**5. Error Analysis**

The upper bound of error for the operational matrix of fractional integration of a Jacobi polynomial of the *i*th degree is given as

$$e\_i^a = I^{(a)} \Psi\_i(\mathbf{x}) - I^a \Psi\_i(\mathbf{x}). \tag{36}$$

From Equation (36), we can write

$$\left\| \left\| \mathbf{c}\_{i}^{a} \right\| \_{2} = \left\| \left| I^{a} \Psi\_{i}(\mathbf{x}) - \sum\_{j=0}^{n} \mu(i,j) \Psi\_{j}(\mathbf{x}) \right\| \_{2} \right\| . \tag{37}$$

Taking the integral operator of order *α* on both sides of Equation (3), we get

$$M^a \Psi\_i(\mathbf{x}) = \sum\_{k=0}^i (-1)^{i-k} \frac{\Gamma(i+b+1)\Gamma(i+k+a+b+1)}{(i-k)!\Gamma(k+b+1)\Gamma(i+a+b+1)\Gamma(a+k+1)} \mathbf{x}^{a+k}.\tag{38}$$

From the construction of the operational matrix we can write

$$\mu(i,j) = \sum\_{k=0}^{i} (-1)^{i-k} \frac{\Gamma(i+b+1)\Gamma(i+k+a+b+1)}{(i-k)!\Gamma(k+b+1)\Gamma(i+a+b+1)\Gamma(a+k+1)} c\_{j,k\prime} \, \, \, j = 0, 1, \ldots, n. \tag{39}$$

Using Theorem 1 we can write

$$\left\lVert \mathbf{x}^{a+k} - \sum\_{j=0}^{n} c\_{j,k} \Psi\_j(\mathbf{x}) \right\rVert\_2 = \left( \frac{T\left(\mathbf{x}^{a+k}; \Psi\_0(\mathbf{x}), \Psi\_1(\mathbf{x}), \dots, \Psi\_n(\mathbf{x})\right)}{T(\Psi\_0(\mathbf{x}), \Psi\_1(\mathbf{x}), \dots, \Psi\_n(\mathbf{x}))} \right)^2. \tag{40}$$

From Equations (37)–(39), we get

$$\begin{split} \|e\_i^a\|\_2 &= \left. \left| \begin{array}{cc} \sum\_{k=0}^i (-1)^{i-k} \frac{\Gamma(i+b+1)\Gamma(i+k+a+b+1)}{\Gamma(i+b+1)\Gamma(i+a+b+1)\Gamma(a+k+1)} \mathbf{x}^{a+k} \\ -\sum\_{j=0}^n \sum\_{k=0}^i (-1)^{i-k} \frac{\Gamma(i+b+1)\Gamma(i+k+a+b+1)}{\Gamma(i+b+1)\Gamma(i+a+b+1)\Gamma(a+k+1)} c\_{j,k} \mathbf{\varprojlim}\_j \mathbf{y}\_j \end{array} \right| \right| \\ &\leq \sum\_{k=0}^i \left| \frac{\Gamma(i+b+1)\Gamma(i+k+a+b+1)}{(i-k)!\Gamma(k+b+1)\Gamma(i+a+b+1)\Gamma(a+k+1)} \right| \left| \left| \mathbf{x}^{a+k} - \sum\_{j=0}^n c\_{j,k} \mathbf{y}\_j(\mathbf{x}) \right| \right|\_2^2. \end{split} \tag{41}$$

Using Equation (40) in Equation (41), we obtain the error bound for the operational matrix of integration of an *i*th-degree polynomial, which is given as

