Legendre spectral projection methods for weakly singular Hammerstein integral equations of mixed type

Abstract

This work provides the Legendre spectral projection (Galerkin and collocation), iterated Legendre spectral projection, Legendre spectral multi-projection and iterated Legendre spectral multi-projection methods to approximate the solution of weakly singular Hammerstein integral equations of mixed type. The convergence rates of approximate solutions to the exact solutions are obtained for all the above four methods in both $L^{2}$ and infinity norm. The comparison of convergence rates for all these methods have been discussed. We also have shown that iterated Galerkin improves over Galerkin, multi-Galerkin improves over iterated Galerkin and iterated multi-Galerkin improves over multi-Galerkin in $L^2$ norm using Legendre polynomial bases.

1 Introduction

We consider the following weakly singular Hammerstein integral equation of mixed type

$$\begin{aligned} u(s)-\sum _{i=1}^{m}\int _{-1}^1 k_{i}(s,t)\,\psi _{i}(t, u(t))\,\mathrm{d}t=f(s),\,\,\, -1\le s\le 1, \end{aligned}$$

(1)

where the source function f, the kernels $k_i(.,.)$ and the nonlinear functions $\psi _i(.,.)$ for $i=1,2,\ldots ,m$ are known and u is the unknown function to be determined in a Banach space $\mathbb {X}$. We consider the kernel $k_i(., .)$ as weakly singular type which is of the form

$$\begin{aligned} k_{i}(s, t)=m_{i}(s, t)g_{\alpha }|s-t|, \end{aligned}$$

(2)

$m_{i}(s, t)\in \mathcal{C}([-1, 1]\times [-1, 1])$ and

$$\begin{aligned} g_{\alpha }(x)=\left\{ \begin{array}{l@{}l} {x^{\alpha -1}},~~~~~~~~~\hbox {if}\;\;0<\alpha <1,\\ {\log x, ~~~~~~~~~\hbox {if}\;\;\alpha =1}. \end{array}\right. \end{aligned}$$

As a reformulation of boundary value problem, this type of problem (1) arises in nonlinear physical phenomenon such as electromagnetic fluid dynamics [3].

Several authors have used numerical methods such as projection methods (Galerkin and collocation), Petrov-Galerkin method, degenerate kernel method and Nyström method ([6,7,8, 11, 12, 15]) to solve the various linear and nonlinear integral equations because these integral equations can’t solve explicitly. Integral equations of type (1) with smooth and weakly singular kernel were solved numerically in ([2, 5, 9, 10, 13, 14, 18]) using piecewise polynomials as bases. In piecewise polynomial based projection methods the number of partitions should be increased to obtain more accurate approximate solution. So, one has to solve a large system of nonlinear equations, which take lots of time to compute. Therefore, many spectral methods have been developed by using global polynomials in last some years. In the global polynomial based projection methods, if $\mathcal {P}_{n}$ denotes either orthogonal or interpolatory projection operator, then $\Vert \mathcal {P}_{n}\Vert _{\infty }$ is unbounded.

We are interested to solve numerically the Hammerstein integral equations of mixed type with weakly singular kernel using Legendre spectral projection, iterated Legendre spectral projection methods. To improve convergence rates further, Legendre multi-projection and iterated Legendre multi-projection method have been used. We evaluate the convergence rates in all the above four methods in both $L^2$ and infinity norm, even if $\Vert \mathcal {P}_{n}\Vert _{\infty }$ is unbounded. We have given a comparison of error bounds in all the methods.

We have organized this paper in the following way. We have discussed the abstract framework for the Legendre spectral projection methods for Hammerstein integral equations of mixed type with the weakly singular kernels in Sect. 2. The convergence rates of approximated solution with exact solution have been discussed using spectral projection, iterated spectral projection, spectral multi-projection and iterated spectral multi-projection methods in Sects. 3, 4, 5 and 6, respectively, in both $L^2$ and infinity norm using Legendre polynomial bases. However, in the end, we have added a remark through which, we have given the comparison of error bounds in all the methods.

Throughout this paper, we assume c is a generic constant which may differ and is independent of n.

2 Hammerstein integral equation of mixed type with weakly singular kernel

In this section, we set up an abstract framework for the Hammerstein integral equation of type (1) with weakly singular kernel of type (2) on the Banach space $\mathbb {X}=\mathcal {C}[-1,1]$ or $L^2[-1,1]$. Throughout the paper, the following assumptions are made on $f,~k_{i}(.,.)$ and $\psi _{i}(.,u(.))$:

• (i) $f \in \mathcal {C}[-1,1],$

• (ii) $s_{i}= \displaystyle \sup _{s,t \in [-1,1]} |m_{i}(s,t)| < \infty$ for $i=1,2,\ldots ,m$ and $M=\sum _{i=1}^{m}s_{i},$

• (iii) $M_{2}=\displaystyle \sup _{s \in [-1,1]}\int _{-1}^{1} \left| g_{\alpha }|s-t|\right| ^{2}\mathrm{d}t < \infty ,$ for $\frac{1}{2}<\alpha \le 1.$

• (iv) The nonlinear functions $\psi _{i}(t,u)$ are continuous on $[-1,1]\times \mathbb {R}$ for $i=1,2,\ldots ,m$. $\psi _{i}(t,u)$ are Lipschitz continuous in u, i.e., for any $u_{1},~u_{2} \in \mathbb {R},~\exists$ constants $c_{i}>0,~i=1,2,\dots ,m$ such that

  $$\begin{aligned} \left| \psi _{i}(t,u_{1})-\psi _{i}(t,u_{2}) \right| \le c_{i} |u_{1}-u_{2}|,~\forall ~t\in [-1,1], \end{aligned}$$

  and $l_{1}=\sup _{i=1,2,\dots ,m}c_{i}.$

• (v) The functions $\psi _{i}^{(0,1)}(t,u(t))$ exist and are Lipschitz continuous in u, i.e., for any $u_{1},~u_{2} \in \mathbb {R}, ~ \exists$ constants $q_{i}>0,~i=1,2,\dots ,m$ such that

  $$\begin{aligned} \left| \psi _{i}^{(0,1)}(t,u_{1})-\psi _{i}^{(0,1)}(t,u_{2}) \right| \le q_{i} |u_{1}-u_{2} |,~\forall ~t\in [-1,1], \end{aligned}$$

  and $l_{2}=\sup _{i=1,2,\dots ,m}q_{i}$. This implies $\psi _{i}^{(0,1)}(.,.) \in \mathcal {C}([-1,1]\times \mathbb {R}).$

Define the integral operator $\mathcal {K}_{i}:\mathbb {X}\rightarrow \mathbb {X},~i=1,2,\dots ,m$ by

$$\begin{aligned} (\mathcal {K}_{i}\psi _{i})u(s)=\int _{-1}^{1}k_{i}(s,t)\psi _{i}(t,u(t))\mathrm{d}t,~ s\in [-1,1]. \end{aligned}$$

Then the equation (1) can be written in the following operator equation

$$\begin{aligned} u-\sum _{i=1}^{m}\mathcal {K}_{i}\psi _{i}(u)=f. \end{aligned}$$

(3)

Next, we define the operator $\mathcal {T}$ on $\mathbb {X}$ by

$$\begin{aligned} \mathcal {T}u=f+\sum _{i=1}^{m}\mathcal {K}_{i}\psi _{i}(u),~u\in \mathbb {X}, \end{aligned}$$

(4)

then the equation (3) can be written as

$$\begin{aligned} u=\mathcal {T}u. \end{aligned}$$

(5)

Define the Fréchet derivatives of $\sum _{i=1}^{m}\mathcal {K}_{i}\psi _{i}(u)$ at $u_0$ by

$$\begin{aligned} \sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u_{0})u(s)=\sum _{i=1}^{m}\int _{-1}^{1}k_{i}(s,t)\psi _{i}^{(0,1)}(t,u_{0}(t))u(t)\mathrm{d}t. \end{aligned}$$

Lemma 2.1

([16]) Let$m_i(s,t)\in \mathcal {C}([-1,1]\times [-1,1])$and$g_{\alpha }|s-t|$be the weakly singular part of the kernel$k_i(s,t)$for$1\le i \le m$. Then for any$s_1,s_2\in [-1,1]$, we have the followings:

• (i) $\displaystyle \lim _{s_1\rightarrow s_2}\int _{-1}^{1}|m_i(s_1,t)-m_i(s_2,t)|^2\mathrm{d}t\rightarrow 0$, for $1\le i\le m$,

• (ii) $\displaystyle \lim _{s_1\rightarrow s_2}\int _{-1}^{1}|g_{\alpha }|s_1-t|-g_{\alpha }|s_2-t||^2\mathrm{d}t\rightarrow 0,$ for $\frac{1}{2}<\alpha \le 1$.

The next theorem shows the existence and uniqueness of the solution of equation (5).

Theorem 2.2

Let$\mathbb {X}=\mathcal {C}[-1,1]$, $f\in \mathbb {X}$and$g_{\alpha }|s-t|$satisfies the assumption (iii) with$m_{i}(.,.) \in \mathcal {C}([-1,1]\times [-1,1])$and$s_{i}=\sup _{s,t\in [-1,1]}|m_{i}(s,t)| < \infty .$Let$\psi _{i}(t,u(t)) \in \mathcal {C}([-1,1]\times \mathbb {R})$satisfy the assumption (iv) with$\sqrt{2M_{2}}Ml_{1} < 1,$where$M=\sum _{i=1}^{m}s_{i}$and$l_{1}=\sup _{i=1,2,\dots ,m}c_{i}.$Then the operator equation$\mathcal {T}u=u$has a unique isolated solution$u_{0}\in \mathbb {X}$, i.e.,$\mathcal {T}u_{0}=u_{0}.$

Proof

The proof follows exactly by using similar technique given in Theorem-2.4 of [12]. $\square$

We will first approximate the space $\mathbb {X}$ by a finite dimensional space $\mathbb {X}_{n}.$ We consider $\mathbb {X}_{n}=\mathrm{span}\left\{ \phi _{0},\phi _{1},\dots , \phi _{n} \right\}$ as the sequence of Legendre polynomial subspaces of $\mathbb {X}$ of degree $\le n$. Define $L_{i}(s)=\sqrt{\dfrac{2i+1}{2}}\phi _{i}(s),\,\,i=0,1,\dots ,n.$ Since $L_{i}$ and $L_{j}$’s are polynomials

$$\begin{aligned} \left\langle L_{i},L_{j} \right\rangle =\int _{-1}^{1}L_{i}(t)\overline{L_{j}(t)}\mathrm{d}t= \int _{-1}^{1}L_{i}(t)L_{j}(t)\mathrm{d}t=\delta _{i,j} \end{aligned}$$

(6)

for $i,j=0,1,\dots ,n.$

Then the Legendre polynomials $\left\{ L_{0},L_{1},\dots ,L_{n} \right\}$ be the orthonormal bases for the subspaces $\mathbb {X}_{n}$ of $\mathbb {X}$ of degree $\le n.$ Now we need to introduce the Legendre orthogonal and Legendre interpolatory projection operator.

