Analytical Solution of Fractional-Order Lane-Emden Type Pantograph Delay Differential Equation

Pantograph differential equations arise in areas such as electrodynamics, epidemiology, and engineering; while Lane-Emden type differential equations appear in mathematical physics, astrophysics, and celestial mechanics. In this paper, a class of fractional-order pantograph delay differential equations of Lane-Emden type is considered using a fractional power series method. The proposed method is applied to a fractional-order version of a generalised pantograph equation, and a recurrence relation for the expansion coefficient is obtained. The main tool used in constructing this explicit recurrence relation is the generalised Cauchy product. The proposed problem is a generalisation of many problems in the literature. Several examples are presented to illustrate the proposed method’s efficiency, reliability, and accuracy. The results obtained agree excellently with the exact solutions and other published results. This excellent agreement of results is an indication that the proposed method is efficient, reliable, and accurate for solving a generalised Lane-Emden type pantograph delay equation of fractional-order.