In this paper we study a one dimensional inverse source problem. Distinguishability according to the source function in time-space fractional equation Db t u(x, t) = D2a x u(x, t)+ f (x), 1 

Demir, Ali

Ozbilge, Ebru

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Appl. Math. Inf. Sci.12, No. 3, 587-591 (2018) 587Applied Mathematics & Information SciencesAn International Journalhttp://dx.doi.org/10.18576/amis/120313An Inverse Source Problem in Time-Space FractionalDifferential EquationsEbru Ozbilge1,∗ and Ali Demir21 American University of the Middle East, Department of Mathematics and Statistics, Egaila, Kuwait2 Kocaeli University, Department of Mathematics, Umuttepe,41380, Izmit-Kocaeli, TurkeyReceived: 30 Feb. 2018, Revised: 19 Apr. 2018, Accepted: 21 Apr. 2018Published online: 1 May 2018Abstract: In this paper we study a one dimensional inverse source problem. Distinguishability according to the source function intime-space fractional equationDβt u(x, t) = D2αx u(x, t)+ f (x),12 < α ≤ 1, 0< x< 1, 0< β ≤ 1 with Dirichlet boundary conditions isinvestigated. In addition to this, the measured output datah(t) determined analytically by a series representation.Keywords: Inverse source problem, Time-space fractional equation1 IntroductionRecent decades have witnessed a fast growingapplications of fractional partial differential equations todiverse scientific and engineering fields regardinganomalous diffusion, constitutive modeling inviscoelasticity, signal processing and control, fluidmechanics, image processing, aerodynamics,electro-dynamics of complex medium and polymerrheology.Compared to integer-order calculus, fractional partialdifferential equations have the capacity of providing amore simple and accurate description of complexmechanical and physical processes featuring historydependency and space non locality and has thus inducedthe occurrences of a series of fractional differentialequations [1,2,3,4,5,6,7].In this paper, we consider time-space fractional inverseproblem with Dirichlet boundary conditions:Dβt u(x, t) = D2αx u(x, t)+ f (x)12 < α ≤ 1,0< x< 1,0< β ≤ 1u(x,0) = g(x) 0≤ x≤ 1u(0, t) =Ψ0(t) u(1, t) =Ψ1(t),0≤ t ≤ T(1)The fractional ordersβ ∈ (0,1) andα ∈(12,1)are relatedto the parameters specifying the large-time behavior ofthe waiting-time distribution or long-range behaviour ofthe particle jump distribution. In hydrological studies, theparameterα is used to characterize the heterogeneity ofporous medium.In theory, these parameters can be determined from theunderlying stochastic model, but often in practice, theyare determined from experimental data.The notation Dβt is the Djrbashian-Caputo derivativeoperator of orderβ ∈ (0,1) in the time variablet, andD2αt denotes the Djrbashian-Caputo derivative of orderα ∈(12,1)in the space variablex.Dβt u(x, t) = (I1−β u′)(t), 0< β ≤ 1, Iβ being the Riemann-Liouville fractional integral,(Iβ f )(t) ={1Γ (α)∫ t0(t − τ)β−1 · f (τ)dτ 0< β ≤ 1f (t) β = 0(2)Ψ0(t) and Ψ1(t) are the left and right boundary valuefunctions, respectively. They belong toC1[0,T].g(x) ∈ C1[0,1] is the function that satisfies the followingconditions:g(0) =Ψ0(0), g(1) =Ψ1(0).