Adaptive Multidomain Numerical Solution for Singularly Perturbed Fractional Differential Equation: Chebyshev Pseudospectral Method

Abstract Singularly perturbed differential equations pose significant challenges for many numerical and semi-analytical solution methods in classical calculus. The complexity of these equations increases when dealing with fractional differential equations, which exhibit unique properties, including the nonlocal nature of the arbitrary non-integer order differential operators. This study presents an adaptive multidomain Chebyshev pseudospectral method to effectively approximate the solutions of singularly perturbed fractional differential equations. In each subdomain, the fractional differential operator in the Caputo sense accounts for both local derivative effects and memory by dividing the discrete fractional operator into two distinct components, one representing local behavior and the other capturing memory effects. The adaptivity of the method is controlled by ensuring that the maximum residual error in each domain remains below a predetermined tolerance value. If the maximum error exceeds the set tolerance, the size of the subinterval is reduced by a specified ratio, and the solution is recomputed within that interval. To assess the performance of the adaptive method, a comparative study with the multidomain pseudospectral method with uniform step length is used. The accuracy of the adaptive multidomain Chebyshev pseudospectral method is validated through a series of numerical tests on various fractional differential equations, thus demonstrating the effectiveness of the approach. The adaptive multidomain pseudospectral method offers a robust solution for handling the challenges posed by singularly perturbed differential equations common in fields such as wave propagation, astrophysics, and quantum mechanics.