Skip to main content

Theory and Modern Applications

A non-local problem with discontinuous matching condition for loaded mixed type equation involving the Caputo fractional derivative

Abstract

This research work is devoted to investigations of the existence and uniqueness of the solution of a non-local boundary value problem with discontinuous matching condition for the loaded equation. Considering parabolic-hyperbolic type equations involves the Caputo fractional derivative and loaded part joins in Riemann-Liouville integrals. The uniqueness of a solution is proved by the method of integral energy and the existence is proved by the method of integral equations.

1 Introduction and formulation of a problem

It is well known that fractional derivatives have been successfully applied to problems in system biology [1], physics [2–5] and hydrology [6, 7]. Physical models fractional differential operators have recently renewed attention from scientist which is mainly due to applications as models for physical phenomena exhibiting anomalous diffusion.

Note that investigations of fractional analogs of main ODE and PDEs appear as a result of the mathematic models for real-life processes [8], and they have recently been proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering [9, 10].

In the monographs of Kilbas et al. [11], Miller and Ross [12], Podlubny [13], and Samko et al. [14] we can see significant development of fractional differential equations.

Very recently some basic theory for the initial boundary value problem (BVP)s of fractional differential equations involving a Riemann-Liouville differential operator of order \(0 < \alpha \le 1\) has been discussed by Lakshmikantham and Vatsala [15, 16]. In a series of papers (see [17, 18]) the authors considered some classes of initial value problems for functional differential equations involving Riemann-Liouville and Caputo fractional derivatives of order \(0 < \alpha \le 1\).

It is well known that most fractional differential equations do not have exact analytic solutions, so approximation and numerical techniques must be used. The numerical solutions based on finite difference methods and several spectral algorithms for fractional differential equations were reported in Refs. [19–25].

It should be noted that problems for a class of fractional differential system and for the non-line differential equations with integral conditions were investigated in [26–30] and BVPs for the mixed type equations involving the Caputo and the Riemann-Liouville fractional differential operators were investigated by many authors; see for instance [31–33].

BVPs discounting matching conditions for the loaded equations with fractional derivative have not been investigated yet.

This paper deals the existence and uniqueness of a solution of the non-local problem with discontinuous matching condition for a loaded mixed type equation:

$$ 0 = \textstyle\begin{cases} u_{xx} - {}_{C}D_{oy}^{\alpha} u + p(x,y)\int_{x}^{1} (t - x)^{\beta - 1}u(t,0)\,dt,& \mbox{at } y > 0, \\ u_{xx} - u_{yy} + q(x,y)\int_{x + y}^{1} (t - x - y)^{\gamma - 1}u(t,0)\,dt,& \mbox{at } y < 0 \end{cases} $$
(1)

involving the Caputo fractional derivative operator [34]:

$$ {}_{C}D_{oy}^{\alpha} f = \frac{1}{\Gamma (1 - \alpha )} \int_{0}^{y} (y - t)^{ - \alpha} f'(t)\,dt, $$
(2)

where \(0 < \alpha, \beta,\gamma < 1\).

Definition

The Riemann-Liouville integral-differential operator of fractional order α (\(\alpha \in R\)), starting from the point a, is represented as follows [34]:

$$\begin{aligned} &D_{ax}^{\alpha} f(x) = \frac{\operatorname{sign}(x - a)}{\Gamma ( - \alpha )} \int_{a}^{x} \frac{f(t)}{\vert x - t \vert ^{\alpha + 1}}\,dt,\quad \alpha < 0; \\ &D_{ax}^{\alpha} f(x) = f(x), \quad\alpha = 0; \\ &D_{ax}^{\alpha} f(x) = \operatorname{sign}^{k}(x - a) \frac{d^{k}}{dt^{k}}D_{ax}^{\alpha - k}f(x),\quad k - 1 < \alpha \le k, k \in N. \end{aligned}$$
(3)

Definition

The Caputo differential operator of fractional order α (\(\alpha > 0\)) is represented as follows [34]:

$${}_{C}D_{ax}^{\alpha} f(x) = \operatorname{sign}^{k}(x - a)D_{ax}^{\alpha - k}f^{(k)}(x),\quad k - 1 < \alpha \le k, k \in N. $$

Let us take Ω, a domain, bounded with segments \(:A_{1}A_{2} = \{ (x,y): x = 1, 0 < y < h\}\), \(B_{1}B_{2} = \{ (x,y): x = 0, 0 < y < h\}\), \(B_{2}A_{2} = \{ (x,y): y = h, 0 < x < 1\}\) at the \(y > 0\), and characteristics: \(A_{1}C: x - y = 1\); \(B_{1}C: x + y = 0\) of equation (1) at \(y < 0\), where \(A_{1} ( 1;0 )\), \(A_{2} ( 1;h )\), \(B_{1} ( 0;0 )\), \(B_{2} ( 0;h )\), \(C ( \frac{1}{2}; - \frac{1}{2} )\).