$$\begin{split} & \| \mathcal{E}\_{i}^{a} \|\_{2} \\ & \leq \sum\_{k=0}^{i} \left| \frac{\Gamma(i+b+1)\Gamma(i+k+a+b+1)}{\Gamma(i-k)!\Gamma(k+b+1)\Gamma(i+a+b+1)\Gamma(a+k+1)} \right| \left( \frac{\Gamma(\mathtt{x}^{a+k};\mathtt{\mathtt{\mathtt{\mathtt{x}}}}\_{\mathtt{\mathtt{\mathtt{\mathtt{\mathtt{x}}}}}{\mathtt{\mathtt{N}}}(\mathtt{x}),\mathtt{\mathtt{\mathtt{\mathtt{x}}}}\_{\mathtt{\mathtt{\mathtt{\mathtt{x}}}}{\mathtt{\mathtt{N}}}(\mathtt{x}),...,\mathtt{\mathtt{\mathtt{\mathtt{x}}}}\_{\mathtt{\mathtt{\mathtt{x}}}}(\mathtt{x})} \right)^{2}, i = \textbf{0},1,2,...,n. \end{split} \tag{42}$$

Now, in particular cases, the error bounds for different orthogonal polynomials are given as follows.

**Case 1:** For Legendre polynomials (S1) the error bound is given as

$$\|e\_i^a\|\_2 \le \sum\_{k=0}^i \left| \frac{\Gamma(i+1)\Gamma(i+k+1)}{(i-k)!\,\Gamma(k+1)\Gamma(i+1)\Gamma(a+k+1)} \right| \left(\frac{T\left(\mathbf{x}^{a+k};\mathsf{Y}\_0(\mathbf{x}),\mathsf{Y}\_1(\mathbf{x}),...,\mathsf{Y}\_n(\mathbf{x})\right)}{T(\mathbf{Y}\_0(\mathbf{x}),\mathsf{Y}\_1(\mathbf{x}),...,\mathsf{Y}\_n(\mathbf{x}))}\right)^2, i = 0,1,2,\ldots,n. \tag{43}$$

**Case 2:** For Chebyshev polynomials of the first kind (S2) the error bound is given as

$$\|e\_{1}^{a}\|\_{2} \leq \sum\_{k=0}^{i} \left| \frac{\Gamma\left(i+\frac{3}{2}\right)\Gamma(i+k+2)}{(i-k)!\,\Gamma\left(k+\frac{3}{2}\right)\Gamma(i+2)\Gamma(a+k+1)} \right| \left(\frac{T\{x^{a+k};\mathsf{Y}\_{0}(x),\mathsf{Y}\_{1}(x),\ldots,\mathsf{Y}\_{n}(x)\}}{T(\mathsf{Y}\_{0}(x),\mathsf{Y}\_{1}(x),\ldots,\mathsf{Y}\_{n}(x))}\right)^{2} \right| = 0,1,2,\ldots,n. \tag{44}$$

**Case 3:** For Chebyshev polynomials of the third kind (S3) the error bound is given as

$$\|\boldsymbol{\varepsilon}\_{i}^{\mathbf{d}}\|\_{2} \leq \sum\_{k=0}^{i} \left| \frac{\Gamma\left(i+\frac{1}{2}\right)\Gamma(i+k+1)}{(i-k)!\,\, \Gamma\left(k+\frac{1}{2}\right)\Gamma(i+1)\Gamma(a+k+1)} \right| \left(\frac{T\{\mathbf{x}^{a+k};\mathbf{\varPsi}\_{0}(\mathbf{x}),\mathbf{\varPsi}\_{1}(\mathbf{x}),\ldots,\mathbf{\varPsi}\_{n}(\mathbf{x})\}}{T(\mathbf{\varPsi}\_{0}(\mathbf{x}),\mathbf{\varPsi}\_{1}(\mathbf{x}),\ldots,\mathbf{\varPsi}\_{n}(\mathbf{x}))}\right)^{2} \mathbf{j} = 0,1,2,\ldots,n. \tag{45}$$

**Case 4:** For Chebyshev polynomials (S4) the error bound is given as

$$\left\|\boldsymbol{\varepsilon}\_{i}^{\mathbf{a}}\right\|\_{2} \leq \sum\_{k=0}^{i} \left| \frac{\Gamma\left(i+\frac{3}{2}\right)\Gamma(i+k1)}{(i-k)!\,\, \Gamma\left(k+\frac{3}{2}\right)\Gamma(i+1)\Gamma(a+k+1)} \right| \left(\frac{T\left(\mathbf{x}^{a+k};\mathbb{W}\_{0}(\mathbf{x}),\mathbb{W}\_{1}(\mathbf{x}),\dots,\mathbb{W}\_{n}(\mathbf{x})\right)}{T(\mathbf{W}\_{0}(\mathbf{x}),\mathbb{W}\_{1}(\mathbf{x}),\dots,\mathbb{W}\_{n}(\mathbf{x}))}\right)^{2} \mathbf{j} = 0,1,2,\ldots,n. \tag{46}$$

