In this section, we compute an error bound for the approximate solution obtained in Sect. “The proposed method”. Let \(\Pi _{N}f\) be the operator that maps \(L^2(\Omega )\) into \(P_N(\Omega )\), defined as follows:$$(\Pi _{N}f-f,v)=0, \forall v\in P_N(\Omega ),\Omega =[0,\tau ],$$where$$P_N(\Omega )=span\{\hat{L}_{\tau ,0},\hat{L}_{\tau ,1},\dots ,\hat{L}_{\tau ,N}\},\, N\in \mathbb {N}.$$in other words,$$(\Pi _Nf)(t)=\sum _{i=0}^{N}a_i\hat{L}_{\tau ,i}(t),$$\(\Pi _{N}f\) is the best approximation of f out of \(\Pi _{N}f\) as mentioned in (41). First, the following theorems need to be proven.
Theorem 1
Suppose\(\Omega =[0,\tau ]\) and\(f \in H^n(\Omega )\) as defined in Definition3. There exists a constant\(\hat{C}_1\) such that:$$\begin{aligned} \left\| \Pi _N f-f\right\| _{L^2(\Omega )} \le \hat{C}_1 N^{-n}\left\| f\right\| _{H^n(\Omega )}, \end{aligned}$$
(34)
Proof
42\(\square\)
Theorem 2
For any\(f\in B^n (\Omega )\) as introduced in Definition3, we have$$\begin{aligned} \begin{aligned} \left\| D_{t}^{n}\left( \Pi _{N} f-f\right) \right\|&\le \hat{C_2} N^{(p-n)}\left\| D_{t}^{n} f\right\| , \\&\le \hat{C_2} N^{(p-n)}\Vert f\Vert, \end{aligned} \end{aligned}$$
(35)
where\(\hat{C_2}\) is constant and\(0 \le p\le n \le N+1,\, n\in \mathbb {N}.\)
In Hilbert space, the following relationship holds:$$\begin{aligned} \left\| D_{t}^{n}\left( \Pi _{N} f-f\right) \right\| \le \hat{C_2}N^{(p-n)}\Vert f\Vert _{H^{n}}. \end{aligned}$$
(36)
Proof
41\(\square\)
Now, we compute an error bound in the estimation of \(D_{t}^\zeta f\) as follows.
Theorem 3
Let\(f\in L^2(\Omega ),p<r\le N+1\), \(p-1<\zeta \le p=\lceil \zeta \rceil\), and\(r\in \mathbb {N}\)then,$$\begin{aligned} \left\| D_t^\zeta \left( \Pi _{M,N} f\right) -D_{t}^\zeta f\right\| _{L^2(\Omega )} \le \frac{\hat{C_\zeta }N^{(p-r)}}{\Gamma (p-\zeta +1)}\left\| f\right\| _{H^{r}(\Omega )}, \end{aligned}$$
(37)
where\(\hat{C_\zeta }\) is constant. Moreover,\(H^{r}\) is defined according to Definition3.
Proof
Using Eq. (5) and the following equation from43:$$\begin{aligned} \Vert f*g\Vert _{L^2(\Omega )} \le \Vert f\Vert _1\Vert g\Vert _{L^2(\Omega )}. \end{aligned}$$
(38)
Consequently, we derive$$\begin{aligned} \begin{aligned} \left\| D_{t}^\zeta \left( \Pi _{M,N} f\right) -D_{t}^\zeta f\right\| _{L^2(\Omega )}^2&=\left\| I^{p-\zeta }\left( D_{t}^{p}\left( \Pi _{M,N} f(x, t)\right) -D_{t}^{p} f(x, t)\right) \right\| _{L^2(\Omega )}^2, \\&=\parallel \frac{1}{t^{1+\zeta -p}\Gamma (p-\zeta )}*\left( D_{t}^{p}\left( \Pi _{M,N}f(x,t) \right) -D_{t}^{p}f(x,t) \right) \parallel _{L^{2}(\Omega )}^{2},\\&\le \Vert \frac{1}{t^{1+\zeta -p}\Gamma (p-\zeta )} \Vert _{1}^2\left\| D_{t}^{p}\left( \Pi _{M,N} f(x,t)\right) -D_{t}^{p} f(x,t)\right\| _{L^2(\Omega )}^2.\\ \end{aligned} \end{aligned}$$
(39)
Based on previous Theorem, we have:$$\begin{aligned} \begin{aligned} \qquad&\le \left( \frac{\hat{C_\zeta }N^{(p-r)}}{\Gamma (p-\zeta +1)}\left\| f\right\| _{H^{r}(\Omega )}\right) ^2. \end{aligned} \end{aligned}$$
(40)
Therefore,$$\begin{aligned} \left\| D_t^\zeta \left( \Pi _{M,N} f\right) -D_{t}^\zeta f\right\| _{L^2(\Omega )} \le \frac{\hat{C_\zeta }N^{(p-r)}}{\Gamma (p-\zeta +1)}\left\| f\right\| _{H^{r}(\Omega )}. \end{aligned}$$\(\square\)
