Open Access

On impulsive partial differential equations with Caputo-Hadamard fractional derivatives

Advances in Difference Equations20162016:281

https://doi.org/10.1186/s13662-016-1008-y

Received: 14 April 2016

Accepted: 20 October 2016

Published: 28 October 2016

Abstract

In this paper, the mixed Caputo-Hadamard fractional derivative is introduced based on the Caputo-type modification of Hadamard fractional derivatives in the existing paper, and impulsive partial differential equations with Caputo-Hadamard fractional derivatives are studied. The formula of a general solution for these impulsive fractional partial differential equations is found by considering some limiting cases (impulses tending to zero), and its validity is shown by an example.

Keywords

impulsive fractional partial differential equationsfractional partial differential equationsimpulsegeneral solution

MSC

34A0834A37

1 Introduction

The fractional calculus was developed within the frame of the Hadamard fractional derivative in [16], and for the general theory of Hadamard fractional calculus we refer the interested reader to [7]. Moreover, some progress was achieved in controllability, some new definitions, some new methods of numerical solution etc. for fractional differential equations [813].

Recently, Jarad et al. presented the definition of Caputo-Hadamard fractional derivative in [14], and developed the fundamental theorem of fractional calculus in the Caputo-Hadamard setting in [14, 15].

Furthermore, Vityuk and Golushkov were concerned with the existence and uniqueness of solution for a kind of fractional partial differential equations in [16]. Next, Abbas and Benchohra first considered fractional partial differential equations with impulses in [17], and the authors gave some results as regards the existence and uniqueness of solution for these impulsive systems in [1722].

Now the equivalent integral equations were found for several fractional-order systems with impulses in [2328], and the obtained results show that there is a general solution for their impulsive fractional-order systems.

Motivated by the above-mentioned work, we will give the definition of a mixed Caputo-Hadamard fractional derivative and seek the equivalent integral equations for a kind of impulsive partial differential equations with Caputo-Hadamard fractional derivatives to find the essential result that there exists a general solution for impulsive fractional differential equations in this paper. We have
$$ \left \{ \textstyle\begin{array}{l} ( {{}_{\mathrm{C}\text{-}\mathrm{H}}D_{(a + ,b + )}^{q} u} )(x,y) = f(x,y,u(x,y)),\quad (x,y) \in J\mbox{ and }x \neq x_{i}\ (i = 1,2,\ldots,m), \\ u(x_{i}^{+} ,y) = u(x_{i}^{-} ,y) + I_{i} ( {u(x_{i}^{-} ,y)} ),\quad i = 1,2,\ldots,m, \\ u(x,b) = \phi(x),\qquad u(a,y) = \psi(y),\quad x \in[a,A], y \in[b,B], \end{array}\displaystyle \right . $$
(1)
where \(J = [a,A] \times[b,B]\) (\(a,b > 0\)), \(q = (q_{1} ,q_{2} )\) (here \(q_{1} ,q_{2} \in\mathbb{C}\) and \((\Re(q_{1} ),\Re(q_{2} ))\in(0,1] \times (0,1]\)), \({}_{\mathrm{C}\text{-}\mathrm{H}}D_{(a + ,b + )}^{q} \) denotes the Caputo-Hadamard fractional derivative of order q. We have the impulsive points \(a = x_{0} < x_{1} < \cdots< x_{m} < x_{m + 1} =A \). \(u(x_{i}^{+} ,y) = \lim_{\varepsilon \to0^{+} } u(x_{i} + \varepsilon,y)\) and \(u(x_{i}^{-} ,y) = \lim_{\varepsilon \to0^{-} } u(x_{i} + \varepsilon,y)\) represent the right and left limits of \(u(x, y)\) at \(x = x_{i}\) (\(i = 1,2,\ldots,m\)), respectively. \(f:J \times\mathbb{C}^{n} \to\mathbb{C}^{n} \) and \(I_{i} : \mathbb{C}^{n} \to\mathbb{C}^{n} \) (\(i = 1, 2,\ldots, m\)) are given functions. \(\phi:[a,A] \to\mathbb{C}^{n} \), \(\psi:[b,B] \to\mathbb {C}^{n} \) are given continuous functions with \(\phi(a) = \psi(b)\).
Consider a limiting case in system (1):
$$\begin{aligned}& \lim_{I_{i} ( {u(x_{i}^{-} ,y)} ) \to0\text{ for all }i \in\{ 1,2,\ldots,m\} } \{ {\text{system (1)}} \} \\& \quad \to \left \{ \textstyle\begin{array}{l} ( {{}_{\mathrm{C}\text{-}\mathrm{H}}D_{(a + ,b + )}^{q} u} )(x,y) = f(x,y,u(x,y)),\quad (x,y) \in J, \\ u(x,b) = \phi(x),\qquad u(a,y) = \psi(y),\quad x \in[a,A], y \in[b,B]. \end{array}\displaystyle \right . \end{aligned}$$
(2)
Therefore,
$$\begin{aligned}& \lim_{I_{i} ( {u(x_{i}^{-} ,y)} ) \to0\text{ for all }i \in\{ 1,2,\ldots,m\} } \{ {\text{the solution of system (1)}} \} \\& \quad =\{ {\text{the solution of system (2)}} \}. \end{aligned}$$
(3)
Next, some preliminaries are given in Section 2, and the equivalent integral equation will be provided for a fractional partial differential system with impulses in Section 3. Finally, an example is presented to illuminate the main result in Section 4.

2 Preliminaries

In this section, we shall present the definition of Caputo-Hadamard fractional partial derivatives according to definition of left-sided Caputo-Hadamard fractional derivatives suggested by Jarad et al. in [14, 15], and we draw a conclusion.

Definition 2.1

Let \(a_{1} \in[a,A]\), \(z^{+} = (a_{1} + ,b + )\), \(J_{z} = [a_{1} ,A] \times[b,B]\), \(q = (q_{1} ,q_{2} )\) (here \(q_{1} ,q_{2} \in \mathbb{C} \) and \((\Re(q_{1} ),\Re(q_{2} )) \in(0,1] \times(0,1]\)). For the function w, the expression
$$\bigl( {{}_{\mathrm{H}}{\mathcal{J}}_{z^{+} }^{q} w} \bigr) (x,y) = \frac{1}{{\Gamma (q_{1} )\Gamma(q_{2} )}} \int_{a_{1} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} w(s,t) \frac {{dt}}{t}\frac{{ds}}{s}} }, $$
where Γ is the gamma function, is called the left-sided mixed Hadamard integral of order q.

Definition 2.2

Let \(q = (q_{1} ,q_{2} )\) (here \(q_{1} ,q_{2} \in \mathbb{C} \) and \((\Re(q_{1} ),\Re(q_{2} )) \in(0,1] \times(0,1]\)). For \(w \in L^{1} (J_{z} ,\mathbb{C}^{n} )\) the mixed Caputo-Hadamard fractional derivative of order q can be defined by the expression
$$\begin{aligned} \begin{aligned} & \bigl( {{}_{\mathrm{C}\text{-}\mathrm{H}}D_{z^{+} }^{q} w} \bigr) (x,y) \\ &\quad = \frac{1}{{\Gamma(1 - q_{1} )\Gamma(1 - q_{2} )}} \int_{a_{1} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac {x}{s}} \biggr)^{ - q_{1} } \biggl( {\ln\frac{y}{t}} \biggr)^{ - q_{2} } \delta_{s} \delta_{t} w(s,t)\frac{{dt}}{t} \frac{{ds}}{s}} } \\ &\quad = \bigl( {{}_{\mathrm{H}}{\mathcal{J}}_{z^{+} }^{1 - q} \delta_{x} \delta_{y} w} \bigr) (x,y), \end{aligned} \end{aligned}$$
where we have the partial differential operator \(\delta_{x} = x{ {\partial \over {\partial x}}}\).

