Skip to main content

Existence of solution for 2-D time-fractional differential equations with a boundary integral condition

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.

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

$$ \partial _{0t}^{\beta } u = -Au+F(t), \qquad u(0)=f, $$
(1.1)

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.

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)

$$ \partial _{0t}^{\beta +1}V- \frac{1}{x}\frac{\partial }{\partial x} \biggl( x\frac{\partial V}{\partial x} \biggr) -\frac{1}{x}\frac{ \partial ^{2} V}{\partial y^{2}} = F(x,y,t),\quad (x,y,t)\in D, $$
(2.1)

associated with the initial data

$$ \ell _{1}V=V(x,y,0)=f(x,y),\qquad \ell _{2}V=V_{t}(x,y,0)=g(x,y). $$
(2.2)

We have the Neumann and Dirichlet boundary conditions

$$ V(c,y,t)=0, \qquad V_{y}(x,d,t)=0, $$
(2.3)

and the integral conditions

$$ \int _{0}^{c}xV(x,y,t)\,dx=0,\qquad \int _{0}^{d}V(x,y,t)\,dy=0. $$
(2.4)

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

$$ \partial _{0t}^{\beta +1}V(x,y,t)=\frac{1}{\varGamma (1-\beta )} \int _{0}^{t}\frac{V_{\tau \tau }(x,y,\tau )}{(t-\tau )^{\beta }}\,d\tau,\quad 0< \beta < 1, $$
(2.5)

\(\varGamma (.) \) is the Gamma function.

\(D_{0t}^{-\beta } \); denotes the Riemann–Liouville integral of order β, defined by

$$ D_{0t}^{-\beta }V(x,y,t)=\frac{1}{\varGamma (\beta )} \int _{0}^{t}\frac{V(x,y, \tau )}{(t-\tau )^{\beta }}\,d\tau,\quad 0< \beta < 1 . $$
(2.6)

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 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:

$$ v(t) \partial _{0t}^{\beta }v(t) \geq \frac{1}{2} \partial _{0t}^{\beta }v^{2}(t),\quad 0< \beta < 1. $$
(2.7)

Lemma 2.2

([16])

Let a nonnegative absolutely continuous function \(y(t) \)satisfy the inequality

$$ \partial _{0t}^{\beta }y(t) \leq cy(t) + k(t),\quad 0< \beta < 1, $$
(2.8)

for almost all \(t\in [0,T] \), wherecis positive and \(k(t) \)is an integrable nonnegative function on \([0,T] \). Then

$$ y(t) \leq y(0)E_{\beta }\bigl(ct^{\beta }\bigr)+\varGamma (\beta )E_{\beta, \beta }\bigl(ct^{\beta }\bigr)D_{0t}^{-\beta } k(t), $$
(2.9)

where

$$ E_{\beta }(x)=\sum_{n=0}^{\infty } \frac{x^{n}}{\varGamma (\beta n+1)} \quad\textit{and}\quad E_{\beta,\alpha }(x)=\sum _{n=0}^{\infty }\frac{x^{n}}{\varGamma ( \beta n+\alpha )}, $$
(2.10)

are the Mittag-Leffler functions.

The Cauchyε-inequality:

$$ AB\leq \frac{\varepsilon }{2}A^{2}+\frac{1}{2\varepsilon }B^{2},\quad \varepsilon >0, $$
(2.11)

where A and B are positive numbers.

A Poincaré type inequalities: [20]

$$ \textstyle\begin{cases} (1) \lVert \Im _{y}(V) \rVert _{L_{p}^{2}(\varOmega )}^{2} \leq \frac{d ^{2}}{2} \lVert V \rVert _{L_{p}^{2}(\varOmega )}^{2}; \\ (2) \lVert \Im _{y}^{2}(V) \rVert _{L_{p}^{2}(\varOmega )}^{2} \leq \frac{d ^{2}}{2} \lVert \Im _{y}(V) \rVert _{L_{p}^{2}(\varOmega )}^{2}, \end{cases} $$
(2.12)

where

$$ \Im _{y}(V)= \int _{0}^{y} V(x,\xi,t)\,d\xi, \qquad \Im _{y}^{2}(V)= \int _{0}^{y} \int _{0}^{\xi } V(x,\eta,t)\,d\eta \,d\xi. $$
(2.13)

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

$$ \lVert V \rVert _{L_{p}^{2}(D)}^{2}= \int _{D} xU^{2}\,dx\,dy\,dt, $$
(2.14)

from the inner product

$$ (U,V)_{ L_{p}^{2}(D)}=(xU,V)_{ L^{2}(D)} $$
(2.15)

Let \(V_{p}^{1,y}(D) \),and \(V^{1}(D) \) be the weighted Sobolev spaces with finite norms

$$\begin{aligned} &\lVert V \rVert _{V_{p}^{1,y}(D)}^{2} =\bigl\lVert \Im _{y}(V) \bigr\rVert _{L_{p}^{2}(D)}^{2}+\bigl\lVert \Im _{y}(V_{x}) \bigr\rVert _{ L_{p}^{2} (D)} ^{2}, \end{aligned}$$
(2.16)
$$\begin{aligned} &\lVert V \rVert _{V^{1}(D)}^{2} =\lVert V \rVert _{L^{2}(D)}^{2}+ \lVert V_{x} \rVert _{ L^{2} (D)}^{2}. \end{aligned}$$
(2.17)

