DSpace at VNU: Regularized solution of an inverse source problem for a time fractional diffusion equation

Nguyen et al., Regularized solution of an inverse source problem for a time fractional... The fractionalinitial boundaryvalue problemwasfirstlyconsidered byNigmatullin [4].After that,seve

journalhomepage:www.elsevier.com/locate/apm

Huy Tuan Nguyena,b, Dinh Long Leb, Van Thinh Nguyenc,∗

a Department of Mathematics and Computer Science, University of Science, Vietnam National University, Ho Chi Minh City, Vietnam

b Institute of Computational Science and Technology, Ho Chi Minh City, Vietnam

c Department of Civil and Environmental Engineering, Seoul National University, Republic of Korea

Inthispaper, westudy onan inverse problemto determine an unknownsource term

inatimefractionaldiffusionequation,whereby thedataareobtainedatthelatertime

Ingeneral,thisproblemisillposed,thereforetheTikhonovregularizationmethodisposedtosolvetheproblem.Inthetheoreticalresults,apriorierrorestimatebetweentheexactsolution and its regularizedsolutions is obtained Wealso proposetwo methods,

pro-aprioriandaposterioriparameterchoicerules,toestimatetheconvergencerateoftheregularizedmethods.Inaddition,theproposedregularizedmethodshavebeenverifiedbynumericalexperimentstoestimatetheerrorsbetweentheregularizedsolutionsand ex-actsolutions.Eventually,fromthenumericalresultsitshowsthattheposterioriparameterchoicerulemethodconvergestotheexactsolutionfasterthantheprioriparameterchoicerulemethod

© 2016ElsevierInc.Allrightsreserved

1 Introduction

Diffusionequationswithfractional orderderivativesplay an importantrole inmodelingofcontaminant diffusioncesses.OneofsuchproblemswasraisingbyAdamandGelhar [1];duringanalyzingthefielddataofdispersioninahetero-geneousaquifer,theycouldnotexplainalong-tailedprofileofspatialdensitydistributionbyaclassicaldiffusion–advectionequationwithintegerorderderivatives.Thelong-tailedprofileisofasymptoticbehavioroffundamentalsolutionneart=0,

pro-whichcanbedescribedbyatime-fractionaldiffusionequationasfollows:

∗ Corresponding author Tel.: +82 28807355

E-mail address: vnguyen@snu.ac.kr (V.T Nguyen)

http://dx.doi.org/10.1016/j.apm.2016.04.009

S0307-904X(16)30210-4/© 2016 Elsevier Inc All rights reserved

Please cite this article as: H.T Nguyen et al., Regularized solution of an inverse source problem for a time fractional

and(.)denotesthestandardGammafunction.Notethatifthefractionalorderα tendstounity,thefractional derivative

D α

t uconverges tothecanonical first-orderderivative du

dt [2], andthusthe problem(∗)reproduces the canonicaldiffusionmodel.See,e.g., [2,3]forthedefinitionandpropertiesofCaputosderivative

The fractionalinitial boundaryvalue problemwasfirstlyconsidered byNigmatullin [4].After that,severalapplications

ofthefractionalcalculus andderivativesinappliedsciencesweredeveloped; andtheresearch ontheoreticalanalysisandnumericalmethodsforsolvingdirectproblems,suchasinitialvalueand/orboundaryvalueproblemsforthetimefractionaldiffusionequationweregrowing.Forawell-posednessanalysis,wereferto [5–9];fornumericalmethodsandsimulations,see [10–16]andreferences therein.Unfortunately,some input dataandparameters ofthediffusionequations inpracticalproblemsmaybeunknown, suchasinitialandboundarydata,diffusioncoefficients,andsourceterms;thereforewehave

todeterminethembyadditionalmeasurementdatawhichcanbeyieldedfromafractionaldiffusioninverseproblem.Thefractional inverseproblem provides an efficient tool for the modeling of the anomalous diffusion processes observed invariousfieldsofscienceandengineering,suchasinbiology [17,18],physics [19,20],chemistry [21],andhydrology [22].Murio [23] considered an inverse problem of recovering boundary functions fromtransient data at an interior pointfor a 1D semi-infinite half-order time-fractional diffusion equation Liu and Yamamoto [24] applied a quasi-reversibilityregularization methodto solve abackward problemfor thetime-fractional diffusionequation Weietal. [25–27]studied

an inversesourceproblemfora spatialfractional diffusionequation usingquasi-boundaryvalue andtruncationmethods.Recently,Kiraneetal. [2,28]studiedconditionalwell-posednesstodetermineaspacedependentsourceforone-dimensionalandtwo-dimensional time-fractionaldiffusionequations.Rundell etal. [29–31]considered an inverseproblemfora one-dimensional time-fractionaldiffusionproblem However,there areonly afew studieson determinationofa sources termdependingonbothtimeandspaceforatimefractionaldiffusionequation

Inthiswork,wefocusonaninverseproblemforthefollowingtime-fractionaldiffusionequation:

