2.1 Distributed and Riesz fractional convection–diffusion equation
To tackle the TDSRFP of the form, GLLCT and GRLCT are suggested.
$$ \int _{0}^{1}\kappa (\mu ){}_{0}^{c} \mathcal{D}^{ \mu }_{t}\mathcal{U}(x,t)\,d\mu +\varepsilon (- \Delta )^{ \frac{\delta }{2}}\mathcal{U}(x,t)=\Lambda \bigl(x,t,\mathcal{U}(x,t) \bigr),\quad (x,t)\in \Lambda ^{\bullet } \times \Lambda ^{\diamond }, $$
(2.1)
where \(\Lambda ^{\bullet }\equiv [-1,1]\) and \(\Lambda ^{\diamond } \equiv [0,t_{\mathrm{end}}]\). Related to
$$ \begin{aligned} &\mathcal{U}(x,0)=\Theta _{1}(x),\quad x\in \Lambda ^{ \bullet }, \\ &\mathcal{U}(0,t)=\Theta _{2}(t), \qquad \mathcal{U}( x_{\mathrm{end}},t)= \Theta _{3}(t),\quad t\in \Lambda ^{\diamond } . \end{aligned} $$
(2.2)
To convert the TDSRFP into a nonlinear algebraic system, the GLLCT and GRSLCT are used. The truncated solution is written as
$$ \mathcal{U}_{\mathcal{N},\mathcal{M}}(x,t)=\sum _{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t), $$
(2.3)
where \(\mathcal{G}_{r_{1}}(x)\) and \(\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t)\) are the Legendre and shifted Legendre polynomials, see [54, 55] for more details. \((-\Delta )^{\frac{\delta }{2}}\mathcal{U}(x,t)\) is calculated as follows:
$$ \begin{aligned} (-\Delta )^{\frac{\delta }{2}} \mathcal{U}(x,t)&=- \frac{1}{2 \cos (\frac{\pi \delta }{2} )} \bigl({}_{-1}^{ c} \mathcal{D}^{\delta }_{x}\mathcal{U}(x,t)+{} _{x}^{c} \mathcal{D}^{\delta }_{1} \mathcal{U}(x,t) \bigr) \\ &=-\frac{1}{2 \cos (\frac{\pi \delta }{2} )} \biggl(\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \bigl({}_{-1}^{ c} \mathcal{D}^{\delta }_{x} \mathcal{G}_{r_{1}}(x)+{} _{x}^{c}\mathcal{D}^{\delta }_{1} \mathcal{G}_{r_{1}}(x) \bigr)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t) \biggr) \\ &=\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \mathcal{G}^{\Delta }_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t), \end{aligned} $$
(2.4)
where \(\mathcal{G}^{\Delta }_{r_{1}}(x)=- \frac{1}{2 \cos (\frac{\pi \delta }{2} )} ({}_{-1}^{ c}\mathcal{D}^{\delta }_{x}\mathcal{G}_{r_{1}}(x)+{} _{x}^{c} \mathcal{D}^{\delta }_{1}\mathcal{G}_{r_{1}}(x) )\), see [56, 57] for Riesz fractional derivative definition.
The Caputo fractional derivative \({}_{0}^{c}\mathcal{D}^{\mu }_{t}\mathcal{U}(x,t)\) is computed as
$$ {}_{0}^{c} \mathcal{D}^{\mu }_{t}\mathcal{U}(x,t)=\sum _{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x) \mathcal{G}^{t_{\mathrm{end}}}_{r_{2}, \mu }(t), $$
(2.5)
where \(\mathcal{G}^{t_{\mathrm{end}}}_{r_{2},\mu }(t)=_{0}^{c}\mathcal{D}^{\mu }_{t} \mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t)\), see [58]. The integrated operator’s treatment provides us with
$$ \begin{aligned} \int _{0}^{1}\kappa (\mu )_{0}^{c} \mathcal{D}^{ \mu }_{t}\mathcal{U}(x,t)\,d\mu &= \int _{0}^{1} \biggl(\kappa (\mu ) \sum _{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x) \mathcal{G}^{t_{\mathrm{end}}}_{r_{2}, \mu }(t) \biggr)\,d\mu \\ &=\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \int _{0}^{1}\kappa (\mu ) \mathcal{G}_{r_{1}}(x) \mathcal{G}^{t_{\mathrm{end}}}_{r_{2},\mu }(t)\,d\mu \\ &=\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M}\\r_{3}=0,\dots ,\mathcal{Q}}} \varsigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(x)\mathcal{F}^{t_{\mathrm{end}}}_{r_{2},r_{3}}(t), \end{aligned} $$
(2.6)
where \(\mathcal{F}^{t_{\mathrm{end}}}_{r_{2},r_{3}}(t)=\varpi _{\mathcal{Q},r_{3}} \kappa (\mu _{\mathcal{Q},r_{3}})\mathcal{G}^{t_{\mathrm{end}}}_{r_{2},\mu _{ \mathcal{Q},r_{3}}}\) and \(\mu _{\mathcal{Q},r_{3}}\), \(r_{3}=0,1,\ldots ,\mathcal{Q}\) are the shifted Legendre Gauss–Lobatto collocation points in the interval \([0,1]\), see [54, 59, 60].