Problem (2.1)–(2.4) can be formulated in operational form:

$$ LV=W, \quad\forall V \in D(L), $$
(2.18)

where \(W=(F,f,g) \), and \(L=(\mathcal{L},\ell _{1},\ell _{2})\) is the operator \(L:X\longrightarrow Y \) with domain of definition

$$ D(L)= \textstyle\begin{cases} V \in L^{2}(D),\partial _{0t}^{\beta +1}V,V_{t},V_{x},V_{y},V_{xx},V _{yy} \in L^{2}(D), \\ V(c,y,t)=0,\qquad V_{y}(x,d,t)=0, \\ \int _{0}^{c}xV(x,y,t)\,dx=0,\qquad \int _{0}^{d}V(x,y,t)\,dy=0,\quad t\in [0,T]. \end{cases} $$
(2.19)

Here X is a Banach space of functions V obtained by enclosing \(D(L) \) with respect to the finite norm

$$ \lVert V \rVert _{X}^{2} = \sup_{0\leq t\leq T} \bigl( D_{0t}^{\beta -1} \bigl\lVert \Im _{y}(V_{t}) \bigr\rVert _{ L_{p}^{2} (\varOmega )}^{2} + \lVert V \rVert _{V_{p}^{1,y}( \varOmega )}^{2} \bigr), $$
(2.20)

and Y is the Hilbert space associated with the finite norm

$$ \lVert W \rVert _{Y}^{2}=\lVert F \rVert _{L_{p}^{2} (D)}^{2}+\lVert f \rVert _{V^{1}(\varOmega )}^{2}+ \lVert g \rVert _{L_{p}^{2}(\varOmega )}^{2} $$
(2.21)

\(L_{p}^{2}(\varOmega )\), \(V_{p}^{1,y}(\varOmega )\) and \(V^{1}(\varOmega )\) the weighted Sobolev spaces on Ω are defined analogously to that on D.

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

$$\begin{aligned} &\sup_{0\leq t\leq T} \bigl( D_{0t}^{\beta -1} \bigl\lVert \Im _{y}(V_{t}) \bigr\rVert _{L_{p}^{2}(\varOmega )}^{2}+ \lVert V \rVert _{V_{p}^{1,y}(\varOmega )}^{2} \bigr) \\ &\quad \leq M_{5} \biggl( \int _{0}^{T}\lVert F \rVert _{L_{p}^{2}(\varOmega )}^{2}\,dt+\lVert f \rVert _{V^{1}(\varOmega )} ^{2}+\lVert g \rVert _{L_{p}^{2}(\varOmega )}^{2} \biggr), \end{aligned}$$
(3.1)

where

$$\begin{aligned} &M_{1} =\max \biggl( 1,\frac{d^{2}}{2} \biggr), \end{aligned}$$
(3.2)
$$\begin{aligned} &M_{2} =\frac{M_{1}\max \lbrace 1,\frac{cd^{2}}{2},\frac{d ^{2}}{2}\frac{T^{1-\beta }}{\varGamma (1-\beta )} \rbrace }{ \min \lbrace 1,\frac{2}{cd^{2}} \rbrace }, \end{aligned}$$
(3.3)
$$\begin{aligned} &M_{3} =\varGamma (\beta )E_{\beta,\beta }\bigl(M_{2}T^{\beta } \bigr)\max \biggl( M_{2}, \frac{ M_{2}T^{\beta +1}}{(\beta +1)\varGamma (\beta +1)} \biggr), \end{aligned}$$
(3.4)
$$\begin{aligned} &M_{4} =M_{2}M_{3}+M_{2}, \end{aligned}$$
(3.5)
$$\begin{aligned} &M_{5} = M_{4} \biggl( 1+\frac{T^{\beta }}{\varGamma (\beta +1)} \biggr). \end{aligned}$$
(3.6)

Proof

Taking the inner product in \(L_{p}^{2}(\varOmega ) \) of Eq. (2.1) and the integro-differential operator

$$ MV=-\Im _{y}^{2}(V_{t})=- \int _{0}^{y} \int _{0}^{\xi } V_{t}(x,\eta,t)\,d \eta \,d\xi. $$
(3.7)

We have

$$\begin{aligned} \biggl( \partial _{0t}^{\beta +1}V- \frac{1}{x}\frac{\partial }{ \partial x} \biggl( x\frac{\partial V}{\partial x} \biggr) - \frac{1}{x}\frac{\partial ^{2} V}{\partial y^{2}},-\Im _{y}^{2}(V _{t}) \biggr)_{L_{p}^{2}(\varOmega )} = \bigl(F,-\Im _{y}^{2}(V_{t}) \bigr) _{L_{p}^{2}(\varOmega )}. \end{aligned}$$
(3.8)

