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Existence of solution for 2-D time-fractional differential equations with a boundary integral condition
Advances in Difference Equations volume 2019, Article number: 511 (2019)
Abstract
In this article, we prove the existence and uniqueness of a solution for 2-dimensional time-fractional differential equations with classical and integral boundary conditions. We start by writing this problem in the operator form and we choose suitable spaces and norms. Then we establish prior estimates from which we deduce the uniqueness of the strong solution. For the existence of solution for the fractional problem, we prove that the range of the operator generated by the considered problem is dense.
1 Introduction
Many physical phenomena bring us back to the study of fractional partial differential equations. We mention, for example, viscoelasticity, signal processing, electro-chemistry, control theory, electrical networks, fluid flow, porous media, rheology, diffusive transport, electromagnetic theory, diffusion phenomena, and a lot of other physical processes are diverse applications of fractional differential equations. For more on this, see [1,2,3,4,5,6].
Fractional diffusion equations appear widely in natural phenomena; these are suggested as mathematical models of physical problems in many fields, like the inhomogeneous fractional diffusion equations of the form
where \(\partial _{0t}^{\beta } \) is the Caputo fractional derivative, A is a positive self-adjoin operator on a Hilbert space H, \(f \in H \), \(F \in C(R^{+}; H) \), and \(0 <\beta \leq 1 \). The Caputo derivative is more suitable and natural for physical models problems, because it enables us to deal with inhomogeneous initial data easily.
Several methods have been used to investigate the existence and uniqueness of solution for fractional-order initial boundary value problems, such as the Laplace transform method, iteration method, the series method, the Fourier transform technique, the operational calculus method, see for example [7,8,9,10,11], but in general, analytical solutions are hard to obtain for most fractional differential equations especially with nonlocal conditions (integral conditions), i.e. when the data cannot be measured on the boundary or on a part of it. In the absence of a precise solution, we often resort to numerical methods such as finite element methods or spectral methods or backward substitution methods (see [12,13,14,15]), which strongly rely on the existence and uniqueness of the solution for a variational problem. The study of existence and uniqueness of a solution for fractional differential equations starts by constructing variational formulation and choosing suitable spaces and norms. Then we choose fixed point theorem or the Lax–Milgram theorem to prove existence results of the solution. For our problem (2.1)–(2.4), we believe that the prior estimate method is the most powerful tool to prove the existence and uniqueness of the solution for fractional differential equation, and is more appropriate with classical and integral boundary conditions. A few papers use the means of the energy inequality method for studying fractional partial differential equations, we cite for example: Alikhanov [16], Akilandeeswari et al. [17], and Mesloub [18]. Our work can be considered as an expansion and generalization of integer order problems such as [19].
We organize our article as follows: In Sect. 2, we give the statement of the problem and the needed functions spaces and the different tools that can be used in other sections. In Sect. 3, we prove an a priori estimate from which we deduce the uniqueness of a strong solution of problem (2.1)–(2.4), and its dependence on the given data. In Sect. 4, we establish the existence of the solution of problem (2.1)–(2.4), by proving that the closure of the range of the operator L generated by the considered problem is dense in the Hilbert space Y.
2 Statement of the problem and associated function spaces
Let \(D=\varOmega \times [0,T] \) be a bounded domain in \(\mathbb{R}^{3} \) with \(\varOmega =(0,c)\times (0,d) \). We consider the 2-dimensional time-fractional partial differential equation (PDE)
associated with the initial data
We have the Neumann and Dirichlet boundary conditions
and the integral conditions
Here F is a known function, where \(F \in C(\overline{D})\).
\({ \partial _{0t}^{\beta +1}} \); denotes the Caputo fractional derivative of order \(\beta +1 \), defined by
\(\varGamma (.) \) is the Gamma function.
\(D_{0t}^{-\beta } \); denotes the Riemann–Liouville integral of order β, defined by
The supposed solution \(V \in C^{2,2,2}(\overline{D}) \), the space of functions together with their partial derivatives of order 2 are continuous on DÌ… for their three variables \((x,y,t) \).
For more information on fractional differential equations, and applications of fractional calculus in physics, see [1,2,3,4,5,6,7,8,9].
We often use the following two lemmas.
Lemma 2.1
([16])
For any absolutely continuous function \(v(t)\)on the interval \([0,T] \), the following inequality holds:
Lemma 2.2
([16])
Let a nonnegative absolutely continuous function \(y(t) \)satisfy the inequality
for almost all \(t\in [0,T] \), wherecis positive and \(k(t) \)is an integrable nonnegative function on \([0,T] \). Then
where
are the Mittag-Leffler functions.
The Cauchyε-inequality:
where A and B are positive numbers.
A Poincaré type inequalities: [20]
where
We now introduce the appropriate function spaces needed in our posed problem. Let \(L_{p}^{2} (D) \) the weighted \(L^{2} \)-space with finite norm
from the inner product
Let \(V_{p}^{1,y}(D) \),and \(V^{1}(D) \) be the weighted Sobolev spaces with finite norms
Problem (2.1)–(2.4) can be formulated in operational form:
where \(W=(F,f,g) \), and \(L=(\mathcal{L},\ell _{1},\ell _{2})\) is the operator \(L:X\longrightarrow Y \) with domain of definition
Here X is a Banach space of functions V obtained by enclosing \(D(L) \) with respect to the finite norm
and Y is the Hilbert space associated with the finite norm
\(L_{p}^{2}(\varOmega )\), \(V_{p}^{1,y}(\varOmega )\) and \(V^{1}(\varOmega )\) the weighted Sobolev spaces on Ω are defined analogously to that on D.
