A collocation method based on Bernoulli-hyperbolic bases for solving the fractional differential equation of electrohydrodynamic flow in a cylindrical conduit

Rahimkhani, Parisa; Keshavarz Hedayati, Elham

doi:10.22052/scj.2026.258074.1332

A collocation method based on Bernoulli-hyperbolic bases for solving the fractional differential equation of electrohydrodynamic flow in a cylindrical conduit

Articles in Press

Document Type : Original Article

Authors

Parisa Rahimkhani ¹

Elham Keshavarz Hedayati ²

¹ Faculty of Science, Mahallat Institute of Higher Education, Mahallat, Iran.

² Department of Mathematics Education‎, ‎Farhangian University‎, ‎P.O‎. ‎Box 14665-889‎, ‎Tehran‎, ‎Iran.

10.22052/scj.2026.258074.1332

Abstract

The present study presents an efficient numerical method for solving the fractional differential equations governing electrohydrodynamic flows. This method is based on an effective combination of Bernoulli polynomial bases and hyperbolic sine functions. In this regard, a collocation method is developed that employs combined Bernoulli–hyperbolic bases for approximation. First, the highest-order fractional derivative appearing in the equation is approximated through an expansion based on fractional-order Bernoulli polynomials and the hyperbolic sine function. Then, by applying the Riemann–Liouville fractional integral operator and imposing the boundary conditions of the problem, approximations for the solution function as well as its lower-order fractional derivatives are obtained. Finally, by forming the residual function and collocating at the nodes corresponding to the roots of Legendre polynomials, a system of nonlinear algebraic equations is derived, which is solved using Newton’s algorithm to obtain the numerical solution of the problem. Also, convergence analysis of the proposed method is presented and evaluation results show that the proposed method achieves accuracy with a small number of collocation points.

Keywords

Fractional differential equations, Electrohydrodynamic flow, Collocation method, Bernoulli&ndash

hyperbolic basis functions, Riemann-Liouville fractional integral, Caputo fractional derivative, Convergence

Subjects