Analytic approximate solutions of diffusion equations arising in oil pollution

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JournalofOceanEngineeringandSciencexxx(xxxx)xxx

www.elsevier.com/locate/joes

Analytic

approximate

solutions

of

diffusion

equations

arising

in

oil

pollution

Hijaz

Ahmad

a,∗

,

Tufail

A. Khan

a

,

Hülya

Durur

b

,

G.M.

Ismail

c,e

,

Asıf

Yokus

d a_Department_of_Basic_Sciences,_University_of_Engineering_and_Technology,_Peshawar,_25000,_Pakistan

b_Department_of_Computer_Engineering,_Faculty_of_Engineering,_Ardahan_University,_Ardahan,_75000,_Turkey c_Department_of_Mathematics,_Faculty_of_Science,_Sohag_University,_Sohag,_82524,_Egypt

d_Department_of_Actuary,_Faculty_of_Science,_Firat_University,_Elazig,_23200,_Turkey

e_Department_of_Mathematics,_Faculty_of_Science,_Islamic_University_of_Madinah,_170,_Madinah,_Saudi_Arabia

Received 18 March 2020; received in revised form 26 April 2020; accepted 5 May 2020 Available online xxx

Abstract

In this article, modified versions of variational iteration algorithms are presented for the numerical simulation of the diffusion of oil pollutions.Threenumericalexamplesaregiventodemonstratetheapplicabilityandvalidityoftheproposedalgorithms.Theobtainedresults are compared withthe existing solutions, which reveal that the proposed methodsare very effective and can be usedfor othernonlinear initial valueproblemsarisingin scienceandengineering.

Thisisanopenaccess articleunderthe CCBY-NC-NDlicense.(http://creativecommons.org/licenses/by-nc-nd/4.0/)

Keywords: Modified variational iteration algorithm-II; Diffusion equation; Allen-Cahn equation; Parabolic equation; MVIA-I.

1. Introduction

Partial differential equations (PDEs) are studied in vari-ous fields such as astrophysics, fluid mechanics, solid-state physics, ocean engineering, plasma physics, optical fibre, wave motion, ocean ecology, and metrology and have been studied by manyscientists for manyyears.

Oil pollutionis the release of afluid oilhydrocarbon into the ocean environment because of human activities such as releases of unrefined petroleum from tankers, drilling rigs, offshore platforms as well as piping and may cause serious damage to the marine ecological environment. Therefore, a precise guess of behaviours of these oils is extremely note-worthytokeeptheseasidenaturalenvironmentalsystem.The zone of oil spreading can be anticipated numerically by the solution of properequations governing theflow fieldandthe

∗_{Corresponding author.}

E-mail addresses: hijaz555@gmail.com (H. Ahmad),

tufailmarwat@uetpeshawar.edu.pk (T.A. Khan), hulyadurur@ardahan.edu.tr

(H. Durur).

diffusion phenomenon. The most reasonable choice is prob-ablythe diffusionequations where the information aboutthe quantityofoil,whichreachedtheoceanoutletcanbetakenas initialandboundaryconditions formodellingof oildiffusion andalteration inthe waters.

To describe oil pollution, consider the general nonlinear diffusionequation of the form:

∂v ∂t =D ∂2_v ∂x2 +αv+βv n_, (1) Where v is concentration, D is diffusion coefficient,

α andβ arerealconstants.Eq. (1) becomesAllen-Cahn equa-tionwhen n=3, α =1and β =−1,whichhasvarious ap-plicationsinquantummechanics,biologyandplasmaphysics

[1] .

There are various analytical and numerical methods for the solution of nonlinear equations i.e. the finite difference method [2 ,3] , Modified Kudryashov method [4] , Sumudu transform method [5] , (1/G)-expansion method [6] , expan-sionmethods[7 ,8] ,finite element method[9] , Hirota’s bilin-ear method [10 ,11] , Decomposition methods [12–15] , varia-tional iteration algorithm-I [16–20] , function transformation

https://doi.org/10.1016/j.joes.2020.05.002

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sub-equation function method [45] , studied nonlınear stabil-ityoftheImplicit-ExplicitmethodsforACequation[46] ,Gui andZhao[47] studiedthe existenceandqualitative properties of travellingwave solutionsof the AC equation andso on.