Introduce the notations: \(\theta (x) = \frac{x + 1}{2} + i \cdot \frac{x - 1}{2}\), \(i^{2} = - 1\). We have

$$\Omega^{ +} = \Omega \cap (y > 0), \quad \Omega^{ -} = \Omega \cap (y < 0),\quad I_{1} = \biggl\{ x: \frac{1}{2} < x < 1 \biggr\} ,\quad I_{2} = \{ y:0 < y < h \}. $$

In the domain of Ω the following problem is investigated.

Problem I

Find a solution \(u(x,y)\) of equation (1) from the following class of functions:

$$W = \bigl\{ u(x,y): u(x,y) \in C(\bar{\Omega} ) \cap C^{2}\bigl( \Omega^{ -} \bigr), u_{xx} \in C \bigl( \Omega^{ +} \bigr), {}_{C}D_{oy}^{\alpha} u \in C \bigl( \Omega^{ +} \bigr) \bigr\} $$

satisfying the boundary conditions

$$\begin{aligned}& u(x,y)|_{{A_{1}A_{2}}} = \varphi (y),\quad 0 \le y \le h, \end{aligned}$$
(4)
$$\begin{aligned}& u(x,y)|_{{B_{1}B_{2}}} = \psi (y),\quad 0 \le y \le h, \end{aligned}$$
(5)
$$\begin{aligned}& \frac{d}{dx}u \bigl( \theta (x) \bigr) = a(x)u_{y}(x,0) + b(x)u_{x}(x,0) + c(x)u(x,0) + d(x),\quad x \in I_{1}, \end{aligned}$$
(6)

and the gluing condition:

$$ \lim_{y \to + 0}y^{1 - \alpha} u_{y}(x,y) = \lambda (x)u_{y}(x, - 0),\quad (x,0) \in A_{1}B_{1}, $$
(7)

where \(\varphi (y)\), \(\psi (y)\), \(a(x)\), \(b(x)\), \(c(x)\), \(d(x)\), and \(\lambda (x)\) (\(\lambda (x) \ne 0\)) are given functions.

2 Results and discussion

The uniqueness of solution of Problem I .

In the sequel, we assume that \(q(x,y) = - q_{1}(x + y)q_{2}(x - y)\). In fact, equation (1) at \(y \le 0\) and on the characteristics coordinate \(\xi = x + y\) and \(\eta = x - y\) in summary looks like:

$$ u_{\xi \eta} = \frac{q_{1}(\xi )q_{2}(\eta )}{4} \int_{\xi}^{1} (t - \xi )^{\gamma - 1}u(t,0)\,dt. $$
(8)

Let us denote \(u(x,0) = \tau (x)\), \(0 \le x \le 1\); \(u_{y}(x, - 0) = \nu^{ -} (x)\), \(0 < x < 1\);

$$\lim_{y \to + 0}y^{1 - \alpha} u_{y}(x,y) = \nu^{ +} (x),\quad 0 < x < 1. $$

It is well known that a solution of the Cauchy problem for equation (1) in the domain \(\Omega^{ -}\) can be represented as follows:

$$\begin{aligned} u(x,y) ={}& \frac{\tau (x + y) + \tau (x - y)}{2} - \frac{1}{2} \int_{x + y}^{x - y} \nu^{ -} (t)\,dt \\ &{}+ \frac{1}{4} \int_{x + y}^{x - y} q_{1}(\xi )\,d\xi \int_{\xi}^{x - y} q_{2}(\eta )\,d\eta \int_{\xi}^{1} (t - \xi )^{\gamma - 1}\tau (t)\,dt. \end{aligned}$$
(9)

After using condition (6) and taking (3) into account from (9) we will get

$$ \bigl( 2a(x) - 1 \bigr)\nu^{ -} (x) = \Gamma (\gamma )q_{1}(x)\tilde{q}_{2}(x)D_{x 1}^{ - \gamma} \tau (x) + \bigl( 1 - 2b(x) \bigr)\tau '(x) - 2c(x)\tau (x) - 2d(x), $$
(10)

where \(\tilde{q}_{2}(x) = \int_{x}^{1} q_{2}(\eta )\,d\eta\).

Considering the notations and gluing condition (7) we have

$$ \nu^{ +} (x) = \lambda (x)\nu^{ -} (x). $$
(11)

Further from equation (1) at \(y \to + 0\) taking (2), (11) into account, and

$$\lim_{y \to 0}D_{0y}^{\alpha - 1}f(y) = \Gamma (\alpha )\lim_{y \to 0}y^{1 - \alpha} f(y) $$

we get [28]

$$ \tau ''(x) - \lambda (x)\Gamma (\alpha ) \nu^{ -} (x) + \Gamma (\beta )p(x,0)D_{x1}^{ - \beta} \tau (x) = 0. $$
(12)

Theorem 1

If the following conditions are satisfied:

$$\begin{aligned}& \frac{\lambda (0)q_{1}(0)\tilde{q}_{2}(0)}{2a(0) - 1} \ge 0,\qquad p(0,0) \le 0,\qquad p'(x,0) \le 0; \end{aligned}$$
(13)
$$\begin{aligned}& \biggl( \frac{q_{1}(x)\tilde{q}_{2}(x)}{2a(x) - 1}\lambda (x) \biggr)^{\prime} \ge 0, \qquad \frac{\lambda (x)c(x)}{2a(x) - 1} \le 0,\qquad \biggl( \frac{1 - 2b(x)}{2a(x) - 1}\lambda (x) \biggr)^{\prime} \le 0, \end{aligned}$$
(14)

then the solution \(u(x,y)\) of Problem I is unique.