∗ Corresponding author e-mail:ebruozbilge@gmail.comc© 2018 NSPNatural Sciences Publishing Cor.588 E. Ozbilge, A. Demir: An Inverse Source Problem in Time-Space...Under the conditions (C1) and (C2), the initial-boundaryvalue problem (1) has the unique solutionu(x, t).Caputo’s fractional derivatives are more widely used ininitial value problems of differential equations and havestronger physical interpretations. The Caputo fractionalderivative can better reconcile the well-established andpolished mathematical theory with practicalneeds.For a real numbern−1< γ < n, n∈ N and f ∈ Hn(0,1)the left-sided Djrbashian-Caputo derivativeDγx f of orderγis defined byDγx f =1Γ (n− γ)∫ x0(x− s)n−1−γ f (n)(s)ds, (3)whereΓ (z) denotes Euler’s Gamma function defined byΓ (z) =∫ ∞0sz−1e−sds ℜ(z)> 0 (4)The Djrbashian-Caputo derivative was first introduced bymathematician Mkhitar M. Djrbashian for studies onspace of analytical functions and integral transforms in1960’s (see [8]). Geophysicist Michele Caputoindependently proposed the use of the derivative formodeling the dynamics of viscoelastic materials in 1967.We note that there are different definitions of fractionalderivatives, notably the Riemann-Liouville fractionalderivative, which formally is obtained from (3) by intercharging the order of integration and differentiation, thatis the left-sided Riemann-Liouville fractional derivativeDγx f of orderγ ∈ (n−1,n), n∈ N is defined byDγx f =dndxn1Γ (n− γ)∫ x0(x− s)n−1−γ f (s)ds (5)In this work, we will focus on the Djrbashian-Caputoderivative since it allows a convenient treatment of theboundary and initial conditions.Consider the inverse problem of determining thedistinguishability of the unknown source functionf (x)from the Neumann type of measured output data at theinner pointx= 12. Measured output data is taken atx=12since it is the middle point but it can be taken at an innerpoint at which output data is measured exactly. It dependson the physical conditions of the systems.Φ[ f ] = D2αx u(x, t; f ) |x= 12 (6)We can formulate the measured output datah(t) asfollows:Φ[ f ] = h h∈C1(0,T] (7)The inverse problem of determining an unknown sourcefunction f (x) is reduced to the problem of invertibility ofΦ[·]. This directs us to investigate the distinguishabilityof the source function via input-output mappings. Themapping Φ[·] : K −→ C1[0,T] has distinguishabilityproperty wheneverΦ[ f1] 6= Φ[ f2] implies f1(x) 6= f2(x).Hence, that means injectivity of the inverse mappingsΦ−1 andΨ−1. Here, measured output data of Neumanntype at the inner pointx = 12 is used in the determinationof the distinguishability of the unknown source functionf (x). As a result of these, in the distinguishability of theunknown source functionf (x), analytical results areobtained. Regardless of the validity of auxiliary functionwhat are the sufficient conditions under which thedistinguishability of the unknown source function isdetermined. It is known that approximate initial andboundary conditions there is a solution but by usingauxiliary solution we check the closeness of the solutionto the right solution.2 Analysis of the Time-Space FractionalInverse Problem with Measured DataDetermine the measured measured output datag(t) at theinner point x = 12. We used the Fourier method in theformulation of problem (1). We need to define anauxiliary functionv(x, t) as follows:v(x, t) = u(x, t)− xΨ1(t)+ (x−1)Ψ0(t) (8)v(x, t) helps to transform problem (1) into a problem withhomogeneous boundary conditions. Therefore (1) can berewritten in terms ofv(x, t) in the following form:Dβt v(x, t)−D2αt v(x, t) = f (x)− xDβt Ψ1(t)+ (x−1)Dβt Ψ0(t)+Ψ1(t)D2αx x−Ψ0(t)D2αx (x−1)v(x,0) = g(x)− xΨ1(0)+ (x−1)Ψ0(t)v(0, t) = v(1, t) = 0(9)We can get the unique solution of the initial boundaryvalue problem by using Fourier method:v(x, t) =∞∑n=1< ζ (θ ),Φn,α(θ )> Eβ ,1(−λn,α tβ )Φn,α (x)+∞∑n=1{∫ t0sβ−1Eβ ,β (−λn,α sβ )(< ξ (θ , t −s),Φn,α (θ )>+< F(θ ),Φn,α(θ )>)ds}Φn,α(x)ζ (x) = g(x)−xΨ1(0)+(x−1)Ψ0(t)ξ (x, t) = f (x)−xDβt Ψ1(t)+(x−1)Dβt Ψ0(t)+Ψ1(t)D2αx x−Ψ0(t)D2αx (x−1)(10)Moreover,< ζ (θ ),Φn,α (θ )>=∫ 10Φn,α(θ )ζ (θ )dθ , (11)c© 2018 NSPNatural Sciences Publishing Cor.Appl. Math. Inf. Sci.12, No. 3, 587-591 (2018) /www.naturalspublishing.com/Journals.asp 589Eα ,β (z) is the two-parameter Mittag-Leffler function (withβ > 0 andα ∈ R) defined byEα ,β (z) =∞∑k=0zkΓ (kβ +α), z∈C. (12)This function withβ = 1 was first introduced by Mittag-Leffler in 1903 [16].It can be verified thatE1,1(z) = ez, E2,1(z) = cosh√z andE2,2(z) =sinh√z√z . As a consequence, it represents ageneralization of the exponential function in thatE1,1(z) = ez. The functions Eβ ,1(−λn,α , tβ ) andsβ−1Eβ ,β (−λn,αsβ ) appear in the kernel of the timefractional diffusion problem with initial data and theright-hand side, respectively. Eigenfunctions also appearin the kernel of fractional Sturm-Liouville problem with azero potential [9,10,11,12,13,14,15].Assume thatΦn,α(x) is the solution of the followingSturm-Liouville problem:D2αΦ(x) = λ Φ(x) 0< x< 1Φ(0) = 0 , Φ(1) = 0(13)where the eigenvalues areλn,α = (nπ)2α , n= 1,2, . . . andthe associated eigenvalues areΦn,α = sin(nΠx).The Neumann type of measured output data atx = 12 canbe written in terms ofv(x, t) in the following formh(t) = D2αx v(12, t)+Ψ1(t)−Ψ0(t) , t ∈ (0,T]. (14)In order to arrange (10), set the followingszn(t) =< ζ (θ ),Φn,α (θ )> Eβ ,1(−λn,αtβ )wn(t) =∫ t0sβ−1Eβ ,β (−λn,αsβ )< ξ (θ , t − s),Φn,α(θ )>yn(t) =∫ t0sβ−1Eβ ,β (−λn,αsβ )< F(θ ),Φn,α (θ )>(15)The solution in terms ofzn(t),wn(t),yn(t) can be writtenin the following form:v(x, t) =∞∑n=1zn(t)Φn,α(x)+∞∑n=1wn(t)Φn,α(x)+∞∑n=1yn(t)Φn,α (x).(16)Now, let us differentiate both sides of (16) with respect toxD2αx v(x, t) =∞∑n=1zn(t)D2αx Φn,α(x)+∞∑n=1wn(t)D2αx Φn,α(x)+∞∑n=1yn(t)D2αx Φn,α(x)(17)By substitutingx= 12 we obtain:D2αx v(12, t)=∞∑n=1zn(t)D2αx Φn,α(12)+∞∑n=1wn(t)D2αx Φn,α(12)+∞∑n=1yn(t)D2αx Φn,α(12) .(18)By considering the over-measured dataD2αx v(12, t)+Ψ1(t)−Ψ0(t) = h(t) (19)we get:h(t) =Ψ1(t)−Ψ0(t)+∞∑n=1zn(t)D2αx Φn,α(12)+∞∑n=1wn(t)D2αx Φn,α(12)+∞∑n=1yn(t)D2αx Φn,α(12)(20). (20) implies that h(t) can be determinedanalytically.The right-hand side of the above identity (20) defines thefollowing input-output mapping:Φ[ f ](t) :=Ψ1(t)−Ψ0(t)+∞∑n=1zn(t)D2αx Φn,α(12)+∞∑n=1wn(t)D2αx Φn,α(12)+∞∑n=1yn(t)D2αx Φn,α(12)(21)So we obtain the relation between the source functionsand the corresponding outputsf j(t) := D2αx u(12, t; f j)byproviding the following lemma.Lemma 1.Let v1(x, t) = v(x, t; f1) and v2(x, t) = v(x, t; f2)be the solutions of the direct problem (9) correspondingto the admissible source functions f1(x), f2(x) ∈ K. Ifh j(t) = D2αx (12, t; f j) +Ψ1(t) −Ψ0(t) j = 1,2 are thecorresponding outputs, the outputs hj(t), j = 