Let $\mathcal {P}_{n}^{G}: \mathbb {X}\rightarrow \mathbb {X}_{n}$ be the orthogonal projection defined by

$$\begin{aligned} \mathcal {P}_{n}^{G}u=\sum _{j=0}^{n}\left\langle u, L_{j} \right\rangle L_{j},\,\,u\in \mathbb {X}, \end{aligned}$$

(7)

where $\left\langle u, L_{j} \right\rangle =\int _{-1}^{1}u(t)L_{j}(t)\mathrm{d}t.$

Let $\left\{ \tau _{0},\tau _{1}, \dots , \tau _{n} \right\}$ be the zeros of Legendre polynomial of degree $n+1$ and define the interpolatory projection $\mathcal {P}_{n}^{C}:\mathbb {X}\rightarrow \mathbb {X}_{n}$ by

$$\begin{aligned} \mathcal {P}_{n}^{C}u \in \mathbb {X}_{n},\,\, \mathcal {P}_{n}^{C}u(\tau _{i})=u(\tau _{i}),\, i=0,1, \dots , n,\, u\in \mathbb {X}. \end{aligned}$$

(8)

Now onwards, we assume that the projection operator $\mathcal {P}_n:\mathbb {X}\rightarrow \mathbb {X}_n$ is either orthogonal projection $\mathcal {P}_{n}^{G}$ or interpolatory projection $\mathcal {P}_n^{C}$ for notational convenience.

Lemma 2.3

([4]) Consider$\mathcal {P}_{n}=\mathcal {P}_{n}^{G}$or$\mathcal {P}_{n}^{C}$as the projection operator is defined to be Legendre orthogonal projection or Legendre interpolatory projection operator. Then the following conditions hold:

• (i) For$u\in \mathcal {C}[-1,1]$, $\Vert \mathcal {P}_{n}u\Vert _{L^{2}}\le p\Vert u\Vert _{\infty }$, wherepis a constant independent ofn.

• (ii) There exists a constant$c>0$such that for any$u\in \mathbb {X}$and any$n\in \mathbb {N}$,

  $$\begin{aligned} \Vert u-\mathcal {P}_{n}u\Vert _{L^{2}}\le c\inf _{\phi \in \mathbb {X}_{n}}\Vert u-\phi \Vert _{L^{2}} \rightarrow 0~as~n\rightarrow \infty . \end{aligned}$$

• (iii) For any$u\in \mathcal {C}^{r}[-1,1],$there exists a constantcindependent ofnsuch that

  $$\begin{aligned}&\Vert \mathcal {P}_{n}u-u\Vert _{L^{2}}\le cn^{-r}\Vert u^{(r)}\Vert _{L^{2}},\\&\Vert \mathcal {P}_{n}^{G}u-u\Vert _{\infty }\le cn^{\frac{3}{4}-r}\Vert u^{(r)}\Vert _{\infty },\\&\Vert \mathcal {P}_{n}^{C}u-u\Vert _{\infty }\le cn^{\frac{1}{2}-r}\Vert u^{(r)}\Vert _{\infty }. \end{aligned}$$

• (iv) For any$u\in \mathcal {C}^{r}[-1,1],$there exists a constantcindependent ofnsuch that

  $$\begin{aligned}&\Vert u-\mathcal {P}_{n}^{G}u\Vert _{\infty }\le cn^{\frac{1}{2}-r}V(u^{(r)}), \end{aligned}$$

  where$V(u^{(r)})$denotes the total variation of$u^{(r)}.$

Note that $\Vert u-\mathcal {P}_{n}u\Vert _{\infty }\nrightarrow 0$ as $n\rightarrow \infty$ for any $u\in \mathcal {C}[-1,1].$

Lemma 2.4

([1]) Let$\mathbb {X}$be a Banach space and$\mathcal{T}$, $\mathcal{T}_n\in \mathbb {BL(X)}$. If$\mathcal{T}_n$is norm convergent to$\mathcal{T}$or$\mathcal{T}_n$is$\nu$-convergent to$\mathcal{T}$and$(\mathcal{I}-\mathcal{T})^{-1}$exists and bounded on$\mathbb {X}$, then$(\mathcal{I}-\mathcal{T}_n)^{-1}$exists and uniformly bounded on$\mathbb {X}$ for sufficiently largen.

3 Legendre spectral projection methods

In this section, Legendre spectral projection methods for weakly singular Hammerstein integral equation of mixed type are being discussed. The convergence rates for approximated solution with exact solution have been evaluated in both $L^2$ and infinity norm.

The Legendre spectral projection methods for the equation (3) is to find an approximate solution $u_{n}\in \mathbb {X}_{n}$ such that

$$\begin{aligned} u_{n}-\sum _{i=1}^{m}\mathcal {P}_{n}\mathcal {K}_{i}\psi _{i}(u_{n})=\mathcal {P}_{n}f. \end{aligned}$$

(9)

If $\mathcal {P}_{n}=\mathcal {P}_n^G$, then the above scheme leads to Legendre Galerkin method, whereas if $\mathcal {P}_{n}=\mathcal {P}_n^C$, we get the Legendre collocation method.

Let $\mathcal {T}_{n}$ be the operator defined by

$$\begin{aligned} \mathcal {T}_{n}u:=\sum _{i=1}^{m}\mathcal {P}_{n}\mathcal {K}_{i}\psi _{i}(u)+\mathcal {P}_{n}f,\,\, u\in \mathbb {X}. \end{aligned}$$

Then equation (9) can be written as

$$\begin{aligned} u_{n}=\mathcal {T}_{n}u_n. \end{aligned}$$

We need the following lemma and theorem for the convergence rates of the approximate solution $u_{n}$ to the exact solution $u_{0}$.

Lemma 3.1

For any$u,~v\in L^{2}[-1,1]$or$\mathcal {C}[-1,1]$, the followings hold

$$\begin{aligned}&\left\| \sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})(u_{0})-\sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})(u)\right\| _{\infty } \le \sqrt{M_{2}}Ml_{1}\Vert u_{0}-u\Vert _{L^{2}}, \end{aligned}$$

(10)

$$\begin{aligned}&\left\| \left[ \sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u_{0})-\sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u)\right] v\right\| _{\infty } \le \sqrt{M_{2}}Ml_{2}\Vert u_{0}-u\Vert _{L^{2}}\Vert v\Vert _{\infty }, \end{aligned}$$

(11)

$$\begin{aligned}&\left\| \left[ \sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u_{0})-\sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u)\right] v\right\| _{\infty } \le \sqrt{M_{2}}Ml_{2}\Vert u_{0}-u\Vert _{\infty }\Vert v\Vert _{L^{2}}. \end{aligned}$$

(12)

Proof

Using Lipschitz continuity of $\psi _{i}(.,u(.))$ and Cauchy-Schwarz inequality, we obtain

$$\begin{aligned}&\left\| \sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})(u_{0})-\sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})(u)\right\| _{\infty }\\&\quad = \sup _{s\in [-1,1]}\left| \sum _{i=1}^{m}\int _{-1}^{1}m_{i}(s,t)g_{\alpha }|s-t|[\psi _{i}(t,u_{0}(t))-\psi _{i}(t,u(t))]\mathrm{d}t\right| \\&\quad \le \sum _{i=1}^{m}\sup _{s,t\in [-1,1]}|m_{i}(s,t)| \sup _{s\in [-1,1]}\int _{-1}^{1} \left| g_{\alpha }|s-t|[\psi _{i}(t,u_{0}(t))-\psi _{i}(t,u(t))] \right| \mathrm{d}t\\&\quad \le \sup _{i=1,2,\ldots ,m}c_{i}\sum _{i=1}^{m}s_{i}\sup _{s\in [-1,1]} \Vert g_{\alpha }|s-t|\Vert _{L^{2}}\left( \int _{-1}^{1}|u_{0}(t)-u(t)|^{2}\mathrm{d}t\right) ^{\frac{1}{2}}\\&\quad \le \sqrt{M_{2}}Ml_{1}\Vert u_{0}-u\Vert _{L^{2}}, \end{aligned}$$

which completes the proof of first inequality. Similarly, using Lipschitz continuity and Cauchy-Schwarz inequality, we obtain

$$\begin{aligned}&\left\| \left[ \sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u_{0})-\sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u) \right] v\right\| _{\infty }\nonumber \\&\quad \le \sup _{s\in [-1,1]}\sum _{i=1}^{m}\left| \int _{-1}^{1}m_{i}(s,t)g_{\alpha }|s-t|\left[ \psi _{i}^{(0,1)}(t,u_{0}(t))-\psi _{i}^{(0,1)}(t,u(t))\right] v(t)\mathrm{d}t\right| \nonumber \\&\quad \le \sum _{i=1}^{m}\sup _{s,t\in [-1,1]}|m_{i}(s,t)|\nonumber \\&\qquad \int _{-1}^{1}\left| g_{\alpha }|s-t|\left[ \psi _{i}^{(0,1)}(t,u_{0}(t))-\psi _{i}^{(0,1)}(t,u(t))\right] v(t)\right| \mathrm{d}t\nonumber \\&\quad \le \sup _{i=1,2,\ldots ,m}q_i\sum _{i=1}^{m} s_{i}\sup _{s\in [-1,1]}\Vert g_{\alpha }|s-t|\Vert _{L^2}\left( \int _{-1}^{1}|u_0(t)-u(t)|^{2}|v(t)|^2\mathrm{d}t\right) ^{\frac{1}{2}}\nonumber \\&\quad \le \sqrt{M_{2}}Ml_{2}\Vert u_{0}-u\Vert _{L^{2}}\Vert v\Vert _{\infty }, \end{aligned}$$

(13)

which completes the proof of (11). From estimate (13), we get

$$\begin{aligned} \left\| \left[ \sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u_{0})-\sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u)\right] v\right\| _{\infty } \le \sqrt{M_{2}}Ml_{2}\Vert u_{0}-u\Vert _{\infty }\Vert v\Vert _{L^2}. \end{aligned}$$

This completes the proof of the lemma. $\square$

Theorem 3.2

Let $\sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u_{0})$ be the Fréchet derivative of $\sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})(u)$ at $u_{0}.$ Then $\left\| (\mathcal {I}-\mathcal {P}_{n})\sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u_{0})\right\| _{L^{2}} \rightarrow 0$ as $n\rightarrow \infty .$

Proof

To prove $\sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u_{0})$ is a compact operator, we have to show that $\sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u_{0})$ is uniformly bounded and equicontinuous.

Now using Cauchy-Schwarz inequality, we obtain

$$\begin{aligned} \left\| \sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u_{0}) u \right\| _{\infty }= & {} \sup _{s\in [-1,1]}\left| \int _{-1}^{1}\sum _{i=1}^{m} m_{i}(s,t)g_{\alpha }|s-t|\psi _{i}^{(0,1)}(t,u_{0}(t))u(t)\mathrm{d}t\right| \nonumber \\\le & {} \sup _{i=1,2,\dots ,m}\sup _{t\in [-1,1]}\left| \psi _{i}^{(0,1)}(t,u_{0}(t))\right| \sum _{i=1}^{m}\sup _{s,t\in [-1,1]}|m_{i}(s,t)| \nonumber \\&\times \int _{-1}^{1}|g_{\alpha }|s-t|u(t)|\mathrm{d}t\nonumber \\\le & {} \sup _{i=1,2,\dots ,m} d_{i} \sum _{i=1}^{m}s_{i}\Vert g_{\alpha }|s-t|\Vert _{L^{2}}\Vert u\Vert _{L^{2}}\nonumber \\\le & {} Md\sqrt{M_{2}}\Vert u\Vert _{L^{2}} \le Md\sqrt{2M_{2}}\Vert u\Vert _{\infty }, \end{aligned}$$

