Numerical solution of Korteweg–de Vries equation using discrete least squares meshless method

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In this study, a meshless approach called Discrete Least Squares Meshless is used to solve the third-order nonlinear Korteweg–de Vries equation numerically. In shallow water, the Korteweg–de Vries equation illustrates the unidirectional propagation of waves. For function approximation, the suggested technique uses moving least squares shape functions. Furthermore, the least squares functional constructed on certain auxiliary points known as sampling points is minimized in order to generate the system of algebraic equations. This approach may be considered meshless since it does not need a mesh for field variable approximation or system of equations creation; it also gives a symmetric and positive definite matrix even for non-self adjoint differential operators. To demonstrate the applicability of the current meshless approach, the propagation of a single soliton and multi-solitons is explored. The approach may be applied directly to the Korteweg–de Vries equation because to the high degrees of continuity of the moving least squares shape function when an exponential weight function is utilized. Several benchmark numerical instances are explored and the resulting results are compared to earlier research to test the capabilities of the suggested approach for the solution of Korteweg–de Vries equation.