AA Kilbas

AM Mathai

F Huang

F Mainardi

H Srivastava

HH Pennes

HJ Haubold

I Podlubny

L Debnath

MN Ozisik

RK Saxena

TAM Langlands

TO Salim

WJ Minkowycz

Y Shang

SpringerPlus

English

R. S. Damor

Sushil Kumar

A. K. Shukla

Crossref

Solution of fractional bioheat equation in terms of Fox’s H‑functionR. S. Damor, Sushil Kumar and A. K. Shukla*BackgroundThe transfer of heat in skin tissue is mainly a heat conduction process, which is cou-pled to several additional complicated physiological process, including blood circula-tion, sweating, metabolic heat generation and sometimes heat dissipation via hair or fur above the skin surface (Ozisik 1985). Accurate description of the thermal interaction between vasculature and tissue is essential for the advancement of medical technology in treating fatal disease such as tumors and skin cancer. Mathematical model has been used significantly in the analysis of hyperthermia in treating tumors, cryosurgery, fatal-placental studies, and many other applications (Minkowycz et al. 2009).Fractals and fractional calculus have been used to improve the modelling accuracy of many phenomena in natural science. The most important advantage of using fractional calculus approach is due to its non-local property. This means that the next state of a system depends not only upon its current state but also upon all of its historical states. Many researchers worked on fractional partial differential equations and gave very important results. Mainardi et al. (2005) obtained the fundamental solution of Cauchy problem for the space–time fractional diffusion equation in terms of H-function, Lang-lands (2006) gave the solution of a modified fractional diffusion equation on an infinite domain, Salim and El-Kahlout (2009) discussed exact solution of time fractional advec-tion dispersion equation with reaction term, Saxena et  al. (2006) obtained solution of generalized fractional kinetic equation in terms of Mittag-Leffler function, Haubold et al. (2011a) obtained solution of a fractional reaction diffusion equation in closed form, Huang and Guo (2010) gave the fundamental solutions to a class of the time fractional Abstract Present paper deals with the solution of time and space fractional Pennes bioheat equation. We consider time fractional derivative and space fractional derivative in the form of Caputo fractional derivative of order α ∈ (0, 1] and Riesz–Feller fractional derivative of order β ∈ (1, 2] respectively. We obtain solution in terms of Fox’s H-func-tion with some special cases, by using Fourier–Laplace transforms.Keywords: Fractional bioheat equation, Caputo derivative, Riesz–Feller derivative, Fox’s H-functionMathematics Subject Classification: 26A33, 35R11, 80A20Open Access© 2016 Damor et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.RESEARCHDamor et al. SpringerPlus  (2016) 5:111 DOI 10.1186/s40064‑016‑1743‑2*Correspondence:   aks@ashd.svnit.ac.in; ajayshukla2@rediffmail.com Department of Applied Mathematics