Robin Type Problem for a Degenerate Equation of Even Order with Caputo Fractional Derivative

Abstract

In this article, for a degenerate inhomogeneous equation of even order with a fractional derivative in the sense of Caputo, a boundary value problem of the Robin problem type is studied. The solution is constructed in the form of a series of eigenfunctions of a one-dimensional spectral problem for a degenerate equation of even order. The existence and positivity of eigenvalues are shown by reducing the spectral problem to an equivalent integral equation with a symmetric kernel. Also, when constructing the posed problem, a boundary value problem for a one-dimensional fractional order equation was studied. Depending on the sign of the constant coefficient of the equation q, the necessary estimates for the solution were obtained. Sufficient conditions for the convergence of the series, which is a solution to the Robin problem, and the series obtained by differentiation are found. The uniqueness of the solution is demonstrated by the spectral method.

Keywords

Differential equation / Even order / Robin problem / Caputo derivative / Eigenfunction / Eigenvalue / Asymptotics / Series / Convergence / Existence / Uniqueness

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Shanghai University