Lemma 2.3

Let \(h \in C(J_{z} ,\mathbb{C}^{n} )\), \(q = (q_{1} ,q_{2} )\) (here \(q_{1} ,q_{2} \in\mathbb{C} \) and \((\Re(q_{1} ),\Re(q_{2} )) \in (0,1] \times(0,1]\)). A function \(u \in C(J_{z} ,\mathbb{C}^{n} )\) is a solution of the differential equation
$$ \bigl( {{}_{\mathrm{C}\text{-}\mathrm{H}}D_{z^{+} }^{q} u} \bigr) (x,y) = h(x,y), \quad (x,y) \in J_{z}, $$
(4)
if and only if
$$\begin{aligned} u(x,y) =& u(x,b) + u \bigl( {a_{1}^{+} ,y} \bigr) - u \bigl( {a_{1}^{+} ,b} \bigr) + \bigl(I_{z^{+} }^{q} h \bigr) (x,y) \\ =& u(x,b) + u \bigl( {a_{1}^{+} ,y} \bigr) - u \bigl( {a_{1}^{+} ,b} \bigr) \\ &{}+ \frac{1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \int_{a_{1} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} h(s,t) \frac{{dt}}{t}\frac{{ds}}{s}} } , \\ &\textit{for }(x,y) \in J_{z}. \end{aligned}$$
(5)

Proof

Let \(u(x, y)\) is a solution of the equation \(( {{}_{\mathrm{C}\text{-}\mathrm{H}}D_{z^{+} }^{q} u} )(x,y) = h(x,y)\), \((x,y) \in J_{z}\). Due to
$$\bigl( {{}_{\mathrm{C}\text{-}\mathrm{H}}D_{z^{+} }^{q} u} \bigr) (x,y) = \bigl( {{}_{\mathrm{H}}{\mathcal {J}}_{z^{+} }^{1 - q} \delta_{x} \delta_{y} u} \bigr) (x,y) $$
we have
$${}_{\mathrm{H}}{\mathcal{J}}_{z^{+} }^{q} \bigl( {{}_{\mathrm{H}}{\mathcal{J}}_{z^{+} }^{1 - q} \delta_{x} \delta_{y} u} \bigr) (x,y) = \bigl( {{}_{\mathrm{H}}{\mathcal{J}}_{z^{+} }^{q} h} \bigr) (x,y), \quad (x,y) \in J_{z} . $$
On the other hand,
$$\begin{aligned}& {}_{\mathrm{H}}{\mathcal{J}}_{z^{+} }^{q} \bigl( {{}_{\mathrm{H}}{\mathcal{J}}_{z^{+} }^{1 - q} \delta_{x} \delta_{y} u} \bigr) (x,y) \\& \quad = {}_{\mathrm{H}}{ \mathcal{J}}_{z^{+} }^{1} ( { \delta_{x} \delta_{y} u} ) (x,y) \\& \quad = u(x,y) - u(x,b) - u \bigl(a_{1}^{+} ,y \bigr) + u \bigl(a_{1}^{+} ,b \bigr),\quad \mbox{for }(x,y) \in J_{z}. \end{aligned}$$
Therefore,
$$u(x,y) = u(x,b) + u \bigl(a_{1}^{+} ,y \bigr) - u \bigl(a_{1}^{+} ,b \bigr) + \bigl( {{}_{\mathrm{H}}{\mathcal {J}}_{z^{+} }^{q} h} \bigr) (x,y), \quad \mbox{for }(x,y) \in J_{z}. $$
Moreover, equation (5) satisfies (4) by Definition 2.2. The proof is completed. □

3 Main results

For convenience, let \(\sum_{i = 1}^{0} {z_{i} } = 0\), \(\Xi(x,y) = \phi (x) + \psi(y) - \phi(a)\), and \(f = f(s,t,u(s,t))\). Define
$$\begin{aligned} \begin{aligned}[b] \bar{u}(x,y) ={}& u(x,b) + u \bigl( {x_{k}^{+},y} \bigr) - u \bigl( {x_{k}^{+},b} \bigr) \\ &{} + \frac{1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \int_{x_{k} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } , \\ & \mbox{for }(x,y) \in (x_{k} ,x_{k + 1} ] \times[b,B], \mbox{and }k \in\{ 1,2,\ldots ,m\}, \end{aligned} \end{aligned}$$
(6)
with \(u(x_{k}^{+},y) = u(x_{k}^{-},y) + I_{k} ( {u(x_{k}^{-},y)} )\).

By Lemma 2.3, it is sure that \(\bar{u}(x,y)\) satisfies the fractional derivative condition and impulsive conditions in system (1). But \(\bar{u}(x,y)\) is not a solution of (1) because it does not satisfy (3). Therefore, \(\bar{u}(x,y)\) will be considered an approximate solution to seek the exact solution of system (1).

Theorem 3.1

Let \(q = (q_{1} ,q_{2} )\), here \(q_{1} ,q_{2} \in\mathbb {C} \) and \((\Re(q_{1} ),\Re(q_{2} )) \in(0,1] \times(0,1]\). \(I_{i} ( {u(x_{i}^{-} ,y)} )\) (\(i = 1, 2,\ldots, m\)) are differentiable functions on y. System (1) is equivalent to the integral equation
$$\begin{aligned} u(x,y) =& \Xi(x,y) + \frac{1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac {y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \\ &{}+ \sum_{i = 1}^{k} { \bigl[ {I_{i} \bigl(u \bigl(x_{i}^{-} ,y \bigr) \bigr) - I_{i} \bigl(u \bigl(x_{i}^{-} ,b \bigr) \bigr)} \bigr]} \\ &{}+ \frac{{\sigma(y)}}{{\Gamma(q_{1} )\Gamma(q_{2} )}}\sum_{i = 1}^{k} I_{i} \bigl(u \bigl(x_{i}^{-} ,y \bigr) \bigr) \biggl[ { \int_{a}^{x_{i} } { \int_{b}^{y} { \biggl( {\ln\frac {{x_{i} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\ &{} + \int_{x_{i} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac {{ds}}{s}} } \\ &{}- \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac {{dt}}{t}\frac{{ds}}{s}} } \biggr], \\ &\textit{for }(x,y) \in (x_{k} ,x_{k + 1} ] \times[b,B]\ \bigl(\textit{here }k \in\{ 0,1,2,\ldots,m\} \bigr), \end{aligned}$$
(7)
provided that the integral in (7) exists, where \(\sigma(y)\) is an arbitrary differentiable function on y.

