Fractional Order Mathematical Model to Investigate Topical Drug Diffusion in Human Skin

Abstract

Objectives:The primary objective of this research paper is to gain a comprehensive understanding of drug diffusion within the human dermal region. Methods: A temporal fractional-order reaction-diffusion equation with Caputo sense is employed to get mathematical insights on the diffusion of drugs in the human dermal region. The explicit finite difference method is employed to numerically solve the modelled problem. A Python-based algorithm is employed to obtain a numerical solution through the finite difference method. Finding: In our research, we focused on examining how fractional-order parameters affect the distribution and concentration profiles of drugs in the dermal region. To convey our findings effectively, we conducted a comprehensive analysis, primarily using graphical representations. These visualizations offer a clear and insightful view of the drug's diffusion rate within the dermal region, taking into account the memory effect associated with the Caputo derivative. In addition to our exploration of fractional-order parameters and drug diffusion profiles, we conducted a comprehensive investigation into the stability and convergence of the explicit finite difference method. Novelty: The fractional order explicit finite difference method can be used to estimate drug concentration in the human skin. An algorithm based on Python provides powerful tool for obtaining numerical solution of fractional order differential equations.

Comparative Study of Solutions of Fractional Order Mixed KdV Burger’s Equation

“…where, 𝐷 𝛼 𝑡 = 𝜕 𝛼 𝜕𝑡 𝛼 , 𝑁 𝑥 = 𝜕 𝜕𝑥 and 𝐿 𝑥 𝑢 = −𝑎 𝜕 2 𝑢 𝜕𝑥 2 +𝑏 𝜕 3 𝑢 𝜕𝑥 3 After applying the operator 𝐽 𝛼 on both sides of the above equations, we get…”

“…(1,2) . Since many fractional order non-linear differential equations are not easy to solve, hence we prefer numerical methods like finite difference method (3) , Adomian Decomposition Method (4)(5)(6) and Homotopy Perturbation Method (7) which gives approximate solutions.…”

“…In this paper, we study following type of nonlinear time fractional mixed KdV Burger's equation (12) 𝜕 𝛼 𝑢 𝜕𝑡 𝛼 + 𝑢 𝜕𝑢 𝜕𝑥 − 𝑎 𝜕 2 𝑢 𝜕𝑥 2 + 𝑏 𝜕 3 𝑢 𝜕𝑥 3 = 0; 𝑡 ≥ 0, 0 < 𝛼 ≤ 1.…”

Solving Time-fractional Order Radon Diffusion Equation in Water by Finite Difference Method

“…To handle this, numerical methods are used to obtain solution of fractional differential equations (3) . Finite difference method is one of the widely used numerical method for solving differential equations in both one and multiple dimensions (4)(5)(6)(7) . They are particularly valuable when analytical solutions are not readily available or https://www.indjst.org/ practical.…”