A kind of improved bivariate even order Bernoulli-type multiquadric quasi-interpolation operator and its application in two-dimensional coupled Burgers’ equations

Multiquadric quasi-interpolation is an efficient high-dimensional approximation algorithm. It can directly obtain the approximation term and its derivatives without solving any large-scale linear equations. Firstly, we study a kind of bivariate even order Bernoulli-type multiquadric quasi-interpolation operator by combining the extended univariate known multiquadric quasi-interpolation operator with the generalized Taylor polynomial as the expansion in the bivariate even order Bernoulli polynomials. Secondly, we modify a kind of multiquadric quasi-interpolation operator with the property of high degree polynomial reproducing to gridded data on multi-dimensional spaces by using multivariate divided difference to approximate the partial derivatives. Thirdly, error bounds of the modulus of continuity of high order and Peano representation for the error are given. Numerical experiments verity the effectiveness and high accuracy of the scheme. Furthermore, applying the proposed quasi-interpolation operator, we present the explicit algorithm of numerical differentiation. Several examples to the fitting of discrete solutions of two-dimensional coupled Burgers’ equations are given.