In this paper, we formulate an initial-value problem of generalized anomalous diffusion equation consisting second order in two dimensional space and fractional order in time derivatives and then solve it by applying a series involving bilinear Eigen functions. Also, we evaluate a numerical approximation formula of this solution and also discuss its some special cases

Kumar, Hemant

Pathan, Mahmood A.

Yadav, Priyanka

Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)

LE MATEMATICHEVol. LXVII (2012) – Fasc. II, pp. 217–229doi: 10.4418/2012.67.2.16SERIES SOLUTIONS FOR INITIAL-VALUE PROBLEMS OFTIME FRACTIONAL GENERALIZED ANOMALOUSDIFFUSION EQUATIONSH. KUMAR - M. A. PATHAN - P. YADAVIn this paper, we formulate an initial-value problem of generalizedanomalous diffusion equation consisting second order in two dimensionalspace and fractional order in time derivatives and then solve it by applyinga series involving bilinear Eigen functions. Also, we evaluate a numericalapproximation formula of this solution and also discuss its some specialcases.1. IntroductionOldham and Spanier [14] presented a general solution of the diffusion equationfor semi-infinite geometries. Oldham [13] again discussed diffusive transportto planar, cylindrical and spherical electrodes through a general equation con-sisting first order in one dimensional space and half order in time derivative andsolved it through Laplace transform method.Oldham and Spanier [15] also obtained various applications of the frac-tional calculus to the problems whose formulations and solutions are normallycouched in terms of integrals or derivatives alone.Entrato in redazione: 26 marzo 2012AMS 2010 Subject Classification: 26A33, 35R11, 60J60, 47B06, 42B05, 33E12, 34K28.Keywords: Initial value problem, Generalized anomalous diffusion equation, Riemann-Liouvillefractional order derivative, Caputo derivative, Bilinear Eigen functions, Numerical approximationformula.218 HEMANT KUMAR - MAHMOOD A. PATHAN - PRIYANKA YADAVThe anomalous diffusion is a phenomenon that occurs in complex and nonhomogeneous mediums. The phenomenon of anomalous diffusion may be basedon generalized diffusion equation which contains fractional order space and / ortime derivatives (see Metzler and Klafter [12]).Turski et al [25] presented the recurrence of anomalous diffusion from thephysical point of view and also described the importance of fractional deriva-tives in space and / or time to diffusion propagation. Agarwal [1] presented ananalytical technique by applying Eigen functions to solve a fractional diffusion-wave system. Again, Agarwal [1] also formulated a general solution by usingfinite sine transform techniques for a fractional diffusion-wave equation in abounded domain.In our investigations, we make an appeal to the following definitions: (seeKilbas et al. [9], Podlubny [16] , Samko et al. [17] and Diethelm [3]).Let n ∈R (set of real numbers). The operator Jna . defined on L1[a,b] := { f :[a,b]→ R; f is measurable on [a,b] and ∫ ba f (x)dx < ∞} byJna f (x) :=1Γ(n)∫ xa(x− t)n−1 f (t)dt (1)for a ≤ x ≤ b, is called the Riemann-Liouville fractional integral operator oforder n.Let n ∈ R, and m = [n]. The operator Dna. is defined byDna f := DmJm−na f ,(D f (x) :=ddxf (x) , Dn f (x) := DDn−1 f (x) :=ddxdn−1dxn−1f (x)), (2)and is called the Riemann- Liouville fractional differential operator of order n,again for n = 0, D0a f = I f = f .For n≥ 0 and f is such that Dna [ f −Tm−1 [ f : a]] exists; where m = [n], thenthere is a function Dn∗a f defined byDn∗a f := Dna [ f −Tm−1 [ f : a]] := Jm−na Dm f (3)The operator Dn∗a is called the Caputo differential operator of order n. Asusual Tm−1 [ f : a] denotes the Taylor polynomial of degree m−1 for the functionf , centered at a; in the case m = 0, Tm−1 [ f : a] := 0 and that isTm−1 [ f : a] (x) =m−1∑k=0f (k) (a)k!(x−a)k (4)Here n denotes the order of the Caputo-type differential operator. We shallonly consider the case n > 0 and n /∈ N (the set of natural numbers) and we useSERIES SOLUTIONS FOR INITIAL-VALUE PROBLEMS 219the notation m = [n] to denote the smallest integer greater than (or equal to) n.For n ∈ N, the Caputo differential operator coincides with the usual differentialoperator of integer order, and for n< 0, the Caputo differential of negative ordercan be interpreted as the Riemann-Liouville differential operator of the sameorder. This Caputo-type differential operator has certain advantages when wetry to model real-world phenomena with fractional differential equations (seealso Podlubny [16] or Diethelm [3]).For t > a, the Caputo time fractional derivative is defined asCt Dβ∗a f (t) :=1Γ(n−β )∫ ta(t− τ)n−β−1(ddτ)nf (τ)dτ (5)where 0 < β < n, n ∈ Z+ (the set of positive integers).The Eigen functions ψn (x) corresponding to the Eigen-values λn satisfy fol-lowing Eigen-value problem (see Hinchey [6, p.541])Dx [ψn (x)]+ iλnψn (x) = 0,(Dx ≡ ddx), x, λn ∈ R (n = 0,1,2,3, . . . .) (6)2. Series of Bilinear Eigen FunctionsWe define a series of bilinear Eigen functions which is a three variable functionF(x,y, t) possesses a formal power series containing bilinear Eigen functions inthe formF (x,y, t) =∞∑n=0fn (t)ψn (x)ψn (y), x,y ∈ R, t > 0. (7)Particularly, if in (7) we put fn (t) = γntn,∀n = 0,1,2,3, . . . ., {γn} is inde-pendent of x,y, t, then F (x,y, t) becomes a bilinear generating function for theset {ψn}∞n=0 (see Srivastava and Manocha [21]).Here, in our work we present an initial-value problem of generalized anoma-lous diffusion equation involving second order in two dimensional space andfractional order in time derivatives which is taken in terms of Caputo derivativeand then solve it on making an application of a three variable function F(x,y, t)given in (7). We find a numerical approximation formula and also discuss someof its special cases.220 HEMANT KUMAR - MAHMOOD A. PATHAN - PRIYANKA YADAV3. Main ProblemIn this section, we formulate following generalized anomalous diffusion prob-lem1kCt Dα∗0u(x,y, t) = D2xu(x,y, t)+D2yu(x,y, t) ,k 6= 0, 0 < α ≤ 1, a, a′,b, b′ ∈ R, t > 0,x ∈ (a,b) and y ∈ (a′,b′) (8)u(x,y, t) = u0 (x,y) , at t = 0 (9)andlimx,y→±∞u(x,y, t) = 0 (10)Particularly, when set α = 1 in the problem (8)-(10), we get the well-knowntwo dimensional diffusion problem of Sneddon [18, p. 305, Misc. Prob. 14].When put α = 1 in the problem (8)-(10) and set u(x,y, t) as independent ofy, we get the well-known one dimensional diffusion problem of Kumar [10].When α is arbitrary and we consider u(x,y, t) as independent of y in theproblem (8)-(10) then it becomes time-fractional diffusion problem of Mathai,Saxena