Proof

It is well known that, if a homogeneous problem has only a trivial solution, then we can state that the original problem has a unique solution. To this aim we assume that Problem I has two solutions, then denoting the difference of these as \(u(x,y)\) we will get an appropriate homogeneous problem.

We multiply equation (12) by \(\tau (x)\) and integrate from 0 to 1:

$$ \int_{0}^{1} \tau ''(x) \tau (x)\,dx - \Gamma (\alpha ) \int_{0}^{1} \lambda (x)\tau (x)\nu^{ -} (x)\,dx + \Gamma (\beta ) \int_{0}^{1} \tau (x)p(x,0)D_{x1}^{ - \beta} \tau (x)\,dx = 0. $$
(15)

We will investigate the integral

$$I = \Gamma (\alpha ) \int_{0}^{1} \lambda (x)\tau (x)\nu^{ -} (x)\,dx - \Gamma (\beta ) \int_{0}^{1} \tau (x)p(x,0)D_{x1}^{ - \beta} \tau (x)\,dx. $$

Taking (10) into account \(d(x) = 0\) we get

$$\begin{aligned} I ={}& \frac{\Gamma (\alpha )\Gamma (\gamma )}{2} \int_{0}^{1} \frac{q_{1}(x)\tilde{q}_{2}(x)}{2a(x) - 1}\lambda (x)\tau (x)D_{x 1}^{ - \gamma} \tau (x)\,dx \\ &{}+ \Gamma (\alpha ) \int_{0}^{1} \frac{ ( 1 - 2b(x) )\lambda (x)}{2a(x) - 1}\tau (x)\tau '(x)\,dx \\ &{}- \Gamma (\alpha ) \int_{0}^{1} \frac{\lambda (x)c(x)}{2a(x) - 1}\tau^{2}(x) \,dx - \Gamma (\beta ) \int_{0}^{1} \tau (x)p(x,0)D_{x1}^{ - \beta} \tau (x)\,dx \\ ={}& \frac{\Gamma (\alpha )}{2} \int_{0}^{1} \frac{q_{1}(x)\tilde{q}_{2}(x)}{2a(x) - 1}\lambda (x)\tau (x) \,dx \int_{x}^{1} (t - x)^{\gamma - 1} \tau (t)\,dt \\ &{}- \frac{\Gamma (\alpha )}{2} \int_{0}^{1} \frac{1 - 2b(x)}{2a(x) - 1}\lambda (x)\,d \bigl( \tau^{2}(x) \bigr) \\ &{}- \Gamma (\alpha ) \int_{0}^{1} \frac{\lambda (x)c(x)}{2a(x) - 1}\tau^{2}(x) \,dx - \int_{0}^{1} \tau (x)p(x,0)\,dx \int_{x}^{1} (t - x)^{\beta - 1} \tau (t)\,dt. \end{aligned}$$
(16)

Considering \(\tau (1) = 0\), \(\tau (0) = 0\) (deduced from the conditions (4), (5) in the homogeneous case) and on the base of the formula [35] we have

$$\vert x - t \vert ^{ - \gamma} = \frac{1}{\Gamma ( \gamma )\cos\frac{\pi \gamma}{2}} \int_{0}^{\infty} z^{\gamma - 1}\cos \bigl[ z ( x - t ) \bigr]\,dz,\quad 0 < \gamma < 1. $$

