In this portion, we Analyze the Hyers-Ulam and general Hyers-Ulam stability analysis32,33. Let us suppose \(\epsilon >0\) with,
$$\begin{aligned} |^{\textbf{C}}\textbf{D}_{t}^{\alpha _{1}}\mathfrak {S}(t)-\mathfrak {P}(t, \mathfrak {S}(t))|\le \epsilon , ~ t\in \textrm{J}, \end{aligned}$$
(13)
for \(\epsilon =\max (\epsilon _{i})^T, i=1,2,3.\)
Definition 3.1
Equation (4) is supposed to be Hyers-Ulam stable such that \(\exists , \mathfrak {H}_{\mathfrak {P}}>0,\) for any \(\epsilon >0\) and for all solution \(\mathfrak {S}\in \mathcal {Y}\) of (4) and (13), has unique solution \(\widetilde{\mathfrak {S}}\in \textbf{Y}\) which satisfy
$$\begin{aligned} |\widetilde{\mathfrak {S}}(t)-\mathfrak {S}(t)|\le \mathfrak {H}_{\mathfrak {P}}\epsilon , ~ t\in \textrm{J}, \end{aligned}$$
(14)
where \(\mathfrak {H}_{\mathfrak {P}}=\max (\mathfrak {H}_{\mathfrak {P}_i})^T,\ i=1,2,3.\)
Now for the generalized Hyers-Ulam stability we have,
Definition 3.2
Equation (4) is generalized Hyers-Ulam stable if for a continuous function \(\varphi :\mathbb {R}^+\rightarrow \mathbb {R}^+\) with \(\varphi (0)=0\), such that for all solution \(\mathfrak {S}\in \mathcal {Y}\) of (4), there is one solution \(\widetilde{\mathfrak {S}}\in \mathcal {Y}\) which holds the given relation
$$\begin{aligned} |\widetilde{\mathfrak {S}}(t)-\mathfrak {S}(t)|\le \varphi _{\mathfrak {P}}\epsilon , ~ t\in \textrm{J}, \end{aligned}$$
(15)
where \(\varphi _{\mathcal {P}}=\max (\varphi _{\mathcal {P}_i})^T,\ ,\ i=1,2,3.\)
Remark 1
An independent function \(\varphi\) of \(\mathfrak {S}\in \mathcal {Y}\), one has
-
(I)
\(|\varphi (t)|\le \epsilon ,\ ~ \varphi =\max (\varphi _i), ~ t\in \textrm{J}, \ i=1,2,3.\)
-
(II)
\(^{\mathcal {C}}\textbf{D}_{t}^{\alpha _{1}}\mathfrak {S}(t)=\mathcal {P}(t, \mathfrak {S}(t))+\varphi (t), ~ t\in \textrm{J}.\)
Lemma 3.2.1
The solution of unsettled problem
$$\begin{aligned} ^{\mathcal {C}}\textbf{D}_{t}^{\alpha _{1}}\mathfrak {S}(t)= & \mathfrak {P}(t, \mathfrak {S}(t))+\varphi (t),\nonumber \\ \mathfrak {S}(0)= & \mathfrak {S}_0, \end{aligned}$$
(16)
holds the inequality
$$\begin{aligned} \bigg |\mathfrak {S}(t)-\bigg (\mathfrak {S}_0(t)-\frac{1}{\Gamma (\alpha _{1})}\int _0^t(t-\nu )^{\alpha _{1}-1}\mathfrak {P}(\nu , \mathfrak {S}(\nu ))d\nu \bigg )\bigg |\le \Omega \epsilon ,\ \text {where}\ \Omega =\frac{\mathfrak {b}^\alpha _{1}}{\Gamma (\alpha _{1}+1)}. \end{aligned}$$
Proof
Utilizing Lemma 3.2.1, the solution of (16), become
$$\begin{aligned} \mathfrak {S}(t)=\mathfrak {S}_0(t)+\frac{1}{\Gamma (\alpha _{1})}\int _0^t(t-\nu )^{\alpha _{1}-1}\mathcal {P}(\nu , \mathfrak {S}(\nu ))d\nu +\frac{1}{\Gamma (\alpha _{1})}\int _0^t(t-\nu )^{\alpha _{1}-1}\varphi (\nu )d\nu . \end{aligned}$$
(17)
From the helpful Remark 1, the result (17) give in
$$\begin{aligned} \bigg |\mathfrak {S}(t)-\bigg (\mathfrak {S}_0(t)+\frac{1}{\Gamma (\alpha _{1})}\int _0^t(t-\nu )^{\alpha _{1}-1}\mathfrak {P}(\nu , \mathfrak {S}(\nu ))d\nu \bigg )\bigg |\le & \frac{1}{\Gamma (\alpha _{1})}\int _0^t(t-\nu )^{\alpha _{1}-1}|\varphi (\nu )|d\nu \nonumber \\\le & \Omega \epsilon . \end{aligned}$$
(18)
\(\square\)
Theorem 1
From supposition \((\mathbf {\mathfrak {C}_1})\) and if \(1-\Psi \mathcal {L}_{\mathfrak {P}}>0\), then form Lemma 3.2.1, the result of (4) is Hyers-Ulam and generalized Hyers-Ulam stable as well.
