Numerical doubly periodic solution of the KdV equation with the initial condition via the decomposition method

In this paper, a scheme is extended to study numerical doubly periodic solutions of the Kortweg-de Vries (KdV) equation with initial conditions using the Adomian decomposition method. As a result, we obtain the numerical doubly periodic solutions of the KdV equation with initial conditions. The numerical doubly periodic solution is compared with the exact doubly periodic solution. Moreover we analyze the absolute error and relative error between them, and also give the contour and density plots of the numerical doubly periodic solutions with two different moduli: m=15,78.