Unique analytical scheme for Fokker-Planck equation by the Matching polynomials of complete graph

The article explores the feasibility of the graph-theoretic polynomial strategy to address the Fokker-Planck equation (FPE) employing a unique Matching Polynomial Collocation Method. Adriaan Fokker and Max Planck invented the FPE in the early twentieth century to characterize Brownian motion, and it has since grown into a cornerstone of stochastic process analysis, featuring significance in physics, biology, and economics. MPCM constructs an innovative functional matrix of integration leveraging the functional basis of matching polynomials of complete graphs, successfully translating the FPE into a system of algebraic equations with equipped collocation points. Newton's Raphson method follows to solve the consequent nonlinear algebraic equations. The proposed approach efficiently fixes technical challenges intrinsic to the FPE, including discretization errors, nonlinear encounters, substantial dimensionality, boundary conditions, stiffness, and computing costs. Illustrative samples spanning linear and nonlinear FPEs reflect MPCM's precision, computational efficacy, and versatility, with findings being consistent with well-established numerical and analytical strategies. The investigation highlights MPCM's potential as a resilient, versatile tool, paving the way for prospective studies into higher-dimensional issues and potential uses in various empirical fields, including quantum physics, demographic dynamics, and economic modeling.