0() Assuming that the source term F=F(x , t) can be split into a product of R(t)f(x), where R(t)

is knowninadvance.We assume thetime-dependentsource termR(t) isobtainedfromobservationdataR (t) in suchawaythatR (t)− R(t) L1(0,T )≤ and isanoiselevelfromameasurement.Thespace-dependentsourcetermf(x)isalsodeterminedfromtheobservationofh(x)atthefinaldatat=T byh ∈L2()withthenoiselevelof andsatisfied:

Itisknownthattheinversesourceproblemmentioned aboveisill-posedingeneral,i.e.,a solutiondoesnotalways exist,andin the caseofexistence of a solution,which doesnot depend continuously on thegiven data.In fact,from a smallnoiseofaphysicalmeasurement,thecorrespondingsolutionsmayhavealargeerror.Thismakesatroublesomenessforthenumericalcomputation,hencearegularizationisrequired

If α=1, the inversesource Problems (1.1) and (1.2) is a classical ill-posed problemand has been studied in[20,32].However,forthefractionalinversesourceproblem,up-to-date,thereareonlyveryfewworks;forexample,SakamotoandYamamoto [9]usedthedatau(x0,t)(x0 ∈) todetermineR(t)oncef(x)wasgiven, wheretheauthorsobtainedaLipschitzstabilityforR(t).ZhangandWei [27]usedtheFouriertruncationmethod tosolveaninversesourceproblemwithR(t)=1

inProblem (1.1)forone-dimensionalproblemwithspecialcoefficients Theinversesourceproblemforthetime-fractionaldiffusion equation with R(t) dependedon time still has a limitedachievement Actually, this problemrecently hasbeen

introduced by JinandRundell on Page 18 of [33], howevera further regularized solution forthisproblem hasnot beenfocusedintheirpaper.

Motivatedby abovereasons,inthisstudy,weapplythe Tikhonovregularizationmethodtosolvethefractional inversesource problemwith variable coefficientsin a general bounded domain We estimate a convergence rate under a prioribound assumption of the exact solution anda priori parameter choice rule Because the priori bound is difficult to ob-tain inpractical application, so we alsoestimate a convergencerateunder theposterioriparameter choice rulewhich isindependentontheprioribound

Thepaperisorganizedasfollows.In Section2,weintroducesomepreliminaryresults.Theill-posednessofthefractionalinversesourceProblem (1.1)andaconditionalstabilityareprovidedin Section3.In Section4,weproposeaTikhonovreg-ularizationmethodandgivetwoconvergenceestimatesunderaprioriassumptionfortheexactsolutionandtworegular-izationparameterchoice rules.Finallytwo numericalexamples toverifyourproposed regularized methodsare showninSection5.Eventually,aconclusionisgivenin Section6

2 Preliminary results

Throughoutthispaper,weusethefollowingdefinitionandlemmas

Definition 2.1(see [3]). TheMittag–Lefflerfunctionis:

E α , β(z)=∞

k=0

z k

( αk+β ), z∈C

whereα>0andβ∈Rarearbitraryconstants

Lemma 2.1(see [3]). Letλ>0,then we have:

Lemma 2.3. For0<α<1,ρ>0,we have0≤ E α , α(−ρ )≤ 1

(α) . Moreover, E α , α(−ρ )is a monotonic decreasing function with

Theproofiscompleted 

3 The inverse source problem

First,we introducea fewpropertiesofthe eigenvaluesofthe operatorA onan open,connectedandboundeddomainwithDirichletboundaryconditions(seealsoinChapter6of [36])

Theorem 2.1. (EigenvaluesoftheLaplaceoperator)

1 Each eigenvalues of A is real The family of eigenvalues{ λp}∞

whereg p=  g(x) φp(x)dx ThenwealsodefinegH ()=∞

p=1(1+λp)2γ|g p|2.Ifγ =0thenH γ()isL2().Thisspace

isintroducedbyBrezis [37](seeChapterV)andFengetal. [38](seepage179)

3.1 The formula and uniqueness of the source termf

Now we use the separation ofvariables to yield the solutionof (1.1).Suppose that the exact u isdefined by Fourierseries:

Theorem 3.1. Let R:[0, T]→Rbe as in Lemma 2.6, then the solution u(x, t),f(x)of Problems (1.1) and (1.2) is unique.

Proof.Letf1andf2 bethesourcefunctionscorrespondingtothefinalvaluesh1 andh2 respectively.Supposethat h1=h2

thenweprovethat f1= f2.Infact,itiswell-knownthatE α , α(−λp(t − s)α)≥ 0for ≤ t.SinceR(t)≥ R0 >0fort∈[0,T],

Theproofiscompleted 

3.2 The ill-posedness of the inverse source problem

Theorem 3.2. The inverse source problem is ill-posed.

