A numerical method for finding solution of the distributed‐order time‐fractional forced Korteweg–de Vries equation including the Caputo fractional derivative

In this paper, for the first time, the distributed‐order time‐fractional forced Korteweg–de Vries equation is studied. We use a numerical method based on the shifted Legendre operational matrix of distributed‐order fractional derivative with Tau method to find approximate solution of distributed‐order forced Korteweg–de Vries equation. This shifted Legendre operational matrix of distributed‐order fractional derivative with Tau method is used to reduce the solution of the distributed‐order time‐fractional forced Korteweg–de Vries equations to a system of algebraic equations. An error analysis and convergence are obtained. Finally, to display the applicability and validity of the numerical method, some examples are implemented.