Abstract

In this paper, we consider an inverse boundary value problem for a mixed type partial differential equation with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain. The differential equation depends from another positive parameter in mixed derivatives. With respect to first variable this equation is a fractional-order nonhomogeneous differential equation in the positive part of the considering segment, and with respect to second variable is a second-order differential equation with spectral parameter in the negative part of this segment. Using the Fourier series method, the solutions of direct and inverse boundary value problems are constructed in the form of a Fourier series. Theorems on the existence and uniqueness of the problem are proved for regular values of the spectral parameter. It is proved the stability of the solution with respect to redefinition functions, and with respect to parameter given in mixed derivatives. For irregular values of the spectral parameter, an infinite number of solutions in the form of a Fourier series are constructed.

中文翻译：

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具有积分重新定义条件的混合型分数阶微分方程的逆边值问题

摘要

在本文中，我们考虑在正矩形域中使用分数积分微分的希尔弗算子和在负矩形域中使用频谱参数的混合型偏微分方程的逆边值问题。微分方程取决于混合导数中的另一个正参数。对于第一个变量，该方程在考虑部分的正部分是分数阶非齐次微分方程，对于第二个变量，在该部分的负部分是具有频谱参数的二阶微分方程。使用傅立叶级数方法，以傅立叶级数的形式构造正和反边值问题的解。对于谱参数的常规值，证明了该问题的存在性和唯一性定理。证明了该解决方案在重新定义函数方面以及在混合导数中给出的参数方面的稳定性。对于光谱参数的不规则值，构造了傅立叶级数形式的无数个解。