International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org Volume 3, Issue 8, August 2014 ISSN 2319 - 4847 The influence of two-temperature fractional order generalized thermoelastic diffusion inside a spherical shell Debarghya Bhattacharya1, Mridula Kanoria2 1 Department of Applied Mathematics, University of Calcutta, India 2 Department of Applied Mathematics, University of Calcutta, India ABSTRACT This problem deals with the two-temperature thermo-elasto-diffusion interaction inside a spherical shell in the context of fractional order generalized thermoelasticity. The inner and outer boundaries of the spherical shell are traction free and subjected to heating. The chemical potential is also assumed to be a function of time on the boundary of the shell. To obtain the general solution the integral transform technique is used. The solution obtained in the Laplace transform domain by using a direct approach. The inversion of the transformed solution is carried out by applying the method of Bellman. Numerical estimates for thermophysical quantities are obtained for copper like material. The computed result for thermoelastic stresses, conductive temperature, displacement, mass concentration and chemical potential are shown graphically and the effect of two-temperature and fractional parameters are discussed. Keywords: Generalized thermo-elasticity, Fractional order heat equation, thermoelastic diffusion and two-temperature. 1. INTRODUCTION Diffusion can be defined as the migration of the particles from region of high concentration to region of lower concentration until equilibrium is reached. It occurs as a result of the second law of thermodynamics which states that the entropy or disorder of any system must always increase with time. Diffusion is important in many life processes. The recent interest in the study of this phenomenon is due to its extensive applications in geophysics and many industrial applications. The phenomenon of diffusion is used to improve the conditions of oil extractions (seeking ways of more efficiently recovering oil from oil deposits). These days, oil companies are interested in the process of thermoelastic diffusion for more efficient extraction of oil from oil deposits. In most of these applications, the concentration is calculated using what is known as Fick’s law. This is a simple law that does not take into consideration the mutual interaction between the introduced substance and the medium into which it is introduced or the effect of the temperature on this interaction. Biot [1] develop the coupled theory of thermo-elasticity to deal with defeat of the uncoupled theory that mechanical cause has no effect on the temperature field. In this theory, the heat equation has a parabolic form which predicts an infinite speed for the propagation of mechanical wave. The theory of generalized thermoelasticity with one relaxation time was introduced by Lord and Shulman [2]. This theory was extended by Dhaliwal and Sherief [3]. In the theory, the Maxwell-Cattaneo law of heat conduction replaces the conventional Fourier’s law. For this theory, Ignaczak [4] studied the uniqueness of solution. Thermodiffusion in the solids is one of the transport processes that have great practical importance. Most of the research associated with the presence of concentration and temperature gradients has been made with metals and alloys. Thermodiffusion in an elastic solid is due to the coupling of the fields of temperature, mass diffusion and strain fields. The first critical review was published in the work of Oriani (1969). Nowcaki [5], [6] developed the theory of thermoelastic diffusion. In this theory, classical coupled thermoelastic model is used. Later on, Gawinecki et al [7] proved a theory on uniqueness and regularity of the solution for a nonlinear parabolic thermoelastic diffusion problem. Sherief et al. [8] and, later on, Kumar and Kansal [9] introduced the generalized theories of thermoelastic diffusion in the frame of LS and GL theories by introducing thermal relaxation time parameters and diffusion relaxation time parameters into the governing equations, which allow the finite speeds of propagation of waves inside the medium. Sherief and Salah [10] investigated the problem of a thermoelastic half space in the context of the theory of generalized thermoelastic diffusion with one relaxation time. Aouadi [11], [12] also gave some attention on thermoelastic diffusion and generalized thermoelastic diffusion. Othman et al. [13] reported some studied on the effects of diffusion on a two-dimensional problem of generalized thermoelasticity in the context of the mechanism of Green-Naghdi theory. The theory introduced by Sherief et al., Kothari and Mukhopadhyay [14], presented the Galerkin-type representation of solutions for thermoelastic diffusion. In the context of the same theory, variational