Using conditions (2.3)–(2.4), then standard integration by parts of each term of the left-hand side in (3.8), leads to

$$\begin{aligned} &{-} \biggl( \partial _{0t}^{\beta +1}V, \int _{0}^{y} \int _{0}^{\xi }V_{t}(x, \eta,t)\,d \eta \,d\xi \biggr)_{L_{p}^{2}(\varOmega )} \\ &\quad = \biggl(\partial _{0t}^{\beta } \biggl( \int _{0}^{y}V_{t}(x,\xi,t)\,d\xi \biggr), \biggl( \int _{0}^{y}V_{t}(x,\xi,t)\,d\xi \biggr) \biggr)_{L_{p}^{2}(\varOmega )}, \end{aligned}$$
(3.9)
$$\begin{aligned} &\biggl(\frac{1}{x}\frac{\partial }{\partial x} \biggl( x \frac{ \partial V}{\partial x} \biggr), \int _{0}^{y} \int _{0}^{\xi }V_{t}(x, \eta,t)\,d \eta \,d\xi \biggr)_{L_{p}^{2}(\varOmega )} \\ &\quad = \biggl( \int _{0}^{y}V _{x}(x,\xi,t)\,d \xi, \int _{0}^{y}V_{xt}(x,\xi,t)\,d\xi \biggr)_{L_{p} ^{2}(\varOmega )}, \end{aligned}$$
(3.10)
$$\begin{aligned} &\biggl(\frac{1}{x}\frac{\partial ^{2} V}{\partial y^{2}}, \int _{0} ^{y} \int _{0}^{\xi }V_{t}(x,\eta,t)\,d\eta \,d\xi \biggr)_{L_{p}^{2}( \varOmega )} = ( V,V_{t} )_{L^{2}(\varOmega )}. \end{aligned}$$
(3.11)

By substitution of (3.9)–(3.11) into (3.8) we obtain

$$\begin{aligned} &2 \bigl( \partial _{0t}^{\beta } \bigl( \Im _{y} ( V_{t} ) \bigr),\Im _{y} ( V_{t} ) \bigr)_{L_{p} ^{2}(\varOmega )}+\frac{\partial }{\partial t} \int _{\varOmega }x \bigl( \Im _{y} (V_{x} ) \bigr)^{2} \,dx\,dy+\frac{\partial }{\partial t} \int _{\varOmega }V^{2}\,dx\,dy \\ &\quad = -2 \biggl(F, \int _{0}^{y} \int _{0}^{\xi }V_{t}(x,\eta,t)\,d\eta \,d \xi \biggr)_{L_{p}^{2}(\varOmega )}. \end{aligned}$$
(3.12)

In the light of Lemma (2.1), the first term on the LHS of (3.12) is estimated as follows:

$$ 2 \bigl( \partial _{0t}^{\beta } \bigl( \Im _{y} ( V_{t} ) \bigr),\Im _{y} ( V_{t} ) \bigr)_{L_{p} ^{2}(\varOmega )}\geq \partial _{0t}^{\beta } \bigl\lVert \Im _{y} ( V_{t} ) \bigr\rVert _{L_{p}^{2}(\varOmega )}^{2}. $$
(3.13)

Combination of inequality (3.13) and equality (3.12) gives

$$\begin{aligned} &\partial _{0t}^{\beta } \bigl\lVert \Im _{y} ( V_{t} ) \bigr\rVert _{L_{p}^{2}(\varOmega )}^{2}+ \frac{\partial }{\partial t}\bigl\lVert \Im _{y} ( V_{x} ) \bigr\rVert _{L_{p}^{2}(\varOmega )}^{2} +\frac{ \partial }{\partial t}\lVert V\rVert _{L^{2}(\varOmega )}^{2} \\ &\quad \leq -2 \biggl(F, \int _{0}^{y} \int _{0}^{\xi }V_{t}(x,\eta,t)\,d\eta \,d\xi \biggr) _{L_{p}^{2}(\varOmega )}. \end{aligned}$$
(3.14)

By using the Cauchy-ε inequality and Poincaré inequality 2), we infer from (3.14) that

$$\begin{aligned} & \partial _{0t}^{\beta } \bigl\lVert \Im _{y} ( V_{t} ) \bigr\rVert _{L_{p}^{2}(\varOmega )}^{2}+ \frac{\partial }{\partial t}\bigl\lVert \Im _{y} ( V_{x} ) \bigr\rVert _{L_{p}^{2}(\varOmega )}^{2} +\frac{ \partial }{\partial t}\lVert V\rVert _{L^{2}(\varOmega )}^{2} \\ &\quad \leq M_{1} \bigl( \lVert F \rVert _{L_{p}^{2}(\varOmega )}^{2}+\bigl\lVert \Im _{y} ( V_{t} ) \bigr\rVert _{L_{p}^{2}(\varOmega )}^{2} \bigr), \end{aligned}$$
(3.15)

where

$$ M_{1}=\max \biggl( 1,\frac{d^{2}}{2} \biggr). $$
(3.16)