(14)

where $d_{i}=\sup _{t\in [-1,1]}|\psi _{i}^{(0,1)}(t,u_{0}(t))|$ and $d=\displaystyle \sup _{i=1,2,\dots ,m}d_{i}$. Hence, $\sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u_{0})$ is uniformly bounded. Next to show equicontinuity, consider

$$\begin{aligned}&\left| \sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u_{0})u(s_{1})- \sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u_{0})u(s_{2}) \right| \nonumber \\&\quad \le \displaystyle \sup _{i=1,2,\dots ,m}\sup _{t\in [-1,1]}|\psi _{i}^{(0,1)}(t,u_{0}(t))|\sum _{i=1}^{m} \int _{-1}^{1}\left| [m_{i}(s_{1},t)g_{\alpha }|s_{1}-t|\right. \nonumber \\&\qquad \left. -m_{i}(s_{2},t)g_{\alpha }|s_{2}-t|]u(t)\right| \mathrm{d}t, \nonumber \\&\quad \le d\sum _{i=1}^{m} \sup _{s,t\in [-1,1]}|m_{i}(s_{1},t)|\int _{-1}^{1}|[g_{\alpha }|s_{1}-t|-g_{\alpha }|s_{2}-t|]u(t)|\mathrm{d}t\nonumber \\&\qquad +d\sum _{i=1}^{m}\int _{-1}^{1}|m_{i}(s_{1},t)-m_{i}(s_{2},t)||g_{\alpha }|s_{2}-t|u(t)|\mathrm{d}t. \end{aligned}$$

(15)

Now using Cauchy-Schwarz inequality and Lemma 2.1 in the above estimate, we obtain

$$\begin{aligned}&\left| \sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u_{0})u(s_{1})- \sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u_{0})u(s_{2}) \right| \nonumber \\&\quad \le dM \left( \int _{-1}^{1}|g_{\alpha }|s_{1}-t|-g_{\alpha }|s_{2}-t||^{2}\mathrm{d}t\right) ^{\frac{1}{2}}\Vert u\Vert _{L^{2}}\nonumber \\&\qquad + d\sum _{i=1}^{m}\left( \int _{-1}^{1}|m_{i}(s_{1},t)-m_{i}(s_{2},t)|^{2}\right) ^{\frac{1}{2}}\Vert g_{\alpha }|s_{2}-t|\Vert _{L^{2}}\Vert u\Vert _{\infty }\\&\quad \rightarrow 0 ~~\mathrm{as}~~s_{1} \rightarrow s_{2}. \end{aligned}$$

This proves that $\sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u_{0})$ is equicontinuous. Hence by Arzelá-Ascoli’s theorem $\sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u_{0})$ is a compact operator.

Let B be a closed unit ball in $\mathbb {X}.$ Thus, $S=\left\{ \sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u_{0})u :u\in B \right\}$ is relatively compact set in $\mathbb {X}.$ By using Lemma 2.3, we get

$$\begin{aligned} \left\| (\mathcal {I}-\mathcal {P}_{n})\sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u_{0}) \right\| _{L^{2}}= & {} \sup \left\{ \left\| (\mathcal {I}-\mathcal {P}_{n})\sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u_{0})u \right\| _{L^{2}}:u\in B\right\} \\= & {} \sup \left\{ \Vert (\mathcal {I}-\mathcal {P}_{n})y\Vert _{L^{2}}:y\in S\right\} \rightarrow 0~~ as ~ n\rightarrow \infty . \end{aligned}$$

Thus, the proof is completed. $\square$

For rest of the paper, we assume that 1 is not an eigenvalue of the operator $\sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u_{0})$.

Theorem 3.3

Let$u_{0}\in \mathcal {C}^{r}[-1,1],~r\ge 1$. Then for sufficiently largen, the operator$(\mathcal {I}- \mathcal {T}_{n}^{\prime }(u_{0}))$is invertible, i.e., there exists a constant$A_{1}>0$such that$\Vert (\mathcal {I}- \mathcal {T}_{n}^{\prime }(u_{0}))^{-1}\Vert _{L^{2}}\le A_{1} <\infty$. Also the equation (9) has a unique solution$u_{n}\in B(u_{0},\delta )=\lbrace u:\Vert u-u_{0}\Vert _{L^{2}}<\delta \rbrace$for some$\delta >0$. Moreover, there exists a constant$0<q<1$, independent ofnsuch that

$$\begin{aligned} \dfrac{\alpha _{n}}{1+q}\le \Vert u_{n}-u_{0}\Vert _{L^{2}}\le \dfrac{\alpha _{n}}{1-q}, \end{aligned}$$

where$\alpha _{n}=\Vert (\mathcal {I}-\mathcal {T}_{n}^{\prime }(u_{0}))^{-1}(\mathcal {T}_{n}(u_{0})-\mathcal {T}(u_{0}))\Vert _{L^{2}}$.

Proof

Using Theorem 3.2, we get

$$\begin{aligned}&\Vert (\mathcal {T}_{n}^{\prime }(u_{0})-\mathcal {T}^{\prime }(u_{0}))(u)\Vert _{L^{2}}\\&\quad = \left\| (\mathcal {P}_{n}-\mathcal {I})\left[ \sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u_{0})\right] (u)\right\| _{L^{2}}\\&\quad \le \left\| (\mathcal {P}_{n}-\mathcal {I})\sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u_{0})\right\| _{L^{2}}\Vert u\Vert _{L^2} \rightarrow 0,~\mathrm{as}~n\rightarrow \infty . \end{aligned}$$

This shows that $\mathcal {T}_{n}^{\prime }(u_{0})$ is norm convergent to $\mathcal {T}^{\prime }(u_{0})$ in $L^{2}$-norm. Since 1 is not an eigenvalue of $\mathcal {T}^{\prime }(u_{0})$, $(\mathcal {I}- \mathcal {T}^{\prime }(u_{0}))$ is invertible. Then by the Lemma 2.4, $(\mathcal {I}- \mathcal {T}_{n}^{\prime }(u_{0}))^{-1}$ exists and is uniformly bounded on $\mathbb {X}$ for sufficiently large n,  i.e., there exists a constant $A_{1}>0$ such that $\Vert (\mathcal {I}- \mathcal {T}_{n}^{\prime }(u_{0}))^{-1}\Vert _{L^{2}}\le A_{1} <\infty .$

Using $\Vert \mathcal {P}_{n}u\Vert _{L^{2}}\le p\Vert u\Vert _{\infty }$ and estimate (11) for any $u\in B(u_{0},\delta )$, we get

$$\begin{aligned} \Vert \mathcal {T}_{n}^{\prime }(u_{0})-\mathcal {T}_{n}^{\prime }(u)\Vert _{L^{2}}= & {} \left\| \sum _{i=1}^{m}\mathcal {P}_{n}(\mathcal {K}_{i}\psi _{i})^{\prime }(u_{0})-\sum _{i=1}^{m}\mathcal {P}_{n}(\mathcal {K}_{i}\psi _{i})^{\prime }(u)\right\| _{L^{2}}\nonumber \\\le & {} \; p\left\| \sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u_{0})-\sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{\prime }(u)\right\| _{\infty }\nonumber \\\le & {} \sqrt{M_{2}}Ml_{2}\Vert u_{0}-u\Vert _{L^{2}} \le \sqrt{M_{2}}Ml_{2}\delta . \end{aligned}$$

(16)

Now using the estimate (16) and $\Vert (\mathcal {I}- \mathcal {T}_{n}^{\prime }(u_{0}))^{-1}\Vert _{L^{2}}\le A_{1}$, we obtain

$$\begin{aligned} \sup _{\Vert u-u_{0}\Vert _{L^{2}} \le \delta } \Vert (\mathcal {I}- \mathcal {T}_{n}^{\prime }(u_{0}))^{-1}(\mathcal {T}_{n}^{\prime }(u_{0})-\mathcal {T}_{n}^{\prime }(u))\Vert _{L^{2}}\le & {} \; A_{1}\Vert \mathcal {T}_{n}^{\prime }(u_{0})-\mathcal {T}_{n}^{\prime }(u)\Vert _{L^{2}}\\\le & {} \; A_{1}\sqrt{M_{2}}Ml_{2}\delta \le q. \end{aligned}$$

Choosing $\delta$ in such a way that $q\in (0,1)$, this proves the equation (4.4) of Theorem-2 of [19].

Now using Lemma 2.3, we get

$$\begin{aligned} \alpha _{n}= & {} \Vert (\mathcal {I}- \mathcal {T}_{n}^{\prime }(u_{0}))^{-1}(\mathcal {T}_{n}(u_{0})-\mathcal {T}(u_{0}))\Vert _{L^{2}}\nonumber \\\le & {} \; A_{1}\Vert (\mathcal {T}_{n}(u_{0})-\mathcal {T}(u_{0}))\Vert _{L^{2}}\nonumber \\= & {} \; A_{1}\left\| (\mathcal {P}_{n}-\mathcal {I})\sum _{i=1}^{m}[(\mathcal {K}_{i}\psi _{i})(u_{0})+f]\right\| _{L^{2}}\nonumber \\= & {} \; A_{1}\Vert (\mathcal {P}_{n}-\mathcal {I})u_{0}\Vert _{L^{2}} \le A_{1}cn^{-r}\Vert u_{0}^{(r)}\Vert _{L^{2}} \rightarrow 0~\mathrm{as}~n\rightarrow \infty . \end{aligned}$$

(17)

Choose n large enough such that $\alpha _{n}\le \delta (1-q)$, then equation (4.5) of Theorem-2 of [19] is satisfied. Then by applying Theorem-2 of [19], we get

$$\begin{aligned} \dfrac{\alpha _{n}}{1+q}\le \Vert u_{n}-u_{0}\Vert _{L^{2}}\le \dfrac{\alpha _{n}}{1-q}, \end{aligned}$$

where $\alpha _{n}=\Vert (\mathcal {I}-\mathcal {T}_{n}^{\prime }(u_{0}))^{-1}(\mathcal {T}_{n}(u_{0})-\mathcal {T}(u_{0}))\Vert _{L^{2}}$. This completes the proof. $\square$

Theorem 3.4

Let$u_{0}\in \mathcal {C}^{r}[-1,1]$. Let$u_n^G$be the Legendre Galerkin and$u_n^C$be the Legendre collocation approximation of the equation (9). Then the following hold

$$\begin{aligned}&\Vert u_n^G-u_0\Vert _{L^2}=\mathcal {O}(n^{-r}),~~\Vert u_n^C-u_0\Vert _{L^2}=\mathcal {O}(n^{-r}),\\&\Vert u_n^G-u_0\Vert _{\infty }=\mathcal {O}\left( n^{\frac{1}{2}-r}\right) ,~~\Vert u_n^C-u_0\Vert _{\infty }=\mathcal {O}\left( n^{\frac{1}{2}-r}\right) . \end{aligned}$$

Proof

The proof of estimates in $L^2$ norm follows directly from Theorem-3.3. Using equations (3) and (9), and $\Vert \mathcal {P}_n\Vert _{\infty }\le c\log n$ (cf., Page-147, [3]), we have

$$\begin{aligned} \Vert u_n-u_0\Vert _{\infty }= & {} \left\| \sum _{i=1}^{m}\mathcal {P}_n(\mathcal {K}_i\psi _i)(u_n)+\mathcal {P}_{n}f-f-\sum _{i=1}^{m}(\mathcal {K}_{i} \psi _{i})(u_0)\right\| _{\infty }\nonumber \\\le & {} \left\| \mathcal {P}_n\left[ \sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_n)-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_0)\right] \right\| _{\infty }\nonumber \\&+\left\| (\mathcal {P}_{n}-\mathcal {I})\Big [\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_0)+f\Big ]\right\| _{\infty }\nonumber \\\le & {} c\log n\left\| \sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_n)-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_0)\right\| _{\infty }\nonumber \\&+\Vert (\mathcal {P}_{n}-\mathcal {I})u_0\Vert _{\infty }. \end{aligned}$$

(18)

Using estimate (10) in the first term of the right hand side of estimate (18), we get

$$\begin{aligned} \Vert u_{n}-u_{0}\Vert _{\infty }\le & {} \sqrt{M_2}Ml_1c\log n\Vert u_n-u_0\Vert _{L^{2}}\nonumber \\&+\Vert (\mathcal {P}_{n}-\mathcal {I}) u_0\Vert _{\infty }. \end{aligned}$$

(19)

Using $u_n=u_n^G$ and $u_n=u_n^C$, then using Lemma 2.3, we obtain the desired results. $\square$

4 Iterated Legendre spectral projection methods

In this section, the iterated Legendre spectral projection methods for Hammerstein integral equations of mixed type with weakly singular kernels have been discussed. The rate of convergence for iterated approximate solution with exact solution have been evaluated for both $L^2$ and infinity norm.

