ABSTRACT

Multivariate Mittag-Leffler functions are a strong generalisation of the univariate and bivariate Mittag-Leffler functions which are known to be important in fractional calculus. We consider the general functional operator defined by an integral transform with a multivariate Mittag-Leffler function in the kernel. We prove an expression for this operator as an infinite series of Riemann–Liouville integrals, thereby demonstrating that it fits into the established framework of fractional calculus, and we show the power of this series formula by straightforwardly deducing many facts, some new and some already known but now more quickly proved, about the original integral operator. We illustrate our work here by calculating some examples both analytically and numerically, and comparing the results on graphs. We also define fractional derivative operators corresponding to the established integral operator. As an application, we consider some Cauchy-type problems for fractional integro-differential equations involving this operator, where existence and uniqueness of solutions can be proved using fixed point theory. Finally, we generalise the theory by applying the same operators with respect to arbitrary monotonic functions.

摘要

多元 Mittag-Leffler 函数是单变量和双变量 Mittag-Leffler 函数的强泛化，已知这些函数在分数微积分中很重要。我们考虑由内核中的多元 Mittag-Leffler 函数的积分变换定义的通用函数算子。我们证明了这个算子的表达式是黎曼-刘维尔积分的无限级数，从而证明它符合已建立的分数阶微积分框架，我们通过直接推导许多事实来展示这个级数公式的威力，一些新的和一些已经已知但现在更快地证明了，关于原始积分算子。我们通过分析和数值计算一些例子来说明我们的工作，并在图表上比较结果。我们还定义了与已建立的积分算子相对应的分数导数算子。作为一个应用，我们考虑了一些涉及该算子的分数积分微分方程的柯西型问题，其中可以使用不动点理论证明解的存在性和唯一性。最后，我们通过对任意单调函数应用相同的算子来推广该理论。