Note that

$$ \int _{0}^{t}\partial _{0t}^{\beta } \bigl\lVert \Im _{y} ( V_{t} ) \bigr\rVert _{L_{p}^{2}(\varOmega )}^{2} \,d\tau =D_{0t}^{\beta -1} \bigl\lVert \Im _{y} ( V_{t} ) \bigr\rVert _{L_{p}^{2}(\varOmega )}^{2}-\frac{t^{1- \beta }}{\varGamma (1-\beta )}\bigl\lVert \Im _{y} \bigl( V_{t}(x,y,0) \bigr) \bigr\rVert _{L_{p}^{2}(\varOmega )}^{2}. $$
(3.17)

By changing t by τ, and integrating both sides of (3.15) with respect to τ on \([0; t] \) we find

$$\begin{aligned} &D_{0t}^{\beta -1} \bigl\lVert \Im _{y} ( V_{t} ) \bigr\rVert _{L_{p}^{2}(\varOmega )}^{2}+ \bigl\lVert \Im _{y} ( V_{x} ) \bigr\rVert _{L_{p}^{2}(\varOmega )}^{2}+\lVert V \rVert _{L^{2}(\varOmega )}^{2} \\ &\quad \le M_{1} \biggl( \int _{0}^{t}\lVert F \rVert _{L_{p}^{2}(\varOmega )} ^{2}\,d\tau + \int _{0}^{t}\bigl\lVert \Im _{y} \bigl( V_{\tau }(x,y,\tau ) \bigr) \bigr\rVert _{L_{p}^{2}(\varOmega )}^{2} \,d\tau \\ &\qquad{}+\frac{T^{1-\beta }}{\varGamma (1- \beta )}\bigl\lVert \Im _{y} ( g ) \bigr\rVert _{L_{p}^{2}(\varOmega )} ^{2} +\bigl\lVert \Im _{y} (f_{x} ) \bigr\rVert _{L_{p}^{2}(\varOmega )} ^{2}+\lVert f \rVert _{L^{2}(\varOmega )}^{2} \biggr). \end{aligned}$$
(3.18)

We consider the following two elementary inequalities:

$$\begin{aligned} & \bigl\lVert \Im _{y} ( V ) \bigr\rVert _{L_{p}^{2}(\varOmega )}^{2} \leq \frac{d^{2}}{2} \lVert V \rVert _{L_{p}^{2}(\varOmega )}^{2} \leq \frac{cd ^{2}}{2} \lVert V \rVert _{L^{2}(\varOmega )}^{2}, \end{aligned}$$
(3.19)
$$\begin{aligned} & \bigl\lVert \Im _{y} ( f_{x} ) \bigr\rVert _{L_{p}^{2}(\varOmega )}^{2} \leq \frac{d^{2}}{2} \lVert f_{x} \rVert _{L_{p}^{2}(\varOmega )}^{2} \leq \frac{cd^{2}}{2} \lVert f_{x} \rVert _{L^{2}(\varOmega )}^{2}. \end{aligned}$$
(3.20)

Inequality (3.18) takes the form

$$\begin{aligned} &D_{0t}^{\beta -1} \bigl\lVert \Im _{y} ( V_{t} ) \bigr\rVert _{L_{p}^{2}(\varOmega )}^{2}+ \bigl\lVert \Im _{y} ( V_{x} ) \bigr\rVert _{L_{p}^{2}(\varOmega )}^{2}+\bigl\lVert \Im _{y} ( V ) \bigr\rVert _{L_{p}^{2}(\varOmega )}^{2} \\ &\quad \le M_{2} \biggl( \int _{0}^{t}\lVert F \rVert _{L_{p}^{2}(\varOmega )} ^{2}\,d\tau + \int _{0}^{t}\bigl\lVert \Im _{y} \bigl( V_{\tau }(x,y,\tau ) \bigr) \bigr\rVert _{L_{p}^{2}(\varOmega )}^{2} \,d\tau \\ &\qquad{}+ \lVert f\rVert _{V^{1}(\varOmega )}^{2}+\lVert g\rVert _{L_{p}^{2}(\varOmega )}^{2} \biggr), \end{aligned}$$
(3.21)

where

$$ M_{2}=\frac{M_{1}\max \lbrace 1,\frac{cd^{2}}{2}, \frac{d^{2}}{2}\frac{T^{1-\beta }}{\varGamma (1-\beta )} \rbrace }{ \min \lbrace 1,\frac{2}{cd^{2}} \rbrace }. $$
(3.22)

Now by omitting the last two terms on the left-hand side of (3.21) we get

$$\begin{aligned} & D_{0t}^{\beta -1} \bigl\lVert \Im _{y} ( V_{t} ) \bigr\rVert _{L_{p}^{2}(\varOmega )}^{2} \\ &\quad \le M_{2} \biggl( \int _{0}^{t}\lVert F \rVert _{L_{p}^{2}(\varOmega )}^{2}\,d\tau + \int _{0}^{t}\bigl\lVert \Im _{y} \bigl( V_{\tau }(x,y,\tau ) \bigr) \bigr\rVert _{L_{p}^{2}(\varOmega )}^{2} \,d \tau \\ &\qquad{}+\lVert f\rVert _{V^{1}(\varOmega )}^{2}+\lVert g \rVert _{L_{p}^{2}(\varOmega )}^{2} \biggr). \end{aligned}$$
(3.23)