After some simplifications from (16) we will get

$$\begin{aligned} I ={}& \frac{\Gamma (\alpha )q_{1}(0)\tilde{q}_{2}(0)\lambda (0)}{4(2a(0) - 1)\Gamma (1 - \gamma )\sin \frac{\pi \gamma}{2}} \int_{0}^{\infty} z^{ - \gamma} \biggl[ \biggl( \int_{0}^{1} \tau (t)\cos zt\,dt \biggr)^{2} \\ &{}+ \biggl( \int_{0}^{1} \tau (t)\sin zt\,dt \biggr)^{2} \biggr]\,dz \\ &{}+ \frac{\Gamma (\alpha )}{4\Gamma (1 - \gamma )\sin \frac{\pi \gamma}{2}} \int_{0}^{\infty} z^{ - \gamma}\,dz \int_{0}^{1} \frac{\partial}{ \partial x} \biggl[ \lambda (x) \frac{q_{1}(x)\tilde{q}_{2}(x)}{2a(x) - 1} \biggr] \\ &{}\times \biggl[ \biggl( \int_{x}^{1} \tau (t)\cos zt\,dt \biggr)^{2} + \biggl( \int_{x}^{1} \tau (t)\sin zt\,dt \biggr)^{2} \biggr]\,dx \\ &{}- \frac{\Gamma (\alpha )}{2} \int_{0}^{1} \tau^{2}(x) \biggl( \lambda (x)\frac{1 - 2b(x)}{2a(x) - 1} \biggr)^{\prime}\,dx - 2\Gamma (\alpha ) \int_{0}^{1} \frac{\lambda (x)c(x)}{2a(x) - 1}\tau^{2}(x) \,dx \\ &{}- \frac{p(0,0)}{2\Gamma (1 - \beta )\sin \frac{\pi \beta}{2}} \int_{0}^{\infty} z^{ - \beta} \biggl[ \biggl( \int_{0}^{1} \tau (t)\cos zt\,dt \biggr)^{2} + \biggl( \int_{0}^{1} \tau (t)\sin zt\,dt \biggr)^{2} \biggr]\,dz \\ &{}- \frac{1}{2\sin \frac{\pi \beta}{2}\Gamma (1 - \beta )} \int_{0}^{\infty} z^{ - \beta}\,dz \int_{0}^{1} \frac{\partial}{\partial x} \bigl[ p(x,0) \bigr] \\ &{}\times \biggl[ \biggl( \int_{x}^{1} \tau (t)\cos zt\,dt \biggr)^{2} + \biggl( \int_{x}^{1} \tau (t)\sin zt\,dt \biggr)^{2} \biggr]\,dx. \end{aligned}$$
(17)

Thus, due to conditions (13), (14) from (17) we infer that \(\tau (x) \equiv 0\). Hence, based on the solution of the first boundary problem for equation (1) [32, 33] by using conditions (4) and (5) we will get \(u(x,y) \equiv 0\) in \(\overline{\Omega}^{ +}\). Further, from the functional relations (10), taking into account \(\tau (x) \equiv 0\), we deduce that \(\nu^{ -} (x) \equiv 0\). Consequently, based on the solution (9) we obtain \(u(x,y) \equiv 0\) in a closed domain \(\overline{\Omega}^{ -}\). □

The existence of a solution of Problem I .

Theorem 2

If conditions (13), (14) are satisfied and

$$\begin{aligned}& \varphi (y), \psi (y) \in C ( \overline{I_{2}} ) \cap C^{1} ( I_{2} ), \quad p(x,0) \in C ( \overline{A_{1}B_{1}} ) \cap C^{2} ( A_{1}B_{1} ), \end{aligned}$$
(18)
$$\begin{aligned}& q(x,y) \in C \bigl( \overline{\Omega^{ -}} \bigr) \cap C^{2} \bigl( \Omega^{ -} \bigr), \quad a(x), b(x), c(x), d(x) \in C^{1} ( \overline{I_{1}} ) \cap C^{2} ( I_{1} ), \end{aligned}$$
(19)

then the solution of the investigated problem exists.

Proof

Taking (10) into account, from equation (12) we will obtain

$$ \tau ''(x) - A(x)\tau '(x) = f(x) - B(x) \tau (x), $$
(20)

where

$$\begin{aligned}& f(x) = \frac{\Gamma (\alpha )\Gamma (\gamma )\lambda (x)q_{1}(x)\tilde{q}_{2}(x)}{2(2a(x) - 1)}D_{x1}^{ - \gamma} \tau (x) - \Gamma ( \beta )p(x,0)D_{x1}^{ - \beta} \tau (x) - \frac{2\Gamma (\alpha )\lambda (x)d(x)}{2a(x) - 1}, \end{aligned}$$
(21)
$$\begin{aligned}& A(x) = \frac{\Gamma (\alpha )\lambda (x)(1 - 2b(x))}{2a(x) - 1}, \qquad B(x) = \frac{2\Gamma (\alpha )\lambda (x)c(x)}{2a(x) - 1}. \end{aligned}$$
(22)

The solution of equation (20) together with the conditions

$$ \tau (0) = \psi (0), \qquad \tau (1) = \varphi (0) $$
(23)

has the form

$$\begin{aligned} \tau (x) ={}& A_{1}(x) \biggl( \int_{x}^{1} \bigl( B(t)\tau (t) - f(t) \bigr) A'_{1}(t)\,dt + \frac{\varphi (0) - \psi (0)}{A_{1}(1)} \biggr) \\ &{}- \frac{A_{1}(x)}{A_{1}(1)} \int_{0}^{1} \bigl( B(t)\tau (t) - f(t) \bigr) \frac{A_{1}(t)}{A'_{1}(t)}\,dt \\ &{} + \int_{0}^{x} \bigl( B(t)\tau (t) - f(t) \bigr) \frac{A_{1}(t)}{A'_{1}(t)}\,dt + \psi (0), \end{aligned}$$
(24)

where

$$ A_{1}(x) = \int_{0}^{x} \exp \biggl( \int_{0}^{t} A(z)\,dz \biggr)\,dt. $$
(25)

