## THERMAL SCIENCE

International Scientific Journal

### DIFFUSION MODELS OF HEAT AND MOMENTUM WITH WEAKLY SINGULAR KERNELS IN THE FADING MEMORIES: HOW THE INTEGRAL-BALANCE METHOD CAN BE APPLIED?

**ABSTRACT**

This work presents an attempt to apply the heat-balance integral approach to diffusion models with fading memories with weakly singular kernels resulting in closed-form solutions. The main steps are exemplified by solutions where the fading memory is represented by Volterra integrals and by a time-fractional Riemann-Liouville derivative. The examples address sole elastic (damping) effects and cases where the viscous diffusivity should be taken into account. As examples polynomial approximation is applied, demonstrating how to avoid problems in determination of the exponent of the general parabolic profile, but without freedom to optimize the final closed-form solution. In general, this is a new implementation of an old idea and related methods to new models and we hope the demonstrated technique could be useful in solutions of practical problems.

**KEYWORDS**

PAPER SUBMITTED: 2013-08-03

PAPER REVISED: 2013-11-09

PAPER ACCEPTED: 2013-11-09

PUBLISHED ONLINE: 2013-11-16

**THERMAL SCIENCE** YEAR

**2015**, VOLUME

**19**, ISSUE

**Issue 3**, PAGES [947 - 957]

- Ferreira, J.A., de Oliveira, P., Qualitative analysis of a delayed non-Fickian model, Applicable Analysis, 87 (2008), 8, pp. 873-886.
- Cattaneo, C., On the conduction of heat (in Italian), Atti Sem. Mat. Fis. Universit´a Modena, 3 (1948),1, pp. 83-101.
- Curtin, M. E, Pipkin, A.C., A general theory of heat conduction with finite wave speeds, Archives of Rational Mathematical Analysis, 31 (1968), 2, pp. 313-332.
- Joseph, D.D., Preciozi, Heat waves , Rev.Mod. Phys., 61 (1989), 1, pp. 41-73
- Lykov, A.V., Some problems of the theory of mass and heat transport, J of Thermophysics, 26 (1974), 5, pp.781-793.
- Berte, A.E., Relaxation function of polycrystalline metals in conditions of pulsed heating, J of Thermophysics, 46 (1974), 6, pp.992-998.
- Kalshnikova , L.S., Mass transport with memory in electrochemical processes, of Thermophysics, 29 (1985), 3, pp.481-486.
- Mikhailov, M.D., Vulchanov, N.L., Solution of the heat-conduction equation for materials with fading memory and high heat conduction, J of Thermophysics, 31 (1975), 2, pp.351-354.
- Hristov, J., A Note on the Integral Approach to non-Linear Heat Conduction with Jeffrey's Fading Memory, Thermal Science, 17 (2013), 3, pp.733-737.
- Olmstead, W.E., Davis, S.H., Rosenblat, S., Kath, W.L., Bifurcation with Memory, SIAM J. Appl. Math., 46 (1986), 2, pp. 171-188.
- Oldham , K.B. , Spanier ,J. , The Fractional calculus , Academic Press, New York, 1974.
- Hayat, T., Asghar, S., Siddiqui, A.M. , Some unsteady unidirectional flows of a non-Newtonian fluid, Int. J. Eng. Sci, 38 (2000), 3, pp.337-346
- Agrawal , O. P. , Application of Fractional Derivatives in Thermal Analysis of Disk Brakes , Nonlinear Dynamics , 38( 2004) ,1-4, pp.191-206.
- Dehghan, M., Solution of a partial integro-differential equation arising from viscoelasticity, Int. J. Computer Math., 83 (2006), 1, pp.123-129.
- Qi , H., Xu, M. , Some unsteady unidirectional flows of a generalized Oldroyd-B fluid with fractional derivative, Appl. Math. Model., 33 (2009),11, 4184-4191. doi:10.1016/j.apm.2009.03.002.
- Siddique, I. , Vieru, D. , Stokes flows of a Newtonian fluid with fractional derivatives and slip at the wall, Int. Rev. Chem. Eng., 3 (2011) ,6, pp.822- 826.
