SOLUTION OF LINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS BASED ON THE OPERATOR MATRIX OF FRACTIONAL BERNSTEIN POLYNOMIALS AND ERROR CORRECTION

This is the final version of the following article: "Dynamic Analysis of the Viscoelastic Pipeline Conveying Fluid with an Improved Variable Fractional Order Model Based on Shifted Legendre Polynomials", which has been published in final form at https://www.mdpi.com/2504-3110/3/4/52International audienceIn this paper, firstly, a new method which makes a modification of the Bern-stein polynomials is introduced to solve the linear fractional partial differential equations (FPDEs). The biggest advantage of the fractional Bernstein polynomials is that the order can be changed with the order of the fractional partial differential equations. For the first time, we try to use this method to solve the linear fractional partial differential equations. Secondly, convergence analysis and error correction are also given to make the calculation results more accurate. The concrete content of this method and error correction are explained briefly and numerical examples are given to demonstrate the validity and accuracy of the method. 1. Introduction. Since 1974, Oldam and Spanier published the first book on the theory of fractional order calculus book [1], and the fractional order calculus is developed quickly. At present, the main definitions of fractional operator include Riemann-Liouvile, Caputo, Grunwald-Letnikov, Weyl, Erdelyi-Kober, Riesz and Marchaud-Hadamard [2, 3]. With the efforts of many scholars, fractional order calculus theory to a certain extent was established. Modeling problem in the field of engineering as well as the complex mechanics, physics problems promoted fractional order calculus theory and applicatio