Abstract
We propose a reproducing kernel method for solving the KdV equation with initial condition based on the reproducing kernel theory. The exact solution is represented in the form of series in the reproducing kernel Hilbert space. Some numerical examples have also been studied to demonstrate the accuracy of the present method. Results of numerical examples show that the presented method is effective.
Highlights
Introduction− ∞ < x < ∞, t > 0, with initial condition u (x, 0) = f (x)
In this paper, we consider the Korteweg-de Vries (KdV) equation of the form ut (x, t) + εu (x, t) ux (x, t) + uxxx (x, t) = 0, (1)− ∞ < x < ∞, t > 0, with initial condition u (x, 0) = f (x) . (2)The constant factor ε is just a scaling factor to make solutions easier to describe
Some mathematicians and physicians investigated the exact solution of the KdV equation without having either initial conditions or boundary conditions [1], while others studied its numerical solution [2, 3]
Summary
− ∞ < x < ∞, t > 0, with initial condition u (x, 0) = f (x) . Some mathematicians and physicians investigated the exact solution of the KdV equation without having either initial conditions or boundary conditions [1], while others studied its numerical solution [2, 3]. The numerical solution of KdV equation is of great importance because it is used in the study of nonlinear dispersive waves. This equation is used to describe many important physical phenomena. Some of these studies are the shallow water waves and the ion acoustic plasma waves [4]
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