Thisresearchpaperaimstoachievenumericalsolutionsof thediffusionandAllen–Cahn (AC)equationsbyusing modi-fiedversionsofvariationaliterationalgorithms,whicharethe advancementsofthevariationaliterationmethodproposedby Ji-Huan He [48] . The organization of the rest of the paper is as follows; in Section 2 , modified versions of variational iteration algorithms are described. In Section 3 , some prob-lems are investigated toshow the accuracy and applicability ofthesuggestedtechniqueandinthelastSection 4 ,adetailed conclusionis discussed.

2. Modified versions of variationaliterationalgorithms

Consider the followingnonlinear differential equation

L[v(ς)]+N[v(ς)]=a(ς), (2)

Where theL[v(ς)]and N[v(ς)] are linear and nonlinear terms respectively, while a(ς) is the source term. For

Eq. (2) correctionfunctioncan bewritten as below:

vn+1(ς)=vn(ς)+ _ς 0 λ( s) L{vn(s)}+N{vn(s)}− a(s) ds, (3) Which is an iterative scheme for variational iteration algorithm-I, where λ is known as the Lagrange multiplier

[49] , its values can be obtained by using optimality condi-tions. where vn(s) is a restricted term which in turn gives δvn(s) =0. Thus

δvn+1(ς)=δv0(ς)+δ

_ς

0 λ(

s)[N{vn(s)}− a(s)]ds, (4)

This givesan exact solution after someiterations

v(ς)=Lim

n_→∞vn(ς). (5)

Summarizing the iterative algorithmfor Eq. (1) is, ⎧

⎨ ⎩

v0(ς)isan init ial approximat ion, vn+1(ς)=v0(ς)+ _ς 0 λ(s)[N{vn(s)}− a(s)]ds n=0, 1, 2, 3,... (6) (7) Thisiterativealgorithmisnamedas MVIA-II.Weemploy this adapted process for the solution of Allen-Cahn and dif-fusion equations.

3. Numerical applications

This section is devoted to the numerical implementation of modified versions of the variational iteration algorithms for different types of Allen-Cahn equations and a diffusion equation.Here,theapproximatesolutionstothediffusionand Allen-Cahnequationsare gainedeasilyandsmartlywith nei-ther useof conversionnorlinearization.

3.1. Example 1 (diffusion equation)

First, Considerthe diffusionequation [55] ∂v

∂t = ∂2_v

∂x2 +cosx, (8)

The givenconditions are:

v(x,0)=0, v(0,t)=1− e−t,

v(1, t )=−1+e−t,

The givenexact solution is:

v(x,t)= cosx1− e−t. (9)

Webegin with the constructionof the correctionfunction of MVIA-I for Eq. (8) as,

vn+1(x, t)= vn(x,t)+ t 0 λ(η) ∂vn(x, η) ∂η − cosx− ∂2_v_n₍_x, _η) ∂x2 dη. (10) Thevalueof λ(η)canbefoundoutmostquicklybyusing variationaltheory. δvn+1(x, t)= δvn(x,t)+ δ t 0 λ(η) ∂vn(x, η) ∂η − cosx− ∂2_v_n₍_x, _η) ∂x2 dη.

Ignoring the restrictedterms

δvn₊₁(x,t)=δvn(x,t)+δ t 0 λ( η) ∂vn(x, η) ∂ η dη, =(1+ λ( η))δvn(x,t)− t 0 λ₍ _η)_δv n(x, η)d η.

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Fig. 1. Absolute error graph in space-time graph form for 10th iteration by VIA-I corresponding to the Example 1.

