*2.1. Shifted Chebyshev Integration Matrices*

We first introduce the definition and some basic properties of shifted Chebyshev polynomials [18], which are used to establish the first- and higher-order shifted Chebyshev integration matrices based on the idea of constructing integration matrices in Reference [17]. However, we slightly modify this idea by instead using a shifted Chebyshev expansion suitable for solving our problem (1) without domain transformation. We give the definition and properties as follows.

**Definition 1.** *The shifted Chebyshev polynomial of degree n* ≥ 0 *is defined by*

$$S\_{\hbar}(\mathbf{x}) = \cos\left(n \arccos\left(\frac{2\mathbf{x}}{L} - 1\right)\right) \text{ for } \mathbf{x} \in [0, L].$$

Note that this shifted Chebyshev polynomial is symmetric, either with respect to the point *x* = *<sup>L</sup>* 2 or the vertical line *x* = *<sup>L</sup>* <sup>2</sup> over [0, *L*], depending on its degree. Next, we provide some important properties of the shifted Chebyshev polynomial, which we use to constructing the shifted Chebyshev integration matrix, as follows.

**Lemma 1.** (i) *For n* <sup>∈</sup> <sup>N</sup>*, the zeros of Sn*(*x*) *are symmetrically distributed over* [0, *<sup>L</sup>*] *and given by*

$$\mathbf{x}\_k = \frac{\mathbf{L}}{2} \left( \cos \left( \frac{2k - 1}{2n} \pi \right) + 1 \right), k \in \{1, 2, 3, \dots, n\}. \tag{2}$$

(ii) *For r* <sup>∈</sup> <sup>N</sup>*, the rth-order derivatives of Sn*(*x*) *at the endpoint b* ∈ {0, *<sup>L</sup>*} *are*

$$\frac{d^r}{dx^r} S\_{\mathfrak{n}}(\mathbf{x}) \Big|\_{\mathbf{x}=\mathbf{b}} = \prod\_{k=0}^{r-1} \left( \frac{n^2 - k^2}{2k + 1} \right) \left( \frac{2b}{L} - 1 \right)^{n+r} . \tag{3}$$

(iii) *For x* ∈ [0, *L*]*, the single-layer integrations of shifted Chebyshev polynomial Sn*(*x*) *are*

$$\begin{aligned} S\_0(\mathbf{x}) &= \int\_0^\mathbf{x} S\_0(\boldsymbol{\xi}) \, d\boldsymbol{\xi} = \mathbf{x}, \\ S\_1(\mathbf{x}) &= \int\_0^\mathbf{x} S\_1(\boldsymbol{\xi}) \, d\boldsymbol{\xi} = \frac{\mathbf{x}^2}{L} - \mathbf{x}, \\ S\_n(\mathbf{x}) &= \int\_0^\mathbf{x} S\_n(\boldsymbol{\xi}) \, d\boldsymbol{\xi} = \frac{L}{4} \left( \frac{S\_{n+1}(\mathbf{x})}{n+1} - \frac{S\_{n-1}(\mathbf{x})}{n-1} - \frac{2(-1)^n}{n^2 - 1} \right), n \in \{2, 3, 4, ...\}. \end{aligned}$$

(iv) *Let* {*xk*}*<sup>n</sup> <sup>k</sup>*=<sup>1</sup> *be a set of zeros of Sn*(*x*)*, the shifted Chebyshev matrix* **S** *is defined by*

$$\mathbf{S} = \begin{bmatrix} S\_0(\mathbf{x}\_1) & S\_1(\mathbf{x}\_1) & \cdots & S\_{n-1}(\mathbf{x}\_1) \\ S\_0(\mathbf{x}\_2) & S\_1(\mathbf{x}\_2) & \cdots & S\_{n-1}(\mathbf{x}\_2) \\ \vdots & \vdots & \ddots & \vdots \\ S\_0(\mathbf{x}\_n) & S\_1(\mathbf{x}\_n) & \cdots & S\_{n-1}(\mathbf{x}\_n) \end{bmatrix}.$$

*Then, it has the multiplicative inverse* **S**−<sup>1</sup> = <sup>1</sup> *<sup>n</sup>*diag(1, 2, 2, ..., 2)**S**.

Next, we use the above definition and properties of shifted Chebyshev polynomials to construct the shifted Chebyshev integration matrices. First, let *N* be a positive integer and *L* be a positive real number. Define an approximate solution *u*(*x*) of a certain differential equation by a linear combination of shifted Chebyshev polynomials *Sn*(*x*); that is,

$$u(\mathbf{x}) = \sum\_{n=0}^{N-1} c\_n \mathbf{S}\_n(\mathbf{x}) \text{ for } \mathbf{x} \in [0, L]. \tag{4}$$

.

