The numerical solution of fractional Korteweg‐de Vries and Burgers' equations via Haar wavelet

Abstract

In this article, Haar wavelet collocation technique is adapted to acquire the approximate solution of fractional Korteweg‐de Vries (KdV), Burgers', and KdV–Burgers' equations. The fractional order derivatives involved are described using the Caputo definition. In the proposed technique, the given nonlinear fractional differential equation is discretized with the help of Haar wavelet and reduced to the nonlinear system of equations, which are solved with Newton's or Broyden's method. The proposed method is semi‐analytic as it involves exact integration of Caputo derivative. The proposed technique is widely applicable and robust. The technique is tested upon many test problems. The results are computed and presented in the form of maximum absolute errors which show the accuracy, efficiency, and simple applicability of the proposed method.

Comparative Study of Solutions of Fractional Order Mixed KdV Burger’s Equation

“…(1,2) . Since many fractional order non-linear differential equations are not easy to solve, hence we prefer numerical methods like finite difference method (3) , Adomian Decomposition Method (4)(5)(6) and Homotopy Perturbation Method (7) which gives approximate solutions.…”

“…Further, in 1969, Su and Gardner derive Kdv Burger's equation to study different physical properties of wave propagation. Many researchers studied numerical methods like the Homogeneous Balance Method, Special Truncated Expansion Method, Exp-function Method, Variational Iterations Method, Element-Free Galerkin Method and New Decomposition Method to find the exact https://www.indjst.org/ solutions of a compound KdVB equation (7)(8)(9)(10)(11) . Fractional derivatives offer a simpler yet more comprehensive description of physical phenomena compared to conventional integer-order derivatives, rendering them more effective and versatile in various applications.…”