Proof

As regards necessity; letting \(I_{i} ( {u(x_{i}^{-},y )} ) \to0\) for all \(i \in\{ 1,2,\ldots,m\} \) in equation (7), we obtain
$$\begin{aligned}& \lim_{ I_{i} ( {u(x_{i}^{-},y )} ) \to0\text{ for all } i \in\{ 1,2,\ldots,m\}} u(x,y) \\& \quad =\Xi(x,y) + \frac{1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \int_{a}^{x} { \int _{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac {y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} }, \\& \qquad {} \mbox{for }(x,y) \in (x_{k} ,x_{k + 1} ] \times[b,B],k \in\{0,1,2,\ldots,m\}. \end{aligned}$$
Therefore, by Lemma 2.3, equation (7) (under conditions \(I_{i} ( {u(x_{i}^{-},y )} ) \to0\) for all \(i \in\{ 1,2,\ldots,m\} \)) is the solution of system (2), that is, equation (7) satisfies condition (3).
Next, for \(\forall x_{i} \) (\(i\in\{1,2,\ldots,m\}\)) in equation (7), we get
$$\begin{aligned} u \bigl(x_{i}^{+} ,y \bigr) - u \bigl(x_{i}^{-} ,y \bigr) =& \Xi \bigl(x_{i}^{+} ,y \bigr) + I_{i} \bigl(u \bigl(x_{i}^{-} ,y \bigr) \bigr) - I_{i} \bigl(u \bigl(x_{i}^{-} ,b \bigr) \bigr) - \Xi \bigl(x_{i}^{-} ,y \bigr) \\ =& I_{i} \bigl(u \bigl(x_{i}^{-} ,y \bigr) \bigr)- I_{i} \bigl(u \bigl(x_{i}^{-} ,b \bigr) \bigr)+\phi \bigl(x_{i}^{+} \bigr)-\phi \bigl(x_{i}^{-} \bigr) \\ =& I_{i} \bigl(u \bigl(x_{i}^{-} ,y \bigr) \bigr). \end{aligned}$$
Therefore, equation (7) satisfies the impulsive conditions in system (1).
Finally, taking fractional derivatives of both sides of equation (7) as \((x,y) \in(x_{k} ,x_{k + 1} ] \times[b,B]\) (here \(k=0,1,2,\ldots,m\)), we obtain
$$\begin{aligned}& \bigl( {{}_{\mathrm{C}\text{-}\mathrm{H}}D_{(a + ,b + )}^{q} u} \bigr) (x,y) \\& \quad = \bigl( {{}_{\mathrm{H}}{\mathcal{J}}_{(a + ,b + )}^{1 - q} \delta_{x} \delta_{y} u} \bigr) (x,y) \\& \quad = {}_{\mathrm{H}}{\mathcal{J}}_{(a + ,b + )}^{1 - q} \delta_{x} \delta_{y} \Biggl\{ {\Xi(x,y) + \sum _{i = 1}^{k} { \bigl[ {I_{i} \bigl(u \bigl(x_{i}^{-} ,y \bigr) \bigr) - I_{i} \bigl(u \bigl(x_{i}^{-} ,0 \bigr) \bigr)} \bigr]} } \\& \qquad {}+ \frac{1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \\& \qquad {}+ \frac{{\sigma(y)}}{{\Gamma(q_{1} )\Gamma(q_{2} )}}\sum_{i = 1}^{k} I_{i} \bigl(u \bigl(x_{i}^{-} ,y \bigr) \bigr) \biggl[ { \int_{a}^{x_{i} } { \int_{b}^{y} { \biggl( {\ln\frac {{x_{i} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\& \qquad {} + \int_{x_{i} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac {{ds}}{s}} } \\& \qquad {}- \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac {{dt}}{t}\frac{{ds}}{s}} } \biggr] \Biggr\} \\& \quad = \Biggl\{ { {f \bigl(x,y,u(x,y) \bigr)} |_{(x,y) \in[a,x_{k + 1} ] \times [b,B]} }+ \frac{1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \\& \qquad {}\times\sum_{i = 1}^{k} {}_{\mathrm{H}}{\mathcal{J}}_{(a + ,b + )}^{1 - q} \delta_{x} \delta_{y} \biggl[ { \int_{x_{i} }^{x} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( { \int _{b}^{y} {\sigma(y)I_{i} \bigl(u \bigl(x_{i}^{-} ,y \bigr) \bigr) \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f\frac{{dt}}{t}} } \biggr)\frac{{ds}}{s}} } \\& \qquad {}- \int_{a}^{x} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( { \int_{b}^{y} {\sigma(y)I_{i} \bigl(u \bigl(x_{i}^{-} ,y \bigr) \bigr) \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f\frac{{dt}}{t}} } \biggr)\frac{{ds}}{s}} \biggr] \Biggr\} _{(x,y) \in (x_{k} ,x_{k + 1} ] \times[b,B]} . \end{aligned}$$
We have
$$\begin{aligned}& {}_{\mathrm{H}}{\mathcal{J}}_{(a + ,b + )}^{1 - q} \delta_{x} \delta_{y} \biggl[ { \int_{x_{i} }^{x} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( { \int _{b}^{y} {\sigma(y)I_{i} \bigl(u \bigl(x_{i}^{-} ,y \bigr) \bigr) \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f\frac{{dt}}{t}} } \biggr)\frac{{ds}}{s}} } \\& \quad {}- \int_{a}^{x} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( { \int _{b}^{y} {\sigma(y)I_{i} \bigl(u \bigl(x_{i}^{-} ,y \bigr) \bigr) \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f\frac{{dt}}{t}} } \biggr)\frac{{ds}}{s}} \biggr] = 0. \end{aligned}$$
(8)
Also, we will give the proof of equation (8) in the Appendix. Thus
$$\bigl( {{}_{\mathrm{C}\text{-}\mathrm{H}}D_{(a + ,b + )}^{q} u} \bigr) (x,y) = {f \bigl(x,y,u(x,y) \bigr)}|_{(x,y) \in (x_{k} ,x_{k + 1} ] \times[b,B]} . $$
So, equation (7) satisfies all conditions of (1).
As regards sufficiency: we will prove that the solution of system (1) satisfies equation (7) by mathematical induction. By Lemma 2.3, the solution of system (1) satisfies
$$\begin{aligned}& u(x,y) = \Xi(x,y) + \frac{1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac {y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } , \\& \quad \mbox{for }(x,y) \in[a,x_{1} ] \times[b,B]. \end{aligned}$$
(9)
Using (9), the approximate solution (as \((x,y) \in(x_{1}, x_{2}] \times [b,B]\)) of system (1) is given by
$$\begin{aligned} \bar{u}(x,y) =& u(x,b) + u \bigl( {x_{1}^{+} ,y} \bigr) - u \bigl( {x_{1}^{+} ,b} \bigr) \\ &{}+ \frac{1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \int_{x_{1} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \\ =& \phi(x) + \phi \bigl(x_{1}^{-} \bigr) + \psi(y) - \phi(a)+ I_{1} \bigl( {u \bigl(x_{1}^{-},y \bigr)} \bigr) \\ &{}+ \frac{1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \int_{a}^{x_{1} } { \int_{b}^{y} { \biggl( {\ln\frac{x_{1}}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \\ &{}- \phi \bigl(x_{1}^{-} \bigr) - \psi(b) + \phi(a)-I_{1} \bigl( {u \bigl(x_{1}^{-},b \bigr)} \bigr) \\ &{}+\frac{1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \int_{x_{1} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \\ =& \Xi(x,y) + I_{1} \bigl( {u \bigl(x_{1}^{-},y \bigr)} \bigr)- I_{1} \bigl( {u \bigl(x_{1}^{-},b \bigr)} \bigr) \\ &{}+ \frac{1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \biggl[ { \int_{a}^{x_{1} } { \int _{b}^{y} { \biggl( {\ln\frac{x_{1}}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac {y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\ &{}+ \int_{x_{1} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac {{ds}}{s}} } \biggr], \\ &\mbox{for }(x,y) \in (x_{1} ,x_{2} ] \times[b,B]. \end{aligned}$$
(10)
Let \(e_{1} (x,y) = u(x,y) - \bar{u}(x,y)\) for \((x,y) \in(x_{1} ,x_{2} ] \times[b,B]\), here \(u(x,y)\) denotes the exact solution of system (1). Moreover, by equation (9), the exact solution \(u(x,y)\) of system (1) satisfies
$$\begin{aligned}& \lim_{I_{1} ( {u(x_{1}^{-} ,y)} ) \to0} u(x,y) = \Xi (x,y) + \frac{1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} }, \\& \quad \mbox{for }(x,y) \in (x_{1} ,x_{2} ] \times[b,B]. \end{aligned}$$
(11)
Thus,
$$\begin{aligned}& \begin{aligned}[b] &\lim_{I_{1} ( {u(x_{1}^{-} ,y)} ) \to0} e_{1} (x,y) \\ &\quad = \lim_{I_{1} ( {u(x_{1}^{-} ,y)} ) \to0} \bigl\{ {u(x,y) - \bar{u}(x,y)} \bigr\} \\ &\quad = \frac{1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \biggl[ { \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\ &\qquad {}- \int_{a}^{x_{1} } { \int_{b}^{y} { \biggl( {\ln\frac{x_{1}}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac {{ds}}{s}} } \\ &\qquad {}- \int_{x_{1} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac {{dt}}{t}\frac{{ds}}{s}} } \biggr]. \end{aligned} \end{aligned}$$
