Numerical solutions of fractional differential equations

In this study, numerical solutions of fractional-order differential equations were obtained. Fractional-order differential equations have been widely used in recent years to model many problems in nature. Finding the exact solution of a fractional differential equation is not as straightforward as in integer-order differential equations, and sometimes it is not possible to reach exact solutions. This situation has led to the need to modify numerical solution methods for fractional-order derivative equations. In this study, the Adams-Bashforth-Moulton method was used as the numerical solution method. Linear interpolation polynomials were used for the formulation of the method. Since consecutive iterations were used in the method and the accurate decimal parts of the numerical results could not be ignored, codes were written in suitable algorithmic languages. In order to evaluate the obtained solutions, the exact solutions of the equations were obtained using Laplace transforms, and the results were compared. The findings showed that the Adams-Bashforth-Moulton method provided results with very low error rates in the solutions of fractional-order differential equations or systems of differential equations. It was observed that when the step size was chosen small enough, the error rates would also be sufficiently low. Solutions were obtained for different values of the fractional order in applications. Based on these results, it was observed that the Adams-Bashforth-Moulton method could be safely used in the numerical solutions of equations or systems of equations that do not have exact solutions or pose computational difficulties.