In this section, we present the definitions of VO RL-integral and Caputo derivative. Then, some basic properties of the SFKCPs are given which are used later.

### 2.1 VO fractional calculus

### Definition 2.1

(See [34])

Let \(p-1<\upsilon (t)\leq p\) and \(z\in C[0,1] \). The RL-integral and Caputo derivative of VO \(\upsilon (t)\) are, respectively, defined by

$$\begin{aligned}& {{}^{R L} I}_{t}^{\upsilon (t)} z(t)= \frac{1}{\Gamma (\upsilon (t))} \int _{0}^{t} (t-\tau )^{\upsilon (t)-1} z(\tau ) \,d \tau , \\& {{}^{C}D^{\upsilon (t)}_{t} } z(t)= \frac{1}{\Gamma (p-\upsilon (t))} \int _{0}^{t} (t-\tau )^{p-\upsilon (t)-1} z^{(p)}(\tau ) \,d\tau . \end{aligned}$$

Two main properties of these operators are given as follows:

$$\begin{aligned}& {{}^{C}D^{\upsilon (t)}_{t}} t^{\zeta }= \textstyle\begin{cases} \frac{\Gamma (\zeta +1)}{\Gamma (\zeta -\upsilon (t)+1)}t^{\zeta - \upsilon (t)},& \zeta \in \mathbb{N} \text{ and } \zeta \geq \lceil \upsilon (t)\rceil \text{ or } \zeta \notin \mathbb{N} \text{ and } \zeta > \lfloor \upsilon (t)\rfloor , \\ 0,& \zeta \in \mathbb{N}\cup \lbrace 0\rbrace \text{ and } \zeta < \lceil \upsilon (t)\rceil , \end{cases}\displaystyle \\& {{}^{C}D^{\upsilon (t)}_{t}}z(t)={{}^{R L} I^{p-\upsilon (t)}_{t}} \bigl(z^{(p)}(t) \bigr). \end{aligned}$$

(3)

### 2.2 Definition of the SFKCPs and function approximation

The SFKCPs on the interval \([0,1] \) are defined by [28, 35]

$$\begin{aligned} \mathcal{C}_{m}^{*}(t)=\mathcal{C}_{m}(2t-1), \quad m=0,1,2,\ldots , \end{aligned}$$

where \(\mathcal{C}_{m}(t) \) is the fifth-kind Chebyshev polynomial defined on \([-1,1] \) as follows:

$$\begin{aligned} \mathcal{C}_{m}(t)=\frac{1}{\sqrt{\delta _{m}}}\overline{ \mathcal{B}}^{(-3,2,-1,1)}_{m}(t), \end{aligned}$$

where

$$ \delta _{m}= \textstyle\begin{cases} \frac{\pi }{2^{2m+1}},& m \text{ is even}, \\ \frac{\pi (m+2)}{m 2^{2m+1}},& m \text{ is odd}, \end{cases} $$

and

$$\begin{aligned}& \overline{\mathcal{B}}_{m}^{(v,w,r,s)}(t)= \Biggl( \prod_{k=0}^{\lfloor \frac{m}{2}\rfloor -1} \frac{(2k+(-1)^{m+1}+2)s+w}{(2k+(-1)^{m+1}+2\lfloor \frac{m}{2}\rfloor )r+v} \Biggr) \mathcal{B}_{m}^{(v,w,r,s)}(t), \end{aligned}$$

with

$$\begin{aligned} \mathcal{B}_{m}^{(v,w,r,s)}(t)= \sum_{j=0}^{\lfloor \frac{m}{2}\rfloor } \Biggl(\binom{\lfloor \frac{m}{2}\rfloor }{j} \Biggl( \prod_{k=0}^{\lfloor \frac{m}{2}\rfloor -j-1} \frac{(2k+(-1)^{m+1}+2\lfloor \frac{m}{2}\rfloor )r+v}{(2k+(-1)^{m+1}+2)s+w} \Biggr) t^{m-2j} \Biggr). \end{aligned}$$

Furthermore, the analytic form of the SFKCPs of degree *m* is given by

$$\begin{aligned}& \mathcal{C}_{m}^{*}(t)=\sum _{l=0}^{m}\varsigma _{l,m} t^{l}, \end{aligned}$$

where

$$\begin{aligned} \varsigma _{l,m}=\frac{2^{2l+\frac{3}{2}}}{\sqrt{\pi }(2l)!} \textstyle\begin{cases} 2 \sum_{k=\lfloor \frac{l+1}{2}\rfloor }^{\frac{m}{2}} \frac{(-1)^{\frac{m}{2}+k-l} k \varepsilon _{k} (2k+l-1)!}{(2k-l)!},& m \text{ is even}, \\ \frac{1}{\sqrt{m(m+2)}} \sum_{k=\lfloor \frac{l}{2} \rfloor }^{\frac{m-1}{2}} \frac{(-1)^{\frac{m+1}{2}+k-l}(2k+1)^{2} (2k+l)!}{(2k-l+1)!},& m \text{ is odd}, \end{cases}\displaystyle \end{aligned}$$