Proof.DefiningalinearoperatorK:L2()→L2()asfollows:

andbyKirsch [39],weconcludethatitisill-posed.Toillustrateanill-posedproblem,weintroducethefollowingexample.Letuschoosetheinputfinaldatah m(x)= φ√m ( x )

Combining (3.23)and (3.25),weconcludethattheinversesourceproblemisill-posed 

3.3 Conditional stability of the source termf

Inthissection,weintroduceaconditionalstabilitybythefollowingtheorem

Theorem 3.3. IffH () ≤ M for M >0then,

Proof. From (3.14)andHölderinequality,wehave:

4 Regularization of the inverse source problem using the Tikhonov method

Asmentionedabove,applyingtheTikhonovregularizationmethodwesolvetheinversesourceproblem,whichminimizesthefunctionfinthefollowingquantityinL2()

4.1 A priori parameter choice

Afterwards,wewillgiveanerrorestimationforf(x)− f β (x) L2()andshow convergencerateunderasuitablechoice

fortheregularizationparameter

Theorem 4.1. Let f be as Theorem 3 and the noise assumption (4.30) hold Then,

Lemma 4.1. Assume that (4.30) holds Then we have the following estimate:

Inordertoobtaintheboundednessofbias,we needtheprioricondition ByTikhonov’stheorem,theL−1 restrictedtothecontinuousimageofacompactsetM.Thus,weassumefisinacompactsubsetofL2().Fromnowon,weassumethat

Case 1:γ≥ 1;inthiscase,wenotethat:

If0≤γ ≤ 1thenfromLemmas4.1and4.2,weget:

4.2 A posteriori parameter choice

Inthissection,weconsideraposterioriregularizationparameterchoiceinMorozov’sdiscrepancyprinciple(seein [35])

Asusual,itfollowsthelemmabelow

If0<<h L2() , then the following results hold:

(a)ρ(β)is a continuous function.

(b)ρ(β)→0asβ→0.

(c)ρ ( β )→h L2() asβ →∞.

(d)ρ(β)is a strictly increasing function.

Theorem 4.2. Assume the priori condition and the noise assumption hold, and there exists k >1such that0< k<h  Then,

we choose a unique regularization parameterβ >0such that:

Firstly,wecanreceivethefollowingestimation:

Secondly,wehaveestimate:

2,wherethevalue

of Mplays a role as the priori condition computedby fH2(0, π ). Similarly, based on the choice of k we can take the

regularization parameter forthe posteriori parameter choice rule,βpos= 12

M{k,R,R  } ,where M{k ,R,R } is dependent on k, }satisfied (4.61)

From (3.14),wecandefinitetheexactsolution,asfollows:

5.1 Example 1

Inthisexample,weconsiderthefunctionfisanexactdatafunction.Weparticularlyconsideranone-dimensionalcase

oftheproblem (1.1)asfollowswithλp=p2 andT=1

Step 1ChooseQandLtogeneratethespatialandtemporaldiscretizationinsuchamanneras:

Step 3Errorestimatebetweentheexactandregularizedsolutions

Relativeerrorestimation:

Fig 1 A comparison between the exact solution and its regularized solution for the priori parameter choice rule in Example 1

Next,we establish the regularized solutionaccording to compositeSimpson’s rule.In this example,we choose P and

Qare large enough, unlike thefirst example,we can have fH2(0, π ) < M with M=112230 fromtheanalytical solution,whichimpliesthatβpri=(1

M)1

2 fortheprioriparameterchoice,andβpos= 12

M{k,R,R  } withM{k ,R,R }=239587fortheoriparameterchoicerulebasedonk=1.8andM{k ,R,R }= supt∈[0,T]|R ( t )|

posteri-2λ2

1 ( k2 −2)inft∈[0,T]|R  ( t )|fH ().In general,the whole numerical

procedureissummarizedinthefollowingsteps:

Fig 2 A comparison between the exact solution and its regularized solution for the posterior parameter choice rule in Example 1

Step 1 ChooseQ and K (in ourcomputations,Q=K=100 are chosen)to discretizespatialandtemporal domain,aslows:

Fig 3 A comparison between exact solution and its regularized solution for the priori parameter choice rule method in Example 2

Relativeerrorestimation:

by Figs. 3and 4combined with Table 2in the second example.Particularly, the error estimate shownin Tables 1and2showthattheposteriorparameterchoicerulemethodconvergestotheexactsolutionwithoneorderfasterthanthepriorparameterchoice rulemethod.Nevertheless,fromtwo examples,it alsoshowsourproposed regularized methodshaveaverygoodconvergenceratetotheexactsolutiononcetendsto0

Fig 4 A comparison between exact solution and its regularized solution for the posteriori parameter choice rule method in Example 2

6 Conclusion

Inthisstudy,we solvedtheinverseproblemtorecoverthesourcetermforthetime fractionaldiffusionequation withthetimedependentcoefficientbyapplyingTikhonovmethod.Inthetheoreticalresults,weobtainedtheerrorestimatesoftwo prioriandposterior parameterchoice rulemethods basedon theprioricondition Inthe numericalresults,it showsthattheproposedregularizedsolutionsareconvergedtotheexactsolutions.Furthermore,italsoshowsthattheposterioriparameterchoice rulemethodisbetter thanthepriori parameterchoicerulemethod intermofthe convergencerate.Inthefuturework,wewillcontinuetostudysomesourcetermsformultiplediffusionequations

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Step 1ChooseQandLtogeneratethespatialandtemporaldiscretizationinsuchamanneras:

Step 3Errorestimatebetweentheexactandregularizedsolutions