Probabilistic methods for nonlinear differential equations based on shift characteristic function

How to find the efficient solution of nonlinear differential equations has long been a challenging problem. To address this issue, this paper proposes a novel method from the perspective of probabilistic evolution. Firstly, the nonlinear differential equation is imposed by symmetric random initial condition, and associated with Liouville equation. Hence, the deterministic problem can be reformulated into a probabilistic evolution framework. Secondly, using the shift characteristic function (SCF) spectral expansion method integrated with Sturm Liouville theory, the nonlinear system with random initial conditions is transformed into a set of linear differential equations governing the evolution of the SCF. Here the evolutionary probability density function can be equivalently reconstructed. Moreover, the linear differential equations are generalized to a constrained differential equations problem, for which three efficient algorithms, i.e., the variable substitution one, SCF projection one, and SCF normalization one, are proposed. The proposed algorithms substantially enhance computational efficiency and accuracy, and establish a foundation for multidimensional extensions of the proposed framework. Two representative examples are provided to demonstrate the accuracy and effectiveness of the proposed methodology.