(4)

and

$$\begin{aligned} \varepsilon _{k}= \textstyle\begin{cases} \frac{1}{2},& k=0, \\ 1,& k>0. \end{cases}\displaystyle \end{aligned}$$

Also, the orthogonality condition is given for these polynomials as follows:

$$\begin{aligned} \int _{0}^{1} w^{*}(t) \mathcal{C}_{r}^{*}(t) \mathcal{C}_{s}^{*}(t) \,dt= \textstyle\begin{cases} 1,&r= s, \\ 0,&r\neq s, \end{cases}\displaystyle \end{aligned}$$

where \(w^{*}(t)= \frac{(2t-1)^{2}}{\sqrt{t-t^{2}}} \).

### Lemma 2.1

(See [35])

*The SFKCPs satisfy the following boundedness property on* \([0,1] \) *for all* \(s\geq 0 \):

$$\begin{aligned} \bigl\vert \mathcal{C}_{s}^{*}(t) \bigr\vert < \sqrt{ \frac{2}{\pi }}(s+2),\quad \forall t\in [0,1]. \end{aligned}$$

Suppose that \(r_{1}, r_{2}\in L^{2}_{w^{*}}(0,1)\). Then the inner product and norm in \(L^{2}_{w^{*}}(0,1)\) are, respectively, defined by

$$\begin{aligned}& \langle r_{1},r_{2}\rangle _{w^{*}}= \int _{0}^{1} w^{*}(t)r_{1}(t)r_{2}(t) \,dt, \\& \Vert r_{1} \Vert _{2}=\sqrt{\langle r_{1},r_{1}\rangle _{w^{*}}}. \end{aligned}$$

Any arbitrary function \(z(t)\in L_{w^{*}}^{2}(0,1)\) can be expanded by the SFKCPs as

$$\begin{aligned} z(t)=\sum_{i=0}^{\infty }z_{i} \mathcal{C}_{i}^{*}(t). \end{aligned}$$

(5)

By considering only the first \(M+1\) terms in (5), we can approximate \(z(t)\) as

$$\begin{aligned} z(t)\simeq z_{M}(t)=\sum _{i=0}^{M}z_{i}\mathcal{C}_{i}^{*}(t)=Z^{T} \varphi (t), \end{aligned}$$

where

$$\begin{aligned} \varphi (t)=\bigl[\mathcal{C}_{0}^{*}(t), \mathcal{C}_{1}^{*}(t),\ldots , \mathcal{C}_{M}^{*}(t) \bigr]^{T}, \end{aligned}$$

and in the vector \(Z=[z_{0},z_{1},\ldots ,z_{M}]^{T}\), the entries \(z_{i}\), \(i=0,1,\ldots ,M \), are given by

$$\begin{aligned} z_{i}= \int _{0}^{1}w^{*}(t) z(t) \mathcal{C}_{i}^{*}(t)\,dt. \end{aligned}$$

(6)

In a similar way, a bivariate function \(f(t,\tau )\in L^{2}_{w^{*}} ((0,1)\times (0,1) ) \) can be approximated based on the SFKCPs as

$$\begin{aligned} f(t,\tau )\simeq \sum_{i=0}^{M}\sum _{j=0}^{M}f_{ij} \mathcal{C}^{*}_{i}(t) \mathcal{C}^{*}_{j}( \tau )=\varphi ^{T}(t) F \varphi (\tau ), \end{aligned}$$

where *F* is an \((M+1)\times (M+1)\) matrix given by

$$\begin{aligned} F= \bigl\langle \varphi (t),\bigl\langle f(t,\tau ),\varphi (\tau ) \bigr\rangle _{w^{*}} \bigr\rangle _{w^{*}}. \end{aligned}$$

We can consider the vector \(\varphi (t)\) in a matrix form as

$$\begin{aligned} \varphi (t)=AT_{M}(t), \end{aligned}$$

(7)

where \(A=[a_{i,j}]\), \(i,j=0,1,\ldots ,M \), with

$$\begin{aligned} a_{i,j}= \textstyle\begin{cases} \varsigma _{i,j}, &i\geq j, \\ 0,&i< j, \end{cases}\displaystyle \end{aligned}$$

\(\varsigma _{i,j} \) are given by (4), and

$$\begin{aligned} T_{M}(t)=\bigl[1,t,\ldots ,t^{M}\bigr]^{T}. \end{aligned}$$

### Theorem 2.1

(See [35])

