**2. Basic Definitions**

The Liouville\_Caputo fractional\_order derivative, shifted Legendre polynomials, and Legendre wavelets are defined below [19,20].

**Definition 1.** *The Liouville\_Caputo fractional derivative of u is defined as [19]*

$$D^{\mathfrak{a}}u(t) = \frac{1}{\Gamma(n-\mathfrak{a})} \int\_0^t \frac{u^{(n)}(\xi)}{(t-\xi)^{n+1-n}} d\xi, \quad n-1 < \mathfrak{a} \le n, \ n \in \mathbb{N}.\tag{1}$$

Some characteristics of the Liouville\_Caputo fractional derivative are as follows:

$$D^a \mathbb{C} = 0,\tag{2}$$

where *C* is a constant. In addition, there is

$$D^{\mathfrak{a}}t^{\mathfrak{f}} = \begin{cases} 0, & \mathfrak{f} \in \mathcal{N}\_0 \text{ and } \mathfrak{f} < \lceil \mathfrak{a} \rceil \\ \frac{\Gamma(\beta+1)}{\Gamma(\beta+1-\mathfrak{a})} \mathfrak{x}^{\mathfrak{f}-\mathfrak{a}}, & \mathfrak{f} \in \mathcal{N}\_0 \text{ and } \mathfrak{f} \ge \lceil \mathfrak{a} \rceil \text{ or } \beta \notin \mathcal{N} \text{ and } \beta > \lfloor \mathfrak{a} \rfloor \end{cases} \tag{3}$$

in which *α* and !*α*" respectively imply that the largest integer is less than or equal to *α*, and the smallest integer is greater than or equal to *α*.

The Liouville\_Caputo fractional order derivative is a linear operation of the integer order derivative

$$D^a(\eta u(t) + \zeta v(t)) = \eta D^a u(t) + \zeta D^a v(t),\tag{4}$$

where *η* and *ζ* are constant.

**Definition 2.** *Let a and b respectively be the parameters of dilation and translation of a single function called the mother wavelet. If a and b change continuously, then we obtain the following family of continuous wavelets [21,22]:*

$$
\psi\_{ab}(t) = |a|^{-1/2} \psi\left(\frac{t-b}{a}\right), \quad a, b \in \mathbb{R}, \ a \neq 0. \tag{5}
$$

**Definition 3.** *Let Pm*(*t*) *imply the shifted Legendre polynomials of order m. Then Pm*(*t*) *can be formulated as [21]*

$$P\_m(t) = \sum\_{k=0}^m (-1)^{m+k} \frac{(m+k)!}{(m-k)!} \frac{t^k}{(k!)^2} \tag{6}$$

*and the orthogonality condition is*

$$\int\_0^1 P\_m(t)P\_n(t)dt = \begin{cases} \frac{1}{2m+1}, & \text{for } m=n\\ 0, & \text{for } m \neq n \end{cases} \tag{7}$$

**Definition 4.** *Let n and k be any positive integer, m be the order of shifted Legendre polynomials, and t be the normalized time. Then the Legendre wavelets ψnm*(*t*) = *ψ*(*k*, *n*, *m*, *t*) *are defined on the interval* [0, 1] *by [21,22]*.

$$\psi\_{nm}(t) = \begin{cases} 2^{\frac{k+1}{2}} \sqrt{m + \frac{1}{2}} P\_m \left( 2^k t - n \right), & \frac{n}{2^k} \le t \le \frac{n+1}{2^k} \\\ 0, & \text{otherwise} \end{cases} \tag{8}$$

*where m* <sup>=</sup> 0, 1, . . . , *<sup>M</sup>*; *<sup>n</sup>* <sup>=</sup> 0, 1, . . . ,(2*<sup>k</sup>* <sup>−</sup> <sup>1</sup>)*. The coefficient <sup>m</sup>*+<sup>1</sup> <sup>2</sup> *is for orthonormality.*

**Definition 5.** *Let u*(*t*) *and v*(*t*) *be functions defined over* [0, 1] *and then expanded in the terms of the Legendre wavelet as [21,22]*

$$\mu(t) = \sum\_{n=0}^{\infty} \sum\_{m=0}^{\infty} c\_{nm} \psi\_{nm}(t),\tag{9}$$

*where cnm* = (*u*(*t*), *ψnm*(*t*))*, in which* (., .) *implies the inner product. If the infinite series in Equation (9) is truncated, then it can be expressed as*

$$u(t) \stackrel{\sim}{=} \sum\_{n=0}^{2^k - 1} \sum\_{m=0}^M c\_{nm} \psi\_{nm}(t) = \mathcal{C}^T \psi(t),\tag{10}$$

