Probabilistic approach to numerical solution of the Cauchy problem for nonlinear parabolic equations

Abstract

Nonlinear partial differential equations (PDEs) of parabolic type are of great interest both in theoretical and applied aspects. To mention just a few, the Burgers equation, Fisher—Kolmogorov—Petrovskii—Piskunov (FKPP) and Ginzburg—Landau equations, equations with nonlinear diffusion and with blow-up solutions, reaction-diffusion systems are examples of parabolic nonlinear PDEs. They are suggested as mathematical models of problems in many fields such as fluid dynamics, filtration, combustion, biochemistry, dynamics of populations, etc. Nonlinear PDEs are usually not susceptible of analytic solution and mostly investigated by means of numerical methods. Their investigation is presented in many publications in which deterministic approaches are applicable (see, e.g., [42, 91,143, 259, 269, 291] and references therein). A few authors only exploit probabilistic approaches (see [70,141,288] and references therein).