At selected nodes, the preceding derivatives are calculated as follows:
$$\begin{aligned}& \bigl((-\Delta )^{\frac{\delta }{2}} \mathcal{U}(x,t) \bigr)^{x=x_{ \mathcal{N},n},}_{t=t^{t_{\mathrm{end}}}_{\mathcal{M},m}}=\sum _{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}}\mathcal{G}^{\Delta }_{r_{1}}(x_{\mathcal{N},n}) \mathcal{G}^{t_{\mathrm{end}}}_{r_{2}} \bigl(t^{t_{\mathrm{end}}}_{\mathcal{M},m} \bigr), \end{aligned}$$
(2.7)
$$\begin{aligned}& \biggl( \int _{0}^{1}\kappa (\mu )_{0}^{c} \mathcal{D}^{\mu }_{t}\mathcal{U}(x,t)\,d\mu \biggr)^{x=x_{\mathcal{N},n},}_{t=t^{t_{\mathrm{end}}}_{ \mathcal{M},m}}=\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M}\\r_{3}=0,\dots ,\mathcal{Q}}} \varsigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x_{\mathcal{N},n}) \mathcal{F}^{t_{\mathrm{end}}}_{r_{2},r_{3}} \bigl(t=t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr), \end{aligned}$$
(2.8)
where \(n=0,1,\ldots , \mathcal{N}\), \(m=0,1,\ldots , \mathcal{M}\) and \(x_{\mathcal{N},n}\), \(t^{t_{\mathrm{end}}}_{\mathcal{M},m}\) are Gauss–Lobatto Legendre collocation and Gauss–Radau shifted Legendre collocation nodes, respectively.
Equation (2.1) is coerced to zero at the \((\mathcal{N}-1)\times (\mathcal{M})\) nodes in the approach.
$$ \Omega \bigl(x_{\mathcal{N},n},t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr)= \Lambda \biggl(x_{ \mathcal{N},n},t^{t{\mathrm{end}}}_{\mathcal{M},m},\sum _{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(x_{\mathcal{N},n}) \mathcal{G}^{t_{\mathrm{end}}}_{r_{2}} \bigl(t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr) \biggr), $$
(2.9)
where
$$\begin{aligned} \Omega \bigl(x_{\mathcal{N},n},t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr) =&\sum _{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \mathcal{G}^{\Delta }_{r_{1}}(x_{\mathcal{N},n}) \mathcal{G}^{t_{\mathrm{end}}}_{r_{2}} \bigl(t^{t_{\mathrm{end}}}_{\mathcal{M},m} \bigr) \\ &{}+ \varepsilon \sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M}\\r_{3}=0,\dots ,\mathcal{Q}}} \varsigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x_{\mathcal{N},n}) \mathcal{F}^{t_{\mathrm{end}}}_{r_{2},r_{3}} \bigl(t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr). \end{aligned}$$
Otherwise, initial-boundary can be obtained by
$$ \begin{aligned} &\sum_{ \substack{ l,r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(0)= \Theta _{1}(x), \\ &\sum_{ \substack{ l,r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(-1)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}} \bigl(t^{t{\mathrm{end}}}_{ \mathcal{M},m} \bigr)=\Theta _{2} \bigl(t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr), \\ &\sum_{ \substack{ l,r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(1)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}} \bigl(t^{t{\mathrm{end}}}_{ \mathcal{M},m} \bigr)=\Theta _{3}(t). \end{aligned} $$
(2.10)
Therefore, adapting (2.1)–(2.10), we get
$$ \Omega \bigl(x_{\mathcal{N},n},t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr)= \Lambda \biggl(x_{\mathcal{N},n},t^{t{\mathrm{end}}}_{\mathcal{M},m},\sum _{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(x_{\mathcal{N},n}) \mathcal{G}^{t_{\mathrm{end}}}_{r_{2}} \bigl(t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr)\biggr), $$
(2.11)
with \(n=1,\ldots ,\mathcal{N}-1\), \(m=1,\ldots ,\mathcal{M}\), additionally
$$ \begin{aligned}&\sum_{ \substack{ l,r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x_{\mathcal{N},n}) \mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(0) =\Theta _{1}(x_{\mathcal{N},n}),\quad k=1,\ldots ,\mathcal{N}-1, \\ &\sum_{ \substack{ l,r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(-1)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}} \bigl(t^{t{\mathrm{end}}}_{ \mathcal{M},m} \bigr) =\Theta _{2} \bigl(t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr),\quad l=0, \ldots ,\mathcal{M}, \\ &\sum_{ \substack{ l,r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(1)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}} \bigl(t^{t{\mathrm{end}}}_{ \mathcal{M},m} \bigr) =\Theta _{3} \bigl(t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr),\quad l=0, \ldots ,\mathcal{M}. \end{aligned} $$
(2.12)
An algebraic equations system is produced by combining Eqs. (2.11) and (2.12), and it is easy to solve.
