# New Iterative Method for Solving a Coupled System of Fractional-Order Drinfeld–Sokolov–Wilson (FDSW) and Fractional Shallow Water (FSW) Equations

Authors: Nasir Ali; Mansour F. Yassen; Saeed Ahmed Asiri; Rashid Nawaz; Laiq Zada; Mohammad Mahtab Alam; Ndolane Sene
Journal: Journal of Nanomaterials
 (2022)

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## Abstract

In this study, the new iterative method has been applied to a coupled system of fractional-order Drinfeld–Sokolov–Wilson (FDSW) and fractional shallow water (FSW) equations. The fractional-order derivatives are taken in the Caputo sense whose order is between 0 and 1. The suggested method is capable to handle the FDEs without any transformation and discretization. The obtained results have been compared with the exact solution and with the q-homotopy analysis transform method. The outcomes show the efficiency and effectiveness of NIM by comparing through tables and graphs.


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## Body

## 1. Introduction

Differential equations (DEs) can be used to model the majority of physical occurrences on the planet. The DEs are divided into many categories. They may be in the form of ordinary differential equations (ODEs) or partial differential equations (PDEs). Due to significant advancements in mathematics, a new discipline known as fractional calculus was introduced which has new concepts and operations for handling derivatives and integrations. Fractional calculus deals with the DEs of noninteger order known as fractional-order differential equations (FDEs). Linear differential equations model the simple phenomenon while nonlinear equations are used in a variety of research and engineering applications including plasma physics, hydrodynamics, fluid dynamics, solid-state physics, acoustics, and optical fibers. In many fields of engineering and biosciences, the DEs occur in the form of coupled systems. The solution of a differential equation may depend on the linearity of the DE. The coupled systems in linear cases may be solved using basic analytical methods. However, due to the higher degree of nonlinearity, solving nonlinear differential equations by simple methods is not always practicable. As a result of the complexity for obtaining a solution of nonlinear DEs, researchers initiated some new approaches for approximating the solution of nonlinear DEs. They may be perturbation methods [1, 2], numerical methods [3, 4], iterative methods [5, 6], etc. Sometimes these techniques apply some transformation to reduce the equations into more simple equations or even a system of equations while some other techniques offer the solution in the form of series that converges to the exact solution [7, 8]. Besides, some other techniques which employ a trial function in an iterative scheme converge quickly. The concept of homotopy from topology and conventional perturbation methods introduced new methods such as the optimal homotopy asymptotic method, homotopy perturbation method, homotopy analysis method suggest a general analytic solution [9–11]. Therefore, these techniques are independent of the availability of a small parameter. On the other hand, a relatively new method known as the new iterative method (NIM) [12, 13] is a modified form of the Adomian decomposition method (ADM) [14] in which the Adomian polynomials are replaced