Investigations of a class of Liouville-Caputo fractional order Pennes bioheat flow partial differential equations through orthogonal polynomials on collocation points

In this study, we give a systematic discussion on convergent approximations of generalized nonlocal form of Pennes bioheat flow type parabolic partial differential equations. These flow problems frequently appear during the examination of the temperature variations in hyperthermia. Here, the nonlocal form involves Caputo-type fractional derivatives. The finite difference approximation in time is used on uniform steps to reduce the nondimensionalized form of the Pennes bioheat flow model into a semi-discrete continuous form in space. Thereafter, this semi-discrete problem is approximated by the third-kind shifted Chebyshev polynomials (TKSCP) on Chebyshev collocation points, at all time levels. This procedure converts the steady-state problem into a system of algebraic equations whose solution is the temperature distribution of the proposed model. In addition to the expected theoretical errors, a uniform convergence of the approximated solution to the exact solution is produced. We also investigated the effect of the order of fractional derivatives on the temperature distribution of living tissues computationally. Graphical results demonstrate that this generalized flow problem maintains a behavior similar to that of classical parabolic problems having integer-order partial derivatives when the fractional parameters tend to a positive integer.