Proof
Assume that \(\mathfrak {S} \in \mathcal {Y}\) be any arbitrary solution and \(\widetilde{\mathfrak {S}}\in \mathcal {Y}\) is one solution of (4), then using \((\textbf{C}_1)\) one has
$$\begin{aligned} \Vert \widetilde{\mathfrak {S}}-\mathfrak {S}\Vert= & \max _{t\in \textrm{J}}\bigg |\widetilde{\mathfrak {S}}(t) -\bigg (\mathfrak {S}_0+\frac{1}{\Gamma (\alpha _{1})}\int _0^t(t-\nu )^{\alpha _{1}-1}\mathfrak {P}(\nu , \mathfrak {S}(\nu ))d\nu \bigg )\bigg |\nonumber \\\le & \max _{t\in \textrm{J}}\bigg |\widetilde{\mathfrak {S}}(t)-\bigg (\widetilde{\mathfrak {S}}_0 +\frac{1}{\Gamma (\alpha _{1})}\int _0^t(t-\nu )^{\alpha _{1}-1}\mathfrak {P}(\nu ,\mathfrak {S}(\nu ))d\nu \bigg )\bigg |\nonumber \\ & +\max _{t\in \textrm{J}}\frac{1}{\Gamma (\alpha _{1})}\int _0^t(t-\nu )^{\alpha _{1}-1}\big |\mathfrak {P}(\nu , \widetilde{\mathfrak {S}}(\nu ))-\mathfrak {P}(\nu ,\mathfrak {S}(\nu ))\big |d\nu \nonumber \\\le & \bigg |\mathfrak {S}(t)-\bigg (\widetilde{\mathfrak {S}}_0+\frac{1}{\Gamma (\alpha _{1})} \int _0^t(t-\nu )^{\alpha _{1}-1}\mathfrak {P}(\nu ,\widetilde{\mathfrak {S}}(\nu ))d\nu \bigg )\bigg |\nonumber \\ & +\frac{\mathcal {L}_{\mathfrak {P}}}{\Gamma (\alpha _{1})}\int _0^t(t-\nu )^{\alpha _{1}-1}\big | \widetilde{\mathfrak {S}}(\nu )-\mathfrak {S}(\nu )\big |d\nu \nonumber \\\le & \Omega \epsilon +\Omega \mathcal {L}_{\mathcal {P}}\Vert \widetilde{\mathfrak {S}}-\mathfrak {S}\Vert . \end{aligned}$$
(19)
After simplifying (19),
$$\begin{aligned} \Vert \widetilde{\mathfrak {S}}-\mathfrak {S}\Vert \le \frac{\Omega \epsilon }{1-\Omega \mathcal {L}_{\mathfrak {P}}}. \end{aligned}$$
(20)
From the above Eq. (20), may be written as
$$\begin{aligned} \mathfrak {H}_{\mathfrak {P}}=\frac{\Omega }{1-\Omega \mathcal {L}_{\mathfrak {P}}}. \end{aligned}$$
(21)
Hence, using \(\varphi _{\mathfrak {P}}(\epsilon )=\mathfrak {H}_{\mathfrak {P}}\epsilon\) with \(\varphi _{\mathfrak {P}}(0)=0\), we summarize that the result of the proposed model (4) is Hyers-Ulam stable and generalized Hyers-Ulam. \(\square\)
Computational investigation of the system (2)
Here we prove the numerical-scheme for the proposed model (2). Our proposed model is non linear, and it is difficult to solve analytically therefore we use numerical approaches of Newton polynomial. Now, consider
$$\begin{aligned} \mathcal {H}(t)=\mathcal {H}_{0}+\frac{1}{\Gamma (\varpi )}\int _{0}^{t}(t-\nu )^{\varpi -1}\mathfrak {D}_{1}(\nu ,\mathcal {H})d\nu ,\\ \mathcal {P}(t)=\mathcal {P}_{0}+\frac{1}{\Gamma (\varpi )}\int _{0}^{t}(t-\nu )^{\varpi -1}\mathfrak {D}_{2}(\nu ,\mathcal {P})d\nu ,\\ \mathcal {I}(t)=\mathcal {I}_{0}+\frac{1}{\Gamma (\varpi )}\int _{0}^{t}(t-\nu )^{\varpi -1}\mathfrak {D}_{3}(\nu ,\mathcal {I})d\nu ,\\ \mathcal {F}(t)=\mathcal {F}_{0}+\frac{1}{\Gamma (\varpi )}\int _{0}^{t}(t-\nu )^{\varpi -1}\mathfrak {D}_{4}(\nu ,\mathcal {F})d\nu ,\\ \mathcal {E}(t)=\mathcal {E}_{0}+\frac{1}{\Gamma (\varpi )}\int _{0}^{t}(t-\nu )^{\varpi -1}\mathfrak {D}_{5}(\nu ,\mathcal {E})d\nu . \end{aligned}$$
At \(t_{{\mu }+1}=({\mu }+1)\nu t,\) the aforementioned system can be presented as:
$$\begin{aligned} \mathcal {H}(t_{{\mu }+1})=\mathcal {H}_{0}+\frac{1}{\Gamma (\varpi )}\int _{0}^{t_{{\mu }+1}}(t_{{\mu }+1}-\nu )^{\varpi -1}\mathfrak {D}_{1}(\nu ,\mathcal {H})d\nu ,\\ \mathcal {P}(t_{{\mu }+1})=\mathcal {P}_{0}+\frac{1}{\Gamma (\varpi )}\int _{0}^{t_{{\mu }+1}}(t_{{\mu }+1}-\nu )^{\varpi -1}\mathfrak {D}_{2}(\nu ,\mathcal {P})d\nu ,\\ \mathcal {I}(t_{{\mu }+1})=\mathcal {I}_{0}+\frac{1}{\Gamma (\varpi )}\int _{0}^{t_{{\mu }+1}}(t_{{\mu }+1}-\nu )^{\varpi -1}\mathfrak {D}_{3}(\nu ,\mathcal {I})d\nu ,\\ \mathcal {F}(t_{{\mu }+1})=\mathcal {F}_{0}+\frac{1}{\Gamma (\varpi )}\int _{0}^{t_{{\mu }+1}}(t_{{\mu }+1}-\nu )^{\varpi -1}\mathfrak {D}_{4}(\nu ,\mathcal {F})d\nu ,\\ \mathcal {E}(t_{{\mu }+1})=\mathcal {E}_{0}+\frac{1}{\Gamma (\varpi )}\int _{0}^{t_{{\mu }+1}}(t_{{\mu }+1}-\nu )^{\varpi -1}\mathfrak {D}_{5}(\nu ,\mathcal {E})d\nu . \end{aligned}$$
Then, we get:
$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {H}(t_{{\mu }+1})=\mathcal {H}_{0}+\frac{1}{\Gamma (\varpi )}\sum _{l=2}^{{\mu }}\int _{t_{l}}^{t_{l+1}}(t_{{\mu }+1}-\nu )^{\varpi -1}\mathfrak {D}_{1}(\nu ,\mathcal {H})d\nu ,\\ \mathcal {P}(t_{{\mu }+1})=\mathcal {P}_{0}+\frac{1}{\Gamma (\varpi )}\sum _{l=2}^{{\mu }}\int _{t_{l}}^{t_{l+1}}(t_{{\mu }+1}-\nu )^{\varpi -1}\mathfrak {D}_{2}(\nu ,\mathcal {P})d\nu ,\\ \mathcal {I}(t_{{\mu }+1})=\mathcal {I}_{0}+\frac{1}{\Gamma (\varpi )}\sum _{l=2}^{{\mu }}\int _{t_{l}}^{t_{l+1}}(t_{{\mu }+1}-\nu )^{\varpi -1}\mathfrak {D}_{3}(\nu ,\mathcal {I})d\nu ,\\ \mathcal {F}(t_{{\mu }+1})=\mathcal {F}_{0}+\frac{1}{\Gamma (\varpi )}\sum _{l=2}^{{\mu }}\int _{t_{l}}^{t_{l+1}}(t_{{\mu }+1}-\nu )^{\varpi -1}\mathfrak {D}_{4}(\nu ,\mathcal {F})d\nu .\\ \mathcal {E}(t_{{\mu }+1})=\mathcal {E}_{0}+\frac{1}{\Gamma (\varpi )}\sum _{l=2}^{{\mu }}\int _{t_{l}}^{t_{l+1}}(t_{{\mu }+1}-\nu )^{\varpi -1}\mathfrak {D}_{5}(\nu ,\mathcal {E})d\nu . \end{array}\right. } \end{aligned}$$
(22)
For the approximation of the kernels \(\mathfrak {D}_{1}(\nu ,\mathcal {H}),\mathfrak {D}_{2}(\nu ,\mathcal {H}),\mathfrak {D}_{3}(\nu ,\mathcal {H}),\mathfrak {D}_{4}(\nu ,\mathcal {H}),\) we will use the Newton polynomial as:
$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {T}_{1{\mu }}(\nu ) & =\mathfrak {D}_{1}\left( t_{{\mu }-2},\mathcal {H}(t_{{\mu }-2})\right) +\frac{\mathfrak {D}_{1}\left( t_{{\mu }-1},\mathcal {H}(t_{{\mu }-1})\right) -\mathfrak {D}_{1}\left( t_{{\mu }-2},\mathcal {H}(t_{{\mu }-2})\right) }{\nu t}\\ & \times (\nu -t_{{\mu }-2})\\ & +\frac{\mathfrak {D}_{1}\left( t_{{\mu }},\mathcal {H}(t_{{\mu }})\right) -2\mathfrak {D}_{1}\left( t_{{\mu }-1},\mathcal {H}(t_{{\mu }-1})\right) +\mathfrak {D}_{1}\left( t_{{\mu }-2},\mathcal {H}(t_{{\mu }-2})\right) }{2(\nu t)^{2}}\\ & \times (\nu -t_{{\mu }-2})\times (\nu -t_{{\mu }-1}),\\ \mathcal {T}_{2{\mu }}(\nu ) & =\mathfrak {D}_{2}\left( t_{{\mu }-2},\mathcal {I}(t_{{\mu }-2})\right) +\frac{\mathfrak {D}_{2}\left( t_{{\mu }-1},\mathcal {I}(t_{{\mu }-1})\right) -\mathfrak {D}_{2}\left( t_{{\mu }-2},\mathcal {I}(t_{{\mu }-2})\right) }{\nu t}\\ & \times (\nu -t_{{\mu }-2})\\ & +\frac{\mathfrak {D}_{2}\left( t_{{\mu }},\mathcal {I}(t_{{\mu }})\right) -2\mathfrak {D}_{2}\left( t_{{\mu }-1},\mathcal {I}(t_{{\mu }-1})\right) +\mathfrak {D}_{2}\left( t_{{\mu }-2},\mathcal {I}(t_{{\mu }-2})\right) }{2(\nu t)^{2}}\\ & \times (\nu -t_{{\mu }-2})\times (\nu -t_{{\mu }-1}),\\ \mathcal {T}_{3{\mu }}(\nu ) & =\mathfrak {D}_{3}\left( t_{{\mu }-2},\mathcal {P}(t_{{\mu }-2})\right) +\frac{\mathfrak {D}_{3}\left( t_{{\mu }-1},\mathcal {P}(t_{{\mu }-1})\right) -\mathfrak {D}_{3}\left( t_{{\mu }-2},\mathcal {P}(t_{{\mu }-2})\right) }{\nu t}\\ & \times (\nu -t_{{\mu }-2})\\ & +\frac{\mathfrak {D}_{3}\left( t_{{\mu }},\mathcal {P}(t_{{\mu }})\right) -2\mathfrak {D}_{3}\left( t_{{\mu }-1},\mathcal {P}(t_{{\mu }-1})\right) +\mathfrak {D}_{3}\left( t_{{\mu }-2},\mathcal {P}(t_{{\mu }-2})\right) }{2(\nu t)^{2}}\\ & \times (\nu -t_{{\mu }-2})\times (\nu -t_{{\mu }-1}),\\ \mathcal {T}_{4{\mu }}(\nu ) & =\mathfrak {D}_{4}\left( t_{{\mu }-2},\mathcal {F}(t_{{\mu }-2})\right) +\frac{\mathfrak {D}_{4}\left( t_{{\mu }-1},\mathcal {F}(t_{{\mu }-1})\right) -\mathfrak {D}_{4}\left( t_{{\mu }-2},\mathcal {F}(t_{{\mu }-2})\right) }{\nu t}\\ & \times (\nu -t_{{\mu }-2})\\ & +\frac{\mathfrak {D}_{4}\left( t_{{\mu }},\mathcal {F}(t_{{\mu }})\right) -2\mathfrak {D}_{4}\left( t_{{\mu }-1},\mathcal {F}(t_{{\mu }-1})\right) +\mathfrak {D}_{4}\left( t_{{\mu }-2},\mathcal {F}(t_{{\mu }-2})\right) }{2(\nu t)^{2}}\\ & \times (\nu -t_{{\mu }-2})\times (\nu -t_{{\mu }-1}),\\ \mathcal {T}_{5{\mu }}(\nu ) & =\mathfrak {D}_{5}\left( t_{{\mu }-2},\mathcal {E}(t_{{\mu }-2})\right) +\frac{\mathfrak {D}_{5}\left( t_{{\mu }-1},\mathcal {E}(t_{{\mu }-1})\right) -\mathfrak {D}_{5}\left( t_{{\mu }-2},\mathcal {E}(t_{{\mu }-2})\right) }{\nu t}\\ & \times (\nu -t_{{\mu }-2})\\ & +\frac{\mathfrak {D}_{5}\left( t_{{\mu }},\mathcal {E}(t_{{\mu }})\right) -2\mathfrak {D}_{5}\left( t_{{\mu }-1},\mathcal {E}(t_{{\mu }-1})\right) +\mathfrak {D}_{5}\left( t_{{\mu }-2},\mathcal {E}(t_{{\mu }-2})\right) }{2(\nu t)^{2}}\\ & \times (\nu -t_{{\mu }-2})\times (\nu -t_{{\mu }-1}). \end{array}\right. } \end{aligned}$$
Upon putting the Newton polynomials in the system (22), we get:
$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {H}^{{\mu }+1} & =\mathcal {H}_{0}+\frac{1}{\Gamma (\varpi )}\sum _{l=2}^{{\mu }}\int _{t_{l}}^{t_{l+1}}\left[ \mathfrak {D}_{1}\left( t_{{\mu }-2},\mathcal {H}^{{\mu }-2}\right) \right. \\ & +\frac{\mathfrak {D}_{1}\left( t_{{\mu }-1},\mathcal {H}^{{\mu }-1}\right) -\mathfrak {D}_{1}\left( t_{{\mu }-2},\mathcal {H}^{{\mu }-2}\right) }{\nu t}(\nu -t_{{\mu }-2})\\ & +\frac{\mathfrak {D}_{1}\left( t_{{\mu }},\mathcal {H}^{{\mu }}\right) -2\mathfrak {D}_{1}\left( t_{{\mu }-1},\mathcal {H}^{{\mu }-1}\right) +\mathfrak {D}_{1}\left( t_{{\mu }-2},\mathcal {H}^{{\mu }-2}\right) }{2(\nu t)^{2}}\\ & \left. \times (\nu -t_{{\mu }-2})\times (\nu -t_{{\mu }-1})\right] (t_{{\mu }+1}-\nu )^{\varpi -1}d\nu ,\\ \mathcal {P}^{{\mu }+1} & =\mathcal {P}_{0}+\frac{1}{\Gamma (\varpi )}\sum _{l=2}^{{\mu }}\int _{t_{l}}^{t_{l+1}}\left[ \mathfrak {D}_{2}\left( t_{{\mu }-2},\mathcal {P}^{{\mu }-2}\right) \right. \\ & +\frac{\mathfrak {D}_{2}\left( t_{{\mu }-1},\mathcal {P}^{{\mu }-1}\right) -\mathfrak {D}_{2}\left( t_{{\mu }-2},\mathcal {P}^{{\mu }-2}\right) }{\nu t}(\nu -t_{{\mu }-2})\\ & +\frac{\mathfrak {D}_{2}\left( t_{{\mu }},\mathcal {P}^{{\mu }}\right) -2\mathfrak {D}_{2}\left( t_{{\mu }-1},\mathcal {P}^{{\mu }-1}\right) +\mathfrak {D}_{2}\left( t_{{\mu }-2},\mathcal {P}^{{\mu }-2}\right) }{2(\nu t)^{2}}\\ & \left. \times (\nu -t_{{\mu }-2})\times (\nu -t_{{\mu }-1})\right] (t_{{\mu }+1}-\nu )^{\varpi -1}d\nu ,\\ \mathcal {I}^{{\mu }+1} & =\mathcal {I}_{0}+\frac{1}{\Gamma (\varpi )}\sum _{l=2}^{{\mu }}\int _{t_{l}}^{t_{l+1}}\left[ \mathfrak {D}_{3}\left( t_{{\mu }-2},\mathcal {P}^{{\mu }-2}\right) \right. \\ & +\frac{\mathfrak {D}_{3}\left( t_{{\mu }-1},\mathcal {I}^{{\mu }-1}\right) -\mathfrak {D}_{3}\left( t_{{\mu }-2},\mathcal {I}^{{\mu }-2}\right) }{\nu t}(\nu -t_{{\mu }-2})\\ & +\frac{\mathfrak {D}_{3}\left( t_{{\mu }},\mathcal {I}^{{\mu }}\right) -2\mathfrak {D}_{3}\left( t_{{\mu }-1},\mathcal {I}^{{\mu }-1}\right) +\mathfrak {D}_{3}\left( t_{{\mu }-2},\mathcal {I}^{{\mu }-2}\right) }{2(\nu t)^{2}}\\ & \left. \times (\nu -t_{{\mu }-2})\times (\nu -t_{{\mu }-1})\right] (t_{{\mu }+1}-\nu )^{\varpi -1}d\nu ,\\ \mathcal {F}^{{\mu }+1} & =\mathcal {F}_{0}+\frac{1}{\Gamma (\varpi )}\sum _{l=2}^{{\mu }}\int _{t_{l}}^{t_{l+1}}\left[ \mathfrak {D}_{4}\left( t_{{\mu }-2},\mathcal {F}^{{\mu }-2}\right) \right. \\ & +\frac{\mathfrak {D}_{4}\left( t_{{\mu }-1},\mathcal {F}^{{\mu }-1}\right) -\mathfrak {D}_{4}\left( t_{{\mu }-2},\mathcal {F}^{{\mu }-2}\right) }{\nu t}(\nu -t_{{\mu }-2})\\ & +\frac{\mathfrak {D}_{4}\left( t_{{\mu }},\mathcal {F}^{{\mu }}\right) -2\mathfrak {D}_{4}\left( t_{{\mu }-1},\mathcal {F}^{{\mu }-1}\right) +\mathfrak {D}_{4}\left( t_{{\mu }-2},\mathcal {F}^{{\mu }-2}\right) }{2(\nu t)^{2}}\\ & \left. \times (\nu -t_{{\mu }-2})\times (\nu -t_{{\mu }-1})\right] (t_{{\mu }+1}-\nu )^{\varpi -1}d\nu ,\\ \mathcal {E}^{{\mu }+1} & =\mathcal {E}_{0}+\frac{1}{\Gamma (\varpi )}\sum _{l=2}^{{\mu }}\int _{t_{l}}^{t_{l+1}}\left[ \mathfrak {D}_{5}\left( t_{{\mu }-2},\mathcal {E}^{{\mu }-2}\right) \right. \\ & +\frac{\mathfrak {D}_{5}\left( t_{{\mu }-1},\mathcal {E}^{{\mu }-1}\right) -\mathfrak {D}_{5}\left( t_{{\mu }-2},\mathcal {E}^{{\mu }-2}\right) }{\nu t}(\nu -t_{{\mu }-2})\\ & +\frac{\mathfrak {D}_{5}\left( t_{{\mu }},\mathcal {E}^{{\mu }}\right) -2\mathfrak {D}_{5}\left( t_{{\mu }-1},\mathcal {E}^{{\mu }-1}\right) +\mathfrak {D}_{5}\left( t_{{\mu }-2},\mathcal {E}^{{\mu }-2}\right) }{2(\nu t)^{2}}\\ & \left. \times (\nu -t_{{\mu }-2})\times (\nu -t_{{\mu }-1})\right] (t_{{\mu }+1}-\nu )^{\varpi -1}d\nu . \end{array}\right. } \end{aligned}$$
(23)
Now, we focusing on the first equation from (23), we have
$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {H}^{{\mu }+1} & =\mathcal {H}_{0}+\frac{1}{\Gamma (\varpi )}\sum _{l=2}^{{\mu }}\mathfrak {D}_{1}\left( t_{{\mu }-2},\mathcal {H}^{{\mu }-2}\right) \int _{t_{l}}^{t_{l+1}}(t_{{\mu }+1}-\nu )^{\varpi -1}d\nu \\ & +\frac{1}{\Gamma (\varpi )}\sum _{l=2}^{{\mu }}\frac{\mathfrak {D}_{1}\left( t_{{\mu }-1},\mathcal {H}^{{\mu }-1}\right) -\mathfrak {D}_{1}\left( t_{{\mu }-2},\mathcal {H}^{{\mu }-2}\right) }{\nu t}\\ & \times \int _{t_{l}}^{t_{l+1}}(\nu -t_{{\mu }-2})(t_{{\mu }+1}-\nu )^{\varpi -1}d\nu \\ & +\frac{1}{\Gamma (\varpi )}\sum _{l=2}^{{\mu }}\frac{\mathfrak {D}_{1}\left( t_{{\mu }},\mathcal {H}(t_{{\mu }})\right) -2\mathfrak {D}_{1}\left( t_{{\mu }-1},\mathcal {H}(t_{{\mu }-1})\right) +\mathfrak {D}_{1}\left( t_{{\mu }-2},\mathcal {H}(t_{{\mu }-2})\right) }{2(\nu t)^{2}}+\\ & \times \int _{t_{l}}^{t_{l+1}}(\nu -t_{{\mu }-2})(\nu -t_{{\mu }-1})(t_{{\mu }+1}-\nu )^{\varpi -1}d\nu . \end{array}\right. } \end{aligned}$$
(24)
Now, we evaluate the integrals of system (24).