and Haubold [11, p. 175, Theorem 6.3]Before solving above problem (8)-(10), we present the following theorem:Theorem 3.1 ( A). If the series of bilinear Eigen functionsu(x,y, t) =∞∑n=0un (t)ψn (x)ψn (y)satisfies the generalized anomalous diffusion problem ((8)-(10)), x ∈ (a,b) andy ∈ (a′,b′),a,a′,b,b′ ∈ R, t > 0 and , ψn (x) satisfy the Eigen value problem (6)for non-zero Eigen values λn,(∀n = 0,1,2,3, . . . .), then there exists a sequenceof integrals and given byun (t) = u0nEα,1(−2kλ 2n tα) ,(n = 0,1,2,3, . . . .) (11)Here, 0 < α ≤ 1 and Eα,1 (.) is the well-known Mittag-Leffler function (seeSrivastava and Manocha [21], Diethelm [3]) andu0n = 1[∫ ba |ψn (x)|2dx][∫ b′a′ |ψn (y)|2dy]∫ ba∫ b′a′u0 (x,y)ψn (x)ψn (y)dxdy(n = 0,1,2,3, . . .)SERIES SOLUTIONS FOR INITIAL-VALUE PROBLEMS 221Proof. Set u(x,y, t) =∞∑n=0un (t)ψn (x)ψn (y) into equation (8) to get1k∞∑n=0Ct Dα∗0un (t)ψn (x)ψn (y)−∞∑n=0un (t)(ψn (y)∂ 2∂x2ψn (x)+ψn (x)∂ 2∂y2ψn (y))= 0 (12)Now making an appeal to (6) and (12), we find that∞∑n=0(1kCt Dα∗0un (t)+2λ2n un (t))ψn (x)ψn (y) = 0 (13)Then the equation (13) gives us1kCt Dα∗0un (t)+2λ2n un (t) = 0(n = 0,1,2,3, . . . .) (14)On taking Laplace transform of (14), and making an appeal to the techniquesdue to Huang and Liu [7], we getL{un (t)}= sα−1un (0)sα +2kλ 2n(n = 0,1,2,3, . . . .) (15)Again on making an appeal to the theorem 4.5 of Diethelm [3, p.72] andequating to the equation (15), we getun (t) = un (0)Eα,1(−2kλ 2n tα)(n = 0,1,2,3, . . . .) (16)where Eα,1 (.) is the well-known Mittag-Leffler function (see Srivastava andManocha [21], Diethelm [3]).Further at t = 0, the series of bilinear Eigen functionsu(x,y, t) =∞∑n=0un (t)ψn (x)ψn (y)gives usu0 (x,y) =∞∑n=0u0nψn (x)ψn (y) (17)Then multiply ψm(x)[∫ ba |ψm(x)|2dx]1/2 .ψm(y)[∫ b′a′ |ψm(y)|2dy]1/2 in both sides of (17) and thenintegrate to that sides with respect to x from a to b and again with respect to yfrom a′ to b′ to get1[∫ ba |ψm (x)|2dx]1/2[∫ b′a′ |ψm (y)|2dy]1/2 ∫ ba∫ b′a′u0 (x,y)ψm (x)ψm (y)dxdy222 HEMANT KUMAR - MAHMOOD A. PATHAN - PRIYANKA YADAV=∞∑n=0u0n∫ ba ψn (x)ψm (x)dx[∫ ba |ψm (x)|2dx]1/2 .∫ b′a′ ψn (y)ψm (y)dy[∫ b′a′ |ψm (y)|2dy]1/2 (18)Now in the right hand side of (18) apply the orthogonal property to find thatu0n =1[∫ ba |ψn (x)|2dx][∫ b′a′ |ψn (y)|2dy] ∫ ba∫ b′a′u0 (x,y)ψn (x)ψn (y)dxdy(n = 0,1,2,3, . . . .) (19)After making some manipulations, we get un (0) = u0n , therefore with thehelp of (16) and (19), we evaluate (11).Solution of the problem stated in the equations (8)-(10)On making an appeal to the theorem A, the solution of the problem statedin the equations (8)-(10) is then evaluated asu(x,y, t) =∞∑n=0un (t)ψn (x)ψn (y) (20)Hereun (t) = u0nEα,1(−2kλ 2n tα) ,(n = 0,1,2,3, . . . .) (21)andu0n = 1[∫ ba |ψn (x)|2dx][∫ b′a′ |ψn (y)|2dy]∫ ba∫ b′a′u0 (x,y)ψn (x)ψn (y)dxdy(n = 0,1,2,3, . . . .) (22)4. Numerical Approximation FormulaIn this section, we present the numerical solution of the problem ((8)-(10)) byapplying Gru¨nwald approximation formula [5] for fractional order derivative(also see Oldham and Spainier [15]) given byDα0 f (x) = limN→∞{[ xN]−α N−1∑j=0Γ( j−α)Γ(−α)Γ( j+1) f(x− jNx)},N ∈ N, N−1 < α < N (23)SERIES SOLUTIONS FOR INITIAL-VALUE PROBLEMS 223Now make an appeal to the equation (23) and the formulae given in theequations (3) and (4), we get the