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This paper applies the concepts of fuzzifying functions to the probability density function of a random variable and introduce a fuzzifying probability to better understand the probability arising from the uncertainties of the probability density function. Using the fuzzifying probability, we derive the fuzzifying expected value and the fuzzifying variance of a random variable with the fuzzifying probability density function. Additionally, we provide examples of a fuzzifying probability density function to validate that the proposed fuzzy concepts generalize crisp expected value and variance in probability theory.

In this article, we provide a comprehensive convergence and stability analysis of a semi-Lagrangian scheme for solving nonlinear Burgers' equations with a high-order spatial discretization. The analysis is for the iteration-free semi-Lagrangian scheme comprising the second-order backward finite difference formula (BDF2) for total derivative and the fourth-order central finite difference for diffusion term along the trajectory. The main highlight of the study is to thoroughly analyze the order of convergence of the discrete $ \ell_2 $-norm error $ \mathcal{O}(h^2+\triangle x^4+ \triangle x^{p+1}/h) $ by managing the relationship between the local truncation errors from each discretization procedure and the interpolation properties with a symmetric high-order discretization of the diffusion term. Furthermore, stability is established by the uniform boundedness of the numerical solution using the discrete Grönwall's Lemma. We provide numerical examples to support the validity of the theoretical convergence and stability analysis for the propounded backward semi-Lagrangian scheme.