$$\begin{aligned} \int _{t_{l}}^{t_{l+1}}(t_{{\mu }+1}-\nu )^{\varpi -1}d\nu= & \frac{(\nu t)^{\varpi }}{\varpi }\left[ ({\mu }-l+1)^{\varpi }-({\mu }-l)^{\varpi }\right] , \end{aligned}$$
$$\begin{aligned} & \int _{t_{l}}^{t_{l+1}}(\nu -t_{{\mu }-2})(t_{{\mu }+1}-\nu )^{\varpi -1}d\nu = \frac{(\nu t)^{\varpi +1}}{\varpi (\varpi +1)}\left[ \begin{array}{cc} ({\mu }-l+1)^{\varpi } & ({\mu }-l+2\varpi +3)\\ -({\mu }-l)^{\varpi } & ({\mu }-l+3\varpi +3) \end{array}\right] , \end{aligned}$$
$$\begin{aligned} \int _{t_{l}}^{t_{l+1}}(\nu -t_{{\mu }-2})(\nu -t_{{\mu }-1})(t_{{\mu }+1}-\nu )^{\varpi -1}d\nu= & \frac{(\nu t)^{\varpi +2}}{\varpi (\varpi +1)(\varpi +2)}+\left[ ({\mu }-l+1)^{\varpi }\right. \\ & \times \left( 2({\mu }-l)^{2}+(3\varpi +10)({\mu }-l)\right. \\ & \left. +2\varpi ^{2}+9\varpi +10\right) -({\mu }-l)^{\varpi }\\ & \times \left( 2({\mu }-l)^{2}+(5\varpi +10)({\mu }-l)\right. \\ & \left. \left. +6\varpi ^{2}+18\varpi +12\right) \right] . \end{aligned}$$
Using the above values of in Eq.(24), we get
$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {H}^{{\mu }+1} & =\mathcal {H}_{0}+\frac{(\nu t)^{\varpi }}{\Gamma (\varpi +1)}\sum _{l=2}^{{\mu }}\mathfrak {D}_{1}\left( t_{{\mu }-2},\mathcal {H}^{{\mu }-2}\right) \left[ ({\mu }-l+1)^{\varpi }-({\mu }-l)^{\varpi }\right] \\ & +\frac{(\nu t)^{\varpi }}{\Gamma (\varpi +2)}\sum _{l=2}^{{\mu }}\frac{\mathfrak {D}_{1}\left( t_{{\mu }-1},\mathcal {H}^{{\mu }-1}\right) -\mathfrak {D}_{1}\left( t_{{\mu }-2},\mathcal {H}^{{\mu }-2}\right) }{\nu t}\\ & \times \left[ \begin{array}{cc} ({\mu }-l+1)^{\varpi } & ({\mu }-l+2\varpi +3)\\ -({\mu }-l)^{\varpi } & ({\mu }-l+3\varpi +3) \end{array}\right] \\ & +\frac{(\nu t)^{\varpi }}{2\Gamma (\varpi +3)}\sum _{l=2}^{{\mu }}\frac{\mathfrak {D}_{1}\left( t_{{\mu }},\mathcal {H}(t_{{\mu }})\right) -2\mathfrak {D}_{1}\left( t_{{\mu }-1},\mathcal {H}(t_{{\mu }-1})\right) +\mathfrak {D}_{1}\left( t_{{\mu }-2},\mathcal {H}(t_{{\mu }-2})\right) }{2(\nu t)^{2}}\\ & \times \left[ ({\mu }-l+1)^{\varpi }\left( 2({\mu }-l)^{2}+(3\varpi +10)({\mu }-l)\right. \right. \\ & \left. +2\varpi ^{2}+9\varpi +10\right) -({\mu }-l)^{\varpi }\times \left( 2({\mu }-l)^{2}+(5\varpi +10)({\mu }-l)\right. \\ & \left. \left. +6\varpi ^{2}+18\varpi +12\right) \right] . \end{array}\right. } \end{aligned}$$
(25)
Likewise, for the remaining classes of Eq.(23), we get
$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {P}^{{\mu }+1} & =\mathcal {P}_{0}+\frac{(\nu t)^{\varpi }}{\Gamma (\varpi +1)}\sum _{l=2}^{{\mu }}\mathfrak {D}_{2}\left( t_{{\mu }-2},\mathcal {P}^{{\mu }-2}\right) \left[ ({\mu }-l+1)^{\varpi }-({\mu }-l)^{\varpi }\right] \\ & +\frac{(\nu t)^{\varpi }}{\Gamma (\varpi +2)}\sum _{l=2}^{{\mu }}\frac{\mathfrak {D}_{2}\left( t_{{\mu }-1},\mathcal {P}^{{\mu }-1}\right) -\mathfrak {D}_{2}\left( t_{{\mu }-2},\mathcal {P}^{{\mu }-2}\right) }{\nu t}\\ & \times \left[ \begin{array}{cc} ({\mu }-l+1)^{\varpi } & ({\mu }-l+2\varpi +3)\\ -({\mu }-l)^{\varpi } & ({\mu }-l+3\varpi +3) \end{array}\right] \\ & +\frac{(\nu t)^{\varpi }}{2\Gamma (\varpi +3)}\sum _{l=2}^{{\mu }}\frac{\mathfrak {D}_{2}\left( t_{{\mu }},\mathcal {P}(t_{{\mu }})\right) -2\mathfrak {D}_{2}\left( t_{{\mu }-1},\mathcal {P}(t_{{\mu }-1})\right) +\mathfrak {D}_{2}\left( t_{{\mu }-2},\mathcal {P}(t_{{\mu }-2})\right) }{2(\nu t)^{2}}\\ & \times \left[ ({\mu }-l+1)^{\varpi }\left( 2({\mu }-l)^{2}+(3\varpi +10)({\mu }-l)\right. \right. \\ & \left. +2\varpi ^{2}+9\varpi +10\right) -({\mu }-l)^{\varpi }\times \left( 2({\mu }-l)^{2}+(5\varpi +10)({\mu }-l)\right. \\ & \left. \left. +6\varpi ^{2}+18\varpi +12\right) \right] . \end{array}\right. } \end{aligned}$$