and Humanities, S. V. National Institute of Technology, Surat 395 007, IndiaPage 2 of 10Damor et al. SpringerPlus  (2016) 5:111 partial differential equation for Cauchy problem in a whole-space domain and signalling problem in a half-space domain. Shang (2015) gave analytic solution of viral infection dynamics in vivo through a time-inhomogeneous Markov chain characterization by Lie algebraic approach.In present study, we consider fractional form of Pennes bioheat equation by replacing first order time derivative by Caputo fractional derivative of order α ∈ (0, 1] and second order space derivative by Riesz–Feller fractional derivative of order β ∈ (1, 2] respec-tively. We make an attempt to solve the fractional model by dividing it into two sections. In section one, time fractional derivative is considered while in section two, space frac-tional derivative is taken into account. We apply Laplace–Fourier transform and obtain the solution in term of Fox H-function.Preliminaries and notationsFractional derivative of order α is denoted as aDαt f (t), the subscripts a and t denote the two limits related to the operation of fractional differentiation, which are called the ter-minal of fractional differentiation. If α is negative then it denotes the fractional integrals of arbitrary order.Definition 1 (Kilbas et al. 2006) The Riemann–Liouville fractional derivative of order α > 0 for Real(α) > 0 and m ∈ N , t > a is defined asDefinition 2 (Kilbas et al. 2006) The Caputo fractional derivative of order α > 0, for Real(α) > 0 and m ∈ N , t > a is defined asDefinition 3 (Kilbas et  al. 2010) The Laplace transform of function f(t) denoted by F(s), s being the complex variable is defined asInverse Laplace transform of F(s) is defined asLaplace transform of Caputo derivative (Podlubny 1999) is given as(1)aDαt f (t) ={dmdtm[1Ŵ(m−α)∫ taf (τ )(t−τ)α+1−mdτ], for m− 1 < α < mdm(f (t))dtm, for α = m.(2)caDαt (f (t)) ={ [1Ŵ(m−α)∫ taf m(τ )(t−τ)α+1−mdτ], for m− 1 < α < mdm(f (t))dtm, for α = m.(3)F(s) = f (s) = L{f (t); s} =∫∞0e−st f (t)dt; t ∈ R+(4)f (t) = L−1{F(s); t} =12π i∫ γ+i∞γ−i∞estF(s)ds; t ∈ R+, γ = R(s).(5)L[c0Dαt f (t)]= sαF(s)−n−1∑k=0sα−k−1f k(0), n− 1 < α ≤ n.Page 3 of 10Damor et al. SpringerPlus  (2016) 5:111 Definition 4 (Debnath and Bhatta 2007) The Fourier transform of a function f(x) is denoted by F∗(k) and defined aswhere, f(x) is continuous and absolutely integrable in (−∞,∞). Inverse Fourier trans-form is defined asFourier cosine transform and Inverse Fourier cosine transform are defined as (Debnath and Bhatta 2007)The Fourier convolution of two functions is defined (Debnath and Bhatta 2007) asDefinition 5 (Kilbas et al. 2010) The Mellin transform of a function φ(t) is defined asand inverse Mellin transform asDefinition 6 (Kilbas et al. 2006) Riesz–Feller partial fractional derivative (Dβθ;xu)(x, t) defined, for 0 < β ≤ 2 and |θ | ≤ min [β , 2− β] via the Fourier transform, byDefinition 7 (Podlubny 1999) Mittag-Leffler function for one parameter is denoted by Eα(z) and defined as(6)F(f (x)) = F∗(k) =∫∞−∞e−ikxf (x)dx,(7)F−1(F∗(k)) =12π∫∞−∞eikxF∗(k)dk .