Improved numerical schemes to solve general fractional diabetes models

Abstract

In this article, we propose a new class of nonlinear fractional differential equations of diabetes disease based on the concept of Caputo fractional derivative. Two numerical techniques are introduced to analyze the solution of the general fractional diabetes model, that describes glucose homeostasis. The first method, which constructed using the nonstandard finite difference technique involves an asymptotically stable difference scheme. This method maintains important properties of the solutions of the studied system, such as the positivity and boundedness. The second method is the Jacobi–Gauss–Lobatto spectral collocation approach, known for its exponential accuracy. By employing this collocation method, the problem is transformed into a set of algebraic nonlinear equations, simplifying the overall task. Numerical simulations were conducted to compare the performance of these two techniques with other standard methods and the analytic solution in specific cases. Our findings show that the Jacobi–Gauss–Lobatto spectral collocation technique provides higher accuracy in solving the fractional diabetes model system, while the nonstandard finite difference approach requires lower computational duration.

Keywords

• Diabetes fractional models
• Caputo fractional derivative
• Nonstandard finite difference technique
• Spectral collocation method
• Jacobi polynomials