by DJ polynomials in the nonlinear terms.In the present work, coupled system of fractional-order differential equations will be solved using NIM. FDEs have been solved using NIM with the help of fractional derivative and integral operators [13, 15–17]. In this paper, we will find the solution of the Fractional Drinfeld–Sokolov–Wilson (FDSW) coupled system and fractional shallow water (FSW) coupled system. The fractional-order DSW equation is used to add memory effects and genetic consequences into the system, and these features let us grasp important physical properties of complex issues. The fractional DSW coupled system is of the form [18]
(1)Dtβφx,t+3ψx,tψx,tx=0,(2)Dtβψx,t+2ψx,txxx+2φx,tψx,tx+φx,txψx,t=0,where 0<β≤1 is the fractional order of derivative of the system and is defined with Caputo’s fractional derivative operator.The second coupled system presented in this paper is the fractional-order shallow water (FSW) equation which describes a thin layer of fluid in hydrostatic equilibrium with a constant density. The equivalent wave motion is the coupled SW equation. The time-fractional SW coupled system is of the form [19]
(3)Dtβφx,t=−ψx,tφx,tx−φx,tψx,tx,(4)Dtβψx,t=−ψx,tψx,tx−φx,tx,where φx,t and ψx,t denote the free surface and the horizontal velocity component.Many researchers pay attention to fractional DSW and fractional SW equations by using different approaches [20–23].The remaining article is planned as follows: Section2 contains some basics from fractional calculus relevant to our study. The essential concepts of the NIM are presented in Section 3. The proposed approach is used to solve the fractional DSW and fractional SW coupled systems in Section 4. The numerical results by NIM are presented in Section 5 with the help of graphs and Tables 1–4. The final paragraph contains the conclusion.Table 1
The absolute error of the 2nd-order NIM and 2nd-order q-HATM solution for φx,t the FDSW coupled system.
xtβ=0.5β=0.7β=0.9β=1.0Exact sol.NIM errorError [18]2.50.0250.08803670.08803670.08803670.08803670.08804296.13333×10−61.23436×10−50.050.09710.09710.09710.09710.09715135.13081×10−51.00990×10−40.0750.1070040.1070040.1070040.1070040.1071851.809556×10−43.48632×10−40.10.1177850.1177850.1177850.1177850.1182334.479405×10−48.45395×10−450.0250.08803670.08803670.08803670.08803670.08804296.13333×10−69.30466×10−80.050.09710.09710.09710.09710.09715135.13081×10−57.63637×10−70.0750.1070040.1070040.1070040.1070040.1071851.809556×10−42.64499×10−60.10.1177850.1177850.1177850.1177850.1182334.479405×10−46.43687×10−67.50.0254.05657×10−64.05657×10−64.05657×10−64.05657×10−64.05689×10−63.215065×10−106.27408×10−100.054.48085×10−64.48085×10−64.48085×10−64.48085×10−64.48356×10−62.702048×10−95.14926×10−90.0754.94552×10−64.94552×10−64.94552×10−64.94552×10−64.95510×10−69.576419×10−91.78358×10−80.15.45240 × 10-65.4524 × 10-65.4524 × 10-65.45240 × 10-65.47623 × 10-62.382852 × 10-84.34062 × 10-8100.0255.37770×10−83.49043×10−82.87695×10−82.7333×10−82.73351×10−82.166307×10−124.22746×10−120.057.37214×10−84.34152×10−83.28010×10−83.01918×10−83.02100×10−81.820637×10−113.46956×10−110.0759.27727×10−85.2151×10−83.71170×10−83.33227×10−83.33872×10−86.452584×10−111.20177×10−100.11.11597×10−76.12942×10−84.17686×10−83.67380×10−83.68986×10−81.605564×10−102.92470×10−10Table 2
The absolute error of the 2nd-order NIM and 2nd-order q-HATM solution for ψx,t the FDSW coupled system.
