Application of a time-fractal fractional derivative with a power-law kernel to the Burke-Shaw system based on Newton's interpolation polynomials

Abstract

This paper proposes some updated and improved numerical schemes based on Newton's interpolation polynomial. A Burke-Shaw system of the time-fractal fractional derivative with a power-law kernel is presented as well as some illustrative examples. To solve the model system, the fractal-fractional derivative operator is used. Under Caputo's fractal-fractional operator, fixed point theory proves Burke-Shaw's existence and uniqueness. Additionally, we have calculated the Lyapunov exponent (LE) of the proposed system. This method is illustrated with a numerical example to demonstrate the applicability and efficiency of the suggested method. To analyze this system numerically, we use a fractal- fractional numerical scheme with a power-law kernel to analyze the variable order fractal- fractional system. Furthermore, the Atangana-Seda numerical scheme, based on Newton polynomials, has been used to solve this problem. This novel method is illustrated with numerical examples. Simulated and analytical results agree. We contribute to real-world mathematical applications. Finally, we applied a numerical successive approximation method to solve the fractional model. • The purpose of this section is to define a mathematical model to study the dynamic behavior of the Burke-Shaw system. • With the Danca algorithm [1,2] and Adams-Bashforth-Moulton numerical scheme, we compute the Lyapunov exponent of the system, which is useful for studying Dissipativity. • In a generalized numerical method, we simulate the solutions of the system using the time-fractal fractional derivative of Atangana-Seda.

Keywords

• Fractional derivatives
• Nonlinear equations
• Simulation
• Numerical results
• Iterative method
• Time varying control system