We apply Lemma (2.2) as follows:

$$\begin{aligned} &y(t)= \int _{0}^{t}\bigl\lVert \Im _{y} \bigl( V_{\tau }(x,y,\tau ) \bigr) \bigr\rVert _{L_{p}^{2}(\varOmega )}^{2} \,d\tau,\qquad y(0)=0, \end{aligned}$$
(3.24)
$$\begin{aligned} &\partial _{0t}^{\beta }y(t)= D_{0t}^{\beta -1} \bigl\lVert \Im _{y} ( V_{t} )\bigr\rVert _{L_{p}^{2}(\varOmega )}^{2}. \end{aligned}$$
(3.25)

Thence (3.23) becomes

$$\begin{aligned} \int _{0}^{t}\bigl\lVert \Im _{y} \bigl( V_{\tau }(x,y,\tau ) \bigr) \bigr\rVert _{L_{p}^{2}(\varOmega )}^{2} \,d\tau \leq M_{3} \bigl( D_{0t}^{- \beta -1}\lVert F \rVert _{L_{p}^{2}(\varOmega )}^{2}+ \lVert f \rVert _{V^{1}(\varOmega )}^{2}+ \lVert g\rVert _{L_{p}^{2}(\varOmega )}^{2} \bigr), \end{aligned}$$
(3.26)

where

$$ M_{3}=\varGamma (\beta )E_{\beta,\beta }\bigl(M_{2}T^{\beta } \bigr)\max \biggl( M_{2},\frac{ M_{2}T^{\beta +1}}{(\beta +1)\varGamma (\beta +1)} \biggr) . $$
(3.27)

Combination of inequalities (3.21) and (3.26) leads to

$$\begin{aligned} & D_{0t}^{\beta -1} \bigl\lVert \Im _{y} ( V_{t} ) \bigr\rVert _{L_{p}^{2}(\varOmega )}^{2}+ \lVert V \rVert _{V_{p}^{1,y}(\varOmega )} ^{2} \\ &\quad \le M_{4} \biggl( D_{0t}^{-\beta -1}\lVert F \rVert _{L_{p}^{2}(\varOmega )}^{2} + \int _{0}^{t}\lVert F \rVert _{L_{p}^{2}(\varOmega )}^{2}\,d\tau +\lVert f\rVert _{V^{1}(\varOmega )} ^{2}+\lVert g\rVert _{L_{p}^{2}(\varOmega )}^{2} \biggr), \end{aligned}$$
(3.28)

where

$$ M_{4}=M_{2}M_{3}+M_{2}. $$
(3.29)

We make use of the inequality

$$ D_{0t}^{-\beta -1}\lVert F \rVert _{L_{p}^{2}(\varOmega )}^{2} \leq \frac{T ^{\beta }}{\varGamma (\beta +1)} \int _{0}^{t}\lVert F \rVert _{L_{p}^{2}(\varOmega )}^{2}\,d\tau. $$
(3.30)

Inequality (3.28) becomes

$$\begin{aligned} & D_{0t}^{\beta -1} \bigl\lVert \Im _{y}(V_{t}) \bigr\rVert _{L_{p}^{2}(\varOmega )} ^{2}+ \lVert V \rVert _{V_{p}^{1,y}(\varOmega )}^{2} \end{aligned}$$
(3.31)
$$\begin{aligned} &\quad\leq M_{5} \biggl( \int _{0}^{t}\lVert F \rVert _{L_{p}^{2}(\varOmega )}^{2}\,d\tau +\lVert f \rVert _{V^{1}(\varOmega )}^{2}+ \lVert g \rVert _{L_{p}^{2}(\varOmega )}^{2} \biggr) \end{aligned}$$
(3.32)
$$\begin{aligned} &\quad\leq M_{5} \biggl( \int _{0}^{T}\lVert F \rVert _{L_{p}^{2}(\varOmega )} ^{2}\,dt+\lVert f \rVert _{V^{1}(\varOmega )}^{2}+\lVert g \rVert _{L_{p}^{2}(\varOmega )}^{2} \biggr). \end{aligned}$$
(3.33)

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

$$ M_{5}= M_{4} \biggl( 1+\frac{T^{\beta }}{\varGamma (\beta +1)} \biggr). $$
(3.34)

 □

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

$$ \overline{L}V=W=(F,f,g) $$
(3.35)

is called a strong solution of problem (2.1)–(2.4) where is the closure of the operator L and \(D( \overline{L}) \) its domain of definition.

The points of the graph of are limits of sequences of points of the graph of L, by passing to the limit, the estimate (3.1) can be extended to

$$ \lVert V \|_{X} \leq \sqrt{M_{5}} \Vert \overline{L}V \Vert _{Y}, \quad\forall V\in D(\overline{L}). $$
(3.36)

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 \(R(\overline{L}) \)ofis closed inY, and \(R(\overline{L})=\overline{R(L)} \).

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

$$ ( \mathcal{L} U,G )_{L_{p}^{2}(D)} = 0. $$
(4.1)

Then \(G=0 \)almost everywhere in the domainD.