Further, considering (21) and using (3) from (24) we will get

$$\begin{aligned} \tau (x) ={}& A_{1}(x) \biggl[ \int_{x}^{1} A'_{1}(t)B(t)\tau (t)\,dt - \frac{\Gamma (\alpha )}{2} \int_{x}^{1} \frac{\lambda (t)q_{1}(t)\tilde{q}_{2}(t)}{2a(t) - 1}A'_{1}(t) \,dt \int_{t}^{1} (s - t)^{\gamma - 1} \tau (s)\,ds \biggr] \\ &{} + A_{1}(x) \int_{x}^{1} A'_{1}(t)p(t,0) \,dt \int_{t}^{1} (s - t)^{\beta - 1}\tau (s)\,ds - \frac{A_{1}(x)}{A_{1}(1)} \int_{0}^{1} \frac{A_{1}(t)}{A'_{1}(t)}B(t)\tau (t)\,dt \\ &{}+ \frac{\Gamma (\alpha )}{2}\frac{A_{1}(x)}{A_{1}(1)} \int_{0}^{1} \frac{A_{1}(t)\lambda (t)q_{1}(t)\tilde{q}_{2}(t)}{(2a(t) - 1)A'_{1}(t)}\,dt \int_{t}^{1} (s - t)^{\gamma - 1} \tau (s)\,ds \\ &{}- \frac{A_{1}(x)}{A_{1}(1)} \int_{0}^{1} \frac{A_{1}(t)}{A'_{1}(t)}p(t,0)\,dt \int_{t}^{1} (s - t)^{\beta - 1}\tau (s)\,ds + \int_{0}^{x} \frac{A_{1}(t)}{A'_{1}(t)}B(t)\tau (t)\,dt \\ &{}- \frac{\Gamma (\alpha )}{2} \int_{0}^{x} \frac{A_{1}(t)\lambda (t)q_{1}(t)\tilde{q}_{2}(t)}{(2a(t) - 1)A'_{1}(t)}\,dt \int_{t}^{1} (s - t)^{\gamma - 1} \tau (s)\,ds \\ &{}+ \int_{0}^{x} \frac{A_{1}(t)}{A'_{1}(t)}p(t,0)\,dt \int_{t}^{1} (s - t)^{\beta - 1}\tau (s)\,ds + f_{1}(x), \end{aligned}$$
(26)

where

$$\begin{aligned} f_{1}(x) ={}& \biggl( 1 - \frac{A_{1}(x)}{A_{1}(1)} \biggr) \int_{0}^{x} \frac{2\Gamma (\alpha )d(t)A_{1}(t)\lambda (t)}{A'_{1}(t)(2a(t) - 1)}\,dt + 2\Gamma ( \alpha )A_{1}(x) \int_{x}^{1} \frac{d(t)A'_{1}(t)\lambda (t)}{2a(t) - 1}\,dt \\ &{}- \frac{A_{1}(x)}{A_{1}(1)} \int_{x}^{1} \frac{2\Gamma (\alpha )d(t)A_{1}(t)\lambda (t)}{A'_{1}(t)(2a(t) - 1)}\,dt - \frac{A_{1}(x)}{A_{1}(1)} \bigl( \psi (0) - \varphi (0) \bigr) + \psi (0). \end{aligned}$$
(27)

After some simplifications (26) we will rewrite our expression in the form

$$\begin{aligned} \tau (x) ={}& A_{1}(x) \biggl[ \int_{x}^{1} A'_{1}(t)B(t)\tau (t)\,dt - \frac{\Gamma (\alpha )}{2} \int_{x}^{1} \tau (s)\,ds \int_{x}^{s} (s - t)^{\gamma - 1} \frac{\lambda (t)q_{1}(t)\tilde{q}_{2}(t)}{2a(t) - 1}A'_{1}(t)\,dt \biggr] \\ &{}+ A_{1}(x) \int_{x}^{1} \tau (s)\,ds \int_{x}^{s} (s - t)^{\beta - 1} A'_{1}(t)p(t,0)\,dt - \frac{A_{1}(x)}{A_{1}(1)} \int_{0}^{1} \frac{A_{1}(t)}{A'_{1}(t)}B(t)\tau (t)\,dt \\ &{}+ \frac{\Gamma (\alpha )}{2}\frac{A_{1}(x)}{A_{1}(1)} \int_{0}^{1} \tau (s)\,ds \int_{0}^{s} (s - t)^{\gamma - 1} \frac{A_{1}(t)\lambda (t)q_{1}(t)\tilde{q}_{2}(t)}{(2a(t) - 1)A'_{1}(t)}\,dt \\ &{}- \frac{A_{1}(x)}{A_{1}(1)} \int_{0}^{1} \tau (s)\,ds \int_{0}^{s} (s - t)^{\beta - 1} \frac{A_{1}(t)}{A'_{1}(t)}p(t,0)\,dt + \int_{0}^{x} \frac{A_{1}(t)}{A'_{1}(t)}B(t)\tau (t)\,dt \\ &{}- \frac{\Gamma (\alpha )}{2}\biggl( \int_{0}^{x} \tau (s)\,ds \int_{0}^{s} \frac{A_{1}(t)\lambda (t)q_{1}(t)\tilde{q}_{2}(t)}{A'_{1}(t)(2a(t) - 1)(s - t)^{1 - \gamma}}\,dt \\ &{} + \int_{x}^{1} \tau (s)\,ds \int_{0}^{x} \frac{A_{1}(t)\lambda (t)q_{1}(t)\tilde{q}_{2}(t)}{A'_{1}(t)(2a(t) - 1)(s - t)^{1 - \gamma}}\,dt \biggr) \\ &{}+ \int_{0}^{x} \tau (s)\,ds \int_{0}^{s} \frac{A_{1}(t)(s - t)^{\beta - 1}}{A'_{1}(t)}p(t,0)\,dt \\ &{}+ \int_{x}^{1} \tau (s)\,ds \int_{0}^{x} \frac{A_{1}(t)(s - t)^{\beta - 1}}{A'_{1}(t)}p(t,0)\,dt + f_{1}(x) \end{aligned}$$