The iterated approximate solution $\widetilde{u}_n$ corresponding to the approximate solution $u_n$ given by equation (9) is defined as

$$\begin{aligned} \widetilde{u}_{n}=\sum _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})(u_{n})+f. \end{aligned}$$

(20)

If $u_n=u_n^{G}$ in equation (20), we get iterated Legendre Galerkin solution and if $u_n=u_n^{C}$, then we get iterated Legendre collocation solution. To discuss the convergence of the iterated approximate solution $\widetilde{u}_{n}$ to the exact solution $u_{0}$, we need the following theorem.

Theorem 4.1

Let$u_{0}\in \mathcal {C}^{r}[-1,1],~r\ge 1$and$\widetilde{u}_{n}$be the iterated Legendre Galerkin approximation$\tilde{u}_n^{G}$or iterated Legendre collocation approximation$u_n^{C}$of$u_{0}$. Then the following holds

$$\begin{aligned} \Vert \widetilde{u}_n-u_0\Vert _{\infty }\le & {} \sqrt{M_2}Ml_2(1+M_3 c\log n)\Vert u_n-u_0\Vert _{L^{2}}\Vert u_n-u_0\Vert _{\infty }\\&+(1+M_3c\log n)\sup _{s\in [-1,1]}|\langle {h(s,.)} ,\,{(\mathcal {P}_n-\mathcal {I})(u_0)(.)} \rangle |, \end{aligned}$$

where$h(s,t)=\sum _{i=1}^{m}k_{i}(s,t)\psi _{i}^{(0,1)}(t,u_0(t))$.

Proof

The steps of the proof follows similarly as in Theorem-4.1 of [16]. So, we omit it. $\square$

Theorem 4.2

Let$u_{0}\in \mathcal {C}^{r}[-1,1],~r\ge 1$and$\widetilde{u}_{n}^{G}$be the iterated Legendre Galerkin approximation of$u_{0}$. Then the following holds

$$\begin{aligned} \Vert \widetilde{u}_{n}^{G}-u_{0}\Vert _{L^{2}}\le \sqrt{2}\Vert \widetilde{u}_{n}^{G}-u_{0}\Vert _{\infty }\le {\left\{ \begin{array}{ll} cn^{\frac{1}{2}-r-\alpha }(1+c\log n),~~~~~~~~\mathrm{for}~\frac{1}{2}<\alpha <1,\\ cn^{-\frac{1}{2}-r}(1+c\log n)\log n,~~~\mathrm{for}~\alpha =1. \end{array}\right. } \end{aligned}$$

Proof

From Theorem-4.1, we have

$$\begin{aligned} \Vert \widetilde{u}_n^G-u_0\Vert _{\infty }\le & {} \sqrt{M_2}Ml_2(1+M_3 c\log n)\Vert u_n^G-u_0\Vert _{L^{2}}\Vert u_n^G-u_0\Vert _{\infty }\nonumber \\&+(1+M_3c\log n)\sup _{s\in [-1,1]}|\langle {h(s,.)} ,\,{(\mathcal {P}_n^G-\mathcal {I})(u_0)(.)} \rangle |. \end{aligned}$$

(21)

Since $\mathcal {P}_n^G$ be the orthogonal projection from the space $\mathbb {X}$ into $\mathbb {X}_n$, then we have

$$\begin{aligned} \langle {w} ,\,{(\mathcal {I}-\mathcal {P}_n^G)u} \rangle =0,~~\forall ~w\in \mathbb {X}_n. \end{aligned}$$

By using Hölder’s inequality, Lemma 2.3, and Theorems 2 and 3 of [17], we obtain

$$\begin{aligned} \sup _{s\in [-1,1]}|\langle {h(s,.)} ,\,{(\mathcal {P}_n^G-\mathcal {I})(u_0)(.)} \rangle |\le & {} \sup _{s\in [-1,1]} |\langle {h(s,.)-\phi (s,.)} ,\,{(\mathcal {P}_n^G-\mathcal {I})(u_0)(.)} \rangle |\nonumber \\\le & {} \Vert h_s-\phi _s\Vert _{L^{1}}\Vert (\mathcal {P}_n^G-\mathcal {I})(u_0) \Vert _{\infty }\nonumber \\\le & {} V(u_{0}^{(r)}){\left\{ \begin{array}{ll} cn^{\frac{1}{2}-r-\alpha },~~~~~~~~\mathrm{for}~\frac{1}{2}<\alpha <1,\\ cn^{-\frac{1}{2}-r}\log n,~~~\mathrm{for}~\alpha =1, \end{array}\right. } \end{aligned}$$

(22)

where $\phi _s\in \mathbb {X}_n$. Now using Theorem-3.4 and estimate (22) in estimate (21), we obtain

$$\begin{aligned} \Vert \widetilde{u}_n^G-u_0\Vert _{\infty }\le & {} (1+c\log n)cn^{-r}n^{\frac{1}{2}-r} \\&+(1+c\log n){\left\{ \begin{array}{ll} cn^{\frac{1}{2}-r-\alpha },~~~~~~~~\mathrm{for}~\frac{1}{2}<\alpha<1,\\ cn^{-\frac{1}{2}-r}\log n,~~~\mathrm{for}~\alpha =1, \end{array}\right. }\\\le & {} {\left\{ \begin{array}{ll} cn^{\frac{1}{2}-r-\alpha }(1+c\log n),~~~~~~~~\mathrm{for}~\frac{1}{2}<\alpha <1,\\ cn^{-\frac{1}{2}-r}(1+c\log n)\log n,~~~\mathrm{for}~\alpha =1. \end{array}\right. } \end{aligned}$$

Hence,

$$\begin{aligned} \Vert \widetilde{u}_{n}^{G}-u_{0}\Vert _{L^{2}}\le \sqrt{2}\Vert \widetilde{u}_{n}^{G}-u_{0}\Vert _{\infty }\le {\left\{ \begin{array}{ll} cn^{\frac{1}{2}-r-\alpha }(1+c\log n),~~~~~~~~\mathrm{for}~\frac{1}{2}<\alpha <1,\\ cn^{-\frac{1}{2}-r}(1+c\log n)\log n,~~~\mathrm{for}~\alpha =1. \end{array}\right. } \end{aligned}$$

Thus, the proof is completed. $\square$

Theorem 4.3

Let$u_{0}\in \mathcal {C}^{r}[-1,1], ~r\ge 1$and$\widetilde{u}_{n}^{C}$be the iterated Legendre collocation approximation of$u_{0}$. Then the following holds

$$\begin{aligned} \Vert \widetilde{u}_{n}^{C}-u_{0}\Vert _{L^{2}}\le \sqrt{2}\Vert \widetilde{u}_{n}^{C}-u_{0}\Vert _{\infty }\le (1+c\log n)cn^{-r}. \end{aligned}$$

Proof

From Theorem-4.1, we have

$$\begin{aligned} \Vert \widetilde{u}_n^C-u_0\Vert _{\infty }\le & {} \sqrt{M_2}Ml_2(1+M_3 c\log n)\Vert u_n^C-u_0\Vert _{L^{2}}\Vert u_n^C-u_0\Vert _{\infty }\nonumber \\&+(1+M_3c\log n)\sup _{s\in [-1,1]}|\langle {h(s,.)} ,\,{(\mathcal {P}_n^C-\mathcal {I})(u_0)(.)} \rangle |. \end{aligned}$$

(23)

Using Cauchy-Schwarz inequality and Lemma 2.3, we get

$$\begin{aligned} \sup _{s\in [-1,1]}|\langle {h(s,.)} ,\,{(\mathcal {P}_n^C-\mathcal {I})(u_0)(.)} \rangle |\le & {} \sup _{s\in [-1,1]}\Vert h_s\Vert _{L^{2}}\Vert (\mathcal {P}_n^C-\mathcal {I})(u_0)\Vert _{L^{2}}\nonumber \\\le & {} \sup _{s\in [-1,1]}\Vert h_s\Vert _{L^{2}}cn^{-r}\Vert u_{0}^{(r)}\Vert _{\infty }. \end{aligned}$$

(24)

Substituting estimate (24) and Theorem-3.4 in equation (23), we obtain

$$\begin{aligned} \Vert \widetilde{u}_n^C-u_0\Vert _{\infty }\le & {} c(1+c\log n)cn^{-r}n^{\frac{1}{2}-r}+c(1+c\log n)cn^{-r}\\\le & {} (1+c\log n)cn^{-r}. \end{aligned}$$

Hence,

$$\begin{aligned} \Vert \widetilde{u}_{n}^{C}-u_{0}\Vert _{L^{2}}\le \sqrt{2}\Vert \widetilde{u}_{n}^{C}-u_{0}\Vert _{\infty }\le (1+c\log n)cn^{-r}. \end{aligned}$$

This completes the proof. $\square$

5 Legendre spectral multi-projection methods

To improve the results further, we use now the Legendre spectral multi-projection methods for weakly singular Hammerstein integral equations of mixed type. Define the multi-projection operator $(\mathcal {K}_{n,i}^{M}\psi _{i}):\mathbb {X}\rightarrow \mathbb {X}$ for $i=1,2,\ldots ,m$ by

$$\begin{aligned} (\mathcal {K}_{n,i}^{M}\psi _{i})(u):=\mathcal {P}_{n}(\mathcal {K}_{i}\psi _i)(u)+(\mathcal {K}_{i}\psi _{i})(\mathcal {P}_{n}u)-\mathcal {P}_n(\mathcal {K}_{i}\psi _{i})(\mathcal {P}_nu). \end{aligned}$$

(25)

The multi-projection method for equation (3) is to find an approximate solution $u_n^M\in \mathbb {X}$ such that

$$\begin{aligned} u_n^M-\sum _{i=1}^{m}(\mathcal {K}_{n,i}^{M}\psi _{i})(u_n^M)=f. \end{aligned}$$

(26)

If $\mathcal {P}_n=\mathcal {P}_n^G$, then equation (26) leads to multi-Galerkin method and if $\mathcal {P}_n=\mathcal {P}_n^C$, then equation (26) leads to multi-collocation method. Let

$$\begin{aligned} \mathcal {T}_n^M(u)=\sum _{i=1}^{m}(\mathcal {K}_{n,i}^{M}\psi _{i})(u)+f,~~u\in \mathbb {X}, \end{aligned}$$

then equation (26) can be written as

$$\begin{aligned} u_n^M=\mathcal {T}_n^M(u_n^M). \end{aligned}$$

The Fréchet derivative of $\mathcal {T}_n^Mu$ at $u_0$ is given by

$$\begin{aligned} \mathcal {T}_n^{M^{\prime }}(u_0)= & {} \sum _{i=1}^{m}(\mathcal {K}_{n,i}^{M} \psi _{i})^{\prime }(u_0)\\= & {} \sum _{i=1}^{m}\mathcal {P}_n(\mathcal {K}_i\psi _i)^{\prime }(u_0)+\sum _{i=1}^{m} (\mathcal {K}_i\psi _i)^{\prime }(\mathcal {P}_nu_0)\mathcal {P}_n\\&- \sum _{i=1}^{m}\mathcal {P}_n(\mathcal {K}_i\psi _i)^{\prime } (\mathcal {P}_{n}u_0)\mathcal {P}_{n}. \end{aligned}$$

To discuss the convergence rates of $u_n^M$ to $u_0$, we need the following lemma and theorem.

