We construct, for the first time, a Mikusiński-type operational calculus structure for partial differential operators of non-integer order. Our operators are of Riemann–Liouville type, and in arbitrary dimensions, although we often focus on the two-dimensional case as a model problem. We establish suitable function spaces, algebraic properties, and interpretations of multi-dimensional fractional integral and derivative operators. As an example application, we consider a fractional differential equation in two dimensions posed on the first quadrant, and find its explicit solution using Wright functions.

中文翻译：

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Mikusiński 的多维分数算子运算演算及其在分数偏微分方程中的应用


我们首次为非整数阶偏微分算子构建了 Mikusiński 型运算演算结构。我们的算子是黎曼-刘维尔类型的，并且是任意维度的，尽管我们经常关注二维情况作为模型问题。我们建立合适的函数空间、代数性质以及多维分数积分和导数算子的解释。作为示例应用，我们考虑第一象限上的二维分数阶微分方程，并使用赖特函数找到其显式解。