987740.955540.31.500001.294241.248871.191701.377131.330621.272010.43.000001.996551.864811.698832.114041.982551.816880.40.11.000000.949450.918970.881720.982200.950220.911140.21.333331.188501.138201.076721.255401.202731.138350.32.000001.555831.462341.348061.662071.566221.449070.44.000002.169001.897541.565762.326582.055631.72447Figure 3
The graphics of approximate solutionu21x,t at t=0.8 and λ=1.16 for various orders α and β.


## 6. Conclusion

In this study, the enhanced version of RPS technique is proposed to establish better approximate solutions of linear and nonlinear space-time fractional problems with Dirichlet boundary conditions by introducing new parameterλ. The illustrated examples present that the best approximate solutions are achieved for specific values of the parameter λ. Moreover, the value of parameter λ which leads to the best approximate solution depends on the fractional mathematical problem. The numerical examples also prove that this new effective and reliable technique is easy to implement.



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