Romanovski–Jacobi spectral collocation schemes for distributed order differential problems

Abstract

The article highlights the growing importance of computational mathematics in tackling intricate real-world phenomena governed by distributed order fractional dynamics. These dynamics, characterized by memory effects and non-local interactions, are pervasive in various fields such as physics, biology, finance, and engineering. Despite their prevalence, analytical solutions for distributed order fractional differential equations remain difficult to find, necessitating the development of robust numerical methods. The paper explores the effectiveness of Romanovski–Jacobi spectral collocation techniques in this context and provides a thorough analysis of how to use them to solve distributed order fractional differential equations. By leveraging these schemes, researchers gain valuable insights into efficiently analyzing and simulating fractional systems, thereby advancing our understanding of complex dynamics across diverse domains. It provides precise numerical findings by using finite expansion to evaluate residuals. The precision of the approach is illustrated by numerical simulations, especially in distributed order fractional differential equations. Additionally, we provide a few numerical tests to demonstrate the method’s ability to preserve the underlying problem’s non-smooth solution.