approximation formula for Caputo derivativein the formCt Dα∗0 f (t) = limN→∞{[ xN]−α N−1∑j=0Γ( j−α)Γ(−α)Γ( j+1) f(t− jNt)}−N−1∑k=0f (k) (0)k!tk−αΓ(k−α+1) (24)Then make an appeal to (14) and (24) to find the numerical approximationof the problem ((2.1)-(10))un (t) =(− 12kλ 2n){[Nt]α N−1∑j=0Γ( j−α)Γ(−α)Γ( j+1)un(t− jNt)−N−1∑k=0un(k) (0)k!tk−αΓ(k−α+1)},N is large, t > 0,λn 6= 0, and k 6= 0(n = 0,1,2,3, . . . .) (25)Particularly, for 0 < α < 1 from the equation (25) and the formula due toOldham and Spainier [15, eq. (3.4.5)], we get the numerical approximationformulaun (t) =(12kλ 2n){un (0)t−αΓ(1−α)− limN→∞[ tN]−1 1∑j=0Γ( j−α)Γ(−α)Γ( j+1)un(t− jNt)},t > 0, λn 6= 0, and k 6= 0 (n = 0,1,2,3, . . . .) . (26)5. Special CasesSet ψn (x) = einx, a, a′ =−pi and b,b′ = pi in (20)-(22) to getu(x,y, t) =∞∑n=0un (t)einxeiny (27)Hereun (t) = Eα,1(−2kλ 2n tα)(1[2pi]2)∫ pi−pi∫ pi−piu0 (x,y)e−in(x+y)dxdy,t > 0, (n = 0,1,2,3, . . . .) . (28)224 HEMANT KUMAR - MAHMOOD A. PATHAN - PRIYANKA YADAVAgain, set ψn (x) = e−inx,a,a′ = 0 and b,b′ = pi in (20)-(22) to getu(x,y, t) =∞∑n=0un (t)e−inxe−iny (29)Hereun (t) = Eα,1(−2kλ 2n tα)(1[pi]2)∫ pi0∫ pi0u0 (x,y)ein(x+y)dxdy,t > 0, (n = 0,1,2,3, . . . .) . (30)Now setu0 (x,y) = (sinx)w1−1(siny)w2−1F E : F ;F′G : H;H ′[(e) : ( f );( f ′);β1(sinx)2ρ1(g) : (h);(h′);β2(siny)2ρ2]×H 0,λ : (µ′,ν ′); . . . ;(µ(r),ν(r))A,C : (B′,D′); . . . ;(B(r),D(r)) [(a) : ξ ′, . . . ,ξ (r)] : [(b′) : φ ′]; . . . ; [(b(r)) : φ (r)];[(c) : η ′, . . . ,η(r)] : [(d′) : δ ′]; . . . ; [(d(r)) : δ (r)];z1(sinx)2σ1(siny)2γ1...zr(sinx)2σr(siny)2γr(31)where,FE : F ;F ′G : H;H ′ [.]is a Kampe´ de Fe´riet function [8] (see also Exton [4], Srivastava and Manocha[21] and Srivastava and Karlsson [20]) andH0,λ : (µ ′,ν ′) ; . . . ;(µ(r),ν(r))A,C : (B′,D′) ; . . . ;(B(r),D(r)) [.]is a multivariable H-function of Srivastava and Panda [22, 23, 24] (see alsoSrivastava, Gupta and Goyal [19] and Mathai, Saxena and Haubold [11]).Then use (30) in (29) and apply the techniques of Chandel, Agarwal andSERIES SOLUTIONS FOR INITIAL-VALUE PROBLEMS 225Kumar [2] to getun (t) =(einpi/22w1+w2−2){Eα,1(−2kλ 2n tα)}∞∑p, p′=0∏Eq=1 (eq)p+p′∏Fq=1 ( fq)p∏Eq=1(f ′q)p′∏Gq=1 (gq)p+p′∏Hq=1 (hq)p∏H ′q=1 (h′q)p′× (β1/4ρ1)pp!(β2/4ρ2)p′p′!H0,λ +2 : (µ ′,ν ′) ; . . . ;(µ(r),ν(r))A+2,C+4 : (B′,D′) ; . . . ;(B(r),D(r)) [[(a) : ξ ′, . . . ,ξ (r)],d1−w1−2pρ1 : 2σ1, . . . ,2σre ,[(c) : η ′, . . . ,η(r)],[12 − w1+2pρ1±n2 : σ1, . . . ,σr],[1−w2−2p′ρ2 : 2γ1, . . . ,2γr] : [(b′) : φ ′] ; . . . ;[(b(r)): φ (r)];[12 − w2+2p′ρ2±n2 : γ1, . . . ,γr]: [(d′) : δ ′] ; . . . ;[(d(r)): δ (r)];z14σ1+γ1...zr4σr+γr],t > 0, (n = 0,1,2,3, . . . .) . (32)provided that∣∣∣ zε4σε+γε∣∣∣< pi24ε ,∀ε = 1,2, . . . ,r and4ε =λ∑J=1ξ (ε)J −A∑J=λ+1ξ (ε)J +ν(ε)∑J=1φ (ε)J−B(ε)∑J=ν(ε)+1φ (ε)J −∑CJ=1η(ε)J +∑µ(ε)J=1 δ(ε)J −∑D(ε)J=µ(ε)+1 δ(ε)J > 0 (33)On making some manipulations in parameters of multivariable H-functionof (30) and (31), we may obtain un (t) in terms of a double series involving manyhypergeometric functions cited in the literature of generalized special functions.6. ConclusionsIf we set ψn (x) = sin(npixb), ψn (y) = sin(npiyb′)and α = 1 , a = 0, and a’ = 0 in(20)-(22) we get the solutionu(x,y, t) =∞∑n=1u0nexp(−2kλ 2n t)sin(npixb )sin(npiyb′ ) (34)Hereu0n =(4bb′)∫ b0∫ b′0u0 (x,y)sin(npixb)sin(npiyb′)dxdy (n = 0,1,2,3, . . . .)226 