*where C and ψ*(*t*) *are matrices, as presented by*

$$\begin{aligned} \mathbf{C} &= \begin{bmatrix} \mathbf{c}\_{0,0}, \mathbf{c}\_{0,1}, \dots, \mathbf{c}\_{0,M\_{1}}, \dots, \mathbf{c}\_{2,M\_{2}}, \dots, \mathbf{c}\_{2^{k}-1,0'}, \mathbf{c}\_{2^{k}-1,1'}, \dots, \mathbf{c}\_{2^{k}-1,M} \end{bmatrix}^{T} \\ \boldsymbol{\upmu} &= \begin{bmatrix} \boldsymbol{\upmu}\_{0,0}, \boldsymbol{\upmu}\_{0,1}, \dots, \boldsymbol{\upmu}\_{0,M\_{1}}, \dots, \boldsymbol{\upmu}\_{2^{k}-1,0'}, \boldsymbol{\upmu}\_{2^{k}-1,1'}, \dots, \boldsymbol{\upmu}\_{2^{k}-1,M} \end{bmatrix}^{T} . \end{aligned} \tag{11}$$

#### **3. Fundamental Relations**

Saadatmandi and Dehghan [11] derived the operational matrix of a fractional derivative by using shifted Legendre polynomials. In this section, we show how we derived the Legendre wavelet operational matrix of fractional derivatives in some special conditions by drawing from Saadatmandi and Dehghan [11]. Additionally, the theorem and corollary related to the Legendre wavelet operational matrix of derivatives illustrated by Mohammadi [21] are cited here as follows.

**Theorem 1.** *Let ψ*(*t*) *be the Legendre wavelet vector introduced in Equation (8). Then ψ*(*t*) *is expressed as [21,22]*

$$\frac{d\psi(t)}{dt} = D\psi(t),\tag{12}$$

*where D is the* 2*k*(*M* + 1) *operational matrix of the derivative, which can be stated as*

$$D = \begin{bmatrix} \mathcal{U} & \mathcal{O} & \cdots & \mathcal{O} \\ \mathcal{O} & \mathcal{U} & \cdots & \mathcal{O} \\ \vdots & \vdots & \ddots & \vdots \\ \mathcal{O} & \mathcal{O} & \cdots & \mathcal{U} \end{bmatrix}' \tag{13}$$

*where U is an* (*M* + 1)(*M* + 1) *matrix and its* (*r*,*s*)*th element is written as*

$$\text{all}\_{\mathbf{r},\mathbf{s}} = \begin{cases} 2^{k+1} \sqrt{(2r-1)(2s-1)}, & r = 2, \dots, (M+1), \text{ s} = 1, \dots, r-1 \text{ and } (\mathbf{r}+\mathbf{s}) \text{ odd} \\\ 0, & \text{otherwise} \end{cases} \text{ (14)}$$

**Corollary 1.** *Using Equation (12), the operational matrix for the nth derivative can be stated as [21]*

$$\frac{d^n \psi(t)}{dt^n} = D^n \psi(t),\tag{15}$$

*where D<sup>n</sup> is the nth power of matrix D*.

**Lemma 1.** *Let ψ*(*t*) *be the Legendre wavelets vector introduced in Equation (8). Assuming that k* = 0*, then*

$$D^{\mathfrak{a}}\psi\_{\mathfrak{r}}(t) = 0, \ \mathfrak{r} = 0, 1, \ldots, \lceil \mathfrak{a} \rceil - 1, \ \mathfrak{a} > 0. \tag{16}$$

**Proof.** The desired result can be obtained by using Equations (2) and (4) in Equation (8).

**Theorem 2.** *Let ψ*(*t*) *be the Legendre wavelets vector introduced in Equation (8). Supposing that k* = 0 *and α* > 0*, then*

$$D^a \psi(t) \cong D^{(a)} \psi(t),\tag{17}$$

*where <sup>D</sup>*(*α*) *is the* (*<sup>M</sup>* <sup>+</sup> <sup>1</sup>)*x*(*<sup>M</sup>* <sup>+</sup> <sup>1</sup>) *operational matrix of the fractional derivative of the order <sup>α</sup>* <sup>&</sup>gt; 0, *<sup>N</sup>* <sup>−</sup> <sup>1</sup> <sup>&</sup>lt; *α* ≤ *N in the Liouville\_Caputo sense and can be stated as*

$$D^{(a)} = \begin{pmatrix} 0 & 0 & \cdots & 0 \\ \vdots & \vdots & \cdots & \vdots \\ 0 & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & 0 \\ \sum\_{h=[a]}^{[a]} \mathfrak{F}\_{[a]} \mathfrak{I}\_{[h]} h & \sum\_{h=[a]}^{[a]} \mathfrak{F}\_{[a]} \mathfrak{I}\_{1} h & \cdots & \sum\_{h=[a]}^{[a]} \mathfrak{F}\_{[a]} \mathfrak{I}\_{m,h} h \\ \vdots & \vdots & \cdots & \vdots \\ \sum\_{h=[a]}^{r} \mathfrak{F}\_{r,0,h} & \sum\_{h=[a]}^{r} \mathfrak{F}\_{r,1,h} & \cdots & \sum\_{h=[a]}^{r} \mathfrak{F}\_{\mathfrak{I},\mathfrak{m},h} \\ \vdots & \vdots & \cdots & \vdots \\ \sum\_{h=[a]}^{\mathfrak{m}} \mathfrak{F}\_{\mathfrak{m},0,h} & \sum\_{h=[a]}^{\mathfrak{m}} \mathfrak{F}\_{\mathfrak{m},1,h} & \cdots & \sum\_{h=[a]}^{\mathfrak{m}} \mathfrak{F}\_{\mathfrak{m},\mathfrak{m},h} \end{pmatrix},\tag{18}$$

