REMARK ON STABILITY OF FRACTIONAL ORDER PARTIAL DIFFERENTIAL EQUATION

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, using a fractional order partial derivative with non-singular kernel we investigate, the stability and its generalization on semi-closed and semi-open interval for the solution of a fractional order partial differential equation with the help of an inequality. In this paper, we will consider the following fractional order partial differential equation


INTRODUCTION
In recent years, the fractional calculus plays an significant role in numerous fields, such as a pure and applied Mathematics, Science and Engineering Technology.
Furthermore, this is concerned with existence of mild solution of evolution with Hilfer fractional derivative generalized the well-known Riemann -Liouville fractional derivative by noncompact measure method and acquire some sufficient conditions to make certain the existence of mild solution [11], An initial value problem for a class of non-linear fractional differential equations concerning Hilfer fractional derivative and prove the existence and uniqueness of universal solutions in the space of weighted continuous functions. Also analyze the stability of the solution for a weighted Cauchy -type problem [12].
They found the existence and the uniqueness of a positive solution in the space of weighted continuous functions and boundary performance of such solution [13], Particularly, sufficient conditions for the existence of solution for a class of initial value problems for impulsive fractional differential equations connecting the Caputo fractional derivative [14].
Additionally, the existence and uniqueness of a solution of a class of initial boundary value problems for implicit fractional differential equations with fractional derivative and the outcome are based upon technique of measures of compactness and the fixed point theorems of Darbo and Monch [15], in addition the existence and uniqueness results for implicit differential equations of Hilfer type fractional order via Schaeder's fixed point theorem and Banach contraction principle [16], Ulam stability and data dependence for fractional differential equations with Caputo fractional derivative of order α and presents four types of Ulam stability results for the fractional differential equation [17], Ulam ψ -Hilfer fractional derivative and present the Hyers -Ulam -Rassias stability and the Hyers-Ulam stability of the fractional Voltera integral -differential equation by means of fixed point method [18].
We organize this paper as : In the second section, we define some basic definitions and notations, in the third section, we investigate the stability and its generalization on semi-closed and semi-open interval for the solution of a fractional order partial differential equation with the help of inequality.

TECHNICAL BACKGROUND
In this section, we use some definitions and notations which are given in [1] with details and present technical preparation needed for further discussion.
In particular, take N = 3, ..., a N and θ 1 , θ 2 ...., θ N are positive constants. Also let u, ψ ∈ C n (Ĩ, R) are two functions such that ψ is increasing and ψ ( We use notation is non -negative and non decreasing then, for any t ≥ α and being E α (.) the one-parameter Mittag-Leffler function.
The right-sided fractional integral is defined in an analogous form.
be two functions such that ψ is increasing and ψ (x) = 0, for all x ∈ I. The left-sided ψ -Hilfer fractional derivative H D α,β ,ψ a + (.) of a function of order α and The right-sided ψ-Hilfer fractional derivative is defined in an analogous form.
For each function y satisfying there is a solution y 0 of the fractional integro-differential equation and a constant C > 0 independent of y and y 0 such that for all x ∈ [a, b], then we say that the integro-differential equation has the Ulam-Hyers stability.
[1] If for each function y satisfying x a K(x, τ, y(τ), y(δ (τ)))dτ) |≤ θ x ∈ [a, b], where θ ≥ 0, there is a solution y 0 of the fractional integro-differential equation and a constant C > 0 independent of y and y 0 such that x ∈ [a, b], for some non-negative function σ defined on [a, b] , then we say that the fractional integro-differential equation has the so-called semi-Ulam-Hyers-Rassias stability.
[1] (Banach) Let (X, d) be a generalized complete metric space and T : X → X a strictly contractive operator with Lipschtiz constant L > 1. If there exist a non-negative integer k such that d(T k+1 x, T k x) < ∞ for some x ∈ X, then the following three propositions hold true: (i) The sequence (T n x) n∈N converges to a fixed point x * of T; (ii) x * is the unique fixed point of T in X * = {y ∈ X; d(T k x, y) < ∞}; (iii) If y ∈ X * , then

MAIN RESULT
In this paper, our aim is to investigate the stability and its generalization on semi-closed and semi-open interval for the solution of a fractional order partial differential equation with the help of inequality.
For our convenience in the calculations, we consider the following set of considerations and and u satisfies (0.1)  real number C 1 f ,φ ,C 2 f ,φ ,C 3 f ,φ and C n f ,φ > 0 such that, for any ∈> 0 and for any solution v to the inequality (3.2), with In the similar manner, we have the inequalities Remark 2: If function v is a solution to the inequality (3.1), if and only if there exist a function is a solution of the following system of integral inequalities.
Then, we have a) for h ∈ C ([0, a), B), g ∈ C ([0, b), B) and k ∈ C ([0, c), B), the equation (3.2) has a unique solution with is a solution to the system.