*Suppose that* \(z(t)\in L^{2}_{w^{*}}(0,1) \) *with* \(\vert z^{(3)}(t)\vert \leq \theta \). *Let* \(\sum_{i=0}^{\infty }z_{i}\mathcal{C}_{i}^{*}(t)\) *be its expansion using the SFKCPs*. *Then*, *for* \(i>3\), *the coefficient* \(z_{i}\) *is bounded as*

$$\begin{aligned} \vert z_{i} \vert < \frac{\sqrt{2\pi } \theta }{2 i^{3}}. \end{aligned}$$

### Lemma 2.2

*Consider the basis vector* \(\varphi (t)\) *defined by* (7). *By applying the first*-*order derivative on this vector we get*

$$\begin{aligned} \frac{d}{dt}\varphi (t)= D\varphi (t), \end{aligned}$$

*where* *D* *is the operational matrix of derivative based on the SFKCPs given by*

$$\begin{aligned} D=A \begin{bmatrix} 0&0&0&\cdots &0&0 \\ 1&0&0&\cdots &0&0 \\ 0&2&0&\cdots &0&0 \\ \vdots &\vdots &\vdots &\vdots &\vdots &\vdots \\ 0&0&0&\cdots &M&0 \end{bmatrix} A^{-1}. \end{aligned}$$

*Also*, *for* \(m\geq 2 \), *we can write*

$$\begin{aligned} \frac{d^{m}}{dt^{m}}\varphi (t)=D^{m} \varphi (t). \end{aligned}$$

(8)

### Proof

It can be easily proved in a similar way as that of the corresponding theorem in [36]. □

### Lemma 2.3

*For the vector* \(\varphi (t) \) *given by* (7), *the dual operational matrix* *Q* *is given by*

$$\begin{aligned} \int _{0}^{1}\varphi (\tau )\varphi ^{T}( \tau )\,d\tau =A \biggl( \int _{0}^{1}T_{M}( \tau )T_{M}^{T}(\tau )\,d\tau \biggr)A^{T}=Q, \end{aligned}$$

(9)

*where* \(Q=AHA^{T}\) *with the well*-*known Hilbert matrix* *H*.

### Proof

The proof process is similar to that given in [36]. □

### Lemma 2.4

*The integral of the vector* \(\varphi (t) \) *given by* (7) *can be approximated as*

$$\begin{aligned} \int _{0}^{t}\varphi (\tau )\,d\tau \simeq P\varphi (t), \end{aligned}$$

(10)

*where* *P* *is called the operational matrix of integration for the SFKCPs*.

### Proof

Using (7), we write

$$\begin{aligned} \int _{0}^{t}\varphi (\tau )\,d\tau =A \int _{0}^{t}T_{M}(\tau )\,d\tau =A B T^{*}(t), \end{aligned}$$

where \(B=[b_{i,j}]\), \(i,j=0,1,\ldots ,M \), is an \((M+1)\times (M+1) \) matrix with elements

$$\begin{aligned} b_{i,j}= \textstyle\begin{cases} \frac{1}{i+1},&i=j, \\ 0,&i\neq j, \end{cases}\displaystyle \end{aligned}$$

and

$$\begin{aligned} T^{*}(t)= \begin{bmatrix} t,t^{2},\ldots ,t^{M+1} \end{bmatrix} ^{T}. \end{aligned}$$

Now, by approximating \(t^{k}\), \(k=1,2,\ldots ,M+1 \), in terms of the SFKCPs using (7), we have

$$\begin{aligned} \textstyle\begin{cases} t^{k}=A^{-1}_{k+1}\varphi (t),&k=1,2,\ldots ,M, \\ t^{M+1}=\mathfrak{L}^{T}\varphi (t), \end{cases}\displaystyle \end{aligned}$$

where \(A^{-1}_{i}\), \(i=2,3,\ldots ,M+1\), is the *i*th row of the matrix \(A^{-1}\), and \(\mathfrak{L}=\langle t^{M+1}, \varphi (t)\rangle _{w^{*}} \). Then, we get

$$\begin{aligned} T^{*}(t)=E\varphi (t), \end{aligned}$$

where \(E= [ A_{2}^{-1},A_{3}^{-1},\ldots ,A_{M+1}^{-1},\mathfrak{L}^{T} ] ^{T} \). Therefore by taking \(P=ABE \), we complete the proof. □

### Lemma 2.5

*Suppose* \(Z= [ z_{0},z_{1},\ldots , z_{M} ] ^{T}\). *Then* *Ẑ* *is the operational matrix of product whenever*

$$\begin{aligned} \varphi (t)\varphi ^{T}(t)Z\simeq \widehat{Z}\varphi (t). \end{aligned}$$