**Case 5:** For Gegenbauer polynomials (S5) the error bound is given as

$$\|e\_i^a\|\_2 \le \sum\_{k=0}^i \left| \frac{\Gamma(i+2)\Gamma(i+k+3)}{(i-k)!\,\Gamma(k+2)\Gamma(i+3)\Gamma(a+k+1)} \right| \left(\frac{T\left(\mathbf{x}^{a+k};\mathsf{Y}\_0(\mathbf{x}),\mathsf{Y}\_1(\mathbf{x}),\ldots,\mathsf{Y}\_n(\mathbf{x})\right)}{T(\mathbf{Y}\_0(\mathbf{x}),\mathsf{Y}\_1(\mathbf{x}),\ldots,\mathsf{Y}\_n(\mathbf{x}))}\right)^2, i = 0,1,2,\ldots,n. \tag{47}$$

Let *e α*,*w <sup>I</sup>*,*<sup>n</sup>* denote the error vector for the operational matrix of integration of order *α* obtained by using (*n* + 1) orthogonal polynomials in *L*<sup>2</sup> *<sup>w</sup>*[0, 1]; then

$$
\varepsilon\_{I,n}^{a,w} = I^{(a)} \Phi\_n(\mathbf{x}) - I^a \Phi\_n(\mathbf{x}). \tag{48}
$$

From Theorems 1 and 2 and from Equations (43)–(47), it is clear that as *n* → ∞ the error vector in Equation (48) tends to zero.

#### **6. Convergence Analysis**

A set of orthogonal polynomials on [0, 1] forms a basis for *L*<sup>2</sup> *<sup>w</sup>*[0, 1]. Let *Sn* be the *n*-dimensional subspace of *L*<sup>2</sup> *<sup>w</sup>*[0, 1] generated by (Φ*i*)0≤*i*≤*n*. Thus, every functional on *Sn* can be written as a linear combination of orthogonal polynomials (Φ*i*)0≤*i*≤*n*. The scalars in the linear combinations can be chosen in such a way that the functional minimizes. Let the minimum value of a functional on space *Sn* be denoted by *mn*. From the construction of *Sn* and *mn*, it is clear that *Sn* ⊂ *Sn*+<sup>1</sup> and *mn*+<sup>1</sup> ≥ *mn*.

**Theorem 4.** *Consider the functional J, then*

$$\lim\_{n \to \infty} m\_n = m = \underbrace{\inf}\_{\ge \iota L\_w^2[0,1]} J[\mathfrak{x}].$$

**Proof .** Using Equation (48) in Equation (23), we have

$$J\left(\mathbf{c}\_{0},\mathbf{c}\_{1},\ldots,\mathbf{c}\_{n}\right) = \int\_{0}^{1} \left(\mathbf{C}^{T}\mathbf{g}(\mathbf{x})\Phi\_{\mathbf{n}}(\mathbf{x}) + \mathbf{C}^{T}\mathbf{l}^{(1)}\mathbf{g}'(\mathbf{x})\Phi\_{\mathbf{n}}(\mathbf{x}) + \mathbf{C}^{T}\mathbf{e}\_{1,n}^{1}\mathbf{g}'(\mathbf{x}) + \mathbf{h}'(\mathbf{x})\right)^{2}d\mathbf{x}.\tag{49}$$