Proof

Equation (4.1) can be written

$$ \biggl( \partial _{0t}^{\beta +1}U- \frac{1}{x}\frac{\partial }{ \partial x} \biggl( x\frac{\partial U}{\partial x} \biggr) - \frac{1}{x}\frac{\partial ^{2} U}{\partial y^{2}}, G \biggr)_{L _{p}^{2}(D)} = 0. $$
(4.2)

Let \(h(x,y,t) \) be a function that satisfies the boundary conditions (2.2)–(2.4) and

$$ h,h_{x},h_{y}, \int _{0}^{t}h(x,y,s)\,ds,x \int _{0}^{t}h_{x}(x,y,s)\,ds,x \int _{0}^{t}h_{y}(x,y,s)\,ds, \partial _{0t}^{\beta +1}h \in L^{2}(D). $$
(4.3)

Then we suppose

$$ U(x,y,t)= \int _{0}^{t} \int _{0}^{s}h(x,y,z)\,dz\,ds. $$
(4.4)

By replacing \(U(x,y,t) \) in (4.2) we have

$$\begin{aligned} &\biggl( \partial _{0t}^{\beta +1} \biggl( x \int _{0}^{t} \int _{0}^{s}h(x,y,z)\,dz\,ds \biggr) - \frac{\partial }{ \partial x} \biggl( x \int _{0}^{t} \int _{0}^{s}h_{x}(x,y,z)\,dz\,ds \biggr) \\ &\quad{}-\frac{\partial }{\partial y} \biggl( \int _{0}^{t} \int _{0} ^{s}h_{y}(x,y,z)\,dz\,ds \biggr), G \biggr) _{L^{2}(D)}=0. \end{aligned}$$
(4.5)

Now, we assume the function

$$ G(x,y,t)=- \int _{0}^{y} \int _{0}^{\xi } \int _{0}^{t}h(x,\eta,s)\,ds\,d \eta \,d\xi. $$
(4.6)

Then Eq. (4.5) becomes

$$\begin{aligned} &{-} \biggl( \partial _{0t}^{\beta +1} \biggl( \int _{0}^{t} \int _{0}^{s}h(x,y,z)\,dz\,ds \biggr), \int _{0}^{y} \int _{0}^{\xi } \int _{0}^{t}h(x,\eta,s)\,ds\,d\eta \,d\xi \biggr) _{L_{p}^{2}(D)} \\ &\quad{}+ \biggl( \frac{\partial }{\partial x} \biggl( x \int _{0}^{t} \int _{0} ^{s}h_{x}(x,y,z)\,dz\,ds \biggr), \int _{0}^{y} \int _{0}^{\xi } \int _{0}^{t}h(x, \eta,s)\,ds\,d\eta \,d\xi \biggr) _{L^{2}(D)} \\ &\quad{}+ \biggl( \frac{\partial }{\partial y} \biggl( \int _{0}^{t} \int _{0} ^{s}h_{y}(x,y,z)\,dz\,ds \biggr), \int _{0}^{y} \int _{0}^{\xi } \int _{0}^{t}h(x, \eta,s)\,ds\,d\eta \,d\xi \biggr) _{L^{2}(D)}= 0. \end{aligned}$$
(4.7)

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

$$\begin{aligned} &{-} \biggl( \partial _{0t}^{\beta +1} \biggl( \int _{0}^{t} \int _{0}^{s}h(x,y,z)\,dz\,ds \biggr), \int _{0}^{y} \int _{0}^{\xi } \int _{0}^{t}h(x,\eta,s)\,ds\,d\eta \,d\xi \biggr) _{L_{p}^{2}(\varOmega )} \\ &\quad = \biggl( \partial _{0t}^{\beta } \biggl( \int _{0}^{y} \int _{0}^{t} h(x, \xi,s)\,ds\,d\xi \biggr), \int _{0}^{y} \int _{0}^{t}h(x,\xi,s)\,ds\,d\xi \biggr) _{L_{p}^{2}(\varOmega )}, \end{aligned}$$
(4.8)
$$\begin{aligned} & \biggl(\frac{\partial }{\partial x} \biggl( x \int _{0}^{t} \int _{0} ^{s}h_{x}(x,y,z)\,dz\,ds \biggr), \int _{0}^{y} \int _{0}^{\xi } \int _{0}^{t}h(x, \eta,s)\,ds\,d\eta \,d\xi \biggr) _{L^{2}(\varOmega )} \\ &\quad= \biggl( \int _{0}^{y} \int _{0}^{t} \int _{0}^{s}h_{x}(x,\xi,z)\,dz \,ds\,d \xi, \int _{0}^{y} \int _{0}^{t}h_{x}(x,\xi,s)\,ds\,d \xi \biggr) _{L_{p} ^{2}(\varOmega )} \\ &\quad=\frac{1}{2}\frac{\partial }{\partial t} \biggl\Vert \int _{0}^{y} \int _{0}^{t} \int _{0}^{s}h_{x}(x,\xi,z)\,dz \,ds\,d\xi \biggr\Vert _{L_{p}^{2}( \varOmega )}^{2}, \end{aligned}$$
(4.9)
$$\begin{aligned} & \biggl( \frac{\partial }{\partial y} \biggl( \int _{0}^{t} \int _{0}^{s}h _{y}(x,y,z)\,dz\,ds \biggr), \int _{0}^{y} \int _{0}^{\xi } \int _{0}^{t}h(x, \eta,s)\,ds\,d\eta \,d\xi \biggr) _{L^{2}(\varOmega )} \\ &\quad= \biggl( \int _{0}^{t} \int _{0}^{s}h(x,y,z)\,dz\,ds, \int _{0}^{t}h(x,y,s)\,ds \biggr) _{L^{2}(\varOmega )} \\ &\quad=\frac{1}{2}\frac{\partial }{\partial t} \biggl\Vert \int _{0}^{t} \int _{0}^{s}h(x,y,z)\,dz\,ds \biggr\Vert _{L^{2}(\varOmega )}^{2}. \end{aligned}$$
(4.10)