and reciprocity theorems have been established by Kumar et al [15]. Differential equations of fractional order have been the focus of many studies due to their frequent appearance in various applications in fluid mechanics, viscoelasticity, biology, heat conduction, diffusion problems etc. It has been examined that the use of fractional order derivatives leads to the formulation of certain physical problems which is more economical and useful than the classical approach. Fractional Volume 3, Issue 8, August 2014 Page 96 International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org Volume 3, Issue 8, August 2014 ISSN 2319 - 4847 calculus has been used successfully to modify many existing models of physical process. The first application of fractional derivatives was given by Abel. He applied the fractional calculus in the solution of an integral equation that arises in the formulation of the Tautochrone problem. Caputo and Mainardi [16] and Caputo [17] found good agreement with experimental results when using fractional derivatives for description of viscoelastic materials and estabished the connection between fractional derivatives and the theory of linear viscoelasticity. Povstenko [18] has constructed a quasistatic uncoupled thermoelasticity model based on the heat conduction equation with a fractional order time derivative. Sur and Kanoria [31] studied the effect of fractional order on two-temperature generalized thermoelasticity with finite wave speed. One dimensional problem of a fractional order two-temperature generalised piezoelasticity has been investigated by M. Islam and M. Kanoria [32] where as fractional order generalized thermo visco-elastic problem has been done by A. Sur and M. Kanoria [33]. The linearized version of the two-temperature theory (2TT) has been studied by many authors. Chen et al. [19] have formulated a theory of heat conduction in deformable bodies, which depends on two distinct temperatures (a) the conductive temperature and (b) the thermodynamic temperature . Lesan [21] has established uniqueness and reciprocity theorems for 2TT. The existence, structural stability and spatial behavior of the solution in 2TT have been discussed by Quintanilla [22]. The key element that sets the two-temperature thermoelasticity (2TT) apart from the classical theory of thermoelasticity (CTE) is the material parameter 0 called the temperature discrepancy. Specifically, if 0 , then and the field equations of the 2TT reduce to those of CTE. It should be pointed out that 2TT suffer from so-called paradox of heat conduction, i.e., the prediction that a thermal disturbance at some point in a body is felt instantly, but unequally, through out the body. The main object of the present work is the investigation of disturbances in a homogeneous isotropic medium in the context of two-temperature generalized thermoelastic diffusion and fractional heat conduction. The formulation is applied to the generalized thermoelasticity based on fractional time derivatives under the effect of diffusion. The analytical expressions for the displacement components, thermoelastic stresses, conductive temperature, thermoelastic temperature, concentration and chemical potential are obtained in the Laplace transform domain. Whose boundaries are traction free, are subjected to a time dependent temperature and chemical potential in the context of two-temperature generalized thermoelasticity and fractional heat conduction. To get the solution in the physical domain, the inversion of the transformed solution is carried out by applying the method of Bellman. The influences inside the shell the theory is analyzed for a copper like material and depicted graphically. Some comparisons have been shown to estimate the effect of the two-temperature and fraction order parameters. 2. DERIVATION OF THE GOVERNING EQUATIONS: In the classical calculus of Newton and Leibniz, Riemann-Liouville reduced the calculation of an n-fold integration of the function f (t ) into a single convolution-type integral, Where I is the -fold Riemann-Liouville integral operator with Here, the definition of fractional derivatives of order is used according to Caputo [28] for an absolute continuous function f (t ) given by Where I is the fractional integral of order of Lebesgue’s integrable function f (t ) defined by Miller and Ross [27]. In the case of absolutely continuous function f (t ) , Now let an arbitrary material occupies the6.volume V and is bounded by a surface A at a time t. Then the law of conservation of energy for V can be written in the form, (2) U is the internal energy per unit mass, σij are the components of stress tensor, ui are the components of displacement vector, is the density, Fi are the components of external forces per unit mass, ni are the Where qi is the heat flux vector, unit outer normal vectors. Using the divergence