A new approach to numerical solution of the time-fractional KdV-Burgers equations using least squares support vector regression

The evolution of the waves on shallow water surfaces is described by a mathematical model given by nonlinear KdV and KdV-Burgers equations. These equations have many other applications and have been simulated by classical numerical methods in recent decades. In this paper, we develop a machine learning algorithm for the time-fractional KdV-Burgers equations. The proposed method implements a linearization of the problem and a time reduction by a Crank-Nicolson scheme. The least squares support vector regression (LS-SVR) is proposed to seek the approximate solution in a finite-dimensional polynomial kernel space. The Bernstein polynomials are used as the kernel of the proposed algorithm to handle the homogeneous boundary conditions easily in the framework of the Petrov-Galerkin spectral method. The proposed LS-SVR implements the orthogonal system of Bernstein-dual polynomials in the learning process, which gives quadratic programming in the primal form and provides a linear system of equations in dual variables with sparse positive definite matrices. It is shown that the involving mass and stiffness matrices are sparse. Some new theorems for the introduced basis are provided. Also, numerical results are presented to support the spectral convergence and accuracy of the method.

Keywords:

Fractional KdV equation , Machine learning , support vector machines , Petrov-Galerkin , least squares support vector regression