*where ξr*,*s*,*<sup>h</sup> is written as*

$$\zeta\_{r,s,h}^{\tau} = \sqrt{2r+1}\sqrt{2s+1}\sum\_{l=0}^{s} \frac{(-1)^{r+s+h+l} (r+h)! (s+l)!}{(r-h)! h! \Gamma(h-a+1) (s-l)! (l!)^2 (h+l-a+1)}.\tag{19}$$

*Consider in D*(*α*) *that the first* !*α*" *rows are all zero*.

**Proof.** Presume that *ψr*(*t*) is the *r*th element of the vector *ψ*(*t*) introduced in Equation (11), where *<sup>r</sup>* <sup>=</sup> *nM* + (*<sup>m</sup>* <sup>+</sup> <sup>1</sup>), *<sup>m</sup>* <sup>=</sup> 0, 1, . . . , *<sup>M</sup>*, *<sup>n</sup>* <sup>=</sup> 0, 1, . . . ,(2*<sup>k</sup>* <sup>−</sup> <sup>1</sup>). Then *<sup>ψ</sup>r*(*t*) can be stated as

$$
\psi\_{\mathbf{r}}(t) = 2^{\frac{k+1}{2}} \sqrt{r + \frac{1}{2}} P\_{\mathbf{r}}(2^k t - n) \chi\_{\left[\frac{n}{2^k r} \frac{n+1}{2^k}\right]}.\tag{20}
$$

Accepting that *k* = 0, and by using the shifted Legendre polynomial, we obtain

$$
\psi\_r(t) = \sqrt{2} \sqrt{r + \frac{1}{2}} \sum\_{h=0}^r \frac{(-1)^{r+h} (r+h)!}{(r-h)! (h!)^2} t^h \chi\_{[0,1]}.\tag{21}
$$

If we use Equations (3), (4), and (21), then we have

$$\begin{split} D^{(a)}\psi\_{l}(t) &= \sqrt{2}\sqrt{r+\frac{1}{2}} \sum\_{h=0}^{r} \frac{(-1)^{r+h}(r+h)!}{(r-h)!(h!)^{2}} D^{a}(t^{h}) \chi\_{[0,1]} \\ &= \sqrt{2r+1} \sum\_{h=\lceil a \rceil}^{r} \frac{(-1)^{r+h}(r+h)!}{(r-h)!(h!)^{\Gamma(h-a+1)}} t^{h-a} \chi\_{[0,1]}, \; r = \lceil a \rceil, \; \dots, m. \end{split} \tag{22}$$

Approximating *t <sup>h</sup>*−*<sup>α</sup>* by (*m* + 1) terms of the Legendre wavelets, then we obtain

$$t^{h-\alpha} \cong \sum\_{s}^{m} b\_{h,s} \psi\_{s}(t),\tag{23}$$

where

$$\begin{split} b\_{h,s} = \int\_0^1 t^{h-\mathfrak{a}} \psi\_s(t) dt &= \sqrt{2} \sqrt{s + \frac{1}{2}} \sum\_{l=0}^s \frac{(-1)^{s+l} (s+l)!}{(s-l)! (l!)^2} \Big| \begin{matrix} 1 \\ t^{h+l-\mathfrak{a}} dt \end{matrix} \\ &= \sqrt{2s+1} \sum\_{l=0}^s \frac{(-1)^{s+l} (s+l)!}{(s-l)! (l!)^2 (h+l-\mathfrak{a}+1)} \end{split} \tag{24}$$

Utilizing Equations (22) and (24), we get

$$\begin{split} D^{\mathfrak{a}}\psi\_{r}(t) &\quad \cong \sqrt{2r+1} \sum\_{h=\left[\mathfrak{a}\right]}^{r} \sum\_{s=0}^{m} \frac{(-1)^{r+h} (r+h)!}{(r-h)!(h!)! \Gamma(h-a+1)} b\_{h,s} \psi\_{s}(t) \chi\_{\left[0,1\right]} \\ &= \sum\_{s=0}^{m} \left( \sum\_{h=\left[\mathfrak{a}\right]}^{r} \mathfrak{z}\_{r,s,h} \right) \psi\_{s}(t) \chi\_{\left[0,1\right]\_{\prime}} \quad r=\left[\mathfrak{a}\right], \dots, m \end{split} \tag{25}$$

in which *ξr*,*s*,*<sup>h</sup>* is presented in Equation (19). In addition, if we use Lemma 1, then we can write

$$D^{\mathfrak{a}}\psi\_r(t) = 0,\ \ r = 0, 1, \ldots, \lceil \mathfrak{a} \rceil - 1,\ \mathfrak{a} > 0. \tag{26}$$

Combining Equations (25) and (27), the result can be obtained.