(26)
$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {I}^{{\mu }+1} & =\mathcal {I}_{0}+\frac{(\nu t)^{\varpi }}{\Gamma (\varpi +1)}\sum _{l=2}^{{\mu }}\mathfrak {D}_{3}\left( t_{{\mu }-2},\mathcal {I}^{{\mu }-2}\right) \left[ ({\mu }-l+1)^{\varpi }-({\mu }-l)^{\varpi }\right] \\ & +\frac{(\nu t)^{\varpi }}{\Gamma (\varpi +2)}\sum _{l=2}^{{\mu }}\frac{\mathfrak {D}_{3}\left( t_{{\mu }-1},\mathcal {I}^{{\mu }-1}\right) -\mathfrak {D}_{3}\left( t_{{\mu }-2},\mathcal {I}^{{\mu }-2}\right) }{\nu t}\\ & \times \left[ \begin{array}{cc} ({\mu }-l+1)^{\varpi } & ({\mu }-l+2\varpi +3)\\ -({\mu }-l)^{\varpi } & ({\mu }-l+3\varpi +3) \end{array}\right] \\ & +\frac{(\nu t)^{\varpi }}{2\Gamma (\varpi +3)}\sum _{l=2}^{{\mu }}\frac{\mathfrak {D}_{3}\left( t_{{\mu }},\mathcal {I}(t_{{\mu }})\right) -2\mathfrak {D}_{3}\left( t_{{\mu }-1},\mathcal {I}(t_{{\mu }-1})\right) +\mathfrak {D}_{3}\left( t_{{\mu }-2},\mathcal {I}(t_{{\mu }-2})\right) }{2(\nu t)^{2}}\\ & \times \left[ ({\mu }-l+1)^{\varpi }\left( 2({\mu }-l)^{2}+(3\varpi +10)({\mu }-l)\right. \right. \\ & \left. +2\varpi ^{2}+9\varpi +10\right) -({\mu }-l)^{\varpi }\times \left( 2({\mu }-l)^{2}+(5\varpi +10)({\mu }-l)\right. \\ & \left. \left. +6\varpi ^{2}+18\varpi +12\right) \right] . \end{array}\right. } \end{aligned}$$
(27)
$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {F}^{{\mu }+1} & =\mathcal {F}_{0}+\frac{(\nu t)^{\varpi }}{\Gamma (\varpi +1)}\sum _{l=2}^{{\mu }}\mathfrak {D}_{4}\left( t_{{\mu }-2},\mathcal {F}^{{\mu }-2}\right) \left[ ({\mu }-l+1)^{\varpi }-({\mu }-l)^{\varpi }\right] \\ & +\frac{(\nu t)^{\varpi }}{\Gamma (\varpi +2)}\sum _{l=2}^{{\mu }}\frac{\mathfrak {D}_{4}\left( t_{{\mu }-1},\mathcal {F}^{{\mu }-1}\right) -\mathfrak {D}_{4}\left( t_{{\mu }-2},\mathcal {F}^{{\mu }-2}\right) }{\nu t}\\ & \times \left[ \begin{array}{cc} ({\mu }-l+1)^{\varpi } & ({\mu }-l+2\varpi +3)\\ -({\mu }-l)^{\varpi } & ({\mu }-l+3\varpi +3) \end{array}\right] \\ & +\frac{(\nu t)^{\varpi }}{2\Gamma (\varpi +3)}\sum _{l=2}^{{\mu }}\frac{\mathfrak {D}_{4}\left( t_{{\mu }},\mathcal {F}(t_{{\mu }})\right) -2\mathfrak {D}_{4}\left( t_{{\mu }-1},\mathcal {F}(t_{{\mu }-1})\right) +\mathfrak {D}_{4}\left( t_{{\mu }-2},\mathcal {F}(t_{{\mu }-2})\right) }{2(\nu t)^{2}}\\ & \times \left[ ({\mu }-l+1)^{\varpi }\left( 2({\mu }-l)^{2}+(3\varpi +10)({\mu }-l)\right. \right. \\ & \left. +2\varpi ^{2}+9\varpi +10\right) -({\mu }-l)^{\varpi }\times \left( 2({\mu }-l)^{2}+(5\nu _{1}+10)({\mu }-l)\right. \\ & \left. \left. +6\varpi ^{2}+18\varpi +12\right) \right] . \end{array}\right. } \end{aligned}$$
(28)
$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {E}^{{\mu }+1} & =\mathcal {E}_{0}+\frac{(\nu t)^{\varpi }}{\Gamma (\varpi +1)}\sum _{l=2}^{{\mu }}\mathfrak {D}_{5}\left( t_{{\mu }-2},\mathcal {E}^{{\mu }-2}\right) \left[ ({\mu }-l+1)^{\varpi }-({\mu }-l)^{\varpi }\right] \\ & +\frac{(\nu t)^{\varpi }}{\Gamma (\varpi +2)}\sum _{l=2}^{{\mu }}\frac{\mathfrak {D}_{5}\left( t_{{\mu }-1},\mathcal {E}^{{\mu }-1}\right) -\mathfrak {D}_{5}\left( t_{{\mu }-2},\mathcal {E}^{{\mu }-2}\right) }{\nu t}\\ & \times \left[ \begin{array}{cc} ({\mu }-l+1)^{\varpi } & ({\mu }-l+2\varpi +3)\\ -({\mu }-l)^{\varpi } & ({\mu }-l+3\varpi +3) \end{array}\right] \\ & +\frac{(\nu t)^{\varpi }}{2\Gamma (\varpi +3)}\sum _{l=2}^{{\mu }}\frac{\mathfrak {D}_{5}\left( t_{{\mu }},\mathcal {E}(t_{{\mu }})\right) -2\mathfrak {D}_{5}\left( t_{{\mu }-1},\mathcal {E}(t_{{\mu }-1})\right) +\mathfrak {D}_{5}\left( t_{{\mu }-2},\mathcal {E}(t_{{\mu }-2})\right) }{2(\nu t)^{2}}\\ & \times \left[ ({\mu }-l+1)^{\varpi }\left( 2({\mu }-l)^{2}+(3\varpi +10)({\mu }-l)\right. \right. \\ & \left. +2\varpi ^{2}+9\varpi +10\right) -({\mu }-l)^{\varpi }\times \left( 2({\mu }-l)^{2}+(5\nu _{1}+10)({\mu }-l)\right. \\ & \left. \left. +6\varpi ^{2}+18\varpi +12\right) \right] . \end{array}\right. } \end{aligned}$$