Substituting (4.8), (4.9), and (4.10) into (4.7) we get

$$\begin{aligned} &2 \biggl( \partial _{0t}^{\beta } \biggl( \int _{0}^{y} \int _{0}^{t} h(x, \xi,s)\,ds\,d\xi \biggr), \int _{0}^{y} \int _{0}^{t}h(x,\xi,s)\,ds\,d\xi \biggr) _{L_{p}^{2}(\varOmega )} \\ &\quad{}+\frac{\partial }{\partial t} \biggl\Vert \int _{0}^{y} \int _{0}^{t} \int _{0}^{s}h_{x}(x,\xi,z)\,dz \,ds\,d\xi \biggr\Vert _{L_{p}^{2}(\varOmega )}^{2} \\ &\quad{}+\frac{\partial }{\partial t} \biggl\Vert \int _{0}^{t} \int _{0}^{s}h(x,y,z)\,dz\,ds \biggr\Vert _{L^{2}(\varOmega )}^{2} =0. \end{aligned}$$
(4.11)

According to Lemma (2.1) the first term on the LHS of (4.11) can be estimated as follows:

$$\begin{aligned} &2 \biggl( \partial _{0t}^{\beta } \biggl( \int _{0}^{y} \int _{0}^{t} h(x, \xi,s)\,ds\,d\xi \biggr), \int _{0}^{y} \int _{0}^{t}h(x,\xi,s)\,ds\,d\xi \biggr) _{L_{p}^{2}(\varOmega )} \\ &\quad \geq \partial _{0t}^{\beta } \biggl\Vert \int _{0} ^{y} \int _{0}^{t} h(x,\xi,s)\,ds\,d\xi \biggr\Vert _{L_{p}^{2}(\varOmega )} ^{2}. \end{aligned}$$
(4.12)

Equation (4.11) can be written

$$\begin{aligned} & \partial _{0t}^{\beta } \biggl\Vert \int _{0}^{y} \int _{0}^{t} h(x,\xi,s)\,ds\,d \xi \biggr\Vert _{L_{p}^{2}(\varOmega )}^{2} + \frac{\partial }{\partial t} \biggl\Vert \int _{0}^{y} \int _{0}^{t} \int _{0}^{s}h_{x}(x,\xi,z)\,dz \,ds\,d\xi \biggr\Vert _{L_{p}^{2}(\varOmega )}^{2} \\ &\quad{}+\frac{\partial }{\partial t} \biggl\Vert \int _{0}^{t} \int _{0}^{s}h(x,y,z)\,dz\,ds \biggr\Vert _{L^{2}(\varOmega )}^{2} \leq 0. \end{aligned}$$
(4.13)

By replacing t by τ and integrating of (4.13) with respect to τ over \([0;t] \) gives

$$\begin{aligned} &D_{0t}^{\beta -1} \biggl\Vert \int _{0}^{y} \int _{0}^{t} h(x,\xi,s)\,ds\,d \xi \biggr\Vert _{L_{p}^{2}(\varOmega )}^{2} \\ &\quad{}+ \biggl\Vert \int _{0}^{y} \int _{0}^{t} \int _{0}^{s}h_{x}(x,\xi,z)\,dz \,ds\,d\xi \biggr\Vert _{L_{p}^{2}( \varOmega )}^{2} \\ &\quad{}+ \biggl\Vert \int _{0}^{t} \int _{0}^{s}h(x,y,z)\,dz\,ds \biggr\Vert _{L^{2}( \varOmega )}^{2} \leq 0. \end{aligned}$$
(4.14)

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

$$ ( \mathcal{L} u,\varphi )_{L_{p}^{2}(D)}+ (\ell _{1}u,g _{1} )_{V^{1}(\varOmega )} + (\ell _{2} u,g_{2} )_{L _{p}^{2}(\varOmega )} = 0. $$
(4.15)

If we put \(u \in D_{0}(L) \) into (4.15) we get

$$ ( \mathcal{L} u,\varphi )_{L_{p}^{2}(D)} = 0, \quad\forall u \in D_{0}(L). $$
(4.16)

By virtue of Theorem 4.1 we deduce that \(\varphi \equiv 0 \), thus (4.15) becomes

$$ (\ell _{1}u,g_{1} )_{V^{1}(\varOmega )} + (\ell _{2} u,g _{2} )_{L_{p}^{2}(\varOmega )} = 0, \quad \forall u \in D(L). $$
(4.17)

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. \(\overline{R(L)}=Y \). □

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.