(12)
Equation (12) means that \(e_{1} (x,y)\) is connected with \(\lim_{I_{1} ( {u(x_{1}^{-} ,y)} ) \to0} e_{1} (x,y)\) and \(I_{1} ( {u(x_{1}^{-} ,y)} )\). Therefore, we suppose
$$\begin{aligned} e_{1} (x,y) =& \kappa \bigl( {I_{1} \bigl( {u \bigl(x_{1}^{-} ,y \bigr)} \bigr)} \bigr) \lim_{I_{1} ( {u(x_{1}^{-} ,y)} ) \to0} e_{1} (x,y) \\ =& \frac{{\kappa ( {I_{1} ( {u(x_{1}^{-} ,y)} )} )}}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \biggl[ { \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\ &{}- \int_{a}^{x_{1} } { \int_{b}^{y} { \biggl( {\ln\frac{x_{1}}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac {{ds}}{s}} } \\ &{}- \int_{x_{1} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac {{dt}}{t}\frac{{ds}}{s}} } \biggr], \end{aligned}$$
(13)
where κ is an undetermined function with \(\kappa(0)=1\). Thus,
$$\begin{aligned} u(x,y) =& \bar{u}(x,y) + e_{1} (x,y) \\ =& \Xi(x,y) + I_{1} \bigl( {u \bigl(x_{1}^{-},y \bigr)} \bigr)- I_{1} \bigl( {u \bigl(x_{1}^{-},b \bigr)} \bigr) \\ &{}+ \frac{1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \\ &{}+ \frac{{1 - \kappa( {I_{1} ( {u(x_{1}^{-} ,y)} )} )}}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \biggl[ { \int_{a}^{x_{1} } { \int_{b}^{y} { \biggl( {\ln\frac{x_{1}}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\ &{}+ \int_{x_{1} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac {{ds}}{s}} } \\ &{}- \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac {{dt}}{t}\frac{{ds}}{s}} } \biggr], \\ &\mbox{for }(x,y) \in (x_{1}, x_{2} ] \times[b,B]. \end{aligned}$$
(14)
Next, using equation (14), the approximate solution (as \((x,y) \in (x_{2}, x_{3}] \times[b,B]\)) of system (1) is provided by
$$\begin{aligned} \bar{u}(x,y) =& u(x,b) + u \bigl( {x_{2}^{+} ,y} \bigr) - u \bigl( {x_{2}^{+} ,b} \bigr) \\ &{}+ \frac{1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \int_{x_{2} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \\ =& \Xi(x,y) + I_{1} \bigl( {u \bigl(x_{1}^{-},y \bigr)} \bigr)- I_{1} \bigl( {u \bigl(x_{1}^{-},b \bigr)} \bigr)+ I_{2} \bigl( {u \bigl(x_{2}^{-},y \bigr)} \bigr)- I_{2} \bigl( {u \bigl(x_{2}^{-},b \bigr)} \bigr) \\ &{}+ \frac{1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \biggl[ { \int_{a}^{x_{2} } { \int _{b}^{y} { \biggl( {\ln\frac{{x_{2} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac {y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\ &{}+ \int_{x_{2} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac {{ds}}{s}} } \biggr] \\ &{}+ \frac{{1 - \kappa ( {I_{1} ( {u(x_{1}^{-} ,y)} )} )}}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \biggl[ { \int_{a}^{x_{1} } { \int_{b}^{y} { \biggl( {\ln\frac{{x_{1} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\ &{}+ \int_{x_{1} }^{x_{2} } { \int_{b}^{y} { \biggl( {\ln\frac{{x_{2} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac {{dt}}{t}\frac{{ds}}{s}} } \\ &{}- \int_{a}^{x_{2} } { \int_{b}^{y} { \biggl( {\ln\frac {{x_{2} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \biggr], \\ &\mbox{for }(x,y) \in (x_{2}, x_{3} ] \times[b,B]. \end{aligned}$$
(15)
Let \(e_{2} (x,y) = u(x,y) - \bar{u}(x,y)\) for \((x,y) \in(x_{2} ,x_{3} ] \times[b,B]\). Moreover, by equation (14), the exact solution of (1) satisfies
$$\begin{aligned}& \lim_{\substack{I_{1} ( {u(x_{1}^{-} ,y)} ) \to0, \\ I_{2} ( {u(x_{2}^{-} ,y)} ) \to0 }} u(x,y) = \Xi(x,y) + \frac {1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac {x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } , \\& \quad \mbox{for }(x,y) \in (x_{2} ,x_{3} ] \times[b,B], \\& \lim_{I_{2} ( {u(x_{2}^{-} ,y)} ) \to0} u(x,y) \\& \quad =\Xi(x,y)+ I_{1} \bigl( {u \bigl(x_{1}^{-},y \bigr)} \bigr)- I_{1} \bigl( {u \bigl(x_{1}^{-},b \bigr)} \bigr) \\& \qquad {}+ \frac{1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \\& \qquad {}+\frac{{1 - \kappa ( {I_{1} ( {u(x_{1}^{-} ,y)} )} )}}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \biggl[ { \int_{a}^{x_{1} } { \int_{b}^{y} { \biggl( {\ln\frac{{x_{1} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\& \qquad {}+ \int_{x_{1} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac {{ds}}{s}} } - \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac {{dt}}{t}\frac{{ds}}{s}} } \biggr], \\& \qquad \mbox{for }(x,y) \in (x_{2} ,x_{3} ] \times[b,B], \\& \lim_{I_{1} ( {u(x_{1}^{-} ,y)} ) \to0} u(x,y) \\& \quad = \Xi(x,y) + I_{2} \bigl( {u \bigl(x_{2}^{-},y \bigr)} \bigr)- I_{2} \bigl( {u \bigl(x_{2}^{-},b \bigr)} \bigr) \\& \qquad {}+ \frac{1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \\& \qquad {}+ \frac{{1 - \kappa ( {I_{2} ( {u(x_{2}^{-} ,y)} )} )}}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \biggl[ { \int_{a}^{x_{2} } { \int_{b}^{y} { \biggl( {\ln\frac{{x_{2} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\& \qquad {}+ \int_{x_{2} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac {{ds}}{s}} } - \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac {{dt}}{t}\frac{{ds}}{s}} } \biggr], \\& \qquad \mbox{for }(x,y) \in (x_{2} ,x_{3} ] \times[b,B]. \end{aligned}$$
Thus,
$$\begin{aligned}& \lim_{\substack{I_{1} ( {u(x_{1}^{-},y)} ) \to0, \\ I_{2} ( {u(x_{2}^{-} ,y)} ) \to0 }} e_{2} (x,y) \\& \quad = \lim_{\substack{I_{1} ( {u(x_{1}^{-} ,y)} ) \to0, \\ I_{2} ( {u(x_{2}^{-} ,y)} ) \to0 }} \bigl\{ {u(x,y) - \bar{u}(x,y)} \bigr\} \\& \quad = \frac{1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \biggl[ { \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\& \qquad {}- \int_{a}^{x_{2} } { \int_{b}^{y} { \biggl( {\ln\frac{{x_{2} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac {{dt}}{t}\frac{{ds}}{s}} } \\& \qquad {}- \int_{x_{2} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac {x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \biggr], \end{aligned}$$
(16)
$$\begin{aligned}& \lim_{I_{2} ( {u(x_{2}^{-} ,y)} ) \to0} e_{2} (x,y) \\& \quad = \lim_{I_{2} ( {u(x_{2}^{-} ,y)} ) \to0} \bigl\{ {u(x,y) - \bar{u}(x,y)} \bigr\} \\& \quad =\frac{{ - \kappa ( {I_{1} ( {u(x_{1}^{-} ,y)} )} )}}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \biggl[ { \int_{a}^{x_{2} } { \int_{b}^{y} { \biggl( {\ln\frac{{x_{2} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\& \qquad {}+ \int_{x_{2} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac {{ds}}{s}} } - \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac {{dt}}{t}\frac{{ds}}{s}} } \biggr] \\& \qquad {}+ \frac{{1 - \kappa ( {I_{1} ( {u(x_{1}^{-} ,y)} )} )}}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \biggl[ { \int_{x_{1} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\& \qquad {}- \int_{x_{1} }^{x_{2} } { \int_{b}^{y} { \biggl( {\ln\frac{{x_{2} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac {{dt}}{t}\frac{{ds}}{s}} } \\& \qquad {}- \int_{x_{2} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac {x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \biggr], \end{aligned}$$
(17)