Thestationaryconditionsare: λ(η)=0,1+ λ( η)=0, wegetascalarvalueofλ(η),whichisλ(η)=−1.Usingthis scalarvalueofλ(η)inEq. (10) givesthefollowingrecurrence relation: vn+1(x, t)= vn(x,t)− t 0 _∂ vn(x, η) ∂η − cosx − ∂2_v n(x, η) ∂x2 dη. (11) Startingtheprocedurebyusinganappropriateinitialguess for the problem,

v0(x,t)=0,

other iterations can be obtained by utilizing the recurrence relation (11) , v1(x,t)=t cos(x), v2(x,t)=− t cos(x)(t− 2) 2 , v3(x,t)= tcos(x)t2_{− 3}_t₊₆ 6 , . . . . . .

we terminate the procedureat v10(x,t). The absoluteerror of v10(x,t) in the domain of solution (x,t) [0, 10]× [0, 5]

can beseen inFig. 1 .

Now, solving the above system of linear diffusion equa-tion byMVIA-I. Using the modified algorithm MVIA-I, the recurrence relationfor Eq. (8) can bewritten as

vn+1(x, t,h)= vn(x,t,h)− h t 0 _∂ vn(x, η,h) ∂η − cosx − ∂2_v n(x, η,h) ∂x2 dη. (12) Starting the procedure by using the appropriate initial guess for the problem v0(x,t)=0, other iterations can be

obtained by utilizingthe recurrence relation(12) ,

v1(x,t, h)=ht cos(x), v2(x,t, h)=− ht cos(x)(2h+ht− 4) 2 , v3(x,t, h)= ht cos(x)h2t2+6h2t+6h2− 9ht− 18h+18 6 , . . . . . .

weterminate the procedureatv10(x,t,h).A residualfunction

for obtaining an optimalvalue of hcan bedefined as

r10(x, t, h)= ∂ v10(x, t,h) ∂t − cos(x)− ∂2_v 10(x, t,h) ∂x2 . (13) The square of residual function for 10th _iteration _with

re-specttoh for (x,t) [0, 10]× [0, 5] is e10(h)= ⎡ ⎣ 1 (11)2 10 i₌₀ 10 j₌₀ r10 i, j 2,h 2⎤ ⎦ 1 2 (14) The residual function (13) can be used to approximate

e10(h) and the proper value of h can be found out by

mak-ing e10(h) minimum. The optimal value of h is determined

tobe0.85683707022111, whene10(h)isminimized.Wemay

getextraaccuratenessbyincreasingthenumeralofiterations. Putting the value ofh in v10(x,t, h) in the domain of

solu-tion (x,t) [0, 10]× [0, 5], theabsolute errorcan beseen inFig. 2 .

ComparingFigs. 1 and2 , we confirmedthat the modified algorithm:MVIA-Igives moreaccurateresultsthanthe clas-sical algorithm. Comparison of absolute errorsof VIA-Iand MVIA-IfordifferentvaluesofparametersisgiveninTable 1 .

3.2. Example 2 (Allen-Cahnequation)

Considerthe AC equation [1] of the form,

∂v ∂t = ∂2_v ∂x2 +v 1− v2, (15)

withthe initialcondition:

v(x,0)=−0.5+0.5 tanh(0.3536x),

The exactsolution is givenby:

v(x,t)= −0.5+0.5 tanh(0.3536x− 0.75t). (16)

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Fig. 2. Absolute error graph in space-time graph form for 10th iteration by MVIA-I corresponding to Example 1. Table 1

Comparison of absolute errors for 10th iteration corresponding to Example 1.

x t Absolute Error in MVIA-I Absolute Error in VIA-I

1 0.5 1.671 × 10 −10 6.344 × 10 −10 2 1.0 3.563 × 10 −10 9.619 × 10 −09 3 1.5 2.683 × 10 −09 1.905 × 10 −06 4 2.0 1.871 × 10 −08 2.870 × 10 −05 5 2.5 5.510 × 10 −08 1.399 × 10 −04 6 3.0 5.522 × 10−07 _3.398_{× 10}−03 7 3.5 2.929 × 10−06 _1.406_{× 10}−02 8 4.0 8.666 × 10−07 _1.141_{× 10}−02 9 4.5 8.689 × 10−06 _2.528_{× 10}−01 10 5.0 1.682 × 10−05 _0.7193

