A dir=rtl align=right After these, what investigated is a hybridized discontinuous Galerkin (HDG) method for solving the third- and fifth-order KdV type equations which is the main part of our work . HDG methods are different from the previous DG methods. These methods were initially developed in order to address the large number of degrees of freedom that more standard DG methods define for the steady-state problems. More precisely, a discontinuous basis in DG methods cause discontinuous solutions along elements faces. So we have multivalued function evaluations at inner-element fluxes. This increases the degree of freedom in contrast to a continuous basis where function evaluations are single-valued along elements faces. In this method, at first it must be expressed the auxiliary variables in order to form the first order system of equations , and then introduced numerical fluxes inside each element in terms of the numerical traces and stabilization parameters which have an important role in stabilizing the method. Related traces are assumed to be single-valued on each face . Next , we impose conservation of the numerical fluxes via two (for third-order KdV equations) or four (for fifth-order KdV equations) extra sets of equations . Using these global flux conservation conditions, a system of linear/nonlinear of equations will be established which can be solved using a suitable solver, e.g. Newton–Raphson for the nonlinear systems . It should be noted that time variable is discretized by using a suitable difference formulae such as the backward Euler method. Stability analysis of the HDG method is investigated by extracting a cell entropy inequality. Some numerical examples are tested , in order to show that for a mesh with k-th order elements , approximate solution and its derivatives have optimal convergence at order k+1, which is comparable with or even superior to the previous methods .