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 are 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.