(29)
Equations (25)–(28) are the solutions of the proposed model. To evaluate the accuracy of the derived scheme, we demonstrate that
$$\begin{aligned} {\left\{ \begin{array}{ll} \left| \mathcal {T}_{1}^{\varpi }\right| \le \frac{(\nu t)^{\varpi +2}}{36\Gamma (\varpi )}\max _{t\in [0,{\mu }]}\left| \mathfrak {D}_{1}^{”}(\nu ,\mathcal {H})\right| _{\nu =\epsilon _{\nu }}({\mu }+1)^{\varpi }({\mu }^{2}+2{\mu }),\\ \left| \mathcal {T}_{2}^{\varpi }\right| \le \frac{(\nu t)^{\varpi +2}}{36\Gamma (\varpi )}\max _{t\in [0,{\mu }]}\left| \mathfrak {D}_{2}^{”}(\nu ,\mathcal {P})\right| _{\nu =\epsilon _{\nu }}({\mu }+1)^{\varpi }({\mu }^{2}+2{\mu }),\\ \left| \mathcal {T}_{3}^{\varpi }\right| \le \frac{(\nu t)^{\varpi +2}}{36\Gamma (\varpi )}\max _{t\in [0,{\mu }]}\left| \mathfrak {D}_{3}^{”}(\nu ,\mathcal {I})\right| _{\nu =\epsilon _{\nu }}({\mu }+1)^{\varpi }({\mu }^{2}+2{\mu }),\\ \left| \mathcal {T}_{4}^{\varpi }\right| \le \frac{(\nu t)^{\varpi +2}}{36\Gamma (\varpi )}\max _{t\in [0,{\mu }]}\left| \mathfrak {D}_{4}^{”}(\nu ,\mathcal {F})\right| _{\nu =\epsilon _{\nu }}({\mu }+1)^{\varpi }({\mu }^{2}+2{\mu }),\\ \left| \mathcal {T}_{5}^{\varpi }\right| \le \frac{(\nu t)^{\varpi +2}}{36\Gamma (\varpi )}\max _{t\in [0,{\mu }]}\left| \mathfrak {D}_{5}^{”}(\nu ,\mathcal {E})\right| _{\nu =\epsilon _{\nu }}({\mu }+1)^{\varpi }({\mu }^{2}+2{\mu }), \end{array}\right. } \end{aligned}$$
(30)
Consider the first equation of considered model as:
$$\begin{aligned} {\left\{ \begin{array}{ll} ^{\mathfrak {C}}\mathfrak {D}_{0,t}^{\mathbb {\beta }_{1}}\mathcal {H}(t)=\mathfrak {D}_{1}(t,\mathcal {H}),\\ \mathcal {H}(0)=\mathcal {H}_{0}. \end{array}\right. } \end{aligned}$$
(31)
After some steps, one can get
$$\begin{aligned} \mathcal {H}(t_{{\mu }+1})=\mathcal {H}_{0}+\frac{1}{\Gamma (\varpi )}\sum _{l=0}^{{\mu }} \int _{t_{l}}^{t_{l+1}}(t_{{\mu }+1}-\nu )\left[ \mathfrak {D}_{1}(\nu ,\mathcal {H}) -\mathfrak {R}_{1}(\nu )\right] d\nu , \end{aligned}$$
(32)
The error term \(\mathcal {T}_{1}^{\varpi }\)can be written as
$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {T}_{1}^{\varpi } & =\frac{1}{\Gamma (\varpi )}\sum _{l=0}^{{\mu }}\int _{t_{l}}^{t_{l+1}} (t_{{\mu }+1}-\nu )^{\varpi -1}\\ & \times \left[ \frac{\mathfrak {D}_{1}^{”}}{2!}(\nu ,\mathcal {H})|_{\nu =\epsilon _{l}}(\nu -t_{l-1})(\nu -t_{l})\right] d\nu . \end{array}\right. } \end{aligned}$$
(33)
Now
$$\begin{aligned} \left| \mathcal {T}_{1}^{\varpi }\right|= & \frac{1}{2!\Gamma (\varpi )}\left| \sum _{l=0}^{\mathfrak {u}}\int _{t_{l}}^{t_{l+1}}(t_{{\mu }+1}-\nu )^{\varpi -1}\left[ \begin{array}{c} \mathfrak {D}_{1}^{”}(\nu ,\mathcal {H})|_{\nu =\epsilon _{l}}\\ \times (\nu -t_{l-1})(\nu -t_{l}) \end{array}\right] d\nu \right| \nonumber \\\le & \frac{1}{2!\Gamma (\varpi )}\sum _{l=0}^{\mathfrak {u}}\int _{t_{l}}^{t_{l+1}}(t_{{\mu }+1}-\nu )^{\varpi -1}\left| \mathfrak {D}_{1}^{”}(\nu ,\mathcal {H})|_{\nu =\epsilon _{l}}(\nu -t_{l-1})(\nu -t_{l})\right| d\nu \nonumber \\ \left| \mathcal {T}_{1}^{\varpi }\right|\le & \frac{1}{2!\Gamma (\varpi )}\max _{\nu \in [0,t_{\mathfrak {u}}]}\left| \mathfrak {D}_{1}^{”}(\nu ,\mathcal {H})\right| _{\nu =\epsilon _{\nu }}\sum _{l=0}^{\mathfrak {u}}\int _{t_{l}}^{t_{l+1}}(t_{{\mu }+1}-\nu )^{\varpi -1}\nonumber \\ & \times (\nu -t_{l-1})(\nu -t_{l})d\nu , \end{aligned}$$