xtβ=0.5β=0.7β=0.9β=1.0Exact sol.NIM errorError [18]2.50.0250.4748050.3856070.3513130.3426210.3426231.595560×10−65.67484×10−60.050.5586660.4291270.3747430.3598970.359911.308247×10−54.57167×10−50.0750.633060.4704740.3984360.3779930.3780384.519525×10−51.55336×10−40.10.7030190.5113050.4227380.3969350.3970441.095138×10−43.70588×10−450.0250.03926220.03193120.02906060.02833160.02833225.654244×10−75.67870×10−70.050.04594250.03554520.03102380.02978010.02978474.581037×10−64.60060×10−60.0750.0516970.0389370.03300330.03129590.03131161.565947×10−51.57255×10−50.10.05697260.04224180.03502460.03287920.03291683.759878×10−53.77553×10−57.50.0250.003222990.002621230.002385560.002325720.002325774.667055×10−84.66719×10−80.050.003771210.00291790.002546730.002444630.002445013.781143×10−73.78125×10−70.0750.004243350.003196320.002709240.002569080.002570371.292494×10−61.29253×10−60.10.004676110.003467560.002875170.002699050.002702153.103259×10−63.10335×10−6100.0250.0002645590.0002151640.0001958190.0001909070.0001909113.831095×10−93.83110×10−90.050.000309560.0002395160.0002090490.0002006680.0002006993.103867×10−83.10387×10−80.0750.0003483150.000262370.0002223880.0002108830.0002109891.060983×10−71.06098×10−70.10.0003838380.0002846350.0002360090.0002215520.0002218062.547404×10−72.54740×10−7Table 3
Numerical comparison of different values ofβ and absolute error of the 2nd-order NIM solution of φx,t for the FSW system.
xtβ=0.5β=0.7β=0.9β=1.0Exact sol.NIM error2.50.0250.3804920.2988030.2699850.2629740.2629851.092004×10−50.050.4618650.3370710.2892790.2769170.2770089.164358×10−50.0750.5366820.3748770.3093670.2918590.2921843.24701×10−40.10.6087010.4133060.3304960.3078330.3086428.08642×10−450.0252.705722.124821.919891.870041.870117.765359×10−50.053.284372.396952.057091.969191.969846.516877×10−40.0753.816412.665792.199942.075442.077752.308985×10−30.14.328542.939072.35022.189042.194795.750343×10−37.50.0257.14485.610855.069724.938074.938272.05054×10−40.058.67286.329455.432015.199885.20161.720863×10−30.07510.07777.039355.809225.480475.486576.097163×10−30.111.437.760976.205985.780435.795611.51845×10−2100.02513.697710.75699.719479.467069.467463.931213×10−40.0516.627112.134610.4149.9699.97233.299169×10−30.07519.320613.495611.137210.506910.51861.168923×10−20.121.913214.87911.897911.08211.11112.911111×10−2Table 4
Numerical comparison of different values ofβ and absolute error of the 2nd-order NIM solution of ψx,t for the FSW system.
xtβ=0.5β=0.7β=0.9β=1.0Exact sol.NIM error2.50.025-1.23094-1.09261-1.03916-1.02563-1.025641.255342×10−50.05-1.35945-1.16031-1.07566-1.05253-1.052631.038012×10−40.075-1.47213-1.22437-1.11257-1.08072-1.081083.623311×10−40.1-1.57702-1.28731-1.15038-1.11022-1.111118.888889×10−450.025-3.2825-2.91362-2.7711-2.73501-2.735043.347578×10−50.05-3.62521-3.09415-2.86844-2.80674-2.807022.768031×10−40.075-3.92569-3.26498-2.96685-2.88192-2.882889.662162×10−40.1-4.20538-3.43283-3.06769-2.96059-2.962962.37037×10−37.50.025-5.33406-4.73464-4.50304-4.44439-4.444445.439815×10−50.05-5.89096-5.028-4.66121-4.56095-4.56144.498051×10−40.075-6.37925-5.30559-4.82113-4.68311-4.684681.570101×10−30.1-6.83374-5.57835-4.98499-4.81096-4.814813.851852×10−3100.025-7.38562-6.55566-6.23498-6.15377-6.153857.532051×10−50.05-8.15671-6.96184-6.45399-6.31517-6.315796.22807×10−40.075-8.83281-7.3462-6.67542-6.48431-6.486492.173986×10−30.1-9.4621-7.72387-6.9023-6.66133-6.666675.333333×10−3


