Optimal systems, series solutions and conservation laws for a time fractional cancer tumor model

Abstract

In this paper, we consider certain time-fractional cancer tumor models where the killing rates are space as well as time- dependent. Applying the Lie symmetry analysis, we obtain the infinitesimal transformations and prove that there is a 1-1 and onto mapping between their symmetries. Further, we construct set of Lie algebras and we study the optimality of those algebras. Considering one of the algebras, we obtain similarity variables and under the invariance condition the given partial differential equation with fractional derivative(FPDE) is reduced to ordinary differential equation with fractional order(FODE). Thereafter, we reduce the FODE to the ordinary differential equation(ODE) and present a series solution of the given FPDE with its convergence analysis. Furthermore, we discuss the effect of β on the nature of the solution graphically. Finally, we present the conservation laws for the given model using Lie symmetry.