A new graph-theoretic approach for the study of the surface of a thin sheet of a viscous liquid model

Abstract

In this study, we considered the model of the surface of a thin sheet of viscous liquid which is known as the Buckmaster Equation (BME), and presented a new graph-theoretic polynomial collocation method named the Hosoya polynomial collocation method (HPCM) for the solution of nonlinear Buckmaster equation. In the literature, the majority of the developed numerical methods considered small time step sizes like 0.01s and 0.05s to obtain relatively accurate approximations for the nonlinear BME. This study focused on optimizing the time step sizes by adopting bigger time steps sizes like 1.0s,3.0s, and 5.0s, etc. without adversely affecting accuracy. First, using the Gram- Schmidt process, we generated the orthonormal functions from the Hosoya polynomial of the path graph. Then developed the functional integration matrix using orthonormal Hosoya polynomials of path graphs. With this active matrix-involved method, the nonlinear BMEs are transformed into a system of nonlinear equations and solved the equations by Newton's method through the Mathematica software for unknown coefficients. The exactness of the proposed strategy is tested with two numerical examples. The acquired results contrasted with the current analytical solutions to these problems. Also provided the convergence analysis, comparison of error norms, graphical plots of correlation of HPCM results, and the results of other numerical methods in the literature to validate the productivity and accuracy of the newly developed HPCM.

A numerical investigation of a well-known nonlinear Newell-Whitehead-Segel equation using the rank polynomial of the star graph

“…Graph polynomials are employed in several other disciplines, such as economics and traffic control. They are also valuable for developing efficient analytical and numerical techniques for solving mathematical models [26][27][28][29]. These uses highlight the adaptability and strength of graph polynomials in the resolution of challenging mathematical models.…”