1 Introduction

Here in this article, we consider the linear Fredholm integral equation of the second kind on the half-line as follows:

$$\begin{aligned} u(\zeta )-\int _{0}^{\infty }k(\zeta , \xi )u(\xi )\text {d}\xi =y(\zeta ),~\zeta \in [0, \infty ). \end{aligned}$$
(1.1)

Integral equations on unbounded intervals are important class of integral equations along with increased interest in the applications to the scientific and technological problems (Atkinson 1969, Anselone and Sloan 1985, 1987, 1988, Chandler and Graham 1987, Amini and Sloan 1989, Guo 1998, Edmond 2008, Parand and Babolghani 2012). Numerical methods for integral equation of type (1.1) have attracted a lot of attention. To find the approximate solution of integral Eq. (1.1), many authors discussed finite section approximation method (Anselone and Sloan 1985, 1987, 1988, Chandler and Graham 1987, Graham and Mendes 1989, Amini and Sloan 1989). In finite section approximation method instead of solving integral Eq. (1.1) on \([0, \infty ),\) one solves the integral equation on \([0, \beta ],\) the bounded subinterval of \([0, \infty ),\) for sufficiently large \(\beta >0,\) using piecewise polynomial basis functions and then letting \(\beta \rightarrow \infty \). In Chandler and Graham (1987), authors approximated the finite section integral operator by Nyström method to solve integral Eq. (1.1) and had shown that the Nyström solution converges to the exact solution with order \(\mathcal {O}(n^{-r}),\) in infinity norm, where r is the order of the rule. However, the finite section approximation methods are computationally very expensive, since approximating basis functions are piecewise polynomials on \([0, \beta ], \beta \rightarrow \infty \) and we need finer grid to get better accuracy. Numerical methods for the half-line integral equations without considering finite section approximation methods have been studied in many papers, for instance (Mastroianni and Monegato 1997, Long et al. 2008, Xuan and Lin 2012, Rahmoune 2013). In Mastroianni and Monegato (1997), discussed the Nyström method using ad hoc product quadrature rules of interpolatory type using the zeros of the Laguerre polynomials of degree n for solving integral Eq. (1.1) under certain assumptions on the kernel function and showed that the approximate solution converges with order \(\mathcal {O}(n^{-\frac{r}{2}})\), in infinity norm, where \(r\ge 1.\) In Long et al. (2008), discussed the Galerkin method with Laguerre polynomials to solve integral Eq. (1.1) for degenerate kernels, but they did not derive any error bounds. In Xuan and Lin (2012), applied Clenshaw–Curtis rational (CCR) quadrature rule and approximate the solution of the Wiener–Hopf integral equation of second kind (1.1). In Xuan and Lin (2012), authors first transformed integral Eq. (1.1) into an integral equation on finite interval by substituting the variables and then apply the Clenshaw–Curtis quadrature to obtain the approximate solution, although the coordinate transformation leads the kernel to a singular kernel. In Rahmoune (2013), the authors discussed the spectral collocation method based on scaled Laguerre polynomials to approximate the solution of Eq. (1.1) under suitable assumptions on the kernel function but no convergence analysis discussed therein.

The main motivation to this article is to reduce the computational complexity and obtaining the superconvergence results for Eq. (1.1) using global polynomials instead of using piecewise polynomials. Since Eq. (1.1) is defined on \([0, \infty )\) and Laguerre polynomials are the only orthogonal polynomials on the half-line \([0, \infty ),\) we use these polynomials as approximating subspace. Laguerre polynomials are one of the most useful class of orthonormal polynomials in the weighted space \(L_{\omega }^{2}(0, \infty )\) with \(\omega (\xi )=e^{-\xi }\) (Guo 1998, Sánchez-Ruiz 2003, Valenciano 2005, Guo et al. 2006, Gülsu et al. 2011). Also these polynomials can be generated recursively using three terms recurrence relation with no difficulty and hold nice property of orthogonality. Due to the orthogonal property of Laguerre polynomials on \([0, \infty )\) with exponential weight functions, it appropriately fits the solution of Eq. (1.1) and it gives global approximation of the solution in the whole domain \([0, \infty ).\) Also use of Laguerre polynomials leads to a smaller algebraic linear system which is highly desirable in computations in comparison with piecewise polynomial-based methods.