i.e., in summary, we have the integral equation

$$ \tau (x) = \int_{0}^{1} K(x,t)\tau (t)\,dt + f_{1}(x). $$
(28)

Here

$$\begin{aligned}& K(x,t) = \textstyle\begin{cases} K_{1}(x,s),& 0 \le t \le x, \\ K_{2}(x,s),& x \le t \le 1, \end{cases}\displaystyle \end{aligned}$$
(29)
$$\begin{aligned}& \begin{aligned}[b] K_{1}(x,s) ={}& \biggl( \frac{A_{1}(x)}{A_{1}(1)} - 1 \biggr) \biggl[ \frac{\Gamma (\alpha )}{2} \int_{0}^{s} (s - t)^{\gamma - 1} \frac{A_{1}(t)\lambda (t)q_{1}(t)\tilde{q}_{2}(t)}{(2a(t) - 1)A'_{1}(t)}\,dt - \frac{A_{1}(s)}{A'_{1}(s)}B(s) \biggr] \\ &{}- \biggl( \frac{A_{1}(x)}{A_{1}(1)} - 1 \biggr) \int_{0}^{s} (s - t)^{\beta - 1} \frac{A_{1}(t)}{A'_{1}(t)}p(t,0)\,dt, \end{aligned} \end{aligned}$$
(30)
$$\begin{aligned}& \begin{aligned}[b] K_{2}(x,s) ={}& A_{1}(x) \biggl( A'_{1}(s)B(s) - \frac{\Gamma (\alpha )}{2} \int_{x}^{s} (s - t)^{\gamma - 1} \frac{\lambda (t)q_{1}(t)\tilde{q}_{2}(t)}{2a(t) - 1}A'_{1}(t)\,dt \biggr) \\ &{}+ A_{1}(x) \int_{x}^{s} (s - t)^{\beta - 1} A'_{1}(t)p(t,0)\,dt - \frac{A_{1}(x)}{A_{1}(1)} \frac{A_{1}(s)}{A'_{1}(s)}B(s) \\ &{}+ \frac{\Gamma (\alpha )}{2}\frac{A_{1}(x)}{A_{1}(1)} \int_{0}^{s} (s - t)^{\gamma - 1} \frac{A_{1}(t)\lambda (t)q_{1}(t)\tilde{q}_{2}(t)}{(2a(t) - 1)A'_{1}(t)}\,dt \\ &{}- \frac{\Gamma (\alpha )}{2} \int_{0}^{x} \frac{A_{1}(t)\lambda (t)q_{1}(t)\tilde{q}_{2}(t)}{A'_{1}(t)(2a(t) - 1)}(s - t)^{\gamma - 1} \,dt \\ &{}+ \biggl( 1 - \frac{A_{1}(x)}{A_{1}(1)} \biggr) \int_{0}^{x} \frac{A_{1}(t)(s - t)^{\beta - 1}}{A'_{1}(t)}p(t,0)\,dt. \end{aligned} \end{aligned}$$
(31)

Due to the class (18), (19) of the given functions and after some evaluations, from (30), (31) and (27), (29) we will conclude that \(\vert K(x,t) \vert \le \mbox{const} \), \(\vert f_{1}(x) \vert \le \mbox{const} \).

Since the kernel \(K(x,t)\) is continuous and the function on the right-hand side \(F(x)\) is continuously differentiable, we can write the solution of integral equation (28) via the resolvent-kernel:

$$ \tau (x) = f_{1}(x) - \int_{0}^{1} \Re (x,t)f_{1}(t) \,dt, $$
(32)

where \(\Re (x,t)\) is the resolvent-kernel of \(K(x,t)\).

