Numerical solution of nonlinear Hunter-Saxton equation, Benjamin-Bona Mahony equation, and Klein-Gordon equation using Hosoya polynomial method

Models of nonlinear partial differential equations are essential in the theoretical sciences, especially Physics and Mathematics. It is exceedingly challenging to arrive at an analytical solution. For this reason, scientists are constantly looking for new computational techniques. For specific nonlinear mathematical models, such as the extensively researched Hunter-Saxton equation (HSE), Benjamin-Bona-Mahony equation (BBME), and Klein-Gordon equation (KGE), we are putting forth a novel and effective graph theoretic polynomial method named the Hosoya polynomial collocation method (HPCM). The orthonormalized Hosoya polynomials of the path graph and their operational matrices serve as the functional foundation for the new HPCM. The considered nonlinear models were transformed into a system of nonlinear algebraic equations using the operational matrix approximation and an appropriate collocation approach to reach the approximate solution. Six numerical examples verified the HPCM's productivity. Compared to the current numerical approaches in the literature, the results are significantly more efficient and almost match the analytical solution.