(8)Fc(f (x)) = F∗c (k) =∫∞0cos(kx)f (x)dx,(9)F−1c (F∗c (k)) = f (x) =2π∫∞0cos(kx)Fc(k)dx.(10)h ∗ φ = (h ∗ φ)(x) =12π∫∞−∞h(x − t)φ(t)dt.(11)(Mφ)(p) = M[φ(t)](p) = φ∗(s) =∫∞0ts−1φ(t)dt; t ∈ R+ = (0,∞), s ∈ C .(12)(M−1g)(x) = M−1[g(s)](x) =12π i∫ γ+i∞γ−i∞x−sg(s)ds; x ∈ R+, γ = R(s).(13)(Fx Dβθ;xu)(σ , t) = |σ |βei(sign(σ ) θπ2)(Fx u)(σ , t); σ ∈ R; t > 0.(14)Eα(z) =∞∑k=0zkŴ(αk + 1), α > 0.Page 4 of 10Damor et al. SpringerPlus  (2016) 5:111 Mittag-Leffler function for two parameter is denoted by Eα ,β (z) and defined asPodlubny (1999) reported the Laplace transform of a derivative of Mittag Leffler func-tion asand inverse Laplace transform of (16) is also existing asDefinition 8 (Mathai et al. 2010) The H- function is defined by means of a Mellin–Barners type integral in the following mannerwhere,and an empty product is interpreted as unity, m, n, p, q ∈ N0 with 0 ≤ n ≤ p, 0 ≤ m ≤ q, Aj ,Bj ∈ R+, aj , bj ∈ C , j = 1, . . . , p; j = 1, . . . , q, such that Aj(bl + k) �= Bl(aj − − 1), k ,  ∈ N0; j= 1, . . . , n; j = 1, . . . ,m. where N0 = 0, 1, 2, . . . ,R = (−∞,∞), R+ =(0,∞) and C being the complex number field. The counter  is infinite contour which sep-arates all the poles of Ŵ(1− aj + sAl), j = 1, . . . , n.The relation between derivative of Mittag-Leffler function and H-Function (Langlands 2006) is given asFractional bioheat equationThe Pennes bioheat model (Pennes 1948) is widely used for study of the heat transfer in skin tissue due to its simplicity, ease application and effectiveness. Pennes (1948) suggested that the rate of heat transfer between blood and tissue is proportional to the product of the volumetric perfusion rate and difference between the arterial blood tem-perature and the local tissue temperature. Pennes equation is employed to describe the heat transfer process as(15)Eα ,β (z) =∞∑k=0zkŴ(αk + β), α > 0,β > 0.(16)L(tαn+β−1Enα ,β (±atα)) =sα−βŴ(n+ 1)(sα ∓ a)n+1,(17)L−1(sα−γ(sα ∓ a)n+1)=1Ŵ(n+ 1)tαn+γ−1Enα ,γ (±atα).H(z) = Hm,np,q(z∣∣∣(ap ,Ap)(bq ,Bq))= Hm,np,q(z∣∣∣(a1,A1),...,(ap ,Ap)(b1,B1),...,(bq ,Bq))=12π i∫L�(s)z−sds,(18)i =√−1, z �= 0, z−s = exp(−s log |z| + i argz),�(s) =∏mj=1 Ŵ(bj + Bjs)∏nj=1 Ŵ(1− aj − Ajs)∏qj=m+1 Ŵ(1− bj − Bjs)∏pj=n+1 Ŵ(aj + Ajs),(19)Emα,β(η) = H1,11,2[−η|(−m,1)(0,1),(1−αm−β ,α)].Page 5 of 10Damor et al. SpringerPlus  (2016) 5:111 where ρ, c and k represent density, specific heat and thermal conductivity respectively. T, t and x represent, temperature, time and distance respectively; the subscript b denotes for blood. Ta and Wb are artillery temperature and blood perfusion rate respectively. Qmet and Qext are metabolic heat generation and external heat source in skin tissue respectively.Considering Eq.  (20) and replacing first order time derivative by Caputo fractional derivative of order α ∈ (0, 1] and second order space derivative by Riesz–Feller fractional derivative of order β ∈ (1, 2]. The fractional form of Pennes bioheat equation is given asInitial and boundary conditions of Eq. (21) are given below asTo find solution of Eq. (21) to (23). We divide it into two parts, in first part, the time frac-tional is considered while in second part, the space fractional is considered.Time fractional bioheat equationOn setting β = 2 in Eq. (21), this reduces to the following equationFor converting Eq. (24) in dimensionless variable, we considerOn making the use of dimensionless variables as Eq.  (25). We can reduce Eqs.  (22) to (24) into Eq. (26) to (28) asSolutionOn replacing fractional derivative by Caputo fractional derivative and taking the Fou-rier–Laplace transform (Haubold et al. 2011b) of Eq. (26) and further using Eq. (5) and Eq. (6), we get(20)ρc∂T∂t= k∂2T∂x2+Wbcb(Ta − T )+ Qmet + Qext ,(21)ρc∂αT∂tα= k∂βT∂xβ+Wbcb(Ta − T )+ Qmet .(22)T (x, 0) = Ta,(23)T (x, t)|x→±∞ = 0.(24)ρc∂αT∂tα= k∂2T∂x2+Wbcb(Ta − T )+ Qmet .(25)ζ =(Wbcbk) 12x, η =(Wbcbρc) 1αt, θ =T − TaT0, φ =QmetT0Wbcb.(26)∂αθ∂ηα+ θ =∂2θ∂ζ 2+ φ,(27)θ(ζ , 0) =0,(28)θ(ζ , η)|ζ→±∞ =0.Page 6 of 10Damor et al. SpringerPlus  (2016) 5:111 where θ∗ is Fourier Laplace transform of θ, on further simplifications, we getOn taking inverse Laplace transform of Eq. (29) and using Eq. (17), we getOn replacing derivative of Mittag-Leffler function by H-function using Eq. (19), Eq. (30) reduces to the following formNow we considerto invert Fourier transform of Eq. (31), we first invert Fourier transform of Eq. (32), by taking its Mellin transform, since Mellin transform of H-function (Srivastava et al. 1982) is given below aswith the following conditionsandFurther Mellin transform of a Fourier transform is given as (Langlands 2006)sαθ∗(ω, s)− θ(ζ , 0)+ θ∗(ω, s) = −ω2θ∗(ω, s)+ φ2πδ(ω)1s,(29)θ∗(ω, s) = 2πφδ(ω)11+ ω2(1s)− 2πφδ(ω)11+ ω2∞∑m=0(−1)msα−1(sα + ω2)m+1.(30)θ∗(ω, η) = 2πφδ(ω)11+ ω2− 2πφδ(ω)11+ ω2∞∑m=0(−1)mηαmm!Emα (−ω2ηα).(31)θ∗(ω, η) = 2πφδ(ω)11+ ω2− 2πφδ(ω)11+ ω2∞∑m=0(−1)mηαmm!H1,11,2[ω2ηα|(−m,1)(0,1),(−αm,α)].(32)G∗m(ω, η) = H1,11,2[ω2ηα|(−m,1)(0,1),(−αm,α)],(33)M{Hm,np,q[ax|(ap ,Ap)(bq ,Bq)]}(z) = a−z{∏mj=1 Ŵ(bj + sBj)}{∏qj=m+1 Ŵ(1− bj − sBj)}{∏nj=1 Ŵ(1− aj − sAj)}{∏pj=n+1 Ŵ(aj + sAj)} ,δ =q∑j=1βj −p∑j=1αj > 0,A =n∑j=1αj −p∑j=n+1αj +m∑j=1βj −q∑j=m+1βj > 0, |arg(a)| <12Aπ ,−min1≤j≤m[R(bjβj)]< R(z) < min1≤j≤n[R(1− ajαj)].(34)M{F [ψ(x)](q)}(z) = 2Ŵ(z)cos(πz2)M{ψ(x)}(1− z),Page 7 of 10Damor et al. SpringerPlus  (2016) 5:111 therefore the Mellin transform of Eq. (32) isOn taking the inverse Fourier transform of Eq. (31) by using Eq. (10), we getSpecial caseOn considering time fractional bioheat equation  (24) in semi infinite space with Qmet = 0(φ = 0) and Neumann boundary condition on the left hand boundary, i.e. Eq. (23) is replaced by the following equationfor dimensionless variable, we considerOn making the use of Eq. (38), then Eqs. (24) and (37) reduce toFurther, on applying the Fourier cosine–Laplace transform to Eq. (39) and using Eq. (5) and Eq. (8), we obtainwhere,further simplification leads us(35)M(Gm(ζ , η))(z) =(12)η−α2Ŵ(z)Ŵ(12 +m−z2)Ŵ(12 +z2)Ŵ(1− α2 +αz2)Ŵ(12 −z2) .