1 research outputs found
Using ANNs Approach for Solving Fractional Order Volterra Integro-differential Equations
Indeed, interesting properties of artificial neural networks approach made this non-parametric model a
powerful tool in solving various complicated mathematical problems. The current research attempts
to produce an approximate polynomial solution for special type of fractional order Volterra integrodifferential
equations. The present technique combines the neural networks approach with the power
series method to introduce an efficient iterative technique. To do this, a multi-layer feed-forward neural
architecture is depicted for constructing a power series of arbitrary degree. Combining the initial conditions
with the resulted series gives us a suitable trial solution. Substituting this solution instead of the
unknown function and employing the least mean square rule, converts the origin problem to an approximated
unconstrained optimization problem. Subsequently, the resulting nonlinear minimization problem
is solved iteratively using the neural networks approach. For this aim, a suitable three-layer feed-forward
neural architecture is formed and trained using a back-propagation supervised learning algorithm which
is based on the gradient descent rule. In other words, discretizing the differential domain with a classical
rule produces some training rules. By importing these to designed architecture as input signals, the
indicated learning algorithm can minimize the defined criterion function to achieve the solution series
coefficients. Ultimately, the analysis is accompanied by two numerical examples to illustrate the ability
of the method. Also, some comparisons are made between the present iterative approach and another
traditional technique. The obtained results reveal that our method is very effective, and in these examples
leads to the better approximations