A nonlocal problem for a mixed equation of high even order with a fractional Caputo derivative

摘要

Abstract This article considers a boundary value problem in a rectangular domain with a nonlocal condition for a high even order mixed type equation with a fractional Caputo derivative. For y>0documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$y >0$$end{document} the order of the fractional derivative is α∈(0,1)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$alpha in (0,1) $$end{document}, and for y<0documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$y <0$$end{document} it is β∈(1,2)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$beta in (1,2) $$end{document}. The problem is solved by the Fourier method. Sufficient conditions for the unique solvability of the formulated problem are found.