Error boundWe are now prepared to determine the error bound of the proposed technique. To accomplish this, let’s rephrase the model described by Eqs. (2) and (3) as follows:$$\begin{aligned} D^\zeta U(t) = g(t,U(t)),t \in [0,\tau ], 0 < \zeta \le 1, \end{aligned}$$
(41)
and$$\begin{aligned} U(0)=U_0, \end{aligned}$$
(42)
where \(U(t) = [H(t),I(t),V(t),L(t)]\) is exact solution and$$\begin{aligned} \begin{aligned}&g(t,U(t))=[g_1 (t,U(t)),g_2 (t,U(t)),g_3 (t,U(t)),g_4 (t,U(t))],\\&U_0=[\nu _1,\nu _2,\nu _3,\nu _4]. \end{aligned} \end{aligned}$$
(43)
Suppose \(U_N(t)\) represents the approximate solution derived from the proposed method for the model (2)-(3), and N is the number of collocation points. Additionally, \(e_N (t)\) denotes the error between U(t) and \(U_N(t)\), expressed as \(e_N (t) =U(t)-U_N (t)\). By replacing \(U_N(t)\) into Eq. (41), we obtain:$$\begin{aligned} Res(t,U_N(t))=D^\zeta U_N (t)- g(t,U_N (t)), \end{aligned}$$
(44)
and$$\begin{aligned} U_N(0)=U_0. \end{aligned}$$
(45)
We have now reached a stage where we are able to compute the upper bound of the residual function. To continue, we present the following theorem.
Theorem 4
Let g( t, U (t)) as defined (43) satisfies the Lipschitz condition, and let\(U(t)\in B^r(\Omega )\). There are constants\(\hat{C_\zeta }\)and C such that:$$\begin{aligned} \begin{aligned} \Vert Res(t,U_N(t))\Vert _{L^2(\Omega )}&\le \frac{\hat{C_\zeta }N^{(p-r)}}{\Gamma (p-\zeta +1)}\left\| f\right\| _{H^{r}(\Omega )}+\\&C N^{-r}\Vert f\Vert _{H^{r}(\Omega )}. \end{aligned} \end{aligned}$$
(46)
where \(n_\zeta <r\le N+1\), \(n_\zeta -1<\zeta \le n_\zeta =\lceil \zeta \rceil\) and \(r\in \mathbb {N}\).
Proof
Since g(t, U(t)) satisfies the Lipschitz condition, we obtain:$$\begin{aligned} \Vert g(t,U(t))-g(t,U_N(t))\Vert _{L^2(\Omega )} \le B\Vert U(t)-U_N(t)\Vert _{L^2(\Omega )}, \end{aligned}$$
(47)
where B is constant. If we subtract Eq. (44) from Eq. (41), then we get.$$\begin{aligned} Res(t,U_N(t))=g(t,U(t))- g(t,U_N(t))+D^\zeta e_N (t), \end{aligned}$$
(48)
with the initial condition$$\begin{aligned} e_{N}(0)=0. \end{aligned}$$
(49)
By applying the norm to both sides of Eq. (48) and using Eq. (47), we obtain:$$\begin{aligned} \begin{aligned} \Vert Res(t,U_N(t))\Vert _{L^2(\Omega )}&=\Vert g(t,U(t))- g(t,U_N (t))+D^\zeta e_N (t) \Vert _{L^2(\Omega )}\\&\le \Vert g(t,U(t))- g(t,U_N (t))\Vert _{L^2(\Omega )}+\Vert D^\zeta e_N (t)\Vert _{L^2(\Omega )}\\&\le B\Vert U(t)-U_N(t)\Vert _{L^2(\Omega )}+|D^\zeta e_N (t)\Vert _{L^2(\Omega )} \end{aligned} \end{aligned}$$
(50)
Based on the results of Theorems 1 and 3, there are constants \(C_{\zeta }\) and \(\lambda\) such that$$\begin{aligned} \begin{aligned} \Vert Res(t,U_N(t))\Vert _{L^2(\Omega )}&\le \frac{\hat{C_\zeta }N^{(p-r)}}{\Gamma (p-\zeta +1)}\left\| f\right\| _{H^{r}(\Omega )}+\\&B\lambda N^{-r}\Vert f\Vert _{H^{r}(\Omega )}. \end{aligned} \end{aligned}$$
(51)
where \(p<r\le N+1\), \(p-1<\zeta \le p=\lceil \zeta \rceil\), and \(r\in \mathbb {N}\). Assuming \(C=B\lambda\), the theorem is proved. \(\square\)
Based on the above theorem, we conclude that \(\left\| Res(t,U_N(t))\right\| \rightarrow 0\) as \(N \rightarrow \infty\). Therefore, \(U_{N}(t)\) approaches U(t).