The Numerical Approximation of Nonlinear Functionals and Functional Differential Equations
It solves a long-standing bottleneck in computational physics by providing a numerical approach to functional differential equations, which previously lacked effective methods.
This work introduces the first effective numerical method for solving functional differential equations, which are crucial in fluid dynamics, quantum field theory, and statistical physics. The method enables direct computation of solutions to equations like the Hopf characteristic functional equation.
The fundamental importance of functional differential equations has been recognized in many areas of mathematical physics, such as fluid dynamics (Hopf characteristic functional equation), quantum field theory (Schwinger-Dyson equations) and statistical physics (equations for generating functionals and effective Fokker-Planck equations). However, no effective numerical method has yet been developed to compute their solution. The purpose of this report is to fill this gap, and provide a new perspective on the problem of numerical approximation of nonlinear functionals and functional differential equations.