2.2 Distributed and Riesz fractional Schrödinger equation
To address the TDSRFSP, GLLCT and GRLCT are developed
$$ \begin{aligned} & i \int _{0}^{1}\kappa (\mu ){}_{0}^{c} \mathcal{D}^{ \mu }_{t}\psi (x,t)\,d\mu +\varepsilon (-\Delta )^{\frac{\delta }{2}} \psi (x,t)+ \bigl\vert \psi (x,t) \bigr\vert ^{2}\psi (x,t) \\ &\quad =\Lambda (x,t), \quad (x,t)\in \Lambda ^{\bullet } \times \Lambda ^{\diamond }, \end{aligned} $$
(2.13)
where \(\Lambda ^{\bullet }\equiv [-1,1]\) and \(\Lambda ^{\diamond } \equiv [0,t_{\mathrm{end}}]\). Related to
$$ \begin{aligned} &\psi (x,0)=\Theta _{1}(x), \quad x\in \Lambda ^{\bullet }, \\ &\psi (0,t)=\Theta _{2}(t), \qquad \psi ( x_{\mathrm{end}},t)=\Theta _{3}(t),\quad t\in \Lambda ^{\diamond } . \end{aligned} $$
(2.14)
Firstly, we split \(\psi (x,t)\) into its real and imaginary functions \(\mathcal{U}(x,t)\) and \(\mathcal{V}(x,t)\) as \(\psi (x,t)=\mathcal{U}(x,t)+i\mathcal{U}(x,t)\). Based on this transformation, we get
$$ \begin{aligned} & \int _{0}^{1}\kappa (\mu ){}_{0}^{c} \mathcal{D}^{ \mu }_{t}\mathcal{U}(x,t)\,d\mu +\varepsilon (- \Delta )^{ \frac{\delta }{2}}\mathcal{V}(x,t)+ \bigl(\mathcal{U}^{2}(x,t)+ \mathcal{V}^{2}(x,t) \bigr) \mathcal{V}(x,t) \\ &\quad =\Lambda _{2}(x,t),\quad (x,t)\in \Lambda ^{\bullet } \times \Lambda ^{\diamond }, \\ &{-} \int _{0}^{1}\kappa (\mu ){}_{0}^{c} \mathcal{D}^{\mu }_{t} \mathcal{V}(x,t)\,d\mu +\varepsilon (- \Delta )^{\frac{\delta }{2}} \mathcal{U}(x,t)+ \bigl(\mathcal{U}^{2}(x,t)+ \mathcal{V}^{2}(x,t) \bigr) \mathcal{U}(x,t) \\ &\quad =\Lambda _{1}(x,t), \quad (x,t)\in \Lambda ^{\bullet } \times \Lambda ^{\diamond }, \end{aligned} $$
(2.15)
where \(\Lambda (x,t)=\Lambda _{2}(x,t)+i\Lambda _{2}(x,t)\), \(\Lambda ^{\bullet }\equiv [-1,1]\), and \(\Lambda ^{\diamond } \equiv [0,t_{\mathrm{end}}]\). Related to
$$ \begin{aligned} &\mathcal{U}(x,0)=\theta _{1}(x),\qquad \mathcal{U}(0,t)= \theta _{2}(t),\qquad \mathcal{U}( x_{\mathrm{end}},t)=\theta _{3}(t), \quad x \in \Lambda ^{\bullet }, t\in \Lambda ^{\diamond }, \\ &\mathcal{U}(x,0)=\vartheta _{1}(x), \qquad \mathcal{U}(0,t)= \vartheta _{2}(t), \qquad \mathcal{U}( x_{\mathrm{end}},t)=\vartheta _{3}(t), \quad x\in \Lambda ^{\bullet }, t\in \Lambda ^{\diamond }, \end{aligned} $$
(2.16)
where \(\Theta \equiv \theta +i\vartheta \).
