A reliable algorithm to compute the approximate solution of KdV-type partial differential equations of order seven

Area of Science:

• Numerical analysis
• Computational mathematics
• Nonlinear partial differential equations

Background:

• Korteweg-de Vries (KdV)-type equations are fundamental in modeling various nonlinear phenomena.
• Solving these higher-order equations analytically is often challenging.
• Numerical methods are crucial for obtaining approximate solutions.

Purpose of the Study:

• To develop and verify an approximate solution for seventh-order KdV-type partial differential equations.
• To adapt the one-dimensional Haar wavelet collocation method for this specific class of equations.
• To assess the accuracy and efficiency of the proposed numerical scheme.

Main Methods:

• The one-dimensional Haar wavelet collocation method was employed.
• The method was applied to standard seventh-order equations: Lax, Sawada-Kotera-Ito, and Kaup-Kuperschmidt.
• Pointwise and maximum absolute errors were calculated to evaluate accuracy.

Main Results:

• The Haar wavelet collocation method provided accurate approximate solutions for the tested seventh-order KdV-type equations.
• Graphical comparisons showed a close agreement between approximate and exact solutions.
• The method proved efficient even with a limited number of grid points.

Conclusions:

• The one-dimensional Haar wavelet collocation method is a reliable and efficient technique for solving seventh-order KdV-type partial differential equations.
• The proposed numerical scheme offers adequate behavior and accurate results.
• This method provides a valuable tool for researchers studying nonlinear wave phenomena.