Approximate Solutions for Time-Fractal Fractional Differential Equations

Abstract

In this Master’s dissertation, we present numerical methods for solving time-fractal fractional differential equations. A new fractal derivative of order 𝛼 ∈ (0, 1] is introduced, and the fractal Laplace transform is developed to obtain exact solutions. For numerical approximations, we implement the fourth-order Runge-Kutta method (RK4) for fractal ordinary differential equations and develop a Legendre collocation method combined with shifted Legendre polynomials for fractal partial differential equations, using the forward Euler scheme for time integration. Numerical results are presented in 2D and 3D plots, with errors computed using 𝐿2 and 𝐿∞ norms, demonstrating high accuracy and convergence of the proposed methods.