Fractional-order Model for Cooling of a Semi-infinite Body by Radiation

Document Type : Original Paper


Department of Mathematics, University of Kurdistan, Sanandaj, Iran


‎In this paper‎, ‎the fractional-order model for cooling of a semi-infinite body by radiation is considered‎.
‎In the supposed semi-infinite body‎, ‎the equation of heat along with an initial condition and an asymptotic boundary condition form an equivalent equation in which the order of derivatives is halved‎.
‎This equation and a boundary condition introduced by the radiation heat transfer give rise to an initial value problem‎, ‎whose differential equation is nonlinear and fractional order‎.
‎The semi-analytical solution to this nonlinear model was determined asymptotically at small and large times‎.
‎Moreover‎, ‎two numerical methods including Grunwald-Letnikov approximation and Muntz-Legendre approximation yield numerical solutions to the problem‎.


Main Subjects

Evans, L.C. (2010).
Partial Differential Equations
, Providence: AMS.
Carslaw, H.S. and Jaeger, J.C. (1959).
Conduction of Heat in Solids
Oxford: Clarendon Press.
Oldham, K.B. and Spanier, J. (1972).
A general solution of the diffusion
equation for semi-infinite geometries
4, Journal of Mathematical Analysis
and Applications,
Holman, J.P. (2009).
Heat Transfer
, New York: McGraw-Hill Education.
5. Podlubny, I. (1999).
Fractional Differential Equations
, San Diego, CA:
Academic Press.
 Baleanu, D., Diethelm, K., Scalas, E. and Trujillo, J.J. (2016).
Calculus: Models and Numerical Methods
, Singapore: World Scientific.
.Diethelm, K. (2010). The Analysis of Fractional Differential Equations,
Berlin: Springer.
.Karniadakis, G.E. (Ed.) (2019). Numerical Methods, Volume 3 of
Handbook of Fractional Calculus with Applications, Berlin: De Gruyter.
Tripathi, N.K. Das, S. Ong, S.H. Hossein, J. and Al Qurashi, M. (2016).
Solution of higher order nonlinear time-fractional reaction diffusion
, 329.
Babenko, Y.I. (1986). Heat and Mass Transfer: The Method of
Calculation for the Heat and Diffusion Flows (in Russian), Leningrad: