To portray the experimentally observable stochastic nature of oxidation of defected graphite,
the piecewise linear, two-stage model was derived in the previous work to which the current
contribution is a sequel. The model takes into account all the major features of the classic
deterministic, graphite oxidation model of Nagle and Strickland-Constable (NSC) including the
simultaneous conversion of the identity of an adjacent basal cluster to an edge cluster. The NSC
model assumes that there are secondary reactions in the second stage due to the decrease in basal
clusters. The model, however, contains a noticeable uncertainty as to the identification of the
breaking point caused by the transition between the two linear stages. This uncertainty is eliminated
by incorporating in the nonlinear stochastic model proposed in the present work a parameter which
renders the extent of the secondary reactions proportional to the concentration of basal clusters in
the carbon matrix. The validity of incorporating this parameter has been amply demonstrated by the
fact that the variances around the means, derived through the method of system-size expansion of
the nonlinear master equation of the current model, are appreciably less than those of the previous
model.

The aim of this work is to study the existence and Ulam stability of solution of Hilfer fractional implicit differential equation with boundary condition of the form

EDα,β

α,β

0+  x(t) = f (t, x(t), D0+ x(t)),       t ∈ [0, T],

(0.1)

I1−γ

1−γ

0+    x(0) = a,    I0+    x(T) = b,     γ = α + β − αβ.

Here Dα,β  is the Hilfer fractional derivative, I1−γ  is the left-sided mixed Riemann-Liouville integral of

0+                                                                       0+

order 1 − γ, α ∈ (0, 1), β ∈ [0, 1] and let X be the Banach space,  f  : J × X × X → X is given continuous

function. The results are established by the application of the contraction mapping principle  and

Schaefer’s fixed point theorem. An example is provided to illustrate the applicability of the results.