(36)θ(ζ , η) = φ exp(−|ζ |)− φ∞∑m=0(−1)mηαmm!∫∞−∞exp(−|ζ |)12ηα/2H2,12,3[|ζ − τ |ηα/2|( 12−m,12 ),(1−α2 ,α2 )(0, 12 ),(12 ,12 ),(12 ,12 )]dτ .(37)−k∂T∂x|x=0 = q0, T (x, t)|x→∞ = 0,(38)ζ =(Wbcbk) 12x, η =(Wbcbρc) 1αt, θ =T − Taq0(kWbcb)12 .(39)∂αθ∂ηα+ θ =∂2θdζ 2,(40)∂θ∂ζ|ζ=0 = −1 θ(ζ , η)|ζ→∞ = 0.sαθ∗(ω, s)− θ(ζ , 0)+ θ∗(ω, s) = −ω2θ∗(ω, s)+ 1 ,θ∗(ω, s) = FcL{θ(ζ , η)} =∫∞0cos(ωζ )∫∞0e−sηθ(ζ , η)dζdη ,(41)θ∗(ω, s) =∞∑m=0(−1)m1(sα + ω2)m+1.Page 8 of 10Damor et al. SpringerPlus  (2016) 5:111 Now taking inverse Laplace transformation to Eq. (41), as in light of Eq. (17), we getAfterwards using Eq. (19), then Eq. (42) reduces to the following formNow we find it’s inverse Fourier cosine transform by using Eq.  (33) and the following result (Langlands 2006) asThus we getThis is the solution of special case for the time fractional bioheat equation in the form of well knows H-function (see in details Mathai et al. 2010 ).Space fractional bioheat equationwe consider α = 1 in Eq. (21) and dimensionless variables assince Riez–Feller derivative of constant is not zero, therefore a new space dependent term f (ζ ) is introduced for simplification.where f (ζ ) = WbcbTaŴ(1−β)T0ζ−β, initial and boundary conditions are given in Eqs.  (27) and  (28) respectively.SolutionOn taking Fourier–Laplace transform of Eq. (47), and using Eq. (13), this leads usfurther simplification gives,(42)θ∗(ω, s) =∞∑m=0(−1)m1Ŵ(m+ 1)ηαm+α−1Emα,α(−ω2ηα).(43)θ∗(ω, s) =∞∑m=0(−1)m1Ŵ(m+ 1)ηαm+α−1H1,11,2[−ω2ηα|(−m,1)(0,1),(1−αm−α,α)].(44)√π2M{Fc[ψ(x)](q)}(z) = Ŵ(z) cos(πz2)M{ψ(x)}(1− z).(45)θ(ζ , η) =∞∑m=0(−1)m1Ŵ(m+ 1)ηαn+α−11(4πηα)1/2H2,12,3[−|ζ |ηα/2|( 12−m,12 ),(1−α/2,α/2)(0, 12 ),(12 ,12 ),(12 ,12 )].(46)ζ =(Wbcbk) 1βx, η =(Wbcbρc)t, θ =T − TaT0, φ =QmetT0Wbcb,(47)∂θ∂η+ θ =∂βθ∂ζβ+ f (ζ ),sθ∗(ω, s)− θ(ζ , 0)+ θ∗(ω, s) = −|ω|βθ∗(ω, s)+ f ∗(ζ )1s,(48)θ∗(ω, s) = f ∗(ζ )11+ |ω|β(1s−1(s + 1+ |ω|β)).Page 9 of 10Damor et al. SpringerPlus  (2016) 5:111 On taking inverse Laplace transform of Eq. (47) and using Eq. (17), this yieldsafterwards taking the inverse Fourier transform of Eq. (48) and employing the following results of Haubold et al. (2011b), which is given asFinally, we arrive atThis is a solution of space fractional bioheat equation.ConclusionIn this paper, we have investigated the temperature distribution of the biological tissue based on fractional bioheat transfer model. The fractional bioheat transfer equation is solved using integral transforms which yields analytical solution. This analytical solution may be useful in measurement of thermal parameters, reconstruction of temperature field, thermal diagnosis and thermal treatments.Authors’ contributionsThe authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.AcknowledgementsThe authors are thankful to anonymous referees for their comments and valuable suggestions which helped us for improving this paper. The first author is also thankful to Technical Education