Lemma 5.1

For any$x,y\in \mathbb {X}$, we have

$$\begin{aligned}&\left\| \sum _{i=1}^{m}(\mathcal {K}_{n,i}^{M}\psi _{i})^{\prime }(x)- \sum _{i=1}^{m}(\mathcal {K}_{n,i}^{M}\psi _{i})^{\prime }(y)\right\| _{L^{2}}\nonumber \\&\quad \le \sqrt{M_2}Ml_2p[1+\sqrt{2}p(1+p)]\Vert x-y\Vert _{L^{2}}. \end{aligned}$$

(27)

Proof

Using estimate (11), we obtain

$$\begin{aligned}&\left\| \sum _{i=1}^{m}(\mathcal {K}_{n,i}^{M}\psi _{i})^{\prime }(x)-\sum _{i=1}^{m}(\mathcal {K}_{n,i}^{M}\psi _{i})^{\prime }(y) \right\| _{L^{2}}\\&\quad \le \left\| \mathcal {P}_n\Big [\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime }(x)-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime }(y)\Big ]\right\| _{L^2}\\&\qquad +\left\| (\mathcal {I}-\mathcal {P}_n)\Big [\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime }(\mathcal {P}_nx)- \sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime }(\mathcal {P}_ny)\Big ]\mathcal {P}_n\right\| _{L^{2}}\\&\quad \le p\left\| \sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime }(x)-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime }(y)\right\| _{\infty }\\&\qquad +\sqrt{2}p(1+p)\left\| \sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime }(\mathcal {P}_nx)-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime }(\mathcal {P}_ny) \right\| _{\infty }\\&\quad \le p\sqrt{M_2}Ml_2\Vert x-y\Vert _{L^{2}}+\sqrt{2}p(1+p)\sqrt{M_2}Ml_2\Vert \mathcal {P}_nx-\mathcal {P}_ny\Vert _{L^{2}}\\&\quad = p\sqrt{M_2}Ml_2[1+\sqrt{2}p(1+p)]\Vert x-y\Vert _{L^{2}}. \end{aligned}$$

The proof is completed. $\square$

Theorem 5.2

Let$u_{0} \in \mathcal {C}^{r}[-1,1], ~r\ge 1$. Then the operator$(\mathcal {I}-\mathcal {T}_{n}^{{M}^{\prime }}(u_{0}))$is invertible on$\mathcal {C}[-1,1]$for sufficiently largen, and there exists a constant$A_{2}>0$independent ofnsuch that$\Vert (\mathcal {I}-\mathcal {T}_{n}^{{M}^{\prime }}(u_{0}))^{-1} \Vert _{L^{2}} \le A_{2} < \infty .$

Proof

For any $u\in \mathbb {X}$, we have

$$\begin{aligned}&\Vert [\mathcal {T}_n^{M^{\prime }}(u_0)-\mathcal {T}^{\prime }(u_0)]u\Vert _{L^2}\nonumber \\&\quad =\left\| \Big [\sum _{i=1}^{m}(\mathcal {P}_n-\mathcal {I})(\mathcal {K}_i\psi _i)^{\prime }(u_0)-\sum _{i=1}^{m}(\mathcal {P}_n-\mathcal {I})(\mathcal {K}_i\psi _i)^{\prime }(\mathcal {P}_nu_0)\mathcal {P}_n\Big ]u\right\| _{L^2}\nonumber \\&\quad =\left\| (\mathcal {P}_n-\mathcal {I})\Big [\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime }(u_0)-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime }(u_0)\mathcal {P}_n\right. \nonumber \\&\qquad \left. +\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime }(u_0)\mathcal {P}_n-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime }(\mathcal {P}_nu_0)\mathcal {P}_n\Big ]u\right\| _{L^2}\nonumber \\&\quad \le \left\| (\mathcal {P}_n-\mathcal {I})\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime }(u_0)(\mathcal {I}-\mathcal {P}_n)u\right\| _{L^2}\nonumber \\&\qquad +\left\| (\mathcal {P}_n-\mathcal {I})\Big [\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime }(u_0)-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime }(\mathcal {P}_nu_0)\Big ]\mathcal {P}_nu\right\| _{L^2}. \end{aligned}$$

(28)

The first term of right hand side of estimate (28) becomes

$$\begin{aligned}&\left\| (\mathcal {P}_n-\mathcal {I})\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime } (u_0)(\mathcal {I}-\mathcal {P}_n)u\right\| _{L^2}\nonumber \\&\quad \le \left\| (\mathcal {P}_n-\mathcal {I})\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime }(u_0)\right\| _{L^2}\Vert (\mathcal {I}-\mathcal {P}_n)u\Vert _{L^2}\nonumber \\&\quad \le (1+p)\left\| (\mathcal {P}_n-\mathcal {I})\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime }(u_0)\right\| _{L^2}\Vert u\Vert _{L^2}. \end{aligned}$$

(29)

Using equation (12) and Lemma 2.3 in the second term of right hand side of estimate (28), we obtain

$$\begin{aligned}&\left\| (\mathcal {P}_n-\mathcal {I})\Big [\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime }(u_0)-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime }(\mathcal {P}_nu_0)\Big ]\mathcal {P}_nu\right\| _{L^2}\nonumber \\&\quad \le \sqrt{2}(1+p)\left\| \Big [\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime }(u_0)-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime }(\mathcal {P}_nu_0)\Big ]\mathcal {P}_nu\right\| _{\infty }\nonumber \\&\quad \le \sqrt{2}(1+p)\sqrt{M_2}Ml_2\Vert u_0-\mathcal {P}_nu_0\Vert _{\infty }\Vert \mathcal {P}_nu\Vert _{L^2}\nonumber \\&\quad \le \sqrt{2M_2}p(1+p)Ml_2cn^{\beta -r}\Vert u_0^{(r)}\Vert _{\infty }\Vert u\Vert _{L^2}. \end{aligned}$$

(30)

Substituting estimates (29) and (30) in estimate (28), and then using Theorem-3.2, we obtain

$$\begin{aligned} \Vert \mathcal {T}_n^{M^{\prime }}(u_0)-\mathcal {T}^{\prime }(u_0)\Vert _{L^2}\le & {} (1+p)\left\| (\mathcal {P}_n-\mathcal {I})\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime }(u_0)\right\| _{L^2}\\&+\sqrt{2M_2}p(1+p)Ml_2cn^{\beta -r}\Vert u_0^{(r)}\Vert _{\infty }\\\rightarrow & {} 0~~\mathrm{as}~n\rightarrow \infty ~\mathrm{and}~\beta <r=1,2,\cdots . \end{aligned}$$

Hence, $\mathcal {T}_{n}^{M^{\prime }}(u_{0})$ is norm convergent to $\mathcal {T}^{\prime }(u_{0})$ in $L^{2}$-norm for $r>\beta$. Since 1 is not an eigenvalue of $(\mathcal {K}\psi )^{\prime }(u_0)$, $(\mathcal {I}-\mathcal {T}^{\prime }(u_{0}))$ is invertible on $\mathbb {X}$. Then by Lemma 2.4, $(\mathcal {I}-\mathcal {T}_{n}^{{M}^{\prime }}(u_{0}))^{-1}$ exists and is uniformly bounded on $\mathbb {X}$, for some sufficiently large n, i.e., there exists some $A_{2}>0$ such that $\Vert (\mathcal {I}-\mathcal {T}_{n}^{{M}^{\prime }}(u_{0}))^{-1} \Vert _{L^{2}} \le A_{2} < \infty .$ Thus, the proof is completed. $\square$

Theorem 5.3

Let$u_{0}\in \mathcal {C}^{r}[-1,1],~r\ge 1$. Then for sufficiently largen, the equation (26) has a unique solution${u}_{n}^{M} \in B(u_{0},\delta )=\lbrace u:\Vert u-u_{0}\Vert _{L^{2}}\le \delta \rbrace$for some$\delta >0$. Further, there exists a constant$0<q<1$, independent ofnsuch that

$$\begin{aligned} \dfrac{\beta _{n}}{1+q} \le \Vert {u}_{n}^{M}-u_{0}\Vert _{L^{2}} \le \dfrac{\beta _{n}}{1-q}, \end{aligned}$$

where$\beta _{n}=\Vert (\mathcal {I}-\mathcal {T}_{n}^{{M}^{\prime }}(u_{0}))^{-1}(\mathcal {T}_{n}^{M}(u_{0})-\mathcal {T}(u_{0}))\Vert _{L^{2}}.$

Proof

Using Lemma 5.1, we obtain

$$\begin{aligned} \Vert \mathcal {T}_n^{M^{\prime }}(u_0)-\mathcal {T}_n^{M^{\prime }}(u)\Vert _{L^2}= & {} \left\| \sum _{i=1}^{m}(\mathcal {K}_{n,i}^{M}\psi _{i})^{\prime }(u_0)-\sum _{i=1}^{m}(\mathcal {K}_{n,i}^{M}\psi _{i})^{\prime }(u)\right\| _{L^{2}}\nonumber \\\le & {} p\sqrt{M_2}Ml_2[1+\sqrt{2}p(1+p)]\Vert u_0-u\Vert _{L^2}\le c\delta . \end{aligned}$$

(31)

Using $\Vert (\mathcal {I}-\mathcal {T}_{n}^{{M}^{\prime }}(u_{0}))^{-1}\Vert _{L^2}\le A_2$ and estimate (31), we obtain

$$\begin{aligned}&\sup _{\Vert u-u_0\Vert _{L^2}\le \delta }\Vert (\mathcal {I}- \mathcal {T}_{n}^{{M}^{\prime }}(u_{0})) ^{-1}(\mathcal {T}_n^{M^{\prime }}(u_0)-\mathcal {T}_n^{M^{\prime }}(u)) \Vert _{L^2}\\&\quad \le A_2\Vert \mathcal {T}_n^{M^{\prime }}(u_0)- \mathcal {T}_n^{M^{\prime }}(u)\Vert _{L^2}\le A_2c\delta \le q. \end{aligned}$$

Choosing $\delta$ in such a way that $0<q<1$. This proves equation (4.4) of Theorem-2 of [19].

