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.
