COMPARISON OF NUMERICAL AND EXACT SOLUTIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS WITH CONSTANT DELAY

Abstract

Since it is well known that the analytical solution of fractional differential equations with delays is challenging, if not impossible, we have compared the exact solutions of two such equations using four numerical methods: Fractional Euler Method (FEM), Backward Euler Method (BEM), Weighted Difference Method (WDM), and Fractional Adams Method (FAM) in this study. The efficiency of each numerical approach is evaluated based on the relative difference between the exact and approximate solutions at predetermined time points, presented graphically and in tabular form. To provide a comprehensive comparison, the execution time is calculated by taking the average of over 10 executions for each method. From the results, we conclude that FAM has the lowest relative errors, is more accurate, and has a lower computational cost. FAM is the best alternative for accurate solutions and short execution times over extended timescales, FEM offers satisfactory accuracy and efficiency at intermediate timescales, WDM shows progress over time but has higher computational costs, and BEM consistently exhibits significant error rates and the variability of the execution time in the two examples.