(11)

### Proof

According to (7) and expanding the function \(\mathcal{C}_{i}^{*}(t)\mathcal{C}_{j}^{*}(t)\), \(i,j=0,1,\ldots ,M \), we have

$$\begin{aligned} \mathcal{C}_{i}^{*}(t)\mathcal{C}_{j}^{*}(t) \simeq \sum_{m=0}^{i+j}c_{m} \mathcal{C}_{m}^{*}(t), \end{aligned}$$

where \(c_{m} \), \(m=0,1,\ldots ,i+j\), can be computed as

$$\begin{aligned} c_{m}= \sum_{k=0}^{i} \sum _{l=0}^{j} \sum _{s=0}^{m}\varsigma _{i,k} \varsigma _{j,l} \varsigma _{m,s} \int _{0}^{1}w^{*}(t)t^{k+l+s} \,dt=\Delta _{i,j,m}, \end{aligned}$$

with

$$\begin{aligned} \Delta _{i,j,m}= \sum_{k=0}^{i} \sum_{l=0}^{j} \sum _{s=0}^{m} \frac{\sqrt{\pi } (3+k^{2}+s(3+s)+k(3+2s) )\Gamma (\frac{3}{2}+k+s)}{\Gamma (4+k+s)} \varsigma _{i,k} \varsigma _{j,l} \varsigma _{m,s}. \end{aligned}$$

By considering \(Z= [ z_{0},z_{1},\ldots ,z_{M}] \) and (6), we have

$$\begin{aligned} \varphi (t)\varphi ^{T}(t) Z\simeq \widehat{Z}\varphi (t), \end{aligned}$$

where the elements of \(\widehat{Z}=[\widehat{z_{i,j}}]\), \(i,j=0,1,\ldots ,M \), are given by

$$\begin{aligned} \widehat{z_{i,j}}=\sum_{m=0}^{M} \Delta _{i,j,m} z_{m}. \end{aligned}$$

□

### Theorem 2.2

*Let* \(\varphi (t) \) *be the SFKCPs vector given in* (7), *and let* \(p-1<\upsilon (t)\leq p \). *Then*

$$\begin{aligned} {{}^{C}D^{\upsilon (t)}_{t} }\varphi (t)=\Upsilon ^{\upsilon (t)} \varphi (t), \end{aligned}$$

(12)

*where* \(\Upsilon ^{\upsilon (t)}= A\Psi ^{\upsilon (t)} A^{-1}\) *with*

$$\begin{aligned} \Psi ^{\upsilon (t)}=\bigl[\rho _{t}^{i,j} \bigr],\quad i,j=0,1,\ldots ,M, \end{aligned}$$

(13)

*and*

$$\begin{aligned} \rho _{t}^{i,j}= \textstyle\begin{cases} \frac{\Gamma (i+1)}{\Gamma (i+1-\upsilon (t))}t^{-\upsilon (t)},&i=j \& i\geq p, \\ 0,&\textit{otherwise}. \end{cases}\displaystyle \end{aligned}$$

### Proof

By employing \({{}^{C}D^{\upsilon (t)}_{t} } \) to both sides of (7), we get

$$\begin{aligned} {{}^{C}D^{\upsilon (t)}_{t} }\varphi (t)={{}^{C}D^{\upsilon (t)}_{t} }\bigl(AT_{M}(t) \bigr)=A\bigl({{}^{C}D^{\upsilon (t)}_{t} }T_{M}(t) \bigr). \end{aligned}$$

(14)

Taking into account that \(p=\lceil \upsilon (t)\rceil \) and using (3), (14) becomes

$$\begin{aligned} \begin{aligned} {{}^{C}D^{\upsilon (t)}_{t} }\varphi (t)&=A \biggl[ 0,0,\ldots ,0,\frac{\Gamma (p+1)}{\Gamma (p+1-\upsilon (t))}t^{p- \upsilon (t)},\ldots ,\frac{\Gamma (M+1)}{\Gamma (M+1-\upsilon (t))}t^{M- \upsilon (t)} \biggr] ^{T} \\ &= A\Psi ^{\upsilon (t)} T_{M}(t), \end{aligned} \end{aligned}$$

where \(\Psi ^{\upsilon (t)}\) is given as (13). Therefore from (7), we get

$$\begin{aligned} {{}^{C}D^{\upsilon (t)}_{t} }\varphi (t)=\Upsilon ^{\upsilon (t)} \upsilon (t), \end{aligned}$$

with

$$\begin{aligned} \Upsilon ^{\upsilon (t)}= A\Psi ^{\upsilon (t)} A^{-1}. \end{aligned}$$

□