Here in this article, we discuss the convergence rates for the approximate solutions of Eq. (1.1) in Galerkin method and its iterated version using Laguerre polynomials as basis functions. We prove that the approximate solution in Galerkin method converges to the exact solution of Eq. (1.1) with order \(\mathcal {O}(n^{-\frac{r}{2}})\) in weighted \(L^{2}-\)norm and the approximate solution in iterated-Galerkin method converges with order \(\mathcal {O}(n^{-r})\) in both infinity and weighted \(L^{2}-\)norms, where r and n denotes the smoothness of the solution and the highest degree of the Laguerre polynomials used in the approximation, respectively. In Kulkarni 2003, Kulkarni and Nelakanti 2004, Chen et al. 2007) a multi-projection method to solve the linear Fredholm integral equation of second kind is proposed to obtain superconvergence results over iterated projection method under the similar assumptions of classical projection methods. The idea behind the multi-projection method is to make an effective method and to define multi-projection operator to get a better approximation of the integral operator. We also obtain the superconvergence results in multi-Galerkin and iterated multi-Galerkin methods. Indeed, we are able to show that the approximate solutions in multi-Galerkin and iterated multi-Galerkin methods converge with orders \(\mathcal {O}(n^{-\frac{3r}{2}})\) and \(\mathcal {O}(n^{-2r}),\) respectively, in weighted \(L^{2}-\)norm which shows superconvergence results over iterated-Galerkin method in weighted \(L^{2}-\)norm. We discuss our theoretical results by numerical examples. We may assume that c is a generic constant which may have different values wherever it appears in the text.

The paper is organized as follows. In Sect. 2, we discuss the Galerkin and iterated Galerkin methods using Laguerre polynomials and discuss the convergence results for the solution of Eq. (1.1). In Sect. 3, we apply the multi-Galerkin, iterated multi-Galerkin methods to integral Eq. (1.1) and give superconvergence results. In Sect. 4, numerical examples are presented to validate the theoretical results.

2 Galerkin and iterated Galerkin methods for linear Fredholm integral equation on half-line

Let \(\mathbb {X}^{+}\) be the Banach space of bounded continuous function on \(\mathbb {R}^{+}=[0, \infty )\) with supremum norm. Consider the linear Fredholm integral equation on the half-line as follows:

$$\begin{aligned} u(\zeta )-\int _{0}^{\infty }k(\zeta , \xi )u(\xi )\text {d}\xi =y(\zeta ),~\zeta \in [0, \infty ), \end{aligned}$$
(2.1)

where the kernel k(., .), and y are known sufficiently smooth functions and u is the unknown function to be approximated. Let

$$\begin{aligned} (\mathcal {K}u)(\zeta )=\int _{0}^{\infty }k(\zeta , \xi )u(\xi )\text {d}\zeta ,~u\in \mathbb {X}^{+} \end{aligned}$$
(2.2)

is a linear operator from \(\mathbb {X}^{+}\) into \(\mathbb {X}^{+}\). Using (2.2), Eq. (2.1) becomes

$$\begin{aligned} u(\zeta )-(\mathcal {K}u)(\zeta )=y(\zeta ),~\zeta \in [0, \infty ). \end{aligned}$$
(2.3)

Throughout the paper, we make the following assumptions:

  1. (i)

    \(\underset{\xi ,\zeta \in [0, \infty )}{\sup }\left| e^{\xi }\frac{\partial ^{r}}{\partial \zeta ^{r}}k(\zeta , \xi )\right| \le c_{1}<\infty , ~\hbox {for}~r=0, 1, 2, \ldots \)

  2. (ii)

    \(\underset{\zeta \in [0, \infty )}{\sup }\left( \int _{0}^{\infty }e^{-\xi }|e^{\xi }\frac{\partial ^{r}}{\partial \zeta ^{r}}k(\zeta ,\xi )|^{2}d\xi \right) ^{\frac{1}{2}}\le c_{2}<\infty ,~\hbox {for}~r=0, 1, 2, \dots \)

  3. (iii)

    \(\underset{\xi ,\zeta \in [0, \infty )}{\sup }\left| \frac{\partial ^{r}}{\partial \xi ^{r}}e^{\xi }k(\zeta , \xi )\right| \le c_{3}<\infty ,~\hbox {for}~r=0, 1, 2, \dots \)