theorem and the equation of motion: We obtain the pointwise form of equation (2) in the form: Volume 3, Issue 8, August 2014 Page 97 International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org Volume 3, Issue 8, August 2014 ISSN 2319 - 4847 (3) Now using the entropy equation: (4) and the equation of conservation of mass (5) We obtain (6) j is the flow of the diffusing mass vector, S is the entropy per unit mass, T is the absolute temperature, C is the concentration and P is the chemical potential. In the above equations, Now, note that: (7) thus we obtain (8) (9) Let us introduce Helmholtz free energy per unit volume F, defined by: It follows that the free energy, as a state variable, is a function of strain tensor, temperature and concentration. Using the chain rule and from equations (6), (8) and (9) we obtain, (10) From this equation the following results are obtained: (11) (12) Where is the temperature of the medium in natural state. cE is the specific heat, cijkm is the tensor of elastic constants, c and natural state, we have: d are measures of the thermo-diffusion effect and diffusive effect, respectively. Now in the Thus, we obtain: (13) Using equation (13) and keeping terms up to the second order only, equation (12) takes the form (14) Using equation (14), relations in (11) yield, (15) (16) (17) For the isotropic case, we have (18) (19) β1, β 2 are the material constants given by where t and c are, respectively, the coefficient of linear thermal expansion and linear diffusion expansion. Substituting from equation (18)-(19) into equation (15)-(17), we get for the isotropic case (20) (21) Volume 3, Issue 8, August 2014 Page 98 International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org Volume 3, Issue 8, August 2014 ISSN 2319 - 4847 (22) Now the linearized form of (4) is, (23) Using (21), equation (23) reduce to: (24) Cattaneo [25] has introduced a law of heat conduction to replace the classical Fourier’s law in the form (25) Where, 0 is the thermal relaxation time and K is the thermal conductivity. Youssef [26] has investigated a two temperature generalized thermoelasticity theory in which Fourier’s law given by equation (25) is replaced by: (26) is the conductive temperature and satisfies the relation: (27) In which 0 is the two temperature parameter. In this paper, the new Taylor series expansion of time-fractional order developed by Jumarie [29] is adopted to expand and the remaining terms upto order in the thermal relaxation time 0 , we get the fractional order Fourier law for two-temperature as: Where 0 1. (28) By taking divergence on both side of (28) and using the equations (24) and (26), we arrive at the equation of heat conduction in our case, namely (29) For the heat flux vector, we assume a similar equation for the mass flux vector of the form: (30) By taking divergence on both side of (30) and using the equation (22), we arrive at (31) 0 Where is the diffusive relaxation time. 3. FORMULATION OF THE PROBLEM We consider an isotropic homogenous thermoelastic spherical shell with inner radius a and outer radius b in a uniform temperature T0 . Let the body be referred to spherical polar coordinate with the origin at the center O of the cavity. Since we consider thermoelastic interactions with the spherical symmetric, so all the functions considered will be function of the radial distance r and the time t only. It follows that the displacement vector u thermodynamic temperature and conductive temperature have the following forms: (32) The strain components are given by: (33) thus the cubical dilation will be: Volume 3, Issue 8, August 2014 Page 99 International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org Volume 3, Issue 8, August 2014 ISSN 2319 - 4847 (34) In the context of two temperature generalized thermoelastic diffusion based on fractional order time derivatives, the equation of motion, the equation of heat conduction and the equation of mass diffusion in absence of body forces for a linearly isotropic generalized thermoelastic solid are, respectively, The relation between the conductive temperature and the thermodynamic temperature is given by, (38) 2 Where is the Laplacian, given in our case by, (39) The constitutive equations are given by For convenience, introducing the following non-dimensional variables Therefore, the governing equations are given by equations. (35)- (38) and (40)-(42) can be expressed in the following forms (where the primes are suppressed for simplicity), as Volume 3, Issue 8, August 2014 Page 100 International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org Volume 3, Issue 8, August 2014 ISSN 2319 - 4847 4. BOUNDARY CONDITIONS From the physical phenomena for the problem, we will take the boundary conditions as: (1) The boundaries of the shell are assumed to be traction free, that is: (50) (2) Boundaries of the shell are subjected to a thermal shock in the form (51) (3) The chemical potential is also assumed to be a known function of time at the boundaries of the shell, that is: (52) 5. SOLUTIONS IN LAPLACE TRANSFORM DOMAIN Applying the Laplace transform defined by the relation: to equation (43) and using homogeneous initial conditions, we get: (53) Now applying Laplace transform on equation (44) and using (53) (46), we get (54) Applying divergence operator on (53), we get: (55) Volume 3, Issue 8, August 2014 Page 101 International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org Volume 3, Issue 8, August 2014 Volume 3, Issue 8, August 2014 ISSN 2319 - 4847 Page 102 International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org Volume 3, Issue 8, August 2014 ISSN 2319 - 4847 We can obtain the A1 ( s ), A2 ( s ), A3 ( s ), B1 ( s ), B2 ( s ), B3 ( s ) by solving the above linear system of equations (82)(86). This completes the solution of the present problem in the Laplace transform domain. 6. NUMERICAL RESULTS AND DISCUSSION In order to illustrate theoretical results in the preceding sections, we now present some numerical results. To get the solutions for the displacement, radial stress, shear stress, conductive temperature, thermodynamic temperature, chemical potential and mass concentration in the physical domain, we have to apply Laplace inversion formula to the equations (81)-(86) respectively. Here we adopt the method of Bellman et al. [23] for inversion and choose a time span given by seven values of time ti , i =1 to 7 at which P and C are evaluated from the negative of logarithms of the roots of the shifted Legendre polynomial of degree 7. For the illustration we consider copper material with material constants. The physical data in SI units for which given in Sherief et al [30]: Also we have taken 0 1, 0 0.01, 0 0.1 and R 1 for computational purposes. The computation were carried out for time t =0.026. The variation of the field is observed when the step input of conductive temperatures with 1 1 and 2 1 applied on the inner boundary a 1 and outer boundary b=2 respectively. The step input of chemical potential with P1 1 and P2 1 are also applied on the boundaries of the shell. Figures 1, 3, 5, 7, 9 and 11 depict the variety of the displacement component u, the stress components rr , , the temperature component , the chemical potential P and the concentration of the diffusive material C for different values of the fractional parameter . The results coincide for 1.0 as in Bhattacharya and Kanoria [24], that are taken in the context of two-temperature generalized thermoelastic diffusion. On the another side, figures 2, 4, 6, 8, 10 and 12 show the difference in all functions for the value of the non dimensional two-temperature variable . The case 0.0 indicates one temperature generalized thermoelastic diffusion while 0.1, 0.15 indicates the new two-temperature case. Volume 3, Issue 8, August 2014 Page 103 International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org Volume 3, Issue 8, August 2014 Fig. 1: Dependence of radial stress on the fractional parameter ISSN 2319 - 4847 Fig. 2: Dependence of radial stress on the two-temperature parameter Figure 1 show the variation of radial stress ( rr ) against radial distance (r), dependence on the fractional parameter inside the spherical shell. At both boundaries the radial stress is noted to be zero, which also agrees with the theoretical boundary conditions. The strain takes negative value throughout the medium ( 1 r 2 ). It is seen from the figure 1 that the presence of fractional parameter has a significant effect on the solution of the stress distribution. The increment for leads to a decrease in its magnitude for fixed r. on the anther side figure 2 show the variation of radial stress for the value of the non dimensional two-temperature parameter . The trend of variation of this field gives significantly lower numerical value with the increment of two-temperature parameter . Fig. 3: Dependence of shear stress on the fractional parameter Fig. 4: Dependence of shear stress on the two-temperature parameter Figures 3 and 4, the variation of shear stress ( ) against radial distance (r) inside the spherical shell for time t =0.026 is displaced under the fractional parameter and the two-temperature parameter , respectively. It is seen from the figures, the hoop stress is fully compressive in nature in all cases. Moreover a significant difference in the hoop stress is Volume 3, Issue 8, August 2014 Page 104 International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org Volume 3, Issue 8, August 2014 ISSN 2319 - 4847 noticed for different value of the fractional parameter and the two-temperature parameter . With the increment of the both parameters the shear stress is decreased in its magnitude throughout the medium. Fig. 5: Dependence of on the