1 Introduction

The fractional calculus was developed within the frame of the Hadamard fractional derivative in [1–6], 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 [8–13].

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 [17–22].

Now the equivalent integral equations were found for several fractional-order systems with impulses in [23–28], 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).