## 2. Fractional Calculus

We will present some essential definitions from fractional calculus that are relevant to our work [24].Definition 1.
The fractional integral operator in Riemann-Liouville (R-L) sense is defined as(5)Itβ=1Γβ∫0tt−γβ−1fγdγifβ>0,t>0,fγifβ=0,where Γ is the gamma function.Definition 2.
The Caputo fractional derivative operator of orderβ is described as follows:
(6)Dtβφt=1Γn−β∫0tt−γn−β−1fnγdγifn−1<β≤n,n∈N.Definition 3.
Relationship of the Caputo’s fractional derivative and the R-L integral is defined as follows:
Ifm−1<β≤m,m∈N, then
(7)ItβDtβφt=φt+∑j=0m−1φjγt−γjΓj+1,t>0,φ∈Cβμ,μ≥−1.


## 3. New Iterative Method

Assume a nonlinear equation of the form [12]
(8)ψχ¯,t=hχ¯,t+Lψχ¯,t+ℵψχ¯,t,whereχ¯=χ1,χ2,⋯,χn,hχ¯,t, L, and ℵ indicate the source term, linear operator, and nonlinear operator, respectively. The solution of Equation (8), according to NIM, can be expanded as follows:
(9)ψχ¯,t=∑m=0∞ψmχ¯,t.Due to the linearity of L, ψχ¯,t is expressed as
(10)L∑m=0∞ψmχ¯,t=∑m=0∞Lψmχ¯,t.The nonlinear operatorN presented by Daftardar-Geiji and Jaffari is expressed as
(11)ℵ∑m=0∞ψmχ¯,t=ℵψ0χ¯,t+∑m=1∞ℵ∑j=0iψjχ¯,t−ℵ∑j=0m−1ψjχ¯,t.Equations (9)–(11) are substituted in Equation (8) to obtain
(12)∑i=1∞ψi=h+∑m=0∞Lψm+ℵψ0+∑m=1∞ℵ∑j=0mψj−ℵ∑j=0m−1ψj.We define the recursive relation as follows:
(13)ψ0χ¯,t=h,ψ1χ¯,t=Lψ0+ℵψ0,ψ2χ¯,t=Lψ1+ℵψ0+ψ1−ℵψ0,⋮ψmχ¯,t=Lψm−1+ℵψ0+ψ1+⋯+ψm−1−ℵψ0+ψ1+⋯+ψm−2,m=1,2,3⋯,and then,
(14)ψmχ¯,t=Lψ0+ψ1+⋯+ψm−1+ℵψ0+ψ1+⋯+ψm−1,m=1,2,3,⋯,∑m=0∞ψmχ¯,t=hχ¯,t+L∑m=0∞ψmχ¯,t+ℵ∑m=0∞ψmχ¯,t.The n-term NIM solution of Equations (8) and (9) is
(15)ψχ¯,t=ψ0+ψ1+⋯+ψn−1.


### 3.1. NIM Convergence

In this section, the conditions for the convergence of NIM are given in the following theorems for the series∑m=0∞ψmχ¯,t.Theorem 4.
Ifℵ is C∞ in a neighborhood of ψ0 and ℵnψ0≤L, for some real L>0 and any nψj≤M<1/e,j=1,2,⋯, then the series ∑n=0∞Gn is convergent, and moreover, Gn≤LMnen−1e−1,n=1,2,⋯.Theorem 5.
Ifℵ is C∞ and ℵnψ0≤M≤e−1∀n, then the series ∑n=0∞Gn is convergent. The detail of NIM convergence can be seen in article written by Bhalekar and Daftardar-Geiji in [25].


## 3.1. NIM Convergence

In this section, the conditions for the convergence of NIM are given in the following theorems for the series∑m=0∞ψmχ¯,t.Theorem 4.
Ifℵ is C∞ in a neighborhood of ψ0 and ℵnψ0≤L, for some real L>0 and any nψj≤M<1/e,j=1,2,⋯, then the series ∑n=0∞Gn is convergent, and moreover, Gn≤LMnen−1e−1,n=1,2,⋯.Theorem 5.
Ifℵ is C∞ and ℵnψ0≤M≤e−1∀n, then the series ∑n=0∞Gn is convergent. The detail of NIM convergence can be seen in article written by Bhalekar and Daftardar-Geiji in [25].


## 4. Implementation of NIM

In this section, we implement NIM firstly to the fractional Drinfeld–Sokolov–Wilson coupled system and then to the fractional shallow water coupled system. The implementation is done by considering the fractional derivative in Caputo’s sense, and the R-L integral is applied to the equations. The method is applied directly for obtaining an approximate solution.


