Studying the stability of variable order caputo fractional differential equations

Introduction:

Fractional calculus is a branch of mathematical analysis that studies the possibility of extending the order of the differentiation and integration operators to a noninteger order. We focus on the stability of the implicit boundary value problem for Caputo fractional differential equations of variable order.

where is a continuous function, , and is the caputo fractional derivative of variable order.

Ulam Hyers stability: Assuming

be a partition of the interval and let be a PWCF with respect to , i.e., where are constants with (). Using (H1) the BVP(1) becomes

Theorem 1:

The (1) is (UH) stable if there exists such that for any , and for every solution of the following inequality , there exists a solution of (1) with

Theorem 2: Assume that (H1) is satisfied and

(H2)

and the inequality holds, then (1) is (UH) stable.

Conclusion:

This study is a valuable contribution to the expanding field of fractional calculus, in which we skillfully proved the stability.