Now using Theorem-5.2 and Lemma 2.3, we obtain

$$\begin{aligned} \beta _n\le & {} A_2\Vert \mathcal {T}_n^M(u_0)-\mathcal {T}(u_0)\Vert _{L^2}\nonumber \\\le & {} A_2\left\| \sum _{i=1}^{m}\mathcal {P}_n(\mathcal {K}_i\psi _i)(u_0)+ \sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(\mathcal {P}_nu_0)\right. \nonumber \\&\left. -\sum _{i=1}^{m}\mathcal {P}_n(\mathcal {K}_i\psi _i)(\mathcal {P}_nu_0)-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_0)\right\| _{L^2}\nonumber \\\le & {} A_2\left\| (\mathcal {P}_n-\mathcal {I})\Big [\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_0)-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(\mathcal {P}_nu_0)\Big ]\right\| _{L^2}\nonumber \\\le & {} A_2\sqrt{2}(p+1)\left\| \sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_0)-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(\mathcal {P}_nu_0)\right\| _{\infty }\\\le & {} A_2\sqrt{2M_2}Ml_1(p+1)\Vert u_0-\mathcal {P}_nu_0\Vert _{L^2}\rightarrow 0~\mathrm{as}~n\rightarrow \infty .\nonumber \end{aligned}$$

(32)

Choosing n sufficiently large such that $\beta _n\le \delta (1-q)$. Then equation (4.5) of Theorem-2 of [19] is satisfied. Hence, by applying Theorem-2 of [19], we obtain

$$\begin{aligned} \dfrac{\beta _{n}}{1+q} \le \Vert {u}_{n}^{M}-u_{0}\Vert _{L^{2}} \le \dfrac{\beta _{n}}{1-q}, \end{aligned}$$

where $\beta _n=\Vert (\mathcal {I}-\mathcal {T}_{n}^{{M}^{\prime }}(u_{0}))^{-1}(\mathcal {T}_{n}^{M}(u_{0})-\mathcal {T}(u_{0}))\Vert _{L^{2}}.$ Thus, the proof of the theorem is completed. $\square$

Lemma 5.4

Let$u_{0}\in \mathcal {C}[-1,1]$. Then the following holds

$$\begin{aligned} \left\| \sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(\mathcal {P}_nu_0)-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_0)\right\| _{\infty }\le & {} Ml_2\sqrt{M_2}\Vert \mathcal {P}_nu_0-u_0\Vert _{L^2}\Vert \mathcal {P}_nu_0-u_0\Vert _{\infty }\\&+\sup _{s\in [-1,1]}|\langle {h(s,.)} ,\,{(\mathcal {P}_n-\mathcal {I})u_0(.)} \rangle |, \end{aligned}$$

where$h(s,t)=\sum _{i=1}^{m}k_{i}(s,t)\psi _i^{(0,1)}(t,u_0(t)).$

Proof

Using Mean value Theorem, we obtain

$$\begin{aligned}&\left| \Big [\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(\mathcal {P}_nu_0)-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_0)\Big ]v(s)\right| \nonumber \\&\quad = \left| \sum _{i=1}^{m}\int _{-1}^{1}k_i(s,t)[\psi _i(t,\mathcal {P}_nu_0(t))-\psi _i(t,u_0(t))]v(t)\mathrm{d}t\right| \nonumber \\&\quad =\left| \sum _{i=1}^{m}\int _{-1}^{1}k_i(s,t)[\psi _i^{(0,1)}(t,u_0+\theta _3(\mathcal {P}_nu_0-u_0)(t))(\mathcal {P}_nu_0-u_0)]v(t)\mathrm{d}t\right| , \end{aligned}$$

where $0<\theta _3<1$. Using Lipschitz continuity of $\psi _{i}^{(0,1)}(t,u_0(.))$ and Cauchy-Schwarz inequality, we have

$$\begin{aligned}&\left| \Big [\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(\mathcal {P}_nu_0)-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_0)\Big ]v(s)\right| \nonumber \\&\quad \le \Big |\sum _{i=1}^{m}\int _{-1}^{1}m_i(s,t)g_{\alpha }|s -t|[\psi _i^{(0,1)}(t,u_0\nonumber \\&\qquad +\theta _3(\mathcal {P}_nu_0-u_0)(t)) -\psi _i^{(0,1)}(t,u_0(t))]\nonumber \\&\qquad \times (\mathcal {P}_nu_0-u_0)v(t)\mathrm{d}t\Big |+\left| \sum _{i=1}^{m}\int _{-1}^{1}k_i(s,t)\psi _i^{(0,1)}(t,u_0(t))(\mathcal {P}_nu_0-u_0)v(t)\mathrm{d}t\right| \nonumber \\&\quad \le \sup _{i=1,2,\ldots ,m}q_i\sum _{i=1}^{m}\sup _{s,t\in [-1,1]}|m_i(s,t)|\Vert v\Vert _{\infty } \int _{-1}^{1}|g_{\alpha }|s-t|(\mathcal {P}_nu_0-u_0)(\mathcal {P}_nu_0-u_0)|\mathrm{d}t\nonumber \\&\qquad +\Vert v\Vert _{\infty }\left| \int _{-1}^{1}h(s,t)(\mathcal {P}_nu_0-u_0)(t)\mathrm{d}t\right| \nonumber \\&\quad \le Ml_2\sqrt{M_2}\Vert \mathcal {P}_{n}u_{0}-u_{0}\Vert _{L^2}\Vert \mathcal {P}_nu_0-u_0\Vert _{\infty }\Vert v\Vert _{\infty } +\Vert v\Vert _{\infty }\left| \int _{-1}^{1}h(s,t)(\mathcal {P}_nu_0-u_0)(t)\mathrm{d}t\right| , \end{aligned}$$

where $h(s,t)=\sum _{i=1}^{m}k_{i}(s,t)\psi _i^{(0,1)}(t,u_0(t))$. Thus, we obtain the desired result.$\square$

Lemma 5.5

Let$u_{0}\in \mathcal {C}^{r}[-1,1], ~r\ge 1$. Let$u_{n}^{M}$be the approximation of$u_{0}$. Then the following holds

$$\begin{aligned} \Vert u_{n}^{M}-u_{0}\Vert _{\infty }\le (c+c\log n)\Vert u_{0}-u_{n}^{M}\Vert _{L^{2}}+ n^{-r}(c+c\log n)\Vert u_{0}^{(r)}\Vert _{\infty }. \end{aligned}$$

Proof

We have

$$\begin{aligned}&\Vert u_{n}^{M}-u_{0}\Vert _{\infty }\\&\quad = \left\| \sum _{i=1}^{m}(\mathcal {K}_{n,i}^M\psi _i)(u_n^M) -\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_0)\right\| _{\infty }\\&\quad =\left\| \sum _{i=1}^{m}\mathcal {P}_n(\mathcal {K}_i\psi _i)(u_n^M)+ \sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(\mathcal {P}_nu_n^M)\right. \\&\qquad \left. -\sum _{i=1}^{m}\mathcal {P}_n(\mathcal {K}_i\psi _i) (\mathcal {P}_nu_n^M)-\sum _{i=1}^{m}(\mathcal {K}_i \psi _i)(u_0)\right\| _{\infty }\\&\quad \le \left\| \mathcal {P}_n\Big [\sum _{i=1}^{m}(\mathcal {K}_i\psi _i) (u_n^M) -\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_0)\Big ]\right\| _{\infty } \\&\qquad +\left\| \sum _{i=1}^{m}(\mathcal {K}_i\psi _i) (\mathcal {P}_nu_n^M)-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(\mathcal {P}_nu_0) \right\| _{\infty }\\&\qquad +\left\| \mathcal {P}_n\Big [\sum _{i=1}^{m}(\mathcal {K}_i\psi _i) (\mathcal {P}_nu_0)-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i) (\mathcal {P}_nu_n^M)\Big ]\right\| _{\infty } \\&\qquad +\left\| \sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(\mathcal {P}_nu_0)- \sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_0)\right\| _{\infty }\\&\qquad +\left\| \mathcal {P}_n\Big [\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_0)-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(\mathcal {P}_nu_0)\Big ]\right\| _{\infty }. \end{aligned}$$

Using $\Vert \mathcal {P}_n\Vert _{\infty }\le c\log n$ (cf., Page-147, [3]) and estimate (10) with Lemma 2.3 in the above estimate, we obtain

$$\begin{aligned} \Vert u_{n}^{M}-u_{0}\Vert _{\infty }\le & {} c\log n M\sqrt{M_2}l_1\Vert u_n^M-u_0\Vert _{L^2}+M\sqrt{M_2}l_1\Vert \mathcal {P}_nu_n^M-\mathcal {P}_nu_0\Vert _{L^2}\\&+c\log nM\sqrt{M_2}l_1\Vert \mathcal {P}_nu_0-\mathcal {P}_nu_n^M\Vert _{L^2}+M\sqrt{M_2}l_1\Vert \mathcal {P}_nu_0-u_0\Vert _{L^2}\\&+c\log nM\sqrt{M_2}l_1\Vert u_0-\mathcal {P}_nu_0\Vert _{L^2}\\\le & {} c\log n\Vert u_n^M-u_0\Vert _{L^2}+c\Vert u_n^M-u_0\Vert _{L^2}+c\log n\Vert u_n^M-u_0\Vert _{L^2}\\&+cn^{-r}\Vert u_0^{(r)}\Vert _{\infty }+cn^{-r}\log n \Vert u_0^{(r)}\Vert _{\infty }\\= & {} (c+c\log n)\Vert u_n^M-u_0\Vert _{L^2}+n^{-r}(c+c\log n)\Vert u_0^{(r)}\Vert _{\infty }. \end{aligned}$$

This completes the proof. $\square$

Theorem 5.6

Let$u_{0} \in \mathcal {C}^{r}[-1,1],~r\ge 1$. Then the Legendre multi-Galerkin approximation${u}_{n}^{M,G}$of$u_{0}$satisfies the followings

$$\begin{aligned}&\Vert {u}_{n}^{M,G}-u_{0}\Vert _{L^{2}}\le {\left\{ \begin{array}{ll} c n^{\frac{1}{2}-r-\alpha },~~~~~~~~~~~~~~~~~ \mathrm{for}~ \frac{1}{2}< \alpha <1,\\ c n^{-\frac{1}{2}-r}\log n, ~~~~~~~~~~~~\mathrm{for} ~ \alpha =1, \end{array}\right. }\\&\Vert {u}_{n}^{M,G}-u_{0}\Vert _{\infty }\le (c+c\log n) cn^{-r}. \end{aligned}$$

Proof

Using Theorem-5.3, and proceeding similarly as in estimate (32), we obtain

$$\begin{aligned} \Vert {u}_{n}^{M,G}-u_{0}\Vert _{L^{2}}\le & {} \dfrac{\beta _{n}}{1-q}\nonumber \\\le & {} \dfrac{1}{1-q}\Vert (\mathcal {I}-\mathcal {T}_{n}^{{M}^{\prime }}(u_{0}))^{-1}(\mathcal {T}_{n}^{M}(u_{0})-\mathcal {T}(u_{0}))\Vert _{L^{2}}\nonumber \\\le & {} c A_2\sqrt{2}(p+1)\left\| \sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_0)-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(\mathcal {P}_n^Gu_0)\right\| _{\infty }. \end{aligned}$$

(33)

Since $\mathcal {P}_n^G$ be the orthogonal projection from the space $\mathbb {X}$ into $\mathbb {X}_n$, then we have

$$\begin{aligned} \langle {w} ,\,{(\mathcal {I}-\mathcal {P}_n^G)u} \rangle =0,~~\forall ~w\in \mathbb {X}_n. \end{aligned}$$

Using Lemma 5.4 with Hölder’s inequality and Lemma 2.3, we get

$$\begin{aligned}&\left\| \sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_0)-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(\mathcal {P}_n^Gu_0)\right\| _{\infty }\nonumber \\&\quad \le Ml_2\sqrt{M_2}\Vert \mathcal {P}_n^Gu_0-u_0\Vert _{L^2}\Vert \mathcal {P}_n^Gu_0-u_0\Vert _{\infty }+\sup _{s\in [-1,1]}|\langle {h(s,.)} ,\,{(\mathcal {P}_n^G-\mathcal {I})u_0(.)} \rangle |\nonumber \\&\quad \le cn^{\frac{1}{2}-2r}\Vert u_0^{(r)}\Vert _{\infty }V(u_0^{(r)})+\sup _{s\in [-1,1]}|\langle {h(s,.)-\phi (s,.)} ,\,{(\mathcal {P}_n^G-\mathcal {I})u_0(.)} \rangle |\nonumber \\&\quad \le cn^{\frac{1}{2}-2r}\Vert u_0^{(r)}\Vert _{\infty }V(u_0^{(r)})+\Vert h_s-\phi _s\Vert _{L^1}\Vert (\mathcal {P}_n^G-\mathcal {I})u_0\Vert _{\infty }, \end{aligned}$$