3 Uniqueness of the solution
In this section, we prove a uniqueness result for problem (2.1)–(2.4), that is, we establish an a priori estimate from which we deduce the uniqueness of the consequences of the solution.
Theorem 3.1
For any \(V \in D(L) \), there exists a positive constant \(M_{5} \)independent ofVsuch that
where
Proof
Taking the inner product in \(L_{p}^{2}(\varOmega ) \) of Eq. (2.1) and the integro-differential operator
We have
Using conditions (2.3)–(2.4), then standard integration by parts of each term of the left-hand side in (3.8), leads to
By substitution of (3.9)–(3.11) into (3.8) we obtain
In the light of Lemma (2.1), the first term on the LHS of (3.12) is estimated as follows:
Combination of inequality (3.13) and equality (3.12) gives
By using the Cauchy-ε inequality and Poincaré inequality 2), we infer from (3.14) that
where
Note that
By changing t by Ï„, and integrating both sides of (3.15) with respect to Ï„ on \([0; t] \) we find
We consider the following two elementary inequalities:
Inequality (3.18) takes the form
where
Now by omitting the last two terms on the left-hand side of (3.21) we get
We apply Lemma (2.2) as follows:
Thence (3.23) becomes
where
Combination of inequalities (3.21) and (3.26) leads to
where
We make use of the inequality
Inequality (3.28) becomes
Then we take the supremum of LHS in (3.31) with respect to t over \([0;T] \), we get the desired estimate (3.1) with
 □
Proposition 3.2
The operator \(L:X\longrightarrow Y \)is closable.
Proof
The proof is analogous to that in [21]. □
Definition
A solution of the operator equation
is called a strong solution of problem (2.1)–(2.4) where L̅ is the closure of the operator L and \(D( \overline{L}) \) its domain of definition.
The points of the graph of LÌ… are limits of sequences of points of the graph of L, by passing to the limit, the estimate (3.1) can be extended to
From this we deduce the following results.
Corollary 3.3
If a strong solution of problem (2.1)–(2.4) exists, it is unique and depends continuously on elements \(W=(F,f,g)\in Y \).
Corollary 3.4
The range of \(R(\overline{L}) \)ofLÌ…is closed inY, and \(R(\overline{L})=\overline{R(L)} \).
4 Existence of the solution
To prove the existence of a strong solution \(V = \overline{L}^{-1}W =\overline{L ^{-1}}W \) of the problem (2.1)–(2.4) \(\forall W=(F,f,g) \in Y \), it suffices to prove that \(\overline{R(L)}=Y \), the density of the range \(R(L) \) in Y is equivalent to the orthogonality of a vector \(W=(F,f,g)\in Y \) to the set \(R(L) \). For this purpose, we begin by the following theorem (the proof of the density in a special case).
Theorem 4.1
For some function \(G \in L_{p}^{2}(D) \), and for all \(U \in D_{0}(L)= \lbrace U /U \in D(L),\ell _{1}U = 0, \ell _{2}U = 0 \rbrace \), we have
Then \(G=0 \)almost everywhere in the domainD.
Proof
Equation (4.1) can be written
Let \(h(x,y,t) \) be a function that satisfies the boundary conditions (2.2)–(2.4) and
Then we suppose
By replacing \(U(x,y,t) \) in (4.2) we have
Now, we assume the function
Then Eq. (4.5) becomes
Taking into account that the function h verifies the conditions (2.2)–(2.4), then integrating by parts each term of (4.7) we have
Substituting (4.8), (4.9), and (4.10) into (4.7) we get
According to Lemma (2.1) the first term on the LHS of (4.11) can be estimated as follows:
Equation (4.11) can be written
By replacing t by Ï„ and integrating of (4.13) with respect to Ï„ over \([0;t] \) gives
We find from inequality (4.14) that \(G\equiv 0 \) almost everywhere in D. □
Theorem 4.2
The range \(R(L) \)of the operatorL, coincides with the whole spaceY.
Proof
Let \(W=(\varphi,g_{1},g_{2})\in R(L)^{\perp } \) such that
If we put \(u \in D_{0}(L) \) into (4.15) we get
By virtue of Theorem 4.1 we deduce that \(\varphi \equiv 0 \), thus (4.15) becomes
The trace operators \(\ell _{1} \) and \(\ell _{2} \) are independent, and \(R(\ell _{1}) \) and \(R(\ell _{2}) \) are everywhere dense in the spaces \(V^{1}(\varOmega ) \) and \(L_{p}^{2}(\varOmega ) \), respectively. Then \(g_{1} = 0 \), \(g_{2} = 0 \). Consequently \(W = 0 \). Hence \(R(L)^{\perp }={0} \) i.e. \(\overline{R(L)}=Y \). □
5 Conclusion
The well posedness of 2-D time-fractional differential equations with boundary integral conditions is proved. The functional analysis method was successfully applied to a fractional-order initial boundary value problem.
Our results develop the traditional functional analysis method which relies on some a priori estimates and some density arguments for a fractional hyperbolic equation with fractional Caputo derivative.
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Kasmi, L., Guerfi, A. & Mesloub, S. Existence of solution for 2-D time-fractional differential equations with a boundary integral condition. Adv Differ Equ 2019, 511 (2019). https://doi.org/10.1186/s13662-019-2444-2
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DOI: https://doi.org/10.1186/s13662-019-2444-2