Department, Government of Gujarat, India for providing the opportunity to carry out this work at SVNIT, Surat, India under quality improvement programme.Competing interestsThe authors declare that they have no competing interests.Received: 27 August 2015   Accepted: 18 January 2016ReferencesDebnath L, Bhatta D (2007) Integral transforms and their applications, 2nd edn. Chapman and Hall/CRC, Taylor and Francis Group, New YorkHaubold HJ, Mathai AM, Saxena RK (2011a) Further solutions of fractional reaction-diffusion equations in terms of H-func-tion. J Comput Appl Math 235(5):1311–1316Haubold HJ, Mathai AM, Saxena RK (2011b) Mittag-Leffler functions and their application. J Appl Math 51. doi:10.1155/2011/298628Huang F, Guo B (2010) General solutions to a class of time fractional partial differential eqautions. Appl Math Mech 31:815–826Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, AmsterdamKilbas AA, Luchko YF, Martinez H, Trujillo JJ (2010) Fractional Fourier transform in the framework of fractional calculus operators. Integral Transforms Spec Funct 21:779–795(49)θ∗(ω, η) = f ∗(ζ )11+ |ω|β− f ∗(ζ )E1,1(−η)11+ |ω|βE1,1(−|ω|βη),(50)F−1{Eβ ,γ (−ηtβ|k|α)}=1α|x|H2,13,3[|x|(ηtβ)1/α|(1, 1α)(γ , βα),(1, 12 )(1,1/α),(1,1),(1, 12 )],F−1{E1,1(−η|ω|β)}=1β|ζ |H2,13,3[|ζ |(η1/β)|(1, 1β)(1, 1β),(1, 12 )(1,1/β),(1,1),(1, 12 )].(51)θ(ω, η) =12π∫∞−∞f (ζ ) exp(−|ζ |β2 )− E1,1(−η)12π∫∞−∞f (ζ ) exp(−|ζ |β2 )1β|ζ |H2,13,3[|ζ − τ |(η1/β)|(1, 1β)(1, 1β),(1, 12 )(1,1/β),(1,1),(1, 12 )]dτ .Page 10 of 10Damor et al. SpringerPlus  (2016) 5:111 Langlands TAM (2006) Solution of a modified fractional diffusion equation. Phys A 367:136–144Mainardi F, Pagnini G, Saxena RK (2005) Fox H function in fractional calculus. J Comput Appl Math 178:321–331Mathai AM, Saxena RK, Haubold HJ (2010) The H-function: theory and applications. Springer, New YorkMinkowycz WJ, Sparrow EM, Abraham JP (2009) Advances in numerical heat transfer. CRC Press, New YorkOzisik MN (1985) Heat transfer: a basic approach. McGraw-Hill Book Company, SingaporePennes HH (1948) Analysis of tissue and arterial blood temperature in the resting human forearm. J Appl Physiol 1:93–122Podlubny I (1999) Fractional differential equations. Academic Press, New YorkSalim TO, El-Kahlout A (2009) Analytical solution of time-fractional advection dispersion equation. Appl Appl Math 4:176–188Saxena RK, Mathai AM, Haubold HJ (2006) Solution of fractional generalized reaction–diffusion equations. Astrophys Space Sci 305:305–313Shang Y (2015) Analytical solution for an in host viral infection model with time in-homogeneous rates. Acta Phys Pol B 46:1567–1577Srivastava H, Gupta K, Goyal S (1982) The H-functions of one and two variables with applications. South Asian Publishers Pvt Ltd, New Delhi

Solution of fractional bioheat equation in terms of Fox’s H-function

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Solution of fractional bioheat equation in terms of Fox’s H-function

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