Tostartwith,wesolvetheabovesystemofACequationby VIA-II.MakingthecorrectionfunctionfortheEq. (15) as,

vn+1(x,t,h) = vn(x,t,h)+ h t 0 λ(ς) ∂vn(x, ς,h) ∂ς − ∂2_v n(x,ς,h) ∂x2 −vn(x,ς,h) 1 − v n(x,ς,h)2 dς, (17) Usingthe valueof λ(ς),whichis -1obtained bythe vari-ational principle [56–58] in Eq. (17) , results in the below correctionfunction: vn+1(x,t,h)=v0(x,t,h)+h t 0 λ( ς) −∂2vn(x,ς,h) ∂x2 − vn( x,ς,h) 1− vn(x,ς,h)2 dς, (18)

Starting with aninitial condition

v0(x,t)=−0.5+0.5tanh(0.3536x),

The followingdifferent iterationscanbeobtained withthe useof recurrence relation(18) ,

v1(x,t,h)= tanh ((221 ∗ x)/625 )/2 − ( h∗ t∗ exp((442 ∗ x)/625 ) ∗(585989 ∗ exp((442 ∗ x)/625 )+585886 ))/(390625 ∗ (exp ((442 ∗ x)/625 )+1 )∧₃_{− 1}_/₂ . . . . . .

Forfinding the optimal valueof h for the estimated solu-tion, the subsequent residual functionis delineated:

r3(x, t, h)= ∂v3(x, t, h) ∂t − ∂2_v 3(x, t, h) ∂x2 −v3( x,t,h) 1− v3(x,t,h)2 . (19)

Thenorm2ofresidualfunction(19) for3rd-order approx-imationfor (x,t) [0, 30]× [0, 1] is e3(h)= ⎡ ⎣ 1 (31)2 30 i₌₀ 30 j₌₀ r3 i, j 30,h 2⎤ ⎦ 1 2 (20)

The residual function (19) can be used to approximate

mak-ing e3(h)minimum.The optimalvalue ofh is determinedto

be 0.956746100698240, when e3(h) is minimized. We may

get extra accurateness by increasing the numeral of itera-tions. Utilizing this optimal value ofh in v3(x,t, h) in the

domain (x,t) [0, 30]× [0, 1], the following results are obtained, whichcanbe seenin Table 2 ,Figs. 3 and 4 .

3.3. Example 3 (ACEquation)

Consider theAC equation [2] of thefollowing form

∂v ∂t =

∂2_v ∂x2 − v

3₊_v_, ₀_<_x_<₁_, ₀_<_t _<₁_, ₍₂₁₎

withthe initial condition:

v(x,0 )= − 3 A1 + 9 A2 1+ 24c A0B1 −1 + t anh 3c1+ √ 2cx 4c 6 A1+ 2 9 A2 1+ 24c A0B1− 6 B1 1 + t anh 3c1+ √ 2cx 4c . WhereA0=−3,c=0.6, B1=−5,A1=−1, c1=0.1

TheexactsolutionofEq. (21) withtheaboveassumptions:

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Fig. 3. Behavior of approximate solution (above) and exact Solution (below) corresponding to Example 2.

Fig. 4. Comparing the curves of absolute errors for different values of t corresponding to Example 2.

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Fig. 5. Comparison of exact and approximate solutions for t= 0.01 corresponding to Example 3.

v(x,t)=− 12[−1+tanh{0.416667(0.3− 1.8t+0.848528x)}]

24+30[1+tanh{0.416667(0.3− 1.8t+0.848528x)}] .

