The time fractional heat conduction equation in the general orthogonal curvilinear coordinate and the cylindrical coordinate systems

In this paper a time fractional Fourier law is obtained from fractional calculus. According to the fractional Fourier law, a fractional heat conduction equation with a time fractional derivative in the general orthogonal curvilinear coordinate system is built. The fractional heat conduction equations in other orthogonal coordinate systems are readily obtainable as special cases. In addition, we obtain the solution of the fractional heat conduction equation in the cylindrical coordinate system in terms of the generalized HH-function using integral transformation methods. The fractional heat conduction equation in the case 0<α≤10<α≤1 interpolates the standard heat conduction equation (α=1)(α=1) and the Localized heat conduction equation (α→0)(α→0). Finally, numerical results are presented graphically for various values of order of fractional derivative.