Lie algebra and conservation laws for the time-fractional heat equation

Abstract

The Lie symmetry method is applied to derive the point symmetries for the N-dimensional fractional heat equation. We find that the numbers of symmetries and Lie brackets are reduced significantly as compared to the integer order for all dimensions. In fact for integer order linear heat equation the number of solution symmetries is equal to the product of the order and space dimension, whereas for the fractional case, it is half of the product on the order and space dimension. The Lie algebras for the integer and fractional order equations are mentioned using the subsequent computations of Lie brackets and by inspection. Interestingly, it is observed that for the one-dimensional fractional heat equation, the Lie algebra obtained by inspection of symmetries is similar to the result obtained by computation of Lie brackets, which is $A_1\oplus A_2\oplus \mathrm{\infty }{A}_1$. The Lie algebra using the symmetries of the two-dimensional heat equation is observed to be $(2A_1{\oplus }_{s}so(2)){\oplus }_{s}2A_1{\oplus }_s\mathrm{\infty }{A}_1$, whereas using the Lie brackets the algebra is deduced to be $A_{3,6}\oplus so(2){\oplus }_s{A}_1{\oplus }_s\mathrm{\infty }{A}_1$. Hence, it can be concluded that the Lie algebra obtained from the nonzero Lie brackets can be conflated to the algebra which is obtained by inspection. Further, the subsequent Lie algebras are mentioned for the three and four-dimensional integer and fractional equations and the conservation laws are explicitly stated.

Keywords:

• lie symmetries
• solutions
• conservation laws
• fractional partial differential equations

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The conservation law for (1(1) $D_t^{\alpha }u={\mathrm{\nabla }}^{2}u,$(1) ) with respect to the Caputo derivative is $C^t{=}_0{I}_t^{n+1-\alpha }{D}_t^{n}u$ (Lukashchuk, 2015).

Additional information

Funding

AKH expresses grateful thanks to NBHM Post-Doctoral Fellowship, Department of Atomic Energy (DAE), Government of India [award no: 0204/3/2021/R&D-II/7242] for financial support. PGLL thanks the University of Kwazulu-Natal, Durban University of Technology for financial support and to the Department of Mathematics, Pondicherry University for gracious hospitality.