Taylor's expansion for fractional matrix functions: theory and applications

Abstract

In this paper, several aims and tasks have been accomplished that can be summarized in the following points. Firstly, we recover some nice results related to the convergence and radii of convergence for the matrix fractional power series formula. Secondly, the Frobenius norm approximations for the matrix fractional derivatives in Caputo sense and fractional integrals in Riemann-Liouville sense are presented. Thirdly, we present the general exact and numerical solutions of four important and interesting matrix fractional differential equations and a new computational technique is also applied for getting the general solutions of the non-linear case in Caputo sense. Finally, some illustrated examples and special cases are also given and considered to show our new approach.

Radu–Miheţ method for UHML stability for a class of ξ‐Hilfer fractional differential equations in matrix valued fuzzy Banach spaces

“…Using the fixed point technique (Alternative theorem), we study HUR stability of the ‐Hilfer fractional Equation () in MVFB space . Our results can apply to improve recent results 23 and by methods used in this paper we can extend some fractional Volterra integro‐differential equations in MVFB spaces 24–29 …”

“…Our results can apply to improve recent results 23 and by methods used in this paper we can extend some fractional Volterra integro-differential equations in MVFB spaces. [24][25][26][27][28][29]…”

Existence–Uniqueness and Wright Stability Results of the Riemann–Liouville Fractional Equations by Random Controllers in MB-Spaces

“…In 2008, Mihet and Radu [16,17] introduced a new method to investigate random stability in MB-spaces and then some authors used this method to get stability results for new equations [18][19][20][21][22][23][24][25][26][27][28][29][30]. Here, we use the Mihet and Radu method and Theorem 1 to investigate random Wright stability of (3) and improve recent results [31]; we can suggest [32][33][34] for more details.…”

Exact and numerical solutions of higher-order fractional partial differential equations: A new analytical method and some applications

“…[ 36 ]. Assume that u ( x , t ) has a multiple fractional power series (FPS) representation at of the following form: If , and can be differentiated times on for , where .…”