Taking *n* → ∞ and using Equations (25)–(27) and (48) in Equation (49), we get

$$\begin{split} I^{\varepsilon} \left( \mathfrak{c}\_{0\prime} \mathfrak{c}\_{1\prime}, \dots, \mathfrak{c}\_{n\prime} \right) &= \int\_{0}^{1} \left( \mathsf{C}^{T} \left( \sum\_{i=0}^{n} \left( \mathsf{E}\_{1}^{i,T} \Phi\_{n}(\mathbf{x}) + \mathfrak{c}\_{\mathbb{E}\_{1}^{i},n}^{w} \right) \right) \\ &+ \mathsf{C}^{T} I^{(1)} \left( \sum\_{i=0}^{n} \left( \mathsf{E}\_{2}^{i,T} \Phi\_{n}(\mathbf{x}) + \mathfrak{c}\_{\mathbb{E}\_{2}^{i},n}^{w} \right) \right) + \mathsf{E}\_{3}^{T} \Phi\_{n}(\mathbf{x}) + \mathfrak{c}\_{\mathbb{E}\_{3},n}^{w} \right)^{2} d\mathbf{x} \end{split} \tag{50}$$

where

$$\begin{aligned} \mathcal{e}\_{E\_1,n}^{w} &= E\_1^{i,T} \Phi(\mathbf{x}) - E\_1^{i,T} \Phi\_n(\mathbf{x}), \\ \mathcal{e}\_{E\_2^i,n}^{w} &= E\_2^{i,T} \Phi(\mathbf{x}) - E\_2^{i,T} \Phi\_n(\mathbf{x}), \\ \mathcal{e}\_{E\_3,n}^{w} &= E\_3^T \Phi(\mathbf{x}) - E\_3^T \Phi\_n(\mathbf{x}), \end{aligned}$$

and *J<sup>e</sup>* is the error term of the functional.

Using Equations (30) and (32) in Equation (50), we get

$$f^{\varepsilon}\left(c\_{0}, c\_{1}, \dots, c\_{n}\right) = \int\_{0}^{1} \left(E^{T}\Phi\_{n}(\mathbf{x}) + \varepsilon\_{n}^{w}\right)^{2} d\mathbf{x} \tag{51}$$

where

$$
\epsilon\_n^{w} = \mathbb{C}^T \sum\_{i=0}^n \epsilon\_{E\_1^i, n}^w + \mathbb{C}^T I^{(1)} \sum\_{i=0}^n \epsilon\_{E\_2^i, n}^w. \tag{52}
$$

Solving Equation (51) similarly to the original functional, Equation (51) reduces to the following form:

$$f^{\varepsilon}\left(c\_{0}, c\_{1}, \dots, c\_{n}\right) = E^{T}PE + c\_{n}^{w}(f^{v}).\tag{53}$$

Using Equation (48) in Equation (34), we get

$$
\mathcal{C}^T I^{(1)} \Phi\_n(1) + \mathcal{C}^T \mathcal{e}\_{I,n}^{1,w} = \epsilon. \tag{54}
$$

Similar to above, by using the Rayleigh-Ritz method on Equation (53) with the boundary condition in Equation (54) we obtain the extreme value of the functional defined in Equation (53). Let this extreme value be denoted by *m*∗ *<sup>n</sup>*(*t*).

Now, from Equation (48), it is obvious that *e<sup>w</sup> Ei* <sup>1</sup>,*<sup>n</sup>* ,*e<sup>w</sup> Ei* 2,*n* , *e<sup>w</sup> <sup>E</sup>*3,*<sup>n</sup>* → 0 as *n* → ∞, which implies that *ew <sup>n</sup>* (*J<sup>e</sup>* ) <sup>→</sup> 0 as *<sup>n</sup>* <sup>→</sup> <sup>∞</sup>. So, it is clear that as *<sup>n</sup>* <sup>→</sup> <sup>∞</sup>, the functional *<sup>J</sup><sup>e</sup>* in Equation (53) comes close to the functional *J* in Equation (23) and the boundary condition in Equation (54) comes close to Equation (34).

$$\text{So, for large values of } n\_\prime$$

$$m\_n^\*(t) \to m\_n(t).\tag{55}$$

From Theorem 4 and Equation (55), we conclude that

$$\lim\_{n \to \infty} m\_n^\*(t) = m(t).$$

Proof completed.
