Numerical solutions of fractional order integro-differential equations by using artificial neural networks and power series method

This study presents a neural network-based framework for finding approximate series solutions to Abel's integral equation (AIE) and fractional order Volterra-type integro-differential equations (FOVIDEs). The suggested method changes the problem by writing the solution as a power series that has been truncated. This makes the original differential equation a problem of estimating parameters. Then, a special neural network is trained to find the coefficients of the series with good accuracy. Numerical tests show that the proposed method works, maintains accuracy, and converges. This shows that it can be used as a strong computational tool for solving fractional order systems.