(34)

where $\phi _s\in \mathbb {X}_n$. Using Theorems 2 and 3 of [17] in estimate (34), we get

$$\begin{aligned}&\left\| \sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_0)- \sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(\mathcal {P}_n^Gu_0)\right\| _{\infty }\nonumber \\&\quad \le cn^{\frac{1}{2}-2r}\Vert u_0^{(r)}\Vert _{\infty }V(u_0^{(r)})\nonumber \\&\qquad +V(u_0^{(r)}){\left\{ \begin{array}{ll} cn^{\frac{1}{2}-r-\alpha },~~~~~~~\mathrm{for}~\frac{1}{2}<\alpha<1,\\ cn^{-\frac{1}{2}-r}\log n,~~\mathrm{for}~\alpha =1, \end{array}\right. } \nonumber \\&\quad \le {\left\{ \begin{array}{ll} cn^{\frac{1}{2}-r-\alpha },~~~~~~~\mathrm{for}~\frac{1}{2}<\alpha <1,\\ cn^{-\frac{1}{2}-r}\log n,~~\mathrm{for}~\alpha =1. \end{array}\right. } \end{aligned}$$

(35)

Combining estimates (33) and (35), we obtain

$$\begin{aligned} \Vert u_{n}^{M,G}-u_{0}\Vert _{L^{2}}\le {\left\{ \begin{array}{ll} cn^{\frac{1}{2}-r-\alpha },~~~~~~~\mathrm{for}~\frac{1}{2}<\alpha <1,\\ cn^{-\frac{1}{2}-r}\log n,~~\mathrm{for}~\alpha =1. \end{array}\right. } \end{aligned}$$

(36)

Again, from Lemma 5.5 and using estimate (36), we get

$$\begin{aligned} \Vert u_{n}^{M,G}-u_{0}\Vert _{\infty }\le & {} (c+c\log n){\left\{ \begin{array}{ll} cn^{\frac{1}{2}-r-\alpha },~~~~~~~\mathrm{for}~\frac{1}{2}<\alpha <1,\\ cn^{-\frac{1}{2}-r}\log n,~~\mathrm{for}~\alpha =1, \end{array}\right. }\\&+ n^{-r}(c+c\log n)\Vert u_{0}^{(r)}\Vert _{\infty }\\\le & {} (c+c\log n) cn^{-r}. \end{aligned}$$

Hence, the proof is completed. $\square$

Theorem 5.7

Let$u_{0} \in \mathcal {C}^{r}[-1,1],~r\ge 1$. Then the Legendre multi-collocation approximation${u}_{n}^{M,C}$of$u_{0}$satisfies the following

$$\begin{aligned} \Vert {u}_{n}^{M,C}-u_{0}\Vert _{L^{2}}\le cn^{-r},\,\,\,\,\,\Vert {u}_{n}^{M,C}-u_{0}\Vert _{\infty }\le c(c+c\log n)n^{-r}. \end{aligned}$$

Proof

As proceeding similarly as in equation (33), we obtain

$$\begin{aligned} \Vert {u}_{n}^{M,C}-u_{0}\Vert _{L^{2}} \le c\left\| \sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_0)-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(\mathcal {P}_n^Cu_0)\right\| _{\infty }. \end{aligned}$$

(37)

Using Lemmas 5.4 and 2.3 with Cauchy-Schwarz inequality, we get

$$\begin{aligned}&\left\| \sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_0)-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(\mathcal {P}_n^Cu_0)\right\| _{\infty }\nonumber \\&\quad \le Ml_2\sqrt{M_2}cn^{\frac{1}{2}-2r}\Vert u_0^{(r)}\Vert _{\infty }^2+\sup _{s\in [-1,1]}\Vert h_s\Vert _{L^2}\Vert (\mathcal {P}_n^C-\mathcal {I})u_0\Vert _{L^2}\nonumber \\&\quad \le Ml_2\sqrt{M_2}cn^{\frac{1}{2}-2r}\Vert u_0^{(r)}\Vert _{\infty }^2+\sup _{s\in [-1,1]}\Vert h_s\Vert _{L^2}cn^{-r}\Vert u_0^{(r)}\Vert _{\infty }\le cn^{-r}. \end{aligned}$$

(38)

Now combining estimates (37) and (38), we obtain

$$\begin{aligned} \Vert u_{n}^{M,C}-u_0\Vert _{L^2}\le cn^{-r}. \end{aligned}$$

(39)

Substituting estimate (39) in Lemma 5.5, we get

$$\begin{aligned} \Vert u_{n}^{M,C}-u_{0}\Vert _{\infty }\le & {} (c+c\log n)cn^{-r}+ n^{-r}(c+c\log n)\Vert u_{0}^{(r)}\Vert _{\infty }\\\le & {} (c+c\log n)cn^{-r}. \end{aligned}$$

This completes the proof. $\square$

6 Iterated Legendre multi-projection methods

In this section, we discuss on the iterated Legendre spectral multi-projection methods for weakly singular Hammerstein integral equations of mixed type in both $L^2$ and infinity norm.

The iterated approximate solution $\widetilde{u}_{n}^{M}$ corresponding to the approximate solution $u^M_n$ given by equation (26) is defined as follows:

$$\begin{aligned} \widetilde{u}_{n}^{M}=\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)({u}_{n}^{M})+f. \end{aligned}$$

(40)

To obtain the supercovergence results for the iterated approximate solution $\widetilde{u}_{n}^{M}$ to the exact solution $u_{0}$, we need the following lemmas.

Lemma 6.1

For any$x,y,z\in \mathbb {X}$, the following holds

$$\begin{aligned}&\left\| \Big [\sum _{i=1}^{m}(\mathcal {K}_{n,i}^{M}\psi _i)^{\prime }(x)- \sum _{i=1}^{m}(\mathcal {K}_{n,i}^{M}\psi _i)^{\prime }(y)\Big ](z) \right\| _{\infty }\\&\quad \le c\log n(c+c\log n)\Vert x-y\Vert _{L^2}\Vert z\Vert _{\infty }. \end{aligned}$$

Proof

Using $\Vert \mathcal {P}_n\Vert _{\infty }\le c\log n$ (cf., Page-147, [3]) and estimate (11), we obtain

$$\begin{aligned}&\left\| \Big [\sum _{i=1}^{m}(\mathcal {K}_{n,i}^{M}\psi _i)^{\prime }(x) -\sum _{i=1}^{m}(\mathcal {K}_{n,i}^{M}\psi _i)^{\prime }(y)\Big ](z)\right\| _{\infty }\nonumber \\&\quad \le \left\| \mathcal {P}_n\Big [\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime }(x)-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime }(y)\Big ]z\right\| _{\infty }\nonumber \\&\qquad +\left\| (\mathcal {I}-\mathcal {P}_n)\Big [\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime }(\mathcal {P}_nx) -\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime }(\mathcal {P}_ny)\Big ]\mathcal {P}_nz\right\| _{\infty }\nonumber \\&\quad \le c\log n \left\| \Big [\sum _{i=1}^{m}(\mathcal {K}_{i}\psi _i)^{\prime }(x)-\sum _{i=1}^{m}(\mathcal {K}_{i}\psi _i)^{\prime }(y)\Big ](z)\right\| _{\infty }\nonumber \\&\qquad +(1+c\log n)\left\| \Big [\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime }(\mathcal {P}_nx)-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)^{\prime }(\mathcal {P}_ny)\Big ] \mathcal {P}_nz\right\| _{\infty }\nonumber \\&\quad \le c\log n\sqrt{M_2}Ml_2\Vert x-y\Vert _{L^2}\Vert z\Vert _{\infty }\nonumber \\&\qquad +(1+c\log n)\sqrt{M_2}Ml_2\Vert \mathcal {P}_nx-\mathcal {P}_ny\Vert _{L^2}\Vert \mathcal {P}_nz\Vert _{\infty }\nonumber \\&\quad \le c\log n\sqrt{M_2}Ml_2\Vert x-y\Vert _{L^2}\Vert z\Vert _{\infty }\nonumber \\&\qquad +(1+c\log n)\sqrt{M_2}Ml_2pc\log n\Vert x-y\Vert _{L^2}\Vert z\Vert _{\infty }\nonumber \\&\quad = c\log n(c+c\log n)\Vert x-y\Vert _{L^2}\Vert z\Vert _{\infty }. \end{aligned}$$

(41)

The proof is completed. $\square$

Theorem 6.2

Let$u_{0}\in \mathcal {C}[-1,1]$. Let$\widetilde{u}_{n}^{M}$be the iterated approximation of$u_{0}.$Then the following holds

$$\begin{aligned}&\Vert \widetilde{u}_n^{M}-u_0\Vert _{\infty }\\&\quad \le [c+c\log n(c+c\log n)]\Vert u_n^M-u_0\Vert _{L^2}\Vert u_n^M-u_0\Vert _{\infty }+(1+c\log n)\nonumber \\&\qquad \times \sup _{s\in [-1,1]} \Big | \langle h(s,.),(\mathcal {P}_n-\mathcal {I})\Big [\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(\mathcal {P}_{n}u_{0})-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_{0})\Big ](.)\rangle \Big |. \end{aligned}$$

Proof

The steps of the proof follows similarly as in Theorem 4.1 of [16]. So, we omit it. $\square$

Theorem 6.3

Let$u_{0}\in \mathcal {C}^{r}[-1,1],~r\ge 1$. Then the iterated Legendre multi-Galerkin approximation$\widetilde{u}_{n}^{M,G}$of$u_{0}$satisfies the followings

$$\begin{aligned} \Vert \widetilde{u}_{n}^{M,G}-u_{0}\Vert _{L^{2}}\le & {} \sqrt{2}\Vert \widetilde{u}_{n}^{M,G}-u_{0}\Vert _{\infty }\le (1+c\log n)^2{\left\{ \begin{array}{ll} cn^{\frac{1}{2}-r-2\alpha },~~~~~~~~~\mathrm{for}~\frac{1}{2}<\alpha <1,\\ cn^{-\frac{3}{2}-r}(\log n)^2,~~\mathrm{for}~\alpha =1. \end{array}\right. } \end{aligned}$$

Proof

From Theorem 6.2, we have

$$\begin{aligned} \Vert \widetilde{u}_{n}^{M,G}-u_{0} \Vert _{\infty }\le & {} [c+c\log n(c+c\log n)] \Vert {u}_{n}^{M,G}-u_{0}\Vert _{L^{2}}\Vert {u}_{n}^{M,G}\nonumber \\&-u_{0}\Vert _{\infty }+(1+c\log n)\nonumber \\&\times \sup _{s\in [-1,1]} \Big | \langle h(s,.),(\mathcal {P}_n^G-\mathcal {I})\nonumber \\&\Big [\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_{0})-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(\mathcal {P}_{n}^Gu_{0})\Big ](.)\rangle \Big |. \end{aligned}$$

(42)

Since $\mathcal {P}_n^G$ be the orthogonal projection from the space $\mathbb {X}$ into $\mathbb {X}_n$, then we have

$$\begin{aligned} \langle {w} ,\,{(\mathcal {I}-\mathcal {P}_n^G)u} \rangle =0,~~\forall ~w\in \mathbb {X}_n. \end{aligned}$$

