New fractional derivatives applied to the Korteweg-de Vries and Korteweg-de Vries-Burger's equations

In this paper, we extend the model of the Korteweg-de Vries (KDV) and Korteweg-de Vries-Burger's (KDVB) to new model time fractional Korteweg-de Vries (TFKDV) and time fractional Korteweg-de Vries-Burger's (TFKDVB) with Liouville-Caputo (LC), Caputo-Fabrizio (CF), and Atangana-Baleanu (AB) fractional time derivative equations, respectively. We utilize the q-homotopy analysis transform method (q-HATM) to compute the approximate solutions of TFKDV and TFKDVB using LC, CF and AB in Liouville-Caputo sense. We study the convergence analysis of q-HATM by computing the Residual Error Function (REF) and finding the interval of the convergence through the h-curves. Also, we find the optimal values of h so that, we assure the convergence of the approximate solutions. The results are very effective and accurate in solving the TFKDV and TFKDVB.