HEMANT KUMAR - MAHMOOD A. PATHAN - PRIYANKA YADAVThis solution is equivalent to the problem of Sneddon [18, p. 305, Misc.Prob. 14] forλ 2n =12n2pi2(1b2+1b′2),(n = 0,1,2,3, . . . .) (35)Again, set y → 0, w2 = 1,α = 1, β1 → 0, β2 → 0, γ1 = 0, . . . , γr = 0 in(29)-(32) to get solution of one dimensional diffusion equation (see Kumar [10])in x-direction of the formu(x, t) =∞∑n=0un (t)e−inx (36)whereun (t) =(einpi/22w1−1){exp(−2kλ 2n t)}×H 0,λ +1 : (µ′,ν ′) ; . . . ;(µ(r),ν(r))A+1,C+2 : (B′,D′) ; . . . ;(B(r),D(r)) [ [(a) : ξ ′, . . . ,ξ (r)] ,[(c) : η ′, . . . ,η(r)],d1−w1 : 2σ1, . . . ,2σre :[12 − w1±n2 : σ1, . . . ,σr]:[(b′) : φ ′] ; . . . ;[(b(r)): φ (r)];[(d′) : δ ′] ; . . . ;[(d(r)): δ (r)];z14σ1...zr4σr , t > 0,(n = 0,1,2,3, . . . .) (37)provided that the conditions due to (33) are satisfied.[5]Again, set x→ 0, w1 = 1,α = 1, β1→ 0, β2→ 0, σ1 = 0, . . . , σr = 0 in(29)-(32), we get solution of one dimensional diffusion equation in y-directionof the formu(y, t) =∞∑n=0un (t)e−iny (38)SERIES SOLUTIONS FOR INITIAL-VALUE PROBLEMS 227whereun (t) =(einpi/22w2−1){exp(−2kλ 2n t)}×H 0,λ +1 : (µ′,ν ′) ; . . . ;(µ(r),ν(r))A+1,C+2 : (B′,D′) ; . . . ;(B(r),D(r)) [ [(a) : ξ ′, . . . ,ξ (r)] ,[(c) : η ′, . . . ,η(r)],d1−w2 : 2γ1, . . . ,2γre :[12 − w2±n2 : γ1, . . . ,γr]:[(b′) : φ ′] ; . . . ;[(b(r)): φ (r)];[(d′) : δ ′] ; . . . ;[(d(r)): δ (r)];z14γ1...zr4γr , t > 0,(n = 0,1,2,3, . . . .) (39)provided that the conditions due to (33) are satisfied.REFERENCES[1] O. P. Agarwal, Solution for a fractional diffusion–wave equation defined in abounded domain, Nonlinear Dynamics 29 (2002), 145–155.[2] R. C. Singh Chandel - R. D. Agarwal - H. Kumar, An integral involving sine func-tions, exponential functions, the Kampe´ de Fe´riet function and the multivariableH-function of Srivastava and Panda, and its applications in a potential problemon a circular disk, PAMS, XXXV, No. 1-2 (1992), 59–69.[3] Kai Diethelm, The Analysis of Fractional Differential Equations: An Ap-plication-Oriented Exposition Using Differential Operators of Caputo Type,Springer, New York, 2010.[4] H. Exton, Multiple Hypergeometric Functions and Applications, John Wiley andSons, New York, 1976.[5] A. K. Gru¨nwald, Uber ‘begrenzte’ Derivationen und deren Anwendung, Z.Angew. Math. Phys. 12 (1867), 441–480.[6] F. A. Hinchey, Introduction to Applicable Mathematics, Part II, Wiley EasternLimited, New Delhi, 1984.[7] F. Huang - F. Liu, The fundamental solution of the space-time fractionaladvective-dispersion equation, J. Appl. Math. & Computing 18 (1-8) (2005),339–350.[8] J. Kampe´ de Fe´riet, Les Functions Hypergeometriques d’Ordre Supe´rieur a` DeuxVariables, C. F. Acad. Sci. Paris 173 (1921), 401–404.[9] A. A. Kilbas - H. M. Srivastava - J. J. Trujillo, Theory and Applications of Frac-tional Differential Equations, Elsevier B. V., the Netherlands, 2006.228 HEMANT KUMAR - MAHMOOD A. PATHAN - PRIYANKA YADAV[10] H. Kumar, Special Functions and Their Applications in Modern Science andTechnolgy, Ph. D. Thesis, Barkatullah University Bhopal (M. P.) India, 1993.[11] A. M. Mathai - R. K. Saxena - H. J. Haubold, The H-function: Theory and Appli-cations, Springer, New York, 2010.[12] R. Metzler - J. Klafter, The random walk’s guide to anomalous diffusion: Afractional dynamic approach, Physics Reports, 339 (2000), 1–77.