(34)
where \(\epsilon _{l}\) represent a constant which would be derive through using Taylor series approximation. Now, we assess the integral:
$$\begin{aligned} \int _{t_{l}}^{t_{l+1}}(t_{{\mu }+1}-\nu )^{\varpi -1}(\nu -t_{l-1})(\nu -t_{l})d\nu= & \int _{t_{l}}^{t_{l+1}}(t_{{\mu }+1}-\nu )^{\varpi -1}\nu ^{2}d\nu \\ & -\int _{t_{l}}^{t_{l+1}}(t_{{\mu }+1}-\nu )^{\varpi -1}(t_{l-1}+t_{l})d\nu \\ & +\int _{t_{l}}^{t_{l+1}}(t_{{\mu }+1}-\nu )^{\varpi -1}t_{l-1}t_{l}d\nu . \end{aligned}$$
Now
$$\begin{aligned} \int _{t_{l}}^{t_{l+1}}(t_{{\mu }+1}-\nu )^{\varpi -1}\nu ^{2}d\nu\le & \int _{0}^{t_{l+1}}(t_{{\mu }+1}-\nu )^{\varpi -1}\nu ^{2}d\nu , \end{aligned}$$
therefore
$$\begin{aligned} \int _{t_{l}}^{t_{l+1}}(t_{{\mu }+1}-\nu )^{\varpi -1}\nu ^{2}d\nu\le & \int _{0}^{t_{l+1}}(t_{{\mu }+1}-\nu )^{\varpi -1}t_{{\mu }+1}^{2}d\nu , \end{aligned}$$
using \(t_{{\mu }+1}>\nu .\) We have
$$\begin{aligned} \int _{t_{l}}^{t_{l+1}}(t_{{\mu }+1}-\nu )^{\varpi -1}\nu ^{2}d\nu\le & t_{{\mu }+1}^{2}\left[ \frac{(t_{{\mu }+1})^{\varpi }}{\varpi }-\frac{(t_{{\mu }+1}-t_{l+1})^{\varpi }}{\varpi }\right] . \end{aligned}$$
Thus
$$\begin{aligned} \int _{t_{l}}^{t_{l+1}}(t_{{\mu }+1}-\nu )^{\varpi -1}\nu ^{2}d\nu\le & (\nu t)^{\varpi +2}\left[ \frac{({\mu }+1)^{\varpi }}{\varpi }-\frac{({\mu }-l)^{\varpi }}{\varpi }\right] . \end{aligned}$$
Then
$$\begin{aligned} \int _{t_{l}}^{t_{l+1}}(t_{{\mu }+1}-\nu )^{\varpi -1}\nu d\nu\le & (\nu t)^{\varpi +1}\left[ \frac{({\mu }+1)^{\varpi }}{\varpi }-\frac{({\mu }-l)^{\varpi }}{\varpi }\right] , \end{aligned}$$
and
$$\begin{aligned} \int _{t_{l}}^{t_{l+1}}(t_{{\mu }+1}-\nu )^{\varpi -1}d\nu\le & (\nu t)^{\varpi }\left[ \frac{({\mu }+1)^{\varpi }}{\varpi }-\frac{({\mu }-l)^{\varpi }}{\varpi }\right] . \end{aligned}$$
Using above equations, we have
$$\begin{aligned}&\sum _{l=0}^{\mathfrak {u}}\int _{t_{l}}^{t_{l+1}}(t_{{\mu }+1}-\nu )^{\varpi -1}(\nu -t_{l-1})(\nu -t_{l})d\nu \le \sum _{l=0}^{\mathfrak {u}}\left\{ \begin{array}{c} (\nu t)^{\varpi +2}\left[ \frac{({\mu }+1)^{\varpi }}{\varpi }-\frac{({\mu }-l)^{\varpi }}{\varpi }\right] \\ -(2l-1)(\nu t)^{\varpi +2}\\ \times \left[ \frac{({\mu }+1)^{\varpi }}{\varpi }-\frac{({\mu }-l)^{\varpi }}{\varpi }\right] \\ +l(l-1)(\nu t)^{\varpi +2}\\ \times \left[ \frac{({\mu }+1)^{\varpi }}{\varpi }-\frac{({\mu }-l)^{\varpi }}{\varpi }\right] \end{array}\right\} \\ \le&\sum _{l=0}^{\mathfrak {u}}\left\{ \begin{array}{c} (\nu t)^{\varpi +2}\left[ \frac{({\mu }+1)^{\varpi }}{\varpi }-\frac{({\mu }-l)^{\varpi }}{\varpi }\right] \\ \times (1+2l-1+l^{2}-l) \end{array}\right\} \le (\nu t)^{\varpi +2}\frac{({\mu }+1)^{\varpi }}{\varpi }\left[ \frac{({\mu }^{2}+{\mu })}{2}+\frac{({\mu }^{2}+{\mu })(2{\mu }+1)}{6}\right] \\ \le&(\nu t)^{\varpi +2}\frac{({\mu }+1)^{\varpi }}{12\varpi }\left( {\mu }+\frac{2{\mu }^{2}+{\mu }}{3}\right) \le (\nu t)^{\varpi +2}\frac{({\mu }+1)^{\varpi }}{18\varpi }\left( {\mu }^{2}+2{\mu }\right) . \end{aligned}$$
Thus, Eq.(34) becomes
$$\begin{aligned} \left| \mathcal {T}_{1}^{\varpi }\right| \le \frac{(\nu t)^{\varpi +2}}{36\Gamma (\varpi )}\max _{\nu \in [0,t_{\mathfrak {u}}]}\left| \mathfrak {D}_{1}^{”}(\nu ,\mathcal {H})\right| _{\nu =\epsilon _{\nu }}({\mu }+1)^{\varpi }\left( {\mu }^{2}+2{\mu }\right) . \end{aligned}$$
(35)
In the same way, we can derive the last three inequalities of Eq.(34).
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