1,2, satisfythe following series identity∆h(t)=∞∑n=1∆wn(t)D2αx Φn,α(12)+∞∑n=1∆yn(t)D2αx Φn,α(12)(22)for each t∈ (0,T] where∆h(t) = h1(t)−h2(t), ∆wn(t) =w1n(t)−w2n(t), ∆ f (t) = f1(t)− f2(t) and∆yn(t) = y1n−y2n.Proof.By using identity (20) we can write the measuredoutput data as follows, respectively [16,17,18,19,20].h j(t) = D2αx v(12, t)+Ψ1(t)−Ψ0(t) j = 1,2 (23)c© 2018 NSPNatural Sciences Publishing Cor.590 E. Ozbilge, A. Demir: An Inverse Source Problem in Time-Space...h1(t) =Ψ1(t)−Ψ0(t)+∞∑n=1z1n(t)D2αx Φn,α(12)+∞∑n=1w1n(t)D2αx Φn,α(12)+∞∑n=1y1n(t)D2αx Φn,α(12)(24)h2(t) =Ψ1(t)−Ψ0(t)+∞∑n=1z2n(t)D2αx Φn,α(12)+∞∑n=1w2n(t)D2αx Φn,α(12)+∞∑n=1y2n(t)D2αx Φn,α(12)(25)Therefore, the desired result is the difference of these twoformulas.Corollary 1.Let the conditions of Lemma hold. In addition< f1(x)− f2(x),Φn,α (x)>= 0 (26)∀ t ∈ (0,T], ∀n= 0,1,2, . . . holds, then h1(t) = h2(t) ∀ t ∈(0,T]SinceΦn,α(x), ∀ n= 0,1,2, . . . form a basis for the spaceand D2αx Φn,α(x) 6= 0 ∀ n = 0,1,2, . . . then f1(x) 6= f2(x)implies that< f1(x)− f2(x),Φn,α (x) > 6= 0 at least forsome n∈ N. Hence by Lemma1 we conclude thath1(t) 6= h2(t) which leads us to the followingconsequence:Φ[ f1] 6= Φ[ f2] =⇒ f1(x) 6= f2(x)Theorem 1.Assume thatΦ[·] : K −→ C1[0,T] is theinput-output mapping defined by (21) and correspondingto the measured output h(t) = D2αx u(12, t). 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Simul., Vol.24,No.1, pp.169-183 (2015).[21] Y.Rossikhin and M.Shitikova, Applications of fractionalcalculus to dynamic problems of linear and nonlinearhereditary mechanics of solids, Appl.Mech.Rev., Vol.50,No.1, pp.15-67 (1997).[22] H.Wei, W.Chen, H.Sun and X.Li, A coupled methodfor inverse source problem of spatial fractional anomalousdiffusion equations, Inv.Prblms.in Sci.Eng., Vol.18, No.7,pp.945-956 (2010).[23] T.Wei and Q.Zhang, Reconstruction of a time-dependentsource term in a time-fractional diffusion equation,Eng.Analy.Boundry.Elmts., Vol.37, No.1, pp.23-31 (2013).[24] Y.Zhang and X.Xiang, Inverse source problem for afractional diffusion equation, Inv.Problms.,Vol.27, No.3,035010 (2011).Ebru Ozbilge obtainedher Ph.D. degree in AppliedMathematics at KocaeliUniversity (Turkey) in 2006.She obtained M.Sc. degreein Mathematics from EasternMediterranean University(Cyprus) in 2002. She earnedher B.Sc. degree in AppliedMathematics and ComputerScience from Eastern Mediterranean University (Cyprus)in 2000. Her main research interests partial differentialequations, numerical analysis, fractional differentialequations and inverse problems. She had worked in IzmirUniversity of Economics (Turkey) between 2006 and2016. She has been working in American University ofthe Middle East (Kuwait) since August, 2016.Ali Demir obtained hisPh.D. degree in AppliedMathematics at the KocaeliUniversity in 2007. Heobtained his M.Sc. degreesin Mathematics fromUniversity of Illinoisat Urbana-Champaing(USA) in 1997. He earned hisB.Sc. degree in MathematicalEngineering from Istanbul Technical University in 1993.His main research interests dynamical systems andinverse problems. He has been working for KocaeliUniversity since 2003.c© 2018 NSPNatural Sciences Publishing Cor.

An Inverse Source Problem in Time-Space Fractional Differential Equations

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