[13] K. B. Oldham, Diffusive transport to planar, cylindrical and spherical elec-trodes, J. Electroanal, Chem. Interfacial Electrochem. 41 (1973), 351–358.[14] K. B. Oldham - J. Spanier, A general solution of the diffusion equation for semi-infinite geometries, J. Math. Anal. Appl. 39 (1972), 655–669.[15] K. B. Oldham - J. Spanier, The Fractional Calculus: Theory and Applications ofDifferentiation and Integration to Arbitrary Order, Dover Publication, Inc. NewYork, 2006.[16] I. Podlubny, Fractional Differential Equations, Academic Press New York, 1999.[17] S. G. Samko - A. A. Kilbas - O. I. Marichev, Fractional Integrals and Deriva-tives: Theory and Applications, Gordon and Breach, Longhorne Pennsylvania,1993.[18] I. N. Sneddon, Elements of Partial Differential Equations, McGraw-Hill BookCompany Inc., New York, 1957.[19] H. M. Srivastava - K. C. Gupta - S. P. Goyal, The H-function of One and TwoVariables with Applications, South Asian Publishers, New Delhi, 1982.[20] H. M. Srivastava - P. W. Karlsson, Multiple Gaussian Hypergeometric Series,Halsted Press, John Wiley and Sons, New York, 1985.[21] H. M. Srivastava - H. L. Manocha, A Treatise on Generating Functions, HalstedPress, John Wiley and Sons, New York, 1984.[22] H. M. Srivastava - R. Panda, Some expansion theorems and generating relationsfor the H-function of several complex variables, Comment. Math. Univ. St. Paul.24 (2) (1975), 119–137.[23] H. M. Srivastava - R. Panda, Some bilateral generating for a class of generalizedhypergeometric polynomials, J. Reine Angew. Math. 283/284 (1976), 265–274.[24] H. M. Srivastava - R. Panda, Expansion theorems for the H-function of severalcomplex variables, J. Reine Angew. Math. 288 (1976), 129–145.[25] A. J. Turski - T. B. Atamaniuk - E. Turska, Application of fractional deriva-tive operators to anomalous diffusion and propagation problems, arXiv: math-ph/0701068v2, 2007.SERIES SOLUTIONS FOR INITIAL-VALUE PROBLEMS 229HEMANT KUMARDepartment of MathematicsD. A-V. (P.G.) CollegeKanpur - 208 001 Indiae-mail: palhemant2007@rediffmail.comMAHMOOD A. PATHANCentre for Mathematical SciencesPala Campus, Arunapuram P.O.Pala-686574, Kerala Indiae-mail: mapathan@gmail.comPRIYANKA YADAVDepartment of MathematicsD. A-V. (P.G.) CollegeKanpur - 208 001 Indiae-mail: yadpriyanka9@yahoo.in

Series splutions for initial-value problems of time fractional generalized anomalous diffusion equations

In this paper, we formulate an initial-value problem of generalized anomalous diffusion equation consisting second order in two dimensional space and fractional order in time derivatives and then solve it by applying a series involving bilinear Eigen functions. Also, we evaluate a numerical approximation formula of this solution and also discuss its some special cases.<br /

Hemant Kumar

Mahmood A. Pathan

Priyanka Yadav

Directory of Open Access Journals

http://doaj.org/search?source=%7B%22query%22%3A%7B%22bool%22%3A%7B%22must%22%3A%5B%7B%22term%22%3A%7B%22id%22%3A%2237aa09d06cfd497f9cad6c95f0f2d949%22%7D%7D%5D%7D%7D%7D

Series splutions for initial-value problems of time fractional generalized anomalous diffusion equations

Abstract

Similar works

Full text

Available Versions

Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)

Directory of Open Access Journals