Coupled Kansa and hybrid optimization methodological approach for Kolmogorov–Feller equations

Abstract

Methodological approaches coupling a meshfree method with multiquadric radial basis function and a hybrid optimization method are elaborated to solve nonlinear stochastic differential equations with a Poisson white noise. The associated probability density function is governed by the Kolmogorov Feller equation named also Generalized Fokker–Planck equation. Actually, the shape parameter of RBFs affects the numerical solution strongly. Choosing arbitrary this parameter may lead to inaccurate solutions and thus parameter should be chosen. A hybrid optimization method is elaborated to predict an efficient shape parameter based on a well adapted residual error. The resulting coupled procedure is applied to the considered stationary and instationary generalized Fokker-Planck equations using an implicit time scheme with a regularized initial condition. The multiquadric radial basis function with optimized parameter is also coupled with the Monte Carlo method. The effectiveness and accuracy of the elaborated methodological approaches are demonstrated and the probability density functions are computed for several nonlinear stochastic differential equations under a Poisson white noise.