The unknown functions \(\nu^{ -} (x)\) and \(\nu^{ +} (x)\) we will find accordingly from (10) and (11):

$$\begin{aligned} \nu^{ -} (x) ={}& \frac{q_{1}(x)\tilde{q}_{2}(x)}{2 ( 1 - 2a(x) )} \int_{x}^{1} (t - x)^{\gamma - 1}\,dt \int_{0}^{1} \Re (t,s)f_{1}(s)\,ds \\ &{}+ \frac{q_{1}(x)\tilde{q}_{2}(x)}{2 ( 2a(x) - 1 )} \int_{x}^{1} (t - x)^{\gamma - 1}f_{1}(t) \,dt \\ &{}+ \frac{1 - 2b(x)}{2a(x) - 1}f'_{1}(x) - \frac{1 - 2b(x)}{2a(x) - 1} \int_{0}^{1} \frac{\partial \Re (x,t)}{\partial x}f_{1}(t) \,dt - \frac{2c(x)}{2a(x) - 1}f_{1}(x) \\ &{}+ \frac{2c(x)}{2a(x) - 1} \int_{0}^{1} \Re (x,t)f_{1}(t)\,dt - \frac{2d(x)}{2a(x) - 1} \end{aligned}$$

and \(\nu^{ +} (x) = \lambda (x)\nu^{ -} (x)\).

Considering the solution of Problem I in the domain \(\Omega^{ +}\) we write our expression as follows [33, 36]:

$$\begin{aligned} u(x,y) ={}& \int_{0}^{y} G_{\xi} (x,y,0,\eta )\psi ( \eta )\,d\eta - \int_{0}^{y} G_{\xi} (x,y,1,\eta )\varphi ( \eta )\,d\eta + \int_{0}^{1} G_{0}(x - \xi,y)\tau (\xi ) \,d\xi \\ &{}- \int_{0}^{y} \int_{0}^{1} G(x,y,0,\eta ) p(\xi )\,d\xi \,d\eta \int_{\xi}^{1} (t - \xi )^{\beta - 1}\tau (t)\,dt. \end{aligned}$$
(33)

Here \(G_{0}(x - \xi,y) = \frac{1}{\Gamma (1 - \alpha )}\int_{0}^{y} \eta^{ - \alpha} G(x,y,\xi,\eta )\,d\eta\),

$$G(x,y,\xi,\eta ) = \frac{(y - \eta )^{\alpha / 2 - 1}}{2}\sum_{n = - \infty}^{\infty} \biggl[ e_{1,\alpha / 2}^{1,\alpha / 2} \biggl( - \frac{\vert x - \xi + 2n \vert }{(y - \eta )^{\alpha / 2}} \biggr) - e_{1,\alpha / 2}^{1,\alpha / 2} \biggl( - \frac{\vert x + \xi + 2n \vert }{(y - \eta )^{\alpha / 2}} \biggr) \biggr]. $$

Here the Green’s function of the first boundary problem equation (1) in the domain \(\Omega^{ +}\) with the Riemanne-Liouville fractional differential operator instead of the Caputo ones [34],

$$e_{1,\delta}^{1,\delta} (z) = \sum_{n = 0}^{\infty} \frac{z^{n}}{n!\Gamma (\delta - \delta n)}, $$

is a Wright type function [36]. □

3 Conclusion

If conditions (13), (14), (18), and (19) are satisfied, then the solution of Problem I is unique and exists, and this solution in the domains \(\Omega^{ -}\) and \(\Omega^{ +}\) will be found by equations (9) and (33), respectively.

References

  1. Diethelm, K, Freed, AD: On the solution of nonlinear fractional order differential equations used in the modeling of viscoelasticity. In: Keil, F, Mackens, W, Voss, H, Werther, J (eds.) Scientific Computing in Chemical Engineering II: Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, pp. 217-224. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  2. El-Sayed, AMA: Fractional order evolution equations. J. Fract. Calc. 7, 89-100 (1995)

    MathSciNet  MATH  Google Scholar 

  3. El-Sayed, AMA: Fractional order diffusion-wave equations. Int. J. Theor. Phys. 35(2), 311-322 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  5. Mainardi, F: Fractional calculus: some basic problems in continuum and statistical mechanics. In: Carpinteri, A, Mainardi, F (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 291-348. Springer, Wien (1997)

    Chapter  Google Scholar 

  6. Glockle, WG, Nonnenmacher, TF: A fractional calculus approach of self-similar protein dynamics. Biophys. J. 68, 46-53 (1995)

    Article  Google Scholar 

  7. Kirchner, JW, Feng, X, Neal, C: Fractal stream chemistry and its implications for contaminant transport in catchments. Nature 403, 524-526 (2000)

    Article  Google Scholar 

  8. Anastasio, TJ: The fractional order dynamics of brainstem vestibule-oculomotor neurons. Biol. Cybern. 72, 69-79 (1994)

    Article  Google Scholar 

  9. Koh, CG, Kelly, JM: Application of fractional derivatives to seismic analysis of base-isolated models. Earthq. Eng. Struct. Dyn. 19, 229-241 (1990)

    Article  Google Scholar 

  10. Magin, R: Fractional calculus in bioengineering. Crit. Rev. Biomed. Eng. 32(1), 1-104 (2004)

    Article  Google Scholar 

  11. Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)

    Book  MATH  Google Scholar 

  12. Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  13. Podlubny, I: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  14. Samko, SG, Kilbas, AA, Marichev, OI: Fractional Integral and Derivatives: Theory and Applications. Gordon & Breach, Longhorne (1993)