By applying Hölder’s inequality, estimate (35), and Theorems 2 and 3 of [17], we obtain

$$\begin{aligned}&\sup _{s\in [-1,1]} \Big | \langle {h(s,.)} ,\,{(\mathcal {P}_n^G-\mathcal {I}) \Big [\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_{0})- \sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(\mathcal {P}_{n}^Gu_{0})\Big ](.)} \rangle \Big |\nonumber \\&\quad \le \Big |\langle {h_s-\phi _s} ,\,{(\mathcal {P}_n^G-\mathcal {I})\Big [\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_{0})-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(\mathcal {P}_{n}^Gu_{0})\Big ](.)} \rangle \Big |\nonumber \\&\quad \le \Vert h_s-\phi _s\Vert _{L^1}\left\| (\mathcal {P}_n^G-\mathcal {I})\Big [\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_{0})-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(\mathcal {P}_{n}^Gu_{0})\Big ]\right\| _{\infty }\nonumber \\&\quad \le (1+c\log n)\left\| \sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_{0})\right. \nonumber \\&\qquad \left. -\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(\mathcal {P}_{n}^Gu_{0})\right\| _{\infty }{\left\{ \begin{array}{ll} cn^{-\alpha },~~~~~~~~\mathrm{for}~\frac{1}{2}<\alpha<1,\\ cn^{-1}\log n,~~\mathrm{for}~\alpha =1, \end{array}\right. }\nonumber \\&\quad \le (1+c\log n){\left\{ \begin{array}{ll} cn^{\frac{1}{2}-r-2\alpha },~~~~~~~~~\mathrm{for}~\frac{1}{2}<\alpha <1,\\ cn^{-\frac{3}{2}-r}(\log n)^2,~~\mathrm{for}~\alpha =1, \end{array}\right. } \end{aligned}$$

(43)

where $\phi _s \in \mathbb {X}_n$. Substituting estimate (43) and Theorem 5.6 in equation (42), we obtain

$$\begin{aligned}&\Vert \widetilde{u}_{n}^{M,G}-u_{0} \Vert _{L^2}\\&\quad \le \sqrt{2}\Vert \widetilde{u}_{n}^{M,G}-u_{0} \Vert _{\infty }\\&\quad \le [c+c \log n(c+c\log n)] {\left\{ \begin{array}{ll} cn^{\frac{1}{2}-r-\alpha },~~~~~~~\mathrm{for}~\frac{1}{2}<\alpha<1,\\ cn^{-\frac{1}{2}-r}\log n,~~\mathrm{for}~\alpha =1. \end{array}\right. }\nonumber \\&\qquad \times (c+c\log n)cn^{-r}+(1+c\log n)^2{\left\{ \begin{array}{ll} cn^{\frac{1}{2}-r-2\alpha },~~~~~~~~~\mathrm{for}~\frac{1}{2}<\alpha<1,\\ cn^{-\frac{3}{2}-r}(\log n)^2,~~\mathrm{for}~\alpha =1, \end{array}\right. }\nonumber \\&\quad \le (1+c\log n)^2{\left\{ \begin{array}{ll} cn^{\frac{1}{2}-r-2\alpha },~~~~~~~~~\mathrm{for}~\frac{1}{2}<\alpha <1,\\ cn^{-\frac{3}{2}-r}(\log n)^2,~~\mathrm{for}~\alpha =1. \end{array}\right. } \end{aligned}$$

This completes the proof. $\square$

Theorem 6.4

Let$u_{0}\in \mathcal {C}^{r}[-1,1],~ r\ge 1$. Then the iterated Legendre multi-collocation approximation$\widetilde{u}_{n}^{M,C}$ of $u_{0}$satisfies the following

$$\begin{aligned} \Vert \widetilde{u}_{n}^{M,C}-u_{0}\Vert _{\infty }\le c(1+c\log n)n^{-r}. \end{aligned}$$

Proof

From Theorem 6.2, we have

$$\begin{aligned} \Vert \widetilde{u}_{n}^{M,C}-u_{0} \Vert _{\infty }\le & {} [c+c\log n(c+c\log n)] \Vert {u}_{n}^{M,C}-u_{0}\Vert _{L^{2}}\Vert {u}_{n}^{M,C}\nonumber \\&-u_{0}\Vert _{\infty }+(1+c\log n)\nonumber \\&\times \sup _{s\in [-1,1]} \Big | \langle h(s,.), (\mathcal {P}_n^C-\mathcal {I})\Big [\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_{0})\nonumber \\&-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(\mathcal {P}_{n}^Cu_{0})\Big ](.)\rangle \Big |. \end{aligned}$$

(44)

Using Cauchy-Schwarz inequality, Lemmas 2.3 and 5.4, we obtain

$$\begin{aligned}&\sup _{s\in [-1,1]} \Big | \langle h(s,.),(\mathcal {P}_n^C-\mathcal {I})\Big [\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_{0})-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(\mathcal {P}_{n}^Cu_{0})\Big ](.)\rangle \Big |\nonumber \\&\quad \le \sup _{s\in [-1,1]} \Vert h_s\Vert _{L^2}\left\| (\mathcal {P}_n^C-\mathcal {I})\Big [\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_{0})-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(\mathcal {P}_{n}^Cu_{0})\Big ]\right\| _{L^2}\nonumber \\&\quad \le \sqrt{2}(1+p)\sup _{s\in [-1,1]} \Vert h_s\Vert _{L^2}\left\| \sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(u_{0})-\sum _{i=1}^{m}(\mathcal {K}_i\psi _i)(\mathcal {P}_{n}^Cu_{0})\right\| _{\infty }\nonumber \\&\quad \le \sqrt{2}(1+p)\sup _{s\in [-1,1]} \Vert h_s\Vert _{L^2}\Big [ Ml_2\sqrt{M_2}\Vert \mathcal {P}_n^Cu_0-u_0\Vert _{L^2}\Vert \mathcal {P}_n^Cu_0-u_0\Vert _{\infty }\nonumber \\&\qquad +\sup _{s\in [-1,1]}|\langle {h(s,.)} ,\,{(\mathcal {P}_n^C-\mathcal {I})u_0(.)} \rangle |\Big ]\nonumber \\&\quad \le cn^{\frac{1}{2}-2r}\Vert u_0^{(r)}\Vert _{\infty }V(u_0^{(r)})+\sup _{s\in [-1,1]}\Vert h_s\Vert _{L^2}\Vert (\mathcal {P}_n^C-\mathcal {I})u_0\Vert _{L^2}\nonumber \\&\quad \le cn^{\frac{1}{2}-2r}\Vert u_0^{(r)}\Vert _{\infty }V(u_0^{(r)})+\sup _{s\in [-1,1]}\Vert h_s\Vert _{L^2}cn^{-r}\Vert u_0^{(r)}\Vert _{\infty }. \end{aligned}$$

(45)

Substituting estimate (45) and Theorem 5.7 in equation (44), we obtain

$$\begin{aligned} \Vert \widetilde{u}_{n}^{M,C}-u_{0} \Vert _{\infty }\le & {} [c+c\log n(c+c\log n)]cn^{-r}cn^{-r}(c+c\log n)+(1+c\log n)\\&\times \Big [ cn^{\frac{1}{2}-2r}\Vert u_0^{(r)}\Vert _{\infty }V(u_0^{(r)})+\sup _{s\in [-1,1]}\Vert h_s\Vert _{L^2}cn^{-r}\Vert u_0^{(r)}\Vert _{\infty }\Big ]\\\le & {} c(1+c\log n)n^{-r}. \end{aligned}$$

Hence, the result. $\square$

Remark 6.5

Let $u_n^{G}$, $u_n^{C}$, $\widetilde{u}_n^{G}$, $\widetilde{u}_n^{C}$ be the Legendre Galerkin, Legendre collocation, iterated Legendre Galerkin and iterated Legendre collocation approximations of u, respectively. Let $u_n^{M,G}$, $u_n^{M,C}$, $\widetilde{u}_n^{M,G}$, $\widetilde{u}_n^{M,C}$ be the Legendre multi-Galerkin, Legendre multi-collocation, iterated Legendre multi-Galerkin, iterated Legendre multi-collocation approximations of u.

From Theorems-3.4, 4.2, 4.3, 5.6, 5.7, 6.3 and 6.4, we observe the following convergence rates in the respective methods.

Legendre projection methods:

$$\begin{aligned}&\Vert u_n^G-u_0\Vert _{L^2} \le cn^{-r}, \quad \qquad \Vert u_n^C-u_0\Vert _{L^2} \le cn^{-r} ,\\&\Vert u_n^G-u_0\Vert _{\infty } \le cn^{\frac{1}{2}-r}, \quad \qquad \Vert u_n^C-u_0\Vert _{\infty } \le cn^{\frac{1}{2}-r}. \end{aligned}$$

Iterated Legendre projection methods:

$$\begin{aligned}&\Vert \widetilde{u}_n^G-u_0\Vert _{L^2} \le {\left\{ \begin{array}{ll} cn^{\frac{1}{2}-r-\alpha }(1+c\log n),~~~~~~~~\mathrm{for}~\frac{1}{2}<\alpha<1,\\ cn^{-\frac{1}{2}-r}(1+c\log n)\log n,~~~\mathrm{for}~\alpha =1, \end{array}\right. }\\&\Vert \widetilde{u}_n^G-u_0\Vert _{\infty } \le {\left\{ \begin{array}{ll} cn^{\frac{1}{2}-r-\alpha }(1+c\log n),~~~~~~~~\mathrm{for}~\frac{1}{2}<\alpha <1,\\ cn^{-\frac{1}{2}-r}(1+c\log n)\log n,~~~\mathrm{for}~\alpha =1, \end{array}\right. }\\&\Vert \widetilde{u}_n^C-u_0\Vert _{L^2}\le cn^{-r}(1+c\log n),\\&\Vert \widetilde{u}_n^C-u_0\Vert _{\infty }\le cn^{-r}(1+c\log n). \end{aligned}$$

Legendre multi-projection methods:

$$\begin{aligned}&\Vert u_n^{M,G}-u_0\Vert _{L^2} \le {\left\{ \begin{array}{ll} c n^{\frac{1}{2}-r-\alpha },\qquad \qquad \mathrm{for}~ \frac{1}{2}< \alpha <1,\\ c n^{-\frac{1}{2}-r}\log n, \qquad ~ \mathrm{for} ~ \alpha =1, \end{array}\right. }\\&\Vert {u}_{n}^{M,G}-u_{0}\Vert _{\infty }\le (c+c\log n) cn^{-r}.\\&\Vert {u}_{n}^{M,C}-u_{0}\Vert _{L^{2}}\le cn^{-r},\\&\Vert {u}_{n}^{M,C}-u_{0}\Vert _{\infty }\le c(c+c\log n)n^{-r}. \end{aligned}$$

Iterated Legendre multi-projection methods:

$$\begin{aligned} \Vert \widetilde{u}_{n}^{M,G}-u_{0}\Vert _{L^{2}}\le & {} \sqrt{2}\Vert \widetilde{u}_{n}^{M,G}-u_{0}\Vert _{\infty }\\\le & {} (1+c\log n)^2{\left\{ \begin{array}{ll} cn^{\frac{1}{2}-r-2\alpha },~~~~~~~~~\mathrm{for}~\frac{1}{2}<\alpha <1,\\ cn^{-\frac{3}{2}-r}(\log n)^2,~~\mathrm{for}~\alpha =1, \end{array}\right. }\\ \Vert \widetilde{u}_{n}^{M,C}-u_{0}\Vert _{L^2}\le & {} \sqrt{2}\Vert \widetilde{u}_{n}^{M,C}-u_{0}\Vert _{\infty }\le c(1+c\log n)n^{-r}. \end{aligned}$$

We observe that

• 1. Iterated Galerkin improves over Galerkin, multi-Galerkin improves over Iterated Galerkin and Iterated multi-Galerkin improves over multi-Galerkin in $L^2$ norm using Legendre polynomial bases.

• 2. However, in infinity norm, multi-Galerkin improves over Galerkin and iterated multi-Galerkin improves over iterated Galerkin using Legendre polynomial bases.

• 3. In collocation method, no improvement recorded from collocation to multi-collocation and iterated collocation to iterated multi-collocation in $L^2$ norm.

• 4. In infinity norm, multi-collocation improves over collocation method. However, there is no improvement from iterated collocation to iterated multi-collocation method.