### 4.1. Fractional Drinfeld–Sokolov–Wilson (FDSW) Coupled System

Consider the FDSW system of the form by rearranging Equation (1), and we write [18]
(16)Dtβφx,t=−3ψx,tψx,tx,(17)Dtβψx,t=−2ψx,txxx−2φx,tψx,tx−φx,txψx,t,together with the initial condition
(18)φx,0=123csech2c2x,ψx,0=csechc2x.Equation (16) has the exact solution for β=1 as
(19)φx,t=123csech2c2x−ct,ψx,t=csechc2x−ct.ApplyingItβ to Equation (16), we have
(20)ItβDtβφx,t=φx,0+Itβ−3ψx,tψx,tx,ItβDtβψx,t=ψx,0+Itβ−2ψx,txxx−2φx,tψx,tx−φx,txψx,t.By substituting the initial condition, we get
(21)φx,t=123csech2c2x−ct+Itβ−3ψx,tψx,tx,ψx,t=csechc2x−ct+Itβ−2ψx,txxx−2φx,tψx,tx−φx,txψx,t.By NIM algorithm, the zeroth-order component of φx,t and ψx,tsolution is as follows:
(22)φ0x,t=123csech2c2xψ0x,t=csechc2x.The first-order component of solution is as follows:
(23)φ1x,t=3c5/2tβtanhcx/2sech2cx/22Γβ+1ψ1x,t=c5/2tβtanhcx/2sechcx/22Γβ+1.The second-order component of solution is as follows:
(24)φ2x,t=3c4sech5cx/216Γβ+122c3/2Γ2β+1t3βsinh3cx/2−7sinhcx/2Γ3β+1+4Γβ+12t2βcosh3cx/2−3coshcx/2Γ2β+1ψ2x,t=c4sech3cx/24Γβ+1232c3/2Γ2β+1t3βsinh3cx/2−6sinhcx/2sech3cx/2Γ3β+1+Γβ+12t2βcosh2cx−3Γ2β+1.By combining the zeroth, first, and second-order components ofφx,t and ψx,t the solution, we obtain the 2nd-order NIM solution as
(25)φx,t=φ0+φ1+φ2=32csech2cx2+3c5/2tβtanhcx/2sech2cx/22Γβ+1+3c4sech5cx/216Γβ+122c3/2Γ2β+1t3βsinh3cx/2−7sinhcx/2Γ3β+1+4Γβ+12t2βcosh3cx/2−3coshcx/2Γ2β+1ψx,t=ψ0+ψ1+ψ2=csechcx2+c5/2tβtanhcx/2sechcx/22Γβ+1+c4sech3cx/24Γβ+1232c3/2Γ2β+1t3βsinh3cx/2−6sinhcx/2sech3cx/2Γ3β+1+Γβ+12t2βcosh2cx−3Γ2β+1.


### 4.2. Fractional Shallow Water (FSW) Coupled System

Consider the nonlinear FSW coupled system by rearranging Equation (3), and we have [19]
(26)Dtβφx,t=−ψx,tφx,tx−φx,tψx,tx,(27)Dtβψx,t=−ψx,tψx,tx−φx,tx,together with the initial condition
(28)φx,0=19x2−2x+1,ψx,0=21−x3,where c is the wave front’s velocity. The exact solution of Equation (26) is given as
(29)φx,t=x−129t−12,ψx,t=2x−13t−1.ApplyingItβ to Equation (30), we have
(30)ItβDtβφx,t=φx,0+Itβ−ψx,tφx,tx−φx,tψx,tx,(31)ItβDtβψx,t=ψx,0+Itβ−ψx,tψx,tx−φx,tx.By substituting the initial condition, we get
(32)φx,t=19x2−2x+1+Itβ−ψx,tφx,tx−φx,tψx,tx,ψx,t=21−x3+Itβ−ψx,tψx,tx−φx,tx.Using the procedure of NIM, we have the zeroth-order component of the solution as
(33)φ0x,t=19x2−2x+1ψ0x,t=21−x3.The first-order component of solution is as follows:
(34)φ1x,t=2x−12tβ9Γβ+1ψ1x,t=−2x−1tβ3Γβ+1.The second-order component of solution is as follows:
(35)φ2x,t=2x−123Γβ+12t2β/Γ2β+1+2Γ2β+1t3β/Γ3β+19Γβ+12ψ2x,t=−4x−13Γβ+12t2β/Γ2β+1+Γ2β+1t3β/Γ3β+19Γβ+12.By NIM algorithm, the zeroth-order component of φx,t and ψx,t solution is as follows:
(36)φx,t=φ0+φ1+φ2=19x2−2x+1+2x−12tβ9Γβ+1+2x−123Γβ+12t2β/Γ2β+1+2Γ2β+1t3β/Γ3β+19Γβ+12ψx,t=ψ0+ψ1+ψ2=21−x3−2x−1tβ3Γβ+1−4x−13Γβ+12t2β/Γ2β+1+Γ2β+1t3β/Γ3β+19Γβ+12.