Next, we consider the weight function \(\omega (\xi )=e^{-\xi }\) and the following weighted space

$$\begin{aligned} L^{2}_{\omega }(0, \infty )=\{v~|~ v\hbox { is measurable on }(0,\infty )\hbox { and}~\Vert v\Vert _{L^{2}_{\omega }}<\infty \}, \end{aligned}$$

with the inner product and norm, respectively,

$$\begin{aligned} (v, z)_{\omega }=\int _{0}^{\infty }\omega (\xi )v(\xi )z(\xi )\text {d}\xi ~~\hbox {and}~~ \Vert v\Vert _{\omega }=(v, v)_{\omega }^{1/2}. \end{aligned}$$

Here we assume that \((\mathcal {I}-\mathcal {K})^{-1}\) exists and bounded in \(\mathbb {X}^{+}\) and in \(B(L^{2}_{\omega }(0, \infty )),\) where \(B(L^{2}_{\omega }(0, \infty ))\) is the set of all bounded linear operators from \(L^{2}_{\omega }(0, \infty )\) into \(L^{2}_{\omega }(0, \infty )\) , i.e., there exist \( M_{1}, M_{2}>0\) s.t \(\Vert (\mathcal {I}-\mathcal {K})^{-1}\Vert _{\infty }\le M_{1}<\infty ,\) and \(\Vert (\mathcal {I}-\mathcal {K})^{-1}\Vert _{\omega }\le M_{2}\), see (Amini and Sloan 1989, Rahmoune 2013) for details.

Next, consider the scaled Laguerre polynomials of degree n defined by

$$\begin{aligned} \mathcal {L}_{n}(\xi )=\frac{1}{n!}e^{\xi }\partial ^{n}_{\xi }(\xi ^{n}e^{-\xi }),~~n=0, 1, 2, \ldots , \end{aligned}$$

where these polynomials satisfy the following recurrence relation:

$$\begin{aligned} (n+1)\mathcal {L}_{n+1}(\xi )=(2n+1-\xi )\mathcal {L}_{n}(\xi )-n\mathcal {L}_{n-1}(\xi ),~n\ge 1, \end{aligned}$$

and the orthogonality

$$\begin{aligned} (\mathcal {L}_{n}(\xi ), \mathcal {L}_{m}(\xi ))_{\omega }=\delta _{nm}. \end{aligned}$$

Using the above Laguerre polynomials, we choose our approximating subspace of \(\mathbb {X}^{+}\) as \(\mathbb {X}_{n}=span\{\mathcal {L}_{0}, \mathcal {L}_{1}, \ldots , \mathcal {L}_{n}\}\), the space of all Laguerre polynomials of degree \(\le n.\)

Next for any integer \(r\ge 0\), we define the non-uniformly weighted space \(A^{r}(0, \infty )\) as follows (Guo et al. 2006):

$$\begin{aligned} A^{r}(0, \infty )=\{v ~|v\hbox { is measurable on }(0,\infty )\hbox { and}~ \Vert v\Vert _{A^{r}(0, \infty )}<\infty \}, \end{aligned}$$

equipped with the following semi-norm and norm, respectively,

$$\begin{aligned} |v|_{A^{r}(0, \infty )}=\Vert \partial ^{r}_{\xi }v\Vert _{\omega _{r}}~~\hbox {and}~~\Vert v\Vert _{A^{r}(0, \infty )}=\left( \displaystyle \sum _{j=0}^{r}|v|^{2}_{A^{j}(0, \infty )}\right) ^{\frac{1}{2}}, \end{aligned}$$

where

$$\begin{aligned} \Vert \partial ^{r}_{\xi }v\Vert _{\omega _{r}}= \left( \int _{0}^{\infty }\xi ^{r}e^{-\xi }|\partial ^{r}_{\xi }v(\xi )|^{2}\text {d}\xi \right) ^{\frac{1}{2}}. \end{aligned}$$

Note that for any \(v \in L^{2}_{\omega }(0, \infty )\)

$$\begin{aligned} \Vert v\Vert _{A^{0}(0, \infty )}=\Vert v\Vert _{\omega }. \end{aligned}$$
(2.4)

