A minimal search method for solving fractional integro-differential equations based on modified Legendre multiwavelets

This paper improves the modified multiwavelets bases of minimal search method for the fractional integro-differential equation. First, it shows the unique solvability of the equation. And then the Legendre multiwavelets are improved and the modified multiwavelets in reproducing kernel space are obtained. Subsequently, it is established a strict theory for obtaining the ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}-approximate solution with minimal search method. Finally, some examples show that the modified continuous multiwavelets method is more effective and stable than the Legendre multiwavelets and other methods.