Abstract
In the paper, a space fractional Benjamin-Bona-Mahony (BBM) equation is proposed. For the direct problem, we develop the Chebyshev-Legendre spectral scheme for the proposed equation. The given approach is based on Legendre Galerkin formulation while the Chebyshev-Gauss-Lobatto (CGL) nodes are used in the computation. By using the presented method, the computational complexity is reduced and both accuracy and efficiency are achieved compared with the Legendre spectral method. Stability and convergence analysis of the numerical method are proven. For the inverse problem, the Bayesian method is developed to estimate some relevant parameters based on the spectral format of the direct problem. The convergence analysis on the Kullback-Leibler distance between the true posterior distribution and the approximation is derived. Some numerical experiments are included to demonstrate the theoretical analysis.
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