Numerical Hilbert space solution of fractional Sobolev equation in 1+1-dimensional space

Abstract

Abstract This paper equipped the time- and space-fractional Sobolev equation with the condition of the Caputo fractional derivative in 1+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left(1+1\right)$$\end{document} -dimensional space to be solved with the help of the reproducing kernel Hilbert space method. The aforesaid method depends on building two Hilbert spaces where the coefficients of fractional expansion are produced by using the generalized Gram Schmidt process. With the use of the Fourier functions expansion theorem, the numeric-analytic solutions are expressed by collection sets of orthonormal functions system in AU\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{A}\left(\mathcal{U}\right)$$\end{document} and BU\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{B}\left(\mathcal{U}\right)$$\end{document} spaces. In this flair, novel steps are fitted for the covering fractional Sobolev equation and the utilized numeric-analytic approach. To display the obtained results and theories, a variety of tables and graphics will be displayed and exhibited. The received effects imply that the technique is shrewd and has numerous capabilities balance for managing many fractional fashions rising in physics and applied mathematics by the Caputo class derivative. Future work and several notes are collected in the final part.