A priori estimate for resolving the boundary fractional problem

1.   Introduction

Recently fractional calculus regained ground with its use in various fields, such as physical and chemical processes that occur in media with fractal geometry, as well as in the mathematical modeling of economic and social-biological phenomena [1, Chap. 5, 6, 11]. The reader can also find in the work of Alikhanov ^[1], Ladyzhenskaya ^[5], and Nakhushev ^[8] other rich domains in terms of applications of fractional calculus such as Abel's integral equation ^[2], fractional-order capacitor models ^[6,7], electroelectrical chemistry ^{[3,4,8,9,10,11]}. Alikhanov ^[1] used the method of energy inequalities to obtain a priori estimates for solutions of boundary value problems for the diffusion-wave equation with the Caputo fractional derivative. Mesloub's ^[6] paper deals with a fractional two-times evolution equation associated with initial and purely boundary integral conditions. Our work was inspired by the latter but with a fractional one-time evolution equation with boundary integral conditions. In the present paper, we use the method of energy inequalities to obtain a priori estimates to prove the uniqueness and existence of the solution using some functional inequalities, the Caputo derivative and Caputo integrals. In Section 1, we formulate and set our nonlocal initial boundary value problem. In Section 2, we introduce some useful preliminaries and notations, and reformulate the problem (1.1)–(1.2) as a problem with homogeneous boundary conditions, (1.3) and (1.4) to simplify the calculations. In Section 3, we apply the a priori estimation method to prove the uniqueness of the solution. We choose a certain functional differential operator multiplier (3.3). In Section 4, we discuss the existence of a solution. We always apply the same method.

1.1.   Position of the problem

In the bounded domain, ${Q^{T} = (0, T)\times (0, \alpha)}$ such that ${0 < \alpha < \infty, 0 < T < \infty}$.

We consider the time fractional order problem with the Caputo-derivative.

The functions $a(x, t), b(x, t)$ and $c(x, t),$\ satisfy the conditions

where

With problem (1.1), we associate the initial conditions

and the boundary conditions

2.   Prelemanaries

For a function $f(t),$ which has absolutely continuous derivatives up to order (n -1), the Riemann-Liouville fractional derivative of arbitrary order $n-1\leq \beta \leq n$ is defined as follows ^[6]:

where $\Gamma$ is the well-known gamma function. The following formula is true:

The Riemann-Liouville fractional integral of order\ $\beta,$ \ $0\leq \beta\leq n,$\ is defined by

The Riemann-Liouville fractional derivative is singular at the origin because of $\left(2.3\right)$. Because of this fact, fractional differential equations in the sense of the Riemann-Liouville fractional derivative require initial conditions of a special form lacking clear physical interpretation. These shortcomings do not occur with the regularized Caputo derivative. The Caputo derivative of order $\beta,$ n-1$\leq \beta \leq n, n\in IN,$ is defined as the integral part of $\left(2.3\right)$; that is,

In the case of our problem, the Caputo and Riemann-Liiouville derivatives of order $\beta, \; 0 \; \leq \beta \leq 1$, and their linked relationship formulas are defined as follows ^[4] $:

(1) The Cauchy $\varepsilon -$inequality:

(2) $Poincar\acute{e}$-type inequalities ^[3] $:

(3) Lemma. For any function $f(t)$ absolutely continuous on [0, T], owe have the inequality

Because the proofs are based on a priori estimates, we first write (1.1)–(1.3) in an equivalent operator form in order to establish the existence and uniqueness of the solution. The solution to (1.1)–(1.3) can be regarded as the solution of the operator equation

where $L = (\mathfrak{L}, l_{1})$, and operator $L$ acts from B to F, with the domain of definition

where B is the Banach space of functions $u$ endowed by the finite norm

$F$ is a Hilbert space equipped with the scalar product:

The associated norm is

3.   Uniqueness of the solution

Theorem 1. In $Q^{T}$ and for sufficiently small $\varepsilon$, there exists a positive constant K independent of $u$ such that

for all $u\in D\left(L\right)$, where

and

Proof. Let $u\in D(L),$ and consider the equality

where

Upon integration by parts and conditions (1.1)–(1.3), we evaluate each term of (3.4) to obtain

Using the same method, we have

Substituting (3.5), (3.6), (3.7) and (3.8) into (3.5), we get

By using $poincar\acute{e}$-type inequalities and a Cauchy-$\varepsilon$ inequality estimate last five terms of the left side of (3.9), we get

Combining of equalities (3.10), (3.14) and (3.9) yields

Consequently,

where

Estimating the first term on the left side of (3.15), we get

Thus, inequality (3.16) takes the form

where

4.   Solvability of the posed problem

To establish the existence of the solution to problems (1.1)–(1.6), we argueusing density argument. That is, we show that $Im L$, the image of the operator L is dense in the space $L^{2}(Q)$ for every element $u$ in the Banach space B. For this, we consider the following theorem:

Theorem 2. For all functions $u\in B,$ and for some function $z\in L^{2}(Q^{T})$,

Then $z$ is zero a.e in the domain $Q^{T}$.

Proof. We see from (4.1) that

Because (4.1) holds for any function $u$ in $B$, it can be expressed in a particular form. Assume that a function $\sigma \left(x, t\right)$ satisfies the conditions (1.3)–(1.4) such that

From the previous discussion, we introduce the function

Equation (4.2) reduces to

where

In this case, equation (4.4) can be written in the form

Integrating by parts, and keeping in mind that the function satisfies (1.2)–(1.4), we deduce the following expressions for each term in (4.5)

Substituting formulas (4.7) and (4.10) into (4.6), we get

By using Poincaré-type inequalities and the Cauchy-$\varepsilon$ inequality estimate for the last four terms of left side of (4.12), we get

Combining equalities (4.12)–(4.15) and (4.12) yields

We have, by lemma 3,

Consequently, in light of (4.18) and (4.17), inequality (4.17) becomes

Take $\varepsilon,$ such that

Consequently,

We conclude that $z$ is zero a.e in the domain $Q^{T}.$

Conflict of interest

The authors declare no conflicts of interest.