- Pfitzenreiter, T., A physical basis for fractional derivatives in constitutive equations, ZAMM, 84 (2004),4, pp.284-287.
- Miller, R.K., An integro-differential equation for rigid heat conductors with memory, J. Math. Anal. Appl., Vol. 66 (1978), 3, pp. 313-332.
- Yoon, J.M. Xie, S., Hrynkiv, V., Two numerical algorithms for solving a partial Integro-Differential equation with a weakly singular kernel, Applications and Applied Mathematics, 7 (2012), 1, pp.133-141.
- Sanz-Serna, J. M. , A numerical method for a partial integro-differential equation, SIAM J. Numer. Anal., Vol. 25, (1988), 2, pp. 319-327
- Lopez-Marcos, J. C., A difference scheme for a nonlinear partial intergro-differential Equation, SIAM J. Numer, Anal., 27 (1990), 1, pp. 20-31.
- Tang, T. A finite difference scheme for partial integro-differential equations with weakly singular kernel, Appl. Num. Math.,11(1993), 4, pp. 309-319.
- Finlayson, B.A., The Method of Weighted Residuals and Variational Principles: With Applications in Fluid Mechanics, Heat and Mass Transfer, Academic Press, 1972,N.Y.
- Hristov, J., The heat-balance integral method by a parabolic profile with unspecified exponent: Analysis and benchmark exercises, Thermal Science, 13 (2009), 2, pp.22-48.
- Myers, T.G., Mitchell, S.L., Application of the Heat-Balance and Refined Integral Methods to the Korteweg-de Vries equation, Thermal Science: 13 (2009), No. 2, pp. 113-119.
- Hristov J., A Short-Distance Integral-Balance Solution to a Strong Subdiffusion Equation: A Weak Power-Law Profile, International Review of Chemical Engineering-Rapid Communications, 2 (2010), 5, pp. 555-563
- Hristov J., Heat-Balance Integral to Fractional (Half-Time) Heat Diffusion Sub-Model, Thermal Science, 14 (2010), 2, pp. 291-316. doi: 10.2298/TSCI1002291H.
- Hristov J., Starting radial subdiffusion from a central point through a diverging medium (a sphere): Heat-balance integral method, ThermalScience,15 (2011), Supl. 1, S5-S20.
- Hristov J., Approximate Solutions to Fractional Subdiffusion Equations: The heat-balance integral method, The European Physical Journal-Special Topics, 193(2011), 1, pp.229-243. DOI:10.1140/epjst/e2011-01394-2
- Hristov, J. Transient Flow of A Generalized Second Grade Fluid Due to a Constant Surface Shear Stress: an approximate Integral-Balance Solution , Int. Rev. Chem. Eng., 3 (2011), 6, pp. 802-809.
- Hristov, J. Integral-Balance Solution to the Stokes' First Problem of a Viscoelastic Generalized Second Grade Fluid, Thermal Science, 16 (2012), 2, pp. 395-410.
- Goodman, T.R., The heat balance integral and its application to problems involving a change of phase, Transactions of ASME, 80 (1958), 1-2, pp. 335-342.
- Goodman T.R., Application of Integral Methods to Transient Nonlinear Heat Transfer, Advances in Heat Transfer, T. F. Irvine and J. P. Hartnett, eds., 1 (1964), Academic Press, San Diego, CA, pp. 51-122.
- Hristov, J., The Heat-Balance Integral: 1. How to Calibrate the Parabolic Profile?, Comptes Rendue Mechanique.340 (2012) 485-492 .
- Hristov, J. The Heat-Balance Integral: 2. A Parabolic profile with a variable exponent: the concept and numerical experiments, Comptes Rendue Mechanique, 340 (212), 7,pp. 493-500.
- Langford D., The heat balance integral method, Int. J. Heat Mass Transfer, 16 (1973), 12, pp. 2424-2428.
- Goodwin, J.W., Hughes, R.W., Rheology for chemists: An Introduction, 2nd ed, RSC Publishing, Cambridge, UK., 2008.
- Myers, T.G., Optimal exponent heat balance and refined integral methods applied to Stefan problem, Int. J. Heat Mass Transfer, 53 (2010), 5-6, pp. 1119-1127.