A NOVEL APPROACH FOR THE NONLINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS USING CLIQUE POLYNOMIALS OF GRAPH

This study proposed an efficient numerical technique for nonlinear elliptic partial differential equations (EPDEs) using the functional matrix generated by Clique polynomials of Complete Graph. Recently, Graph theory has attracted many mathematicians' attention due to its wide applications. Here, Three nonlinear problems have been considered to examine the proposed scheme proficiency. Some theorems on convergence are discussed. Here, the nonlinear elliptic PDEs are rehabilitated into a nonlinear algebraic equation system using the operational matrix of Clique polynomials and collocation technique. Using the Newton-Raphson method, we numerically solved this system of algebraic equations to the desired results. The proposed scheme results are compared with the literature's analytical and other method solutions through tables and graphs. Tables and graphs are used to support the proposed technique's efficacy and accuracy. The obtained results reveal that the current approach is more accurate than other methods. The theorems are used to draw the convergent analysis for the suggested approach. From the obtained results, we can conclude that to find a numerical solution for these kinds of nonlinear EPDEs, the method is extremely effective, requires less computational effort, and is easy to implement.