To convert the TDSRFP into a nonlinear algebraic system, the GLLCT and GRSLCT are used. The truncated solution is written as follows:
$$ \begin{aligned}& \mathcal{U}_{\mathcal{N},\mathcal{M}}(x,t)= \sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t), \\ &\mathcal{V}_{\mathcal{N},\mathcal{M}}(x,t)=\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \sigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t). \end{aligned} $$
(2.17)
Therefore, according to the previous analysis, we get
$$ \begin{aligned} &\Upsilon _{1} \bigl(x_{\mathcal{N},n},t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr)= \Lambda _{2} \bigl(x_{\mathcal{N},n},t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr),\quad n=1, \ldots ,\mathcal{N}-1, m=1,\ldots ,\mathcal{M}, \\ &\Upsilon _{2} \bigl(x_{\mathcal{N},n},t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr)=\Lambda _{1} \bigl(x_{ \mathcal{N},n},t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr),\quad n=1,\ldots ,\mathcal{N}-1, m=1,\ldots ,\mathcal{M}, \end{aligned} $$
(2.18)
where
$$\begin{aligned}& \begin{aligned}\Upsilon _{1}(x,t)&=\sum _{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M}\\r_{3}=0,\dots ,\mathcal{Q}}} \varsigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x) \mathcal{F}^{t_{\mathrm{end}}}_{r_{2},r_{3}}(t)+ \varepsilon (-\Delta )^{\frac{\delta }{2}}\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \sigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t) \\ &\quad {}+ \biggl(\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t) \biggr)^{2} \biggl(\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \sigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t) \biggr) \\ &\quad {}+ \biggl(\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \sigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t) \biggr)^{2} \biggl(\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \sigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t) \biggr), \end{aligned} \\& \begin{aligned} \Upsilon _{2}(x,t)&=-\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M}\\r_{3}=0,\dots ,\mathcal{Q}}} \sigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x)\mathcal{F}^{t_{\mathrm{end}}}_{r_{2},r_{3}}(t)+ \varepsilon (-\Delta )^{\frac{\delta }{2}}\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t) \\ &\quad {}+ \biggl(\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t) \biggr)^{2} \biggl(\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t) \biggr) \\ &\quad {} + \biggl(\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \sigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t) \biggr)^{2} \biggl(\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t) \biggr). \end{aligned} \end{aligned}$$
In addition to
$$\begin{aligned}& \sum_{ \substack{ l,r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x_{\mathcal{N},n}) \mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(0) =\theta _{1}(x_{\mathcal{N},n}),\quad k=1,\ldots ,\mathcal{N}-1, \\& \sum_{ \substack{ l,r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(-1)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}} \bigl(t^{t{\mathrm{end}}}_{ \mathcal{M},m} \bigr) =\theta _{2} \bigl(t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr),\quad l=0, \ldots ,\mathcal{M}, \\& \sum_{ \substack{ l,r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(1)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}} \bigl(t^{t{\mathrm{end}}}_{ \mathcal{M},m} \bigr) =\theta _{3} \bigl(t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr),\quad l=0, \ldots ,\mathcal{M}, \\& \sum_{ \substack{ l,r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \sigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(x_{\mathcal{N},n}) \mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(0) = \vartheta _{1}(x_{\mathcal{N},n}) ,\quad k=1,\ldots ,\mathcal{N}-1, \\& \sum_{ \substack{ l,r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \sigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(-1)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}} \bigl(t^{t{\mathrm{end}}}_{ \mathcal{M},m} \bigr) =\vartheta _{2} \bigl(t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr),\quad l=0, \ldots ,\mathcal{M}, \\& \sum_{ \substack{ l,r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \sigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(1)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}} \bigl(t^{t{\mathrm{end}}}_{ \mathcal{M},m} \bigr) =\vartheta _{3} \bigl(t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr),\quad l=0, \ldots ,\mathcal{M}, \end{aligned}$$
(2.19)
when Eqs. (2.18) and (2.19) are combined, we have a linear system of algebraic equations that is simple to solve.