Throughout the paper, we consider that the unique solution say u of integral Eq. (1.1) is in \(A^{r}(0, \infty )\) for \(r\ge 1,\) i.e.,

$$\begin{aligned} |u|_{A^{r}(0, \infty )}=\left( \int _{0}^{\infty }\xi ^{r}e^{-\xi }|\partial ^{r}_{\xi }u(\xi )|^{2}\text {d}\xi \right) ^{\frac{1}{2}}<\infty . \end{aligned}$$

2.1 Orthogonal projection

Define the orthogonal projection \(\pi _{n}^{\omega }:L^{2}_{\omega }(0, \infty )\rightarrow \mathbb {X}_{n}\) by

$$\begin{aligned} (\pi _{n}^{\omega }u-u, \phi )_{\omega }=0,~\hbox {for all} ~\phi \in \mathbb {X}_{n}, ~ u \in L^{2}_{\omega }(0, \infty ). \end{aligned}$$
(2.5)

The crucial properties of \(\pi _{n}^{\omega }\) (Guo 1998, Guo et al. 2006), which we need in our analysis are

for any \(v \in A^{r}(0, \infty )\),

  1. (i)

    \(\Vert \pi _{n}^{\omega }v-v\Vert _{A^{r'}(0, \infty )}\le cn^{\frac{r'-r}{2}}|v|_{A^{r}(0, \infty )}, 0\le r'<r\);

  2. (ii)

    \(\Vert \pi _{n}^{\omega }\Vert _{\omega }\le c\).

From Eq. (2.4) and (i),  it follows that

$$\begin{aligned} \Vert \pi _{n}^{\omega }v-v\Vert _{A^{0}(0, \infty )}=\Vert \pi _{n}^{\omega }v-v\Vert _{\omega }\le cn^{-\frac{r}{2}}|v|_{A^{r}(0, \infty )}, ~\hbox {for}~r\ge 1. \end{aligned}$$
(2.6)

Below we give the following Lemma, which is very useful for our convergence results.

Lemma 1

(Mandal and Nelakanti 2017) Let X be a Banach space and T be a bounded linear operator on X, such that \((I-T)^{-1} \in B(X\)), where B(X) be the set of all bounded linear operators from X into X. Consider \(\{T_{n}\} \in B(X)\) and \(\Vert T-T_{n}\Vert \rightarrow 0\), then for large n,  \((I-T_{n})^{-1} \in B(X)\).

The Galerkin method for integral Eq. (2.3) is to find \(u_{n}\in \mathbb {X}_{n},\) such that

$$\begin{aligned} u_{n}-\pi _{n}^{\omega }\mathcal {K}u_{n}=\pi _{n}^{\omega }y. \end{aligned}$$
(2.7)

Next define the iterated solution by

$$\begin{aligned} \widetilde{u}_{n}=\mathcal {K}u_{n}+y. \end{aligned}$$
(2.8)

Applying \(\pi _{n}^{\omega }\) on both sides of Eq. (2.8), we have

$$\begin{aligned} \pi _{n}^{\omega }\widetilde{u}_{n}=\pi _{n}^{\omega }\mathcal {K}u_{n}+\pi _{n}^{\omega }y. \end{aligned}$$
(2.9)

From (2.7) and (2.9), we obtain

$$\begin{aligned} \pi _{n}^{\omega }\widetilde{u}_{n}=u_{n}, \end{aligned}$$
(2.10)

so \(u_{n}\) is the projection of \(\widetilde{u}_{n}\) into \(\mathbb {X}_{n}.\) Substituting Eq. (2.10) into (2.8), we obtain that \(\widetilde{u}_{n}\) satisfies the following equation:

$$\begin{aligned} (\mathcal {I}-\mathcal {K}\pi _{n}^{\omega })\widetilde{u}_{n}=y. \end{aligned}$$
(2.11)

2.2 Convergence results

Here in this, we discuss the convergence analysis of Galerkin and iterated Galerkin solutions. For this, we denote

\(l_{r}(\zeta , \xi )=e^{\xi }\frac{\partial ^{r}}{\partial \zeta ^{r}}k(\zeta , \xi ),\) \(\xi , \zeta \in \mathbb {R}^{+}\), for \(r=0, 1, 2, \ldots \)

Note that, \(\left( \int _{0}^{\infty }e^{-\xi }\text {d}\xi \right) =1\), hence for any \(v \in \mathbb {X}^{+}\)

$$\begin{aligned} \Vert v\Vert _{\omega }=\left( \int _{0}