Let *xk* for *k* ∈ {1, 2, 3, ..., *N*} be the interpolated points which are meshed by the zeros of *SN*(*x*) defined in (2). Substituting each *xk* into (4), it can be expressed (in matrix form) as

$$
\begin{bmatrix} u(\mathbf{x}\_1) \\ u(\mathbf{x}\_2) \\ \vdots \\ u(\mathbf{x}\_N) \end{bmatrix} = \begin{bmatrix} S\_0(\mathbf{x}\_1) & S\_1(\mathbf{x}\_1) & \cdots & S\_{N-1}(\mathbf{x}\_1) \\ S\_0(\mathbf{x}\_2) & S\_1(\mathbf{x}\_2) & \cdots & S\_{N-1}(\mathbf{x}\_2) \\ \vdots & \vdots & \ddots & \vdots \\ S\_0(\mathbf{x}\_N) & S\_1(\mathbf{x}\_N) & \cdots & S\_{N-1}(\mathbf{x}\_N) \end{bmatrix} \begin{bmatrix} c\_0 \\ c\_1 \\ \vdots \\ c\_{N-1} \end{bmatrix}'
$$

which is denoted by **u** = **Sc**. The unknown coefficient vector can be performed by **c** = **S**−1**u**. Let us consider the single-layer integration of *u*(*x*) from 0 to *xk*, which is denoted by *U*(1)(*xk*); we obtain

$$\iota L^{(1)}(\mathbf{x}\_k) = \int\_0^{\mathbf{x}\_k} \mu(\boldsymbol{\xi}) \, d\boldsymbol{\xi} = \sum\_{\boldsymbol{n}=0}^{N-1} c\_{\boldsymbol{n}} \int\_0^{\mathbf{x}\_k} S\_{\boldsymbol{n}}(\boldsymbol{\xi}) \, d\boldsymbol{\xi} = \sum\_{\boldsymbol{n}=0}^{N-1} c\_{\boldsymbol{n}} S\_{\boldsymbol{n}}(\mathbf{x}\_k)$$

for *k* ∈ {1, 2, 3, ..., *N*} or, in matrix form,

$$
\begin{bmatrix}
\mathcal{U}^{(1)}(\mathbf{x}\_{1}) \\
\mathcal{U}^{(1)}(\mathbf{x}\_{2}) \\
\vdots \\
\mathcal{U}^{(1)}(\mathbf{x}\_{N})
\end{bmatrix} = \begin{bmatrix}
\mathcal{S}\_{0}(\mathbf{x}\_{1}) & \mathcal{S}\_{1}(\mathbf{x}\_{1}) & \cdots & \mathcal{S}\_{N-1}(\mathbf{x}\_{1}) \\
\mathcal{S}\_{0}(\mathbf{x}\_{2}) & \mathcal{S}\_{1}(\mathbf{x}\_{2}) & \cdots & \mathcal{S}\_{N-1}(\mathbf{x}\_{2}) \\
\vdots & \vdots & \ddots & \vdots \\
\mathcal{S}\_{0}(\mathbf{x}\_{N}) & \mathcal{S}\_{1}(\mathbf{x}\_{N}) & \cdots & \mathcal{S}\_{N-1}(\mathbf{x}\_{N})
\end{bmatrix} \begin{bmatrix}
c\_{0} \\
c\_{1} \\
\vdots \\
c\_{N-1}
\end{bmatrix}
$$

We denote the above matrix by **<sup>U</sup>**(1) <sup>=</sup> **Sc**¯ <sup>=</sup> **SS**¯ <sup>−</sup>1**<sup>u</sup>** :<sup>=</sup> **Au**, where **<sup>A</sup>** <sup>=</sup> **SS**¯ <sup>−</sup><sup>1</sup> := [*aki*]*N*×*<sup>N</sup>* is called the first-order shifted Chebyshev integration matrix for the FIM-SCP; that is,

$$\mathcal{U}^{(1)}(\mathbf{x}\_k) = \int\_0^{\mathbf{x}\_k} \mu(\xi) \, d\xi = \sum\_{i=1}^N a\_{ki}\mu(\mathbf{x}\_i).$$

*Symmetry* **2020**, *12*, 497

Next, consider the double-layer integration of *u*(*x*) from 0 to *xk*, which denoted by *U*(2)(*xk*). We have

$$\mathcal{U}^{(2)}(\mathbf{x}\_k) = \int\_0^{\mathbf{x}\_k} \int\_0^{\mathbf{\tilde{z}}\_2} u(\mathbf{\tilde{z}}\_1) \, d\mathbf{\tilde{z}}\_1 d\mathbf{\tilde{z}}\_2 = \sum\_{i=1}^N a\_{ki} \int\_0^{\mathbf{x}\_i} u(\mathbf{\tilde{z}}\_1) \, d\mathbf{\tilde{z}}\_1 = \sum\_{i=1}^N \sum\_{j=1}^N a\_{ki} a\_{ij} u(\mathbf{x}\_j).$$

for *<sup>k</sup>* ∈ {1, 2, 3, ..., *<sup>N</sup>*}. It can be written, in matrix form, as **<sup>U</sup>**(2) <sup>=</sup> **<sup>A</sup>**2**u**. Similarly, we can calculate the *n*-layer integration of *u*(*x*) from 0 to *xk*, which is denoted by *U*(*n*)(*xk*). Then, we have

$$\iota L^{(n)}(\mathbf{x}\_k) = \int\_0^{\mathbf{x}\_k} \cdots \int\_0^{\mathbf{\tilde{x}}\_2} \mu(\mathbf{\tilde{y}}\_1) \, d\mathbf{\tilde{y}}\_1 \cdots \, d\mathbf{\tilde{y}}\_n = \sum\_{i\_n=1}^N \cdots \sum\_{j=1}^N a\_{ki\_n} \cdots \, a\_{i\_1 j} \mu(\mathbf{x}\_j),$$

for *<sup>k</sup>* ∈ {1, 2, 3, ..., *<sup>N</sup>*}, which can be expressed, in matrix form, as **<sup>U</sup>**(*n*) <sup>=</sup> **<sup>A</sup>***n***u**.
