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In this short monograph Newton-likeand other similar numerical methods with applications to solving multivariateequations are developed, which involve Caputotype fractional mixed partial derivatives and multivariate fractional Riemann-Liouvilleintegral operators. These are studied for the first time in the literature. Thechapters are self-contained and can be read independently. An extensive list ofreferences is given per chapter. The book's results are expected to findapplications in many areas of applied mathematics, stochastics, computerscience and engineering. As such this short monograph is suitable forresearchers, graduate students, to be used in graduate classes and seminars ofthe above subjects, also to be in all science and engineering libraries.