(22) Tostartwith,wesolvetheabovesystemofACequationby VIA-II.MakingthecorrectionfunctionfortheEq. (21) as,

vn+1(x, t,h)=vn(x,t,h)+h t 0 λ( ς) _∂ vn(x, ς,h) ∂ς −∂2vn(x,ς,h) ∂x2 − vn(x,ς,h)+ vn(x,ς,h)3 dς, (23)

Usingthe valueof λ(ς),whichis -1obtained bythe vari-ational principle [56–58] in Eq. (23) , results in the below correctionfunction: vn+1(x, t,h)=vn(x,t,h)− h t 0 ∂vn(x, ς,h) ∂ς −∂2vn( x,ς,h) ∂x2 − vn(x,ς,h)+vn(x,ς,h) 3 dς. (24)

Starting with aninitial condition

v0(x,t)=−

12[−1+tanh{0.416667(0.3+0.848528x)}] 24+30[1+tanh{0.416667(0.3+0.848528x)}], The otherdifferentiterationscanbeobtained withthe use ofrecurrencerelation(24) ,Forfindingoptimalvalueof hfor

estimated solution, the subsequent residual function is delin-eated: r4(x, t, h)= ∂v4(x, t,h) ∂t − ∂2_v 4(x, t,h) ∂x2 −v4( x,t,h)+v4(x,t,h)3 (25)

Thenorm2of residualfunction(25) for4th-order approx-imationfor (x,t) [0, 1]× [0, 1] is e4(h) = ⎡ ⎣ 1 (11)2 10 i₌₀ 10 j₌₀ r4 i 10, j 10,h 2⎤ ⎦ 1 2 (26) The residual function (25) can be used to approximate

mak-ing e4(h)minimum.The optimalvalue ofh is determinedto

be 1.00971373400743, when e4(h) is minimized. We may

get extra accurateness by increasing the numeral of itera-tions. Utilizing this optimal value ofh in v4(x,t, h) in the

domain(x,t) [0, 1]× [0, 1],moreaccurateresultsare ob-tained, which are reported in Table 3 , while comparison of exact andapproximate solutionscan beseenin Fig. 5

Thenumericalresultsfordiffusionequationarereportedin

Table 1 , while for AC equations are reportedin Table 2 and

Table 3 .Toprovetheeffectivenessoftheplannedtechniques, absolute errors are reported along with the results of other methods;finitedifference method[2] ,Adomian’s decomposi-tion [47] , multiquadric quasi-interpolation methods [47] and

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Table 3

Comparison of numerical results for different values of t and x for Example 3.

x t

Exact Solution

Approximate Solution Absolute Errors

Present FDM [2] Present FDM [2] 0.00 0.01 0.1842577027 0.1842576837 0.184247 1.908E −08 1.06626E −05 0.01 0.01 0.18319724455 0.1831972253 0.183187 1.924E −08 1.06500E −05 0.02 0.01 0.1821415268 0.1821415074 0.182131 1.939E −08 1.06370E −05 0.03 0.01 0.18109054409 0.1810905245 0.181080 1.954E −08 1.06236E −05 0.04 0.01 0.1800442906 0.1800442709 0.180034 1.969E −08 1.06098E −05 0.05 0.01 0.1790027607 0.1790027408 0.178992 1.984E −08 1.05956E −05 0.06 0.01 0.1779659481 0.1779659281 0.177955 1.999E −08 1.05810 E −05

LLWM[1] .In comparisonwithothertechniquesresults, one canensurethat theresultsof modifiedvariationaliteration al-gorithms are moreprecise. It iscleared from figuresthat the proposed algorithms canhandle the problemsaccurately and will be applicable in ocean engineering for studying linear andnonlinear waterwaves.

4. Conclusions

In this study, approximate solutions of diffusion equation arisinginoilpollutionanddifferenttypesofACequationsare obtained byusing two modified algorithms.Basedonthe re-sults, ithasbeen foundthatthe appliedmethodswillbeable to use without using discretization, shape parameter, Ado-mian polynomials, transformation, linearization or restrictive assumptions and thus are particularly perfect with the ex-panded andflexible nature of the physical problemsandcan be easily extended tofractal calculus [59–62] arise inocean engineeringand science.

Declaration of CompetingInterest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared toinfluence thework reportedinthis paper.

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