Numerical Algorithm For Three Kinds Of Calculus Equations With Variable Coefficients Based On Legendre Polynomial Approximation

The polynomial approximation has played an important role in the mathematicalanalysis and numerical approximation theory. It has been widely used in engineeringcalculation and the actual life. The study of numerical methods for solution of calculusequations is always a significant issue in all kinds of fields. This paper based onLegendre polynomial studies the numerical algorithm of three kinds of calculusequations with variable coefficients. The method is to use Legendre orthogonalpolynomial approximation to function, combined with the definition of derivative, theoperational property of matrix, and the discrete of variable coefficient and the knownfunction to transform the original equation into a system of linear algebraic equationswhich is convenient for the Matlab programming. And the convergence speed andaccuracy of the algorithm can be improved.Firstly, this paper introduces the research background and significance of theapproximation of functions by polynomials, as well as the historical background andcurrent research status of fractional order calculus. Then the basic knowledge of thecommon fractional calculus and orthogonal polynomial is given.Secondly, in this study, a numerical method is researched to solve high-orderlinear Fredholm integro-differential equations under the mixed conditions in terms ofLegendre polynomials. Converting the equation and conditions to matrix equations bymeans of polynomial approximation and collocation method and making dense matrixtranslate into sparse matrix in the process of calculation, we can give a computingformat which solves the numerical solution of the type of integro-differentialequations.Thirdly, the matrix form of using Legendre operator matrix to represent thefractional calculus is derived by combining the shifted Legendre polynomial with theidea of operational matrix. We have discussed the numerical method of the fractionalorder Fredholm differential equation with variable coefficients by using the Legendreoperational matrix. Finally, this paper may obtain the numerical solution of fractionalconvection-diffusion equation with variable coefficients by using Legendrepolynomial. The original problem which solves the fractional partial differentialequation is translated into the problem which solves Sylvester equations. Numericalexamples prove that the method is effective.