Non-differentiable solutions for non-linear local fractional heat conduction equation

Abstract

Fractional calculus has many advantages. Under consideration of this paper is a (2+1)-dimensional non-linear local fractional heat conduction equation with arbitrary degree non-linearity. Backlund transformation of a reduced form of the local fractional heat conduction equation is constructed by Painleve analysis. Based on the Backlund transformation, some exact non-differentiable solutions of the local fractional heat conduction equation are obtained. To gain more insights of the obtained solutions, two solutions are constrained to a Cantor set and then two spatio-temporal fractal structures with profiles of these two solutions are shown. This paper further reveals by local fractional heat conduction equation that fractional calculus plays important role in dealing with non-differentiable problems.