fractional parameter Fig. 6: Dependence of on the two-temperature parameter In the figure 5 and 6, the effect of two-temperature and fractional order parameters on temperature field are observed. We find that initially the conductive temperature field shows the maximum value at both the boundaries and it decreases with the increase of radial distance towards the middle, becoming minimum at the middle. At both the boundaries the temperature field is noted to be in agreement with the boundary condition. With the increment of fractional parameter α, the variation of conductive temperature field is almost similar throughout the medium, whereas with the advancement of two-temperature parameter ω, the conductive temperature is also increased. Fig. 7: Dependence of ur on the fractional parameter Fig. 8: Dependence of ur on the two-temperature parameter In figure 7 and 8, some comparisons have been shown to estimate the effect of two-temperature and fractional order parameters on displacement field inside the spherical shell. From figure 7, it has been observed that in the interval 1.2< r Volume 3, Issue 8, August 2014 Page 105 International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org Volume 3, Issue 8, August 2014 ISSN 2319 - 4847 <2, as the value of fractional order parameter α increase, the peak of thermal displacement (ur) also increase. We note that ur for α=1 has the maximum magnitude when compared to the other two values of ur for α=0.2 and α=0.7, whereas in figure 8, ω=0.15 corresponds to the maximum of ur. Thus we note that both the parameters takes a significant role on displacement field inside the spherical shell. Fig. 9: Dependence of concentration on the fractional parameter Fig. 10: Dependence of concentration on the two-temperature parameter Figure 9 and 10 are ploted to show the variation of concentration (C) against the radial distance r when the fractional parameter α has the values 0.2, 0.7, 1.0 (fig.-9) and the two-temperature parameter ω has the value 0.0, 0.1 (fig.-10), respectively. From both these figures it is noted that, the mass concentration shows the maximum value near the boundaries and it decreases with the increase of radial distance towards the middle and becomes minimum at the middle of the shell. The presence of both the parameters has a significant effect on the solution of mass concentration. With the increment of both the parameters (α and ω) the mass concentration is increased in its magnitude throughout the medium. Fig. 11: Dependence of chemical potential on the fractional parameter Fig. 12: Dependence of chemical potential on the two-temperature parameter Volume 3, Issue 8, August 2014 Page 106 International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org Volume 3, Issue 8, August 2014 ISSN 2319 - 4847 In figures 11 and 12, some comparisons have been shown to estimate the effect of two-temperature and fractional order parameters on chemical potential (P) inside the spherical shell. The chemical ptential is noted to be zero at both the boundaries, which agrees the initial boundary conditions. It is also observed from these figures that with the increase of fractional order parameter and dimensionless two-temperature parameter, the chemical potential leads to a increase in its magnitude for a fixed r. 7. Conclusions According to the analysis above and from the numerical results presented in the figure 1-12, we can conclude the following important points: 1. The presence of diffusion plays a significant role on the displacement components, thermoelastic stresses, conductive temperature, thermoelastic temperature, concentration and chemical potential. 2. The generalized theory of thermoelasticity of fractional order heat transfer describes the behavior of an elastic body more realistic then the theory of generalized thermoelasticity with integer order. 3. The theory of two-temperature generalized thermoelasticity describes the behaviour of the particles of the elastic body more real then the theory of one-temperature generalized thermoelasticity. This study is very important for microscale problems, because in these cases the materal parameters are temperature dependent. Acknowledgements We are grateful to Professor S. C. Bose of the Department of Applied mathematics, University of Calcutta, for his kind help and guidance in preparation of the paper. References [1] Biot, M. A. (1956). Thermoelasticity and irreversible thermodynamics. J Appl Phys, 27: 240-253. [2] Lord, H., Shulman, Y. A. (1967). Generlaized theory of thermoelasticity. J Mech Phys Solid. 15: 299-309. [3] Dhaliwal, R. S., Sherief, H. H. (1980). A uniqueness theorem and a variational principle for generalized thermoelasticity. J Therm Stress. 3: 223-230. [4] Ignaczak, J. (1982). A note on uniqueness in thermoelasticity with one relaxation time. J Therm Stress. 