References

  1. Hilfer, R.: Applications of Fractional Calculus in Physics. Applications of Fractional Calculus in Physics. World Scientific Publishing, Singapore (2000)

    Book  Google Scholar 

  2. Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  3. Mainardi, F.: Fractional relaxation–oscillation and fractional diffusion–wave phenomena. Chaos Solitons Fractals 7(9), 1461–1477 (1996)

    Article  MathSciNet  Google Scholar 

  4. He, J.: Some applications of nonlinear fractional differential equations and their approximations. Bull. Sci. Technol. 15(2), 86–90 (1999)

    Google Scholar 

  5. He, J.-H.: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Methods Appl. Mech. Eng. 167(1–2), 57–68 (1998)

    Article  MathSciNet  Google Scholar 

  6. Giusti, A., Mainardi, F.: A dynamic viscoelastic analogy for fluid-filled elastic tubes. Meccanica 51(10), 2321–2330 (2016)

    Article  MathSciNet  Google Scholar 

  7. Podlubny, I.: The Laplace transform method for linear differential equations of the fractional order (1997). arXiv preprint funct-an/9710005

  8. Odibat, Z., Momani, S.: Application of variational iteration method to nonlinear differential equations of fractional order. Int. J. Nonlinear Sci. Numer. Simul. 7(1), 27–34 (2006)

    Article  MathSciNet  Google Scholar 

  9. Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198. Elsevier, Amsterdam (1998)

    MATH  Google Scholar 

  10. Namias, V.: The fractional order Fourier transform and its application to quantum mechanics. IMA J. Appl. Math. 25(3), 241–265 (1980)

    Article  MathSciNet  Google Scholar 

  11. Hilfer, R., Luchko, Y., Tomovski, Z.: Operational method for the solution of fractional differential equations with generalized Riemann–Liouville fractional derivatives. Fract. Calc. Appl. Anal. 12(3), 299–318 (2009)

    MathSciNet  MATH  Google Scholar 

  12. Ford, N., Xiao, J., Yan, Y.: A finite element method for time fractional partial differential equations. Fract. Calc. Appl. Anal. 14(3), 454–474 (2011)

    Article  MathSciNet  Google Scholar 

  13. Li, X., Xu, C.: Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation. Commun. Comput. Phys. 8(5), 1016 (2010)

    Article  MathSciNet  Google Scholar 

  14. Reutskiy, S.Y., Lin, J.: A semi-analytic collocation method for space fractional parabolic PDE. Int. J. Comput. Math. 95(6–7), 1326–1339 (2018)

    Article  MathSciNet  Google Scholar 

  15. Lin, J., Xu, Y., Zhang, Y.: Simulation of linear and nonlinear advection–diffusion–reaction problems by a novel localized scheme. Appl. Math. Lett. 99, 106005 (2020)

    Article  MathSciNet  Google Scholar 

  16. Alikhanov, A.: A priori estimates for solutions of boundary value problems for fractional-order equations. Differ. Equ. 46(5), 660–666 (2010)

    Article  MathSciNet  Google Scholar 

  17. Akilandeeswari, A., Balachandran, K., Annapoorani, N.: Solvability of hyperbolic fractional partial differential equations. J. Appl. Anal. Comput. 7(4), 1570–1585 (2017)

    MathSciNet  MATH  Google Scholar 

  18. Mesloub, S., Bachar, I.: On a nonlocal 1-d initial value problem for a singular fractional-order parabolic equation with Bessel operator. Adv. Differ. Equ. 2019(1), 254 (2019)

    Article  MathSciNet  Google Scholar 

  19. Mesloub, S.: On a singular two dimensional nonlinear evolution equation with nonlocal conditions. Nonlinear Anal., Theory Methods Appl. 68(9), 2594–2607 (2008)

    Article  MathSciNet  Google Scholar 

  20. Mesloub, S.: A nonlinear nonlocal mixed problem for a second order pseudoparabolic equation. J. Math. Anal. Appl. 316(1), 189–209 (2006)

    Article  MathSciNet  Google Scholar 

  21. Mesloub, S., Bouziani, A.: On a class of singular hyperbolic equation with a weighted integral condition. Int. J. Math. Math. Sci. 22(3), 511–519 (1999)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors wish to thank the anonymous referees for their comments and valuable suggestions.

Availability of data and materials

The authors emphasize the existence of all data and materials in their manuscript and everything that supports their findings.

Funding

The authors stress that there are no sources of funding.

Author information

Authors and Affiliations

Authors

Contributions

Both authors contributed equally to the writing of this paper. Both authors read and approved the final manuscript.

Corresponding author

Correspondence to Lotfi Kasmi.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13662-019-2444-2

MSC

  • 35R11
  • 35B45
  • 35D35
  • 35L20

Keywords

  • Fractional differential equations
  • A priori estimate method
  • Boundary integral condition
  • Existence and uniqueness