Studies on Variable Order Fractional Differential Equations and Their Applications

Abstract

The derivatives and integrals of fractional (non-integer) order have gained popularity as tools for modeling phenomena with persistent memory effects. Schaefer s fixed point theorem is used to prove the existence of solutions to the proposed problems, while the Banach fixed point theorem yields unique results. The Adams-Bashforth-Moulton technique computes the numerical solutions. To show the method s effectiveness, computational simulations of chaotic behaviors of delayed systems with different variable orders are analyzed. The numerical solution to the suggested problem provides high-precision approximations. In addition, a system of variable order integro-differential equations is considered. The computing implications of the presented problem are investigated and evaluated utilizing modified Euler and Runge-Kutta 4th-order techniques. The Atangana-Baleanu-Caputo derivative, the Caputo Fabrizio derivative, the Atangana-Baleanu fractal-fractional derivative, and the Caputo derivative are the many fractional derivative types that we investigated using a new numerical technique. These numerical approaches rely on fractional calculus and Newton s polynomial interpolation. This method was used to compute the solving of Wang-Sun, Rucklidge, and Rikitake systems with variable order. Numerical examples to demonstrate the significance and effectiveness of this unique technique are provided. For Chen, Logistic, Samardzija-Greller, and Lorenz, Lotka-Volterra predator-prey system involving time-varying delay systems are also analyzed. Furthermore, the RLC circuit system is studied as the application of variable order fractional calculus in engineering. newline