$$\begin{aligned}& \lim_{I_{1} ( {u(x_{1}^{-} ,y)} ) \to0} e_{2} (x,y) \\& \quad = \lim_{I_{1} ( {u(x_{1}^{-} ,y)} ) \to0} \bigl\{ {u(x,y) - \bar{u}(x,y)} \bigr\} \\& \quad = \frac{{- \kappa ( {I_{2} ( {u(x_{2}^{-} ,y)} )} )}}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \biggl[ { \int_{a}^{x_{2} } { \int_{b}^{y} { \biggl( {\ln\frac{{x_{2} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\& \qquad {}+ \int_{x_{2} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac {{ds}}{s}} } \\& \qquad {}- \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac {{dt}}{t}\frac{{ds}}{s}} } \biggr]. \end{aligned}$$
(18)
By (16)-(18), we get
$$\begin{aligned} e_{2} (x,y) =& \frac{{1 - \kappa ( {I_{2} ( {u(x_{2}^{-} ,y)} )} ) - \kappa ( {I_{1} ( {u(x_{1}^{-} ,y)} )} )}}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \biggl[ { \int_{a}^{x_{2} } { \int_{b}^{y} { \biggl( {\ln\frac{{x_{2} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\ &{}+ \int_{x_{2} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac {{ds}}{s}} } \\ &{}- \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac {{dt}}{t}\frac{{ds}}{s}} } \biggr] \\ &{}+ \frac{{1 - \kappa ( {I_{1} ( {u(x_{1}^{-} ,y)} )} )}}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \biggl[ { \int_{x_{1} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\ &{}- \int_{x_{1} }^{x_{2} } { \int_{b}^{y} { \biggl( {\ln\frac{{x_{2} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac {{dt}}{t}\frac{{ds}}{s}} } \\ &{}- \int_{x_{2} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac {x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \biggr]. \end{aligned}$$
(19)
Therefore, by (15) and (19), we have
$$\begin{aligned} u(x,y) =& \bar{u}(x,y) + e_{2} (x,y) \\ =& \Xi(x,y) + I_{1} \bigl(u \bigl(x_{1}^{-} ,y \bigr) \bigr) - I_{1} \bigl(u \bigl(x_{1}^{-} ,b \bigr) \bigr) + I_{2} \bigl(u \bigl(x_{2}^{-} ,y \bigr) \bigr) - I_{2} \bigl(u \bigl(x_{2}^{-} ,b \bigr) \bigr) \\ &{}+ \frac{1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \\ &{}+ \frac{{1 - \kappa ( {I_{1} ( {u(x_{1}^{-} ,y)} )} )}}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \biggl[ { \int_{a}^{x_{1} } { \int_{b}^{y} { \biggl( {\ln\frac{{x_{1} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\ &{}+ \int_{x_{1} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac {{ds}}{s}} } \\ &{}- \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac {{dt}}{t}\frac{{ds}}{s}} } \biggr] \\ &{}+ \frac{{1 - \kappa ( {I_{2} ( {u(x_{2}^{-} ,y)} )} )}}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \biggl[ { \int_{a}^{x_{2} } { \int_{b}^{y} { \biggl( {\ln\frac{{x_{2} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\ &{}+ \int_{x_{2} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac {{ds}}{s}} } \\ &{}- \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac {{dt}}{t}\frac{{ds}}{s}} } \biggr], \\ &\mbox{for }(x,y) \in (x_{2} ,x_{3} ] \times[b,B]. \end{aligned}$$
(20)
On the other hand, for system (1), we have
$$\begin{aligned}& \lim_{x_{2} \to x_{1} } \left \{ \textstyle\begin{array}{l} ( {{}_{\mathrm{C}\text{-}\mathrm{H}}D_{(a + ,b + )}^{q} u} )(x,y) = f(x,y,u(x,y)),\quad (x,y) \in J\mbox{ and }x \ne x_{1} ,x_{2} , \\ u(x_{i}^{+} ,y) = u(x_{i}^{-} ,y) + I_{i} ( {u(x_{i}^{-} ,y)} ),\quad i = 1,2, \\ u(x,b) = \phi(x),\qquad u(a,y) = \psi(y),\quad x \in[a,A], y \in[b,B] \end{array}\displaystyle \right . \end{aligned}$$
(21)
$$\begin{aligned}& \quad = \left \{ \textstyle\begin{array}{l} ( {{}_{\mathrm{C}\text{-}\mathrm{H}}D_{(a + ,b + )}^{q} u} )(x,y) = f(x,y,u(x,y)),\quad (x,y) \in J\mbox{ and }x \ne x_{1} , \\ u(x_{1}^{+} ,y) = u(x_{1}^{-} ,y) + I_{1} ( {u(x_{1}^{-} ,y)} ) + I_{2} ( {u(x_{1}^{-} ,y)} ), \\ u(x,b) = \phi(x),\qquad u(a,y) = \psi(y), \quad x \in[a,A], y \in[b,B]. \end{array}\displaystyle \right . \end{aligned}$$
(22)
Using (20) and (14) to (21) and (22), respectively, we get
$$\begin{aligned}& 1 - \kappa \bigl[ {I_{1} \bigl( {u \bigl(x_{1}^{-} ,y \bigr)} \bigr) + I_{2} \bigl( {u \bigl(x_{1}^{-} ,y \bigr)} \bigr)} \bigr] = 1 - \kappa \bigl( {I_{1} \bigl( {u \bigl(x_{1}^{-} ,y \bigr)} \bigr)} \bigr) + 1 - \kappa \bigl( {I_{2} \bigl( {u \bigl(x_{1}^{-} ,y \bigr)} \bigr)} \bigr), \\& \quad \mbox{for }\forall I_{1} \bigl( {u \bigl(x_{1}^{-} ,y \bigr)} \bigr)\mbox{ and }I_{2} \bigl( {u \bigl(x_{1}^{-} ,y \bigr)} \bigr). \end{aligned}$$
(23)
Therefore, \(1 - \kappa ( {I_{i} ( {u(x_{i}^{-} ,y)} )} )=\sigma(y)I_{i} ( {u(x_{i}^{-} ,y)} )\), here \(\sigma(y)\) is a differentiable function on y. Thus, (14) and (20) can be rewritten into
$$\begin{aligned}& u(x,y) = \Xi(x,y) + I_{1} \bigl( {u \bigl(x_{1}^{-},y \bigr)} \bigr)- I_{1} \bigl( {u \bigl(x_{1}^{-},b \bigr)} \bigr) \\& \hphantom{u(x,y) ={}}{}+ \frac{1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \\& \hphantom{u(x,y) ={}}{}+ \frac{{\sigma(y) {I_{1} ( {u(x_{1}^{-} ,y)} )}}}{{\Gamma (q_{1} )\Gamma(q_{2} )}} \biggl[ { \int_{a}^{x_{1} } { \int_{b}^{y} { \biggl( {\ln\frac {x_{1}}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\& \hphantom{u(x,y) ={}}{}+ \int_{x_{1} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac {{ds}}{s}} } \\& \hphantom{u(x,y) ={}}{}- \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac {{dt}}{t}\frac{{ds}}{s}} } \biggr], \\& \hphantom{u(x,y) ={}}\mbox{for }(x,y) \in (x_{1}, x_{2} ] \times[b,B], \end{aligned}$$
(24)
$$\begin{aligned}& u(x,y) = \Xi(x,y) + I_{1} \bigl(u \bigl(x_{1}^{-} ,y \bigr) \bigr) - I_{1} \bigl(u \bigl(x_{1}^{-} ,b \bigr) \bigr) + I_{2} \bigl(u \bigl(x_{2}^{-} ,y \bigr) \bigr) - I_{2} \bigl(u \bigl(x_{2}^{-} ,b \bigr) \bigr) \\& \hphantom{u(x,y) ={}}{}+ \frac{1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \\& \hphantom{u(x,y) ={}}{}+ \frac{{ \sigma(y){I_{1} ( {u(x_{1}^{-} ,y)} )} }}{{\Gamma (q_{1} )\Gamma(q_{2} )}} \biggl[ { \int_{a}^{x_{1} } { \int_{b}^{y} { \biggl( {\ln\frac {{x_{1} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\& \hphantom{u(x,y) ={}}{}+ \int_{x_{1} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac {{ds}}{s}} } \\& \hphantom{u(x,y) ={}}{}- \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac {{dt}}{t}\frac{{ds}}{s}} } \biggr] \\& \hphantom{u(x,y) ={}}{}+ \frac{{ \sigma(y){I_{2} ( {u(x_{2}^{-} ,y)} )} }}{{\Gamma (q_{1} )\Gamma(q_{2} )}} \biggl[ { \int_{a}^{x_{2} } { \int_{b}^{y} { \biggl( {\ln\frac {{x_{2} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\& \hphantom{u(x,y) ={}}{}+ \int_{x_{2} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac {{ds}}{s}} } \\& \hphantom{u(x,y) ={}}{}- \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac {{dt}}{t}\frac{{ds}}{s}} } \biggr], \\& \hphantom{u(x,y) ={}}\mbox{for }(x,y) \in (x_{2} ,x_{3} ] \times[b,B]. \end{aligned}$$
(25)
For \((x,y) \in(x_{n} ,x_{n + 1} ] \times[b,B]\), suppose
$$\begin{aligned} u(x,y) =& \Xi(x,y) + \frac{1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac {y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \\ &{}+ \sum_{i = 1}^{n} { \bigl[ {I_{i} \bigl(u \bigl(x_{i}^{-} ,y \bigr) \bigr) - I_{i} \bigl(u \bigl(x_{i}^{-} ,b \bigr) \bigr)} \bigr]} \\ &{}+ \frac{{\sigma(y)}}{{\Gamma(q_{1} )\Gamma(q_{2} )}}\sum_{i = 1}^{n} I_{i} \bigl(u \bigl(x_{i}^{-} ,y \bigr) \bigr) \biggl[ { \int_{a}^{x_{i} } { \int_{b}^{y} { \biggl( {\ln\frac {{x_{i} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\ &{}+ \int_{x_{i} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac {{ds}}{s}} } \\ &{}- \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac {{dt}}{t}\frac{{ds}}{s}} } \biggr], \\ &\mbox{for }(x,y) \in (x_{n} ,x_{n + 1} ] \times[b,B]. \end{aligned}$$
(26)
Using (26), the approximate solution (when \((x,y) \in(x_{n+1},x_{n + 2} ] \times[b,B]\)) of (1) can be given by
$$\begin{aligned} \bar{u}(x,y) =& u(x,b) + u \bigl( {x_{n+1}^{+} ,y} \bigr) - u \bigl( {x_{n+1}^{+},b} \bigr) \\ &{}+ \frac{1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \int_{x_{n+1} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } , \\ =& \Xi(x,y) + \sum_{i = 1}^{n + 1} { \bigl[ {I_{i} \bigl(u \bigl(x_{i}^{-} ,y \bigr) \bigr) - I_{i} \bigl(u \bigl(x_{i}^{-} ,b \bigr) \bigr)} \bigr]} \\ &{}+ \frac{1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \biggl[ { \int_{a}^{x_{n + 1} } { \int_{b}^{y} { \biggl( {\ln\frac{{x_{n + 1} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\ &{}+ \int_{x_{n + 1} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac {{dt}}{t}\frac{{ds}}{s}} } \biggr] \\ &{}+ \frac{{\sigma(y)}}{{\Gamma(q_{1} )\Gamma(q_{2} )}}\sum_{i = 1}^{n} I_{i} \bigl(u \bigl(x_{i}^{-} ,y \bigr) \bigr) \biggl[ { \int_{a}^{x_{i} } { \int_{b}^{y} { \biggl( {\ln\frac {{x_{i} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\ &{}+ \int_{x_{i} }^{x_{n + 1} } { \int_{b}^{y} { \biggl( {\ln\frac{{x_{n + 1} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \\ &{}- \int_{a}^{x_{n + 1} } { \int_{b}^{y} { \biggl( {\ln\frac{{x_{n + 1} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \biggr], \\ &\mbox{for }(x,y) \in (x_{n+1} ,x_{n + 2} ] \times[b,B]. \end{aligned}$$
(27)
Let \(e_{n+1} (x,y) = u(x,y) - \bar{u}(x,y)\) for \((x,y) \in(x_{n+1} ,x_{n + 2} ] \times[b,B]\), here \(u(x,y)\) denotes the exact solution of system (1). Moreover, by equation (26), the exact solution satisfies
$$\begin{aligned}& \lim_{\substack{I_{i} (u(x_{i}^{-} ,y)) \to0, \\ \text{for all }i \in\{ 1,2, \ldots,n + 1\}}} u(x,y) = \Xi(x,y) + \frac {1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac {x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } , \\& \quad \mbox{for }(x,y) \in (x_{n + 1} ,x_{n + 2} ] \times[b,B], \end{aligned}$$
(28)
$$\begin{aligned}& \lim_{\substack{I_{j} (u(x_{j}^{-} ,y)) \to0,\\ \text{here }j \in\{ 1,2, \ldots,n + 1\}}} u(x,y) \\& \quad = \Xi(x,y) + \sum_{\substack{1 \le i \le n + 1, \\ \text{and }i \ne j }} { \bigl[ {I_{i} \bigl(u \bigl(x_{i}^{-} ,y \bigr) \bigr) - I_{i} \bigl(u \bigl(x_{i}^{-} ,b \bigr) \bigr)} \bigr]} \\& \qquad {} + \frac{1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \\& \qquad {} + \frac{{\sigma(y)}}{{\Gamma(q_{1} )\Gamma(q_{2} )}}\sum_{\substack {1 \le i \le n + 1, \\ \text{and }i \ne j }} I_{i} \bigl(u \bigl(x_{i}^{-} ,y \bigr) \bigr) \biggl[ { \int_{a}^{x_{i} } { \int_{b}^{y} { \biggl( {\ln\frac{{x_{i} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\& \qquad{} + \int_{x_{i} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac {{ds}}{s}} } \\& \qquad {}- \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac {{dt}}{t}\frac{{ds}}{s}} } \biggr], \\& \qquad \mbox{for }(x,y) \in (x_{n + 1} ,x_{n + 2} ] \times[b,B]. \end{aligned}$$
(29)
Thus,
$$\begin{aligned}& \lim_{\substack{I_{i} (u(x_{i}^{-} ,y)) \to0, \\ \text{for all }i \in\{ 1,2, \ldots,n + 1\} }} e_{n + 1} (x,y) \\& \quad = \lim_{\substack{I_{i} (u(x_{i}^{-} ,y)) \to0, \\ \text{for all }i \in\{ 1,2, \ldots,n + 1\}}} \bigl\{ {u(x,y) - \bar{u}(x,y)} \bigr\} \\& \quad = \frac{1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \biggl[ { \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\& \qquad {}- \int_{a}^{x_{n + 1} } { \int_{b}^{y} { \biggl( {\ln\frac{{x_{n + 1} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac {{dt}}{t}\frac{{ds}}{s}} } \\& \qquad {} - \int_{x_{n + 1} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac {{dt}}{t}\frac{{ds}}{s}} } \biggr], \end{aligned}$$
(30)
$$\begin{aligned}& \lim_{\substack{I_{j} (u(x_{j}^{-} ,y)) \to0, \\[-1pt] \text{here }j \in\{ 1,2, \ldots,n + 1\} }} e_{n + 1} (x,y) \\ & \quad = \lim_{\substack{I_{j} (u(x_{j}^{-} ,y)) \to0, \\ \text{here }j \in\{ 1,2, \ldots,n + 1\}}} \bigl\{ {u(x,y) - \bar{u}(x,y)} \bigr\} \\& \quad = \frac{{1 - \sigma(y)\sum_{\substack{1 \le i \le n + 1, \\ \text{and }i \ne j }} {I_{i} (u(x_{i}^{-} ,y))} }}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \biggl[ { \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac {{ds}}{s}} } } \\& \qquad {}- \int_{a}^{x_{n + 1} } { \int_{b}^{y} { \biggl( {\ln\frac{{x_{n + 1} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \\& \qquad {}- \int_{x_{n + 1} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \biggr] \\& \qquad {}+ \frac{{\sigma(y)}}{{\Gamma(q_{1} )\Gamma(q_{2} )}}\sum_{\substack {1 \le i \le n + 1, \\ \text{and }i \ne j}} I_{i} \bigl(u \bigl(x_{i}^{-} ,y \bigr) \bigr) \biggl[ { \int_{x_{i} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\& \qquad {} - \int_{x_{i} }^{x_{n + 1} } { \int_{b}^{y} { \biggl( {\ln\frac{{x_{n + 1} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \\& \qquad {}- \int_{x_{n + 1} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \biggr]. \end{aligned}$$
(31)
By (30) and (31), we obtain
$$\begin{aligned} e_{n + 1} (x,y) =& \frac{{1 - \sigma(y)\sum_{1 \le i \le n + 1} {I_{i} (u(x_{i}^{-} ,y))} }}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \biggl[ { \int_{a}^{x} { \int _{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac {y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\ &{}- \int_{a}^{x_{n + 1} } { \int_{b}^{y} { \biggl( {\ln\frac{{x_{n + 1} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \\ &{}- \int_{x_{n + 1} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \biggr] \\ &{}+ \frac{{\sigma(y)}}{{\Gamma(q_{1} )\Gamma(q_{2} )}}\sum_{i = 1}^{n + 1} I_{i} \bigl(u \bigl(x_{i}^{-} ,y \bigr) \bigr) \biggl[ { \int_{x_{i} }^{x} { \int_{b}^{y} { \biggl( {\ln \frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\ &{}- \int_{x_{i} }^{x_{n + 1} } { \int_{b}^{y} { \biggl( {\ln\frac{{x_{n + 1} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \\ &{}- \int_{x_{n + 1} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \biggr]. \end{aligned}$$
(32)
Therefore, by (27) and (32), we get
$$\begin{aligned} u(x,y) =& \bar{u}(x,y) + e_{n + 1} (x,y) \\ =& \Xi(x,y) + \sum_{i = 1}^{n + 1} { \bigl[ {I_{i} \bigl(u \bigl(x_{i}^{-} ,y \bigr) \bigr) - I_{i} \bigl(u \bigl(x_{i}^{-} ,b \bigr) \bigr)} \bigr]} \\ &{}+ \frac{1}{{\Gamma(q_{1} )\Gamma(q_{2} )}} \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } \\ &{}+\frac{{\sigma(y)}}{{\Gamma(q_{1} )\Gamma(q_{2} )}}\sum_{i = 1}^{n + 1} I_{i} \bigl(u \bigl(x_{i}^{-} ,y \bigr) \bigr) \biggl[ { \int_{a}^{x_{i} } { \int_{b}^{y} { \biggl( {\ln\frac {{x_{i} }}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac{{ds}}{s}} } } \\ &{}+ \int_{x_{i} }^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac{{dt}}{t}\frac {{ds}}{s}} } \\ &{} - \int_{a}^{x} { \int_{b}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{q_{1} - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{q_{2} - 1} f \frac {{dt}}{t}\frac{{ds}}{s}} } \biggr], \\ &\mbox{for }(x,y) \in (x_{n + 1} ,x_{n + 2} ] \times[b,B]. \end{aligned}$$
(33)
Therefore, the solution of system (1) satisfies equation (7). Thus, by necessity and sufficiency, system (1) is equivalent to equation (7). The proof is completed. □

4 Examples

In this section, we will give an example to reveal that there exists a general solution for impulsive fractional partial differential equations.

Example 4.1

Let us consider the impulsive fractional system
$$ \left \{ \textstyle\begin{array}{l} ( {{}_{\mathrm{C}\text{-}\mathrm{H}}D_{(1 + ,1 + )}^{q} u} )(x,y) = \ln x\ln y,\quad (x,y) \in[1,3] \times[1,3]\mbox{ and }x \ne2 \\ u(2^{+} ,y) = u(2^{-} ,y) + ly, \\ u(x,1) = u(1,y) \equiv0,\quad x \in[1,3], y \in[1,3], \end{array}\displaystyle \right . $$
(34)
where \(q = ({ {1 \over 2}} + j,{ {1 \over 2}} + j)\) (here j denotes the imaginary unit) and l is a constant. By Theorem 3.1, the general solution of (34) is given by
$$\begin{aligned}& u(x,y) = \frac{1}{{\Gamma({ {1 \over 2}} + j)\Gamma({ {1 \over 2}} + j)}} \int_{1}^{x} { \int_{1}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{{ {1 \over 2}} + j - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{{ {1 \over 2}} + j - 1} \ln t \ln s\frac{{dt}}{t}\frac{{ds}}{s}} } \\& \hphantom{u(x,y)}=\frac{1}{{\Gamma({ {5 \over 2}} + j)\Gamma({ {5 \over 2}} + j)}}( {\ln x} )^{{ {3 \over 2}} + j} ( {\ln y} )^{{ {3 \over 2}} + j}, \quad \mbox{for }(x,y) \in (1,2 ] \times (1,3 ], \end{aligned}$$
(35)
$$\begin{aligned}& u(x,y) = ly + \frac{1}{{\Gamma({ {1 \over 2}} + j)\Gamma({ {1 \over 2}} + j)}} \int_{1}^{x} { \int_{1}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{{ {1 \over 2}} + j - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{{ {1 \over 2}} + j - 1} \ln t \ln s\frac{{dt}}{t}\frac{{ds}}{s}} } \\& \hphantom{u(x,y) ={}}{}+ \frac{{\sigma(y)ly}}{{\Gamma({ {1 \over 2}} + j)\Gamma({ {1 \over 2}} + j)}} \biggl[ { \int_{1}^{2} { \int_{1}^{y} { \biggl( {\ln\frac{2}{s}} \biggr)^{{ {1 \over 2}} + j - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{{ {1 \over 2}} + j - 1} \ln t \ln s\frac{{dt}}{t}\frac{{ds}}{s}} } } \\& \hphantom{u(x,y) ={}}{}+ \int_{2}^{x} { \int_{1}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{{ {1 \over 2}} + j - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{{ {1 \over 2}} + j - 1} \ln t \ln s\frac{{dt}}{t}\frac{{ds}}{s}} } \\& \hphantom{u(x,y) ={}}{}- \int_{1}^{x} { \int_{1}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{{ {1 \over 2}} + j - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{{ {1 \over 2}} + j - 1} \ln t \ln s\frac{{dt}}{t}\frac{{ds}}{s}} } \biggr] \\& \hphantom{u(x,y)}= ly + \frac{1}{{\Gamma({ {5 \over 2}} + j)\Gamma({ {5 \over 2}} + j)}} {( {\ln x} )^{{ {3 \over 2}} + j} } \bigg|_{x > 1} {( {\ln y} )^{{ {3 \over 2}} + j} } \bigg|_{y > 1} \\& \hphantom{u(x,y) ={}}{}+ \frac{{\sigma(y)ly}}{{\Gamma({ {5 \over 2}} + j)\Gamma({ {5 \over 2}} + j)}} \biggl[ {( {\ln2} )^{{ {3 \over 2}} + j} + { \biggl[ {\ln x + \biggl( {{ {1 \over 2}} + j} \biggr)\ln2} \biggr] \biggl( {\ln\frac{x}{2}} \biggr)^{{ {1 \over 2}} + j} } \bigg|_{x > 2} } \\& \hphantom{u(x,y) ={}}{}- {( {\ln x} )^{{ {3 \over 2}} + j} } \bigg|_{x > 1} \biggr] {( {\ln y} )^{{ {3 \over 2}} + j} } \bigg|_{y > 1} ,\quad \mbox{for }(x,y) \in (2,3 ] \times (1,3 ], \end{aligned}$$
(36)
where \(\sigma(y)\) is a differentiable function on y in equation (36). Next, we will verify that the general solution (35)-(36) satisfies all conditions in system (34). By the Appendix, we have
$$\begin{aligned}& (\mathrm{i})\quad {}_{\mathrm{C}\text{-}\mathrm{H}}D_{(1 + ,1 + )}^{q} \biggl\{ \frac{{\sigma(y)y}}{{\Gamma ({ {1 \over 2}} + j)\Gamma({ {1 \over 2}} + j)}} \biggl[ \int_{2}^{x} { \int _{1}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{{ {1 \over 2}} + j - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{{ {1 \over 2}} + j - 1} \ln t \ln s\frac {{dt}}{t}\frac{{ds}}{s}} } \\& \hphantom{(\mathrm{i})\quad}\qquad {}- \int_{1}^{x} { \int_{1}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{{ {1 \over 2}} + j - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{{ {1 \over 2}} + j - 1} \ln t \ln s\frac{{dt}}{t}\frac{{ds}}{s}} } \biggr]\biggr\} \\& \hphantom{(\mathrm{i})\quad}\quad = \frac{1}{{\Gamma({ {1 \over 2}} + j)\Gamma({ {1 \over 2}} + j)}} \\& \hphantom{(\mathrm{i})\quad}\qquad {}\times{}_{\mathrm{H}}{\mathcal{J}}_{(1 + ,1 + )}^{1 - q} \delta_{x} \delta_{y}\biggl\{ \biggl[ { \int_{2}^{x} { \int_{1}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{{ {1 \over 2}} + j - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{{ {1 \over 2}} + j - 1} \ln t \ln s\frac{{dt}}{t}\frac{{ds}}{s}} } } \\& \hphantom{(\mathrm{i})\quad}\qquad {} - \int_{1}^{x} \int_{1}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{{ {1 \over 2}} + j - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{{ {1 \over 2}} + j - 1} \ln t \ln s\frac{{dt}}{t}\frac{{ds}}{s}} \biggr]\sigma(y)y \biggr\} \\& \hphantom{(\mathrm{i})\quad}\quad \equiv0. \end{aligned}$$
Taking fractional derivatives of the two sides of equations (35)-(36), we have
$$\begin{aligned}& \bigl( {{}_{\mathrm{C}\text{-}\mathrm{H}}D_{(1 + ,1 + )}^{q} u} \bigr) (x,y) \\& \quad = {}_{\mathrm{C}\text{-}\mathrm{H}}D_{(1 + ,1 + )}^{q} \biggl( { \frac{1}{{\Gamma({ {5 \over 2}} + j)\Gamma({ {5 \over 2}} + j)}} { ( {\ln x} )^{{ {3 \over 2}} + j} } \bigg|_{x > 1} { ( {\ln y} )^{{ {3 \over 2}} + j} } \bigg|_{y > 1} } \biggr) \\& \quad = \ln x\ln y, \quad \mbox{for }(x,y) \in (1,2 ] \times (1,3 ], \\& \bigl( {{}_{\mathrm{C}\text{-}\mathrm{H}}D_{(1 + ,1 + )}^{q} u} \bigr) (x,y) \\& \quad = {}_{\mathrm{C}\text{-}\mathrm{H}}D_{(1 + ,1 + )}^{q} \biggl\{ {ly + \frac{1}{{\Gamma({ {1 \over 2}} + j)\Gamma({ {1 \over 2}} + j)}} \int_{1}^{x} { \int_{1}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{{ {1 \over 2}} + j - 1} \biggl( {\ln\frac {y}{t}} \biggr)^{{ {1 \over 2}} + j - 1} \ln t\ln s\frac{{dt}}{t}\frac {{ds}}{s}} } } \\& \qquad {}+ \frac{{\sigma(y)ly}}{{\Gamma({ {1 \over 2}} + j)\Gamma({ {1 \over 2}} + j)}} \biggl[ { \int_{1}^{2} { \int_{1}^{y} { \biggl( {\ln\frac{2}{s}} \biggr)^{{ {1 \over 2}} + j - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{{ {1 \over 2}} + j - 1} \ln t \ln s\frac{{dt}}{t}\frac{{ds}}{s}} } } \\& \qquad {}+ \int_{2}^{x} { \int_{1}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{{ {1 \over 2}} + j - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{{ {1 \over 2}} + j - 1} \ln t \ln s\frac{{dt}}{t}\frac{{ds}}{s}} } \\& \qquad {} - \int_{1}^{x} { \int_{1}^{y} { \biggl( {\ln\frac{x}{s}} \biggr)^{{ {1 \over 2}} + j - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{{ {1 \over 2}} + j - 1} \ln t \ln s\frac{{dt}}{t}\frac{{ds}}{s}} } \biggr] \biggr\} \quad ( \mbox{using (i)}) \\& \quad =\ln x\ln y,\quad \mbox{for }(x,y) \in (2,3 ] \times (1,3 ]. \end{aligned}$$
Therefore, equations (35)-(36) satisfy the fractional derivative condition in system (34).
Next, by equations (35)-(36), we have
$$\begin{aligned}& { \bigl[ {u \bigl(2^{+} ,y \bigr) - u \bigl(2^{-} ,y \bigr)} \bigr]}_{y \in (1,3 ]} \\& \quad = ly + \frac{1}{{\Gamma({ {1 \over 2}} + j)\Gamma({ {1 \over 2}} + j)}} \int_{1}^{2} { \int_{1}^{y} { \biggl( {\ln\frac{2}{s}} \biggr)^{{ {1 \over 2}} + j - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{{ {1 \over 2}} + j - 1} \ln t \ln s\frac{{dt}}{t}\frac{{ds}}{s}} } \\& \qquad {}- \frac{1}{{\Gamma({ {1 \over 2}} + j)\Gamma({ {1 \over 2}} + j)}} \int_{1}^{2} { \int_{1}^{y} { \biggl( {\ln\frac{2}{s}} \biggr)^{{ {1 \over 2}} + j - 1} \biggl( {\ln\frac{y}{t}} \biggr)^{{ {1 \over 2}} + j - 1} \ln t \ln s\frac{{dt}}{t}\frac{{ds}}{s}} } \\& \qquad (\mbox{using (35) and (36)} ) \\& \quad = ly |_{y \in (1,3 ]} . \end{aligned}$$
Therefore, equations (35)-(36) satisfy the impulsive conditions in system (34).
Finally, for system (34), we have
$$\begin{aligned}& \lim_{l \to0} \left \{ \textstyle\begin{array}{l} ( {{}_{\mathrm{C}\text{-}\mathrm{H}}D_{(1 + ,1 + )}^{q} u} )(x,y) = \ln x\ln y,\quad (x,y) \in[1,3] \times[1,3]\mbox{ and }x \ne2, \\ u(2^{+} ,y) = u(2^{-} ,y) + ly, \\ u(x,1) = u(1,y) \equiv0,\quad x \in[1,3], y \in[1,3] \end{array}\displaystyle \right . \\& \quad = \left \{ \textstyle\begin{array}{l} ( {{}_{\mathrm{C}\text{-}\mathrm{H}}D_{(1 + ,1 + )}^{q} u} )(x,y) = \ln x\ln y,\quad (x,y) \in[1,3] \times[1,3], \\ u(x,1) = u(1,y) \equiv0, \quad x \in[1,3], y \in[1,3]. \end{array}\displaystyle \right . \end{aligned}$$
(37)
On the other hand, by equations (35)-(36), we have
$$\begin{aligned}& \lim_{l \to0} \{ {\text{equations (35)-(36)}} \} \\& \quad \Rightarrow\quad \lim_{l \to0} u(x,y) \\& \hphantom{\quad \Rightarrow\quad }\quad = \lim_{l \to0} \left \{ \textstyle\begin{array}{l@{\quad}l} \frac{1}{{\Gamma({ {5 \over 2}} + j)\Gamma({ {5 \over 2}} + j)}} ( {\ln x} )^{{ {3 \over 2}} + j} ( {\ln y} )^{{ {3 \over 2}} + j},& \mbox{for }(x,y) \in (1,2 ] \times (1,3 ], \\ ly + \frac{1}{{\Gamma({ {5 \over 2}} + j)\Gamma({ {5 \over 2}} + j)}} { ( {\ln x} )^{{ {3 \over 2}} + j} } |_{x > 1} { ( {\ln y} )^{{ {3 \over 2}} + j} } |_{y > 1} \\ \quad {}+ \frac{{\sigma(y)ly}}{{\Gamma({ {5 \over 2}} + j)\Gamma({ {5 \over 2}} + j)}} [ ( {\ln2} )^{{ {3 \over 2}} + j} \\ \quad {}+ { [ {\ln x + ( {{ {1 \over 2}} + j} )\ln2} ] ( {\ln\frac{x}{2}} )^{{ {1 \over 2}} + j} } |_{x > 2} \\ \quad {}- { ( {\ln x} )^{{ {3 \over 2}} + j} } |_{x > 1} ] { ( {\ln y} )^{{ {3 \over 2}} + j} } |_{y > 1} , &\mbox{for }(x,y) \in (2,3 ] \times (1,3 ] \end{array}\displaystyle \right . \\& \quad \Rightarrow\quad u(x,y) = \left \{ \textstyle\begin{array}{l@{\quad}l} \frac{1}{{\Gamma({ {5 \over 2}} + j)\Gamma({ {5 \over 2}} + j)}}( {\ln x} )^{{ {3 \over 2}} + j} ( {\ln y} )^{{ {3 \over 2}} + j}, &\mbox{for }(x,y) \in (1,2 ] \times (1,3 ], \\ \frac{1}{{\Gamma({ {5 \over 2}} + j)\Gamma({ {5 \over 2}} + j)}} {( {\ln x} )^{{ {3 \over 2}} + j} } |_{x > 1} {( {\ln y} )^{{ {3 \over 2}} + j} } |_{y > 1} ,& \mbox{for }(x,y) \in (2,3 ] \times (1,3 ]. \end{array}\displaystyle \right . \end{aligned}$$
(38)
By Lemma 2.3, equation (38) is equivalent to system (37). Therefore, equations (35)-(36) satisfy the corresponding condition (3) of system (34). Thus, equations (35)-(36) satisfy all conditions of system (34), that is, equations (35)-(36) is the general solution of system (34).

Declarations

Acknowledgements

The work described in this paper is financially supported by the National Natural Science Foundation of China (Grants Nos. 21576033, 21636004).

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.

Authors’ Affiliations

(1)
School of Electronic Engineering, Jiujiang University

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© Zhang 2016