    MATH  Google Scholar 

  15. Lakshmikantham, V, Vatsala, AS: Basic theory of fractional differential equations. Nonlinear Anal. 69(8), 2677-2682 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lakshmikantham, V, Vatsala, AS: Theory of fractional differential inequalities and applications. Commun. Appl. Anal. 11(3-4), 395-402 (2007)

    MathSciNet  MATH  Google Scholar 

  17. Belarbi, A, Benchohra, M, Ouahab, A: Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces. Appl. Anal. 85(12), 1459-1470 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Benchohra, M, Henderson, J, Ntouyas, SK, Ouahab, A: Existence results for fractional order functional differential equations with infinite delay. J. Math. Anal. Appl. 338(2), 1340-1350 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ding, Z, Xiao, A, Li, M: Weighted finite difference methods for a class of space fractional partial differential equations with variable coefficients. J. Comput. Appl. Math. 233, 1905-1914 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wang, H, Du, N: Fast alternating-direction finite difference methods for three-dimensional space-fractional diffusion equations. J. Comput. Phys. 258, 305-318 (2014)

    Article  MathSciNet  Google Scholar 

  21. Bhrawy, AH, Alhamed, Y, Baleanu, D, Al-Zahrani, A: New spectral techniques for systems of fractional differential equations using fractional-order generalized Laguerre orthogonal functions. Fract. Calc. Appl. Anal. 17, 1137-1157 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Bhrawy, AH, Taha, TM, Alzahrani, E, Baleanu, D, Alzahrani, A: Correction: New operational matrices for solving fractional differential equations on the half-line. PLoS ONE 10(9), e0138280 (2015). doi:10.1371/journal.pone.0138280

    Article  Google Scholar 

  23. Bhrawy, AH, Doha, EH, Baleanu, D, Ezz-eldein, SS: A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations. J. Comput. Phys. 293, 142-156 (2015)

    Article  MathSciNet  Google Scholar 

  24. Bhrawy, AH, Zaky, MA, Gorder, RAV: A space-time Legendre spectral tau method for the two-sided space-time Caputo fractional diffusion-wave equation. Numer. Algorithms 71, 151-180 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  25. Bhrawy, AH, Doha, EH, Ezz-Eldien, SS, Abdelkawy, MA: A numerical technique based on the shifted Legendre polynomials for solving the time-fractional coupled KdV equation. Calcolo 53, 1-17 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. Cabada, A, Wang, G: Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl. 389, 403-411 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Caballero, J, Cabrera, I, Sadarangani, K: Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. Abstr. Appl. Anal. 2012, Article ID 303545 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Harjani, J, Rocha, J, Sadarangani, K: Existence and uniqueness of solutions for a class of fractional differential coupled system with integral boundary conditions. Appl. Math. Inf. Sci. 9(2L), 401-405 (2015)

    MathSciNet  Google Scholar 

  29. Baleanu, D, Mehdi, M, Hakimeh, B: A fractional derivative inclusion problem via an integral boundary condition. J. Comput. Anal. Appl. 21(3), 504-514 (2016)

    MathSciNet  MATH  Google Scholar 

  30. Selvaraj, S, Baleanu, D, Palaniyappan, K: On fractional neutral integro-differential systems with state-dependent delay and non-instantaneous impulses. Adv. Differ. Equ. 2015, Article ID 372 (2015)

    Article  MathSciNet  Google Scholar 

  31. Kadirkulov, BJ, Turmetov, BK: On a generalisation of the heat equation. Uzbek. Mat. Zh. 3, 40-46 (2006) http://uzmj.mathinst.uz/files/uzmj-2006_3.pdf

    MathSciNet  Google Scholar 

  32. Kadirkulov, BJ: Boundary problems for mixed parabolic-hyperbolic equations with two lines of changing type and fractional derivative. Electron. J. Differ. Equ. 2014, 57 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Karimov, ET, Akhatov, J: A boundary problem with integral gluing condition for a parabolic-hyperbolic equation involving the Caputo fractional derivative. Electron. J. Differ. Equ. 2014, 14 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  34. Pskhu, AV: Uravneniye v chasnykh proizvodnykh drobnogo poryadka [Partial differential equation of fractional order]. Nauka, Moscow (2005) (in Russian), 200 pp.

    Google Scholar 

  35. Smirnov, MM: Mixed Type Equations. Nauka, Moscow (2000)

    Google Scholar 

  36. Pskhu, AV: Solution of boundary value problems fractional diffusion equation by the Green function method. Differ. Equ. 39(10), 1509-1513 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the reviewers for useful suggestions, which improved the contents of this paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Obidjon Khayrullayevich Abdullaev.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The two authors have participated into the results obtained. The collaboration of each one cannot be separated in different parts of the paper. Both of them have made substantial contributions to the theoretical results. The two authors have been involved in drafting the manuscript and revising it critically for important intellectual content. Both authors have given final approval of the version to be published.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sadarangani, K., Abdullaev, O.K. A non-local problem with discontinuous matching condition for loaded mixed type equation involving the Caputo fractional derivative. Adv Differ Equ 2016, 241 (2016). https://doi.org/10.1186/s13662-016-0969-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13662-016-0969-1

MSC

Keywords