5: 257-263. [5] Nowacki, W. (1974). Dynamical problems of thermoelastic diffusion in solids I. Bull Acad Pol Sci Tech. 22: 55-64. [6] Nowacki, W. (1974). Dynamical problems of thermoelastic diffusion in solids II. Bull Acad Pol Sci Tech. 22: 129135. [7] Gawinecki, J., Kacprzyk, P., Bar-Yoseph, P. (2000). Intial boundary value problem for some coupled nonlinear parabolic system of partial differential equations appearing in thermoelastic diffusion in solid body. J.Math Appl. 19: 121-130. [8] Sherief, H. H., Hamaza, F. A., Salah, H. A. (2004). The theory of generalized thermoelastic diffusion. Int J Eng Sci. 42: 591-608. [9] Kumar, R., Kansal, T. (2008). Propagation of lamd waves in transversely isotropic thermoelastic diffusion plate. Int J Solid Struct, 45: 5890-5913. [10] Sherief, H. H., Salah, H. A. (2005). A half space problem in the theory of generalized thermoelastic diffusion. Int J Solids Struct. 42: 4484-4493. [11] Aouadi, M. (2007). Uniqueness and reciprocity theorem in the theory of generalized thermoelastic diffusion. J Therm Stresses. 30: 665-678. [12] Aouadi, M. (2009). Spatial stability for quasi-state problem in thermoelastic diffusion theory. Acta Appl Math. 106: 307-323. [13] Othman, M. I. A., Atwa, S. Y., Farouk, R. M. (2009). The effect of on two-dimensional problem of generalized thermoelasticity with Green Naghdi theory. Int Commun Heat Mass. 36: 857-864. [14] Kothari, S., Mukhopadhyay, S. (2011). On the representation of solutions in the theory of generalized thermoelastic diffusion. Math Solids. 17(2): 120-130. [15] Kumer, R., Kothari, S., Mukhopadhyay, S. (2011). Some theorems on generalized thermoelastic diffusion. Acta Mech. 217: 287-296. [16] Caputo, M., Mainardi, F. (1971). Linear model of dissipation in elastic solid. Rivista Del Nuovo Cimento. 1: 161198. [17] Caputo, M. (1974). Vibration on an infinite viscoelastic layer with a dissipative memory. J Acoust Soc America. 56: 897-904. [18] Povstenko, Y. Z. (2005). Fractional heat conduction equation and associated thermal stresses. J Therm Stress. 28: 83-102. [19] Chen, P. J., Gurtin, M. E., Williams, W. O. (1968). A note on non-simple heat conduction,. Zeitschrift für angewandte Mathematik und Physik. 19: 969-985. [20] Chen, P. J., Gurtin, M. E., Williams, W. O. (1969). Zeitschrift für angewandte Mathematik und Physik, 20: 107-121. [21] Lesan, D. (1970). Journal of Applied Mathematics and Physics. 21: 583-601. [22] Quintanilla, R. (2004). Acta Mechanica. 168: 61-73. Volume 3, Issue 8, August 2014 Page 107 International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org Volume 3, Issue 8, August 2014 ISSN 2319 - 4847 [23] Bellman, R.; Kolaba, R. E.; Lockette, J. A. (1966). Numerical Inversion of the Laplace Transform. American Elsevier Publishing Company, New York. [24] Bhattacharya, D., Kanoria, M. (2014). The influence of two temperature generalized thermoelastic diffusion inside a spherical shell. Int J of Engg and Tech Research. 2(5): 151-159. [25] Cattaneo, C. (1948). On the conduction of heat. Atti Semin Mat Fis Univ Modena. 2: 3-21. [26] Youssef, H. M. (2006). Theory of two-temperature-generalized thermoelasticity. IMA J Appl Math. 71: 383-390. [27] Miller, K. S., Ross, B. (1993). An introduction to the fractional calculus and fractional differential equations. Wiley, New York. [28] Caputo, M. (1967). Linear models of dissipation whose Q is almost frequency independent II. Geophys J R Astron Soc. 13: 529-539. [29] Jumarie, G. (2010). Derivation and solutions of some fractional black schools equations in coarse-grained space and time application to Merton’s optimal portfolio. Comput Math Appl. 59: 1112-1164. [30] Sherief, H. H., Salah, H. A. (2005). A half space problem in the theory of generalized thermoelastic diffusion. Int J Solids Struct. 42: 4484-4493. [31] Sur, A., Kanoria, M. (2012). Fractional order two-temperature thermoelasticity with finite wave speed. Acta Mechanica. 223(12):2685-2701. [32] Islam, M., Kanoria, M. (2013). One dimensional problem of a fractional order two-temperature generalised thermopiezoelasticity. Mathematics and Mechanics of Solids. 1-22, DOI:10.1177/1081286513482605. [33] Sur, A., Kanoria, M. (2014). Fractional order generalized thermo-visco-elastic problem of a spherical shell with three-phase-lag effect. Latin American Journal of Solid and Structures. 11: 1132-1162. AUTHOR Debarghya Bhattacharya received M.Sc degree in Applied Mathematics from University of Calcutta in 2011. Currently he is a Ph.D student in Department of Applied Mathematics, University of Calcutta, India. Dr. Mridula Kanoria is currently a professor of Applied Mathematics at the Department of Applied Mathematics, University of Calcutta, 92 A.P.C. Road, Kol-700009, West Bengal, India. She received her Ph.D degree from University of Calcutta. She has 110 scientific publications in the field of solid mechenics. Volume 3, Issue 8, August 2014 Page 108