## 4.1. Fractional Drinfeld–Sokolov–Wilson (FDSW) Coupled System

Consider the FDSW system of the form by rearranging Equation (1), and we write [18]
(16)Dtβφx,t=−3ψx,tψx,tx,(17)Dtβψx,t=−2ψx,txxx−2φx,tψx,tx−φx,txψx,t,together with the initial condition
(18)φx,0=123csech2c2x,ψx,0=csechc2x.Equation (16) has the exact solution for β=1 as
(19)φx,t=123csech2c2x−ct,ψx,t=csechc2x−ct.ApplyingItβ to Equation (16), we have
(20)ItβDtβφx,t=φx,0+Itβ−3ψx,tψx,tx,ItβDtβψx,t=ψx,0+Itβ−2ψx,txxx−2φx,tψx,tx−φx,txψx,t.By substituting the initial condition, we get
(21)φx,t=123csech2c2x−ct+Itβ−3ψx,tψx,tx,ψx,t=csechc2x−ct+Itβ−2ψx,txxx−2φx,tψx,tx−φx,txψx,t.By NIM algorithm, the zeroth-order component of φx,t and ψx,tsolution is as follows:
(22)φ0x,t=123csech2c2xψ0x,t=csechc2x.The first-order component of solution is as follows:
(23)φ1x,t=3c5/2tβtanhcx/2sech2cx/22Γβ+1ψ1x,t=c5/2tβtanhcx/2sechcx/22Γβ+1.The second-order component of solution is as follows:
(24)φ2x,t=3c4sech5cx/216Γβ+122c3/2Γ2β+1t3βsinh3cx/2−7sinhcx/2Γ3β+1+4Γβ+12t2βcosh3cx/2−3coshcx/2Γ2β+1ψ2x,t=c4sech3cx/24Γβ+1232c3/2Γ2β+1t3βsinh3cx/2−6sinhcx/2sech3cx/2Γ3β+1+Γβ+12t2βcosh2cx−3Γ2β+1.By combining the zeroth, first, and second-order components ofφx,t and ψx,t the solution, we obtain the 2nd-order NIM solution as
(25)φx,t=φ0+φ1+φ2=32csech2cx2+3c5/2tβtanhcx/2sech2cx/22Γβ+1+3c4sech5cx/216Γβ+122c3/2Γ2β+1t3βsinh3cx/2−7sinhcx/2Γ3β+1+4Γβ+12t2βcosh3cx/2−3coshcx/2Γ2β+1ψx,t=ψ0+ψ1+ψ2=csechcx2+c5/2tβtanhcx/2sechcx/22Γβ+1+c4sech3cx/24Γβ+1232c3/2Γ2β+1t3βsinh3cx/2−6sinhcx/2sech3cx/2Γ3β+1+Γβ+12t2βcosh2cx−3Γ2β+1.


## 4.2. Fractional Shallow Water (FSW) Coupled System

Consider the nonlinear FSW coupled system by rearranging Equation (3), and we have [19]
(26)Dtβφx,t=−ψx,tφx,tx−φx,tψx,tx,(27)Dtβψx,t=−ψx,tψx,tx−φx,tx,together with the initial condition
(28)φx,0=19x2−2x+1,ψx,0=21−x3,where c is the wave front’s velocity. The exact solution of Equation (26) is given as
(29)φx,t=x−129t−12,ψx,t=2x−13t−1.ApplyingItβ to Equation (30), we have
(30)ItβDtβφx,t=φx,0+Itβ−ψx,tφx,tx−φx,tψx,tx,(31)ItβDtβψx,t=ψx,0+Itβ−ψx,tψx,tx−φx,tx.By substituting the initial condition, we get
(32)φx,t=19x2−2x+1+Itβ−ψx,tφx,tx−φx,tψx,tx,ψx,t=21−x3+Itβ−ψx,tψx,tx−φx,tx.Using the procedure of NIM, we have the zeroth-order component of the solution as
(33)φ0x,t=19x2−2x+1ψ0x,t=21−x3.The first-order component of solution is as follows:
(34)φ1x,t=2x−12tβ9Γβ+1ψ1x,t=−2x−1tβ3Γβ+1.The second-order component of solution is as follows:
(35)φ2x,t=2x−123Γβ+12t2β/Γ2β+1+2Γ2β+1t3β/Γ3β+19Γβ+12ψ2x,t=−4x−13Γβ+12t2β/Γ2β+1+Γ2β+1t3β/Γ3β+19Γβ+12.By NIM algorithm, the zeroth-order component of φx,t and ψx,t solution is as follows:
(36)φx,t=φ0+φ1+φ2=19x2−2x+1+2x−12tβ9Γβ+1+2x−123Γβ+12t2β/Γ2β+1+2Γ2β+1t3β/Γ3β+19Γβ+12ψx,t=ψ0+ψ1+ψ2=21−x3−2x−1tβ3Γβ+1−4x−13Γβ+12t2β/Γ2β+1+Γ2β+1t3β/Γ3β+19Γβ+12.


## 5. Numerical Results and Discussion

The fractional DSW and fractional SW coupled systems of PDEs have been solved by NIM. We calculated the approximate solution up to 2nd order and observed the convergence of the method. The results have been plotted with the help of 2D and 3D graphs and also shown in the tables through a numerical comparison for different values. The following discussion shows the detail of figures and tables.Figures1 and 2 show the 2nd-order NIM and the exact solution, respectively, in 3D plots φx,t while Figures 3 and 4 show the 2nd-order NIM and the exact solution, respectively, ψx,t for the fractional DSW equations. In Figures 5 and 6, the different fractional values of β of the 2nd-order NIM solution are compared for φx,t and ψx,t, respectively. In Figure 7, the 2D plot shows the absolute error for the 2nd-order NIM solution for both φx,t and ψx,t of the fractional DSW equation. Similarly, the fractional SW coupled system of equations is discussed in Figures 8–14. The 3D plots of Figures 8 and 9 represent the 2nd-order NIM solution and the exact solution, respectively, φx,t. Figures 10 and 11 show the 2nd-order approximate solution and exact solution for ψx,t of the fractional SW equations. The 2D plots in Figures 12 and 13 compare the different fractional values of β for the 2nd-order approximate solution of φx,t and ψx,t, respectively. In Figure 14, the absolute errors are compared for the 2nd-order NIM solution of the fractional SW equations. In all these figures, we noted that as the fractional order of differential equation tends to 1, the approximate solution converges to exact solution and for β=1, the approximate solution overlaps the exact solution which verifies the accuracy of our proposed method.Figure 1
2nd-order NIM solution of φx,t FDSW system.Figure 2
Exact solution ofφx,t FDSW system.Figure 3
2nd-order NIM solution of ψx,t FDSW system at c=2.Figure 4
Exact solution ofψx,t FDSW system at c=2.Figure 5
2nd-order NIM and exact solution of φx,t at t=0.1 of FDSW equation and for fractional values of β.Figure 6
2nd-order NIM and exact solution ofψx,t at t=0.1 of FDSW equation and for fractional values of β.Figure 7
Comparison of the absolute error of the 2nd-order NIM solution for φx,t and ψx,t at t=0.01 for the fractional DSW equations.Figure 8
2nd-order NIM solution of φx,t FSW system.Figure 9
Exact solution ofφx,t DSW system.Figure 10
2nd-order NIM solution of φx,t DSW system.Figure 11
Exact solution ofψx,t DSW system.Figure 12
2nd-order NIM solution and exact solution of φx,t at t=0.1 of FSW equation and for fractional values of β.Figure 13
2nd-order NIM solution and exact solution of ψx,t at t=0.1 of FSW equation and for fractional values of β.Figure 14
Comparison of the absolute error of the 2nd-order NIM solution for φx,t and ψx,t at t=0.01 for the fractional SW equations.


## 6. Conclusion

We implemented new iterative method (NIM) for the solution of the fractional Drinfeld–Sokolov–Wilson equations and fractional-order shallow water equations. The numerical comparison is made with the q-homotopy analysis transform method. The results show that NIM is conveniently convergent and provides an accurate approximate solution. The tables and figures show that as the value ofβ approaches the classical value (1 for these systems) of the differential equation, the approximate solution converges to the exact solution. Comparisons in tables and graphs verify that NIM converges more rapidly and is widely useful for obtaining the approximate solution of differential equations.



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*Source: 8370107-2022-04-07.xml*