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. □