Fractional Galilei invariant advection–diffusion (GIADE) equation, along with its more general version that is the GIADE equation with nonlinear source term, is discretized by coupling weighted and shifted Grünwald difference approximation formulae and Crank–Nicolson technique. The new version of the backward substitution method, a well-established class of meshfree methods, is proposed for a numerical approximation of the consequent equation. In the present approach, the final approximation is given by the summation of the radial basis functions, the primary approximation, and the related correcting functions. Then, the approximation is substituted back to the governing equations where the unknown parameters can be determined. The polynomials, trigonometric functions, multiquadric, or the Gaussian radial basis functions are used in the approximation of the GIADE. Moreover, a quasilinearization technique is employed to transform a nonlinear source term into a linear source term. Finally, three numerical experiments in one and two dimensions are presented to support the method.

We show the boundedness for the commutator of Bochner-Riesz operator on some weighted H1 space

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Small and medium-sized technological enterprises in Hebei Province are featuring in great innovation, high increase; large cost, high risk and high profits, and difficulty of financing as well as great risks bring obstacles for those private enterprises to industrialize their high-tech products. Thus, the enterprises cannot be truly developed and their economic potential cannot be fully tapped. This paper, against this backdrop, adopts the combination of qualitative analysis and quantitative analysis to make evaluation upon financing risks and builds up a risk pre-warning evaluation systems of corporate financing. Based on analytical hierarchy process, this paper evaluates the influences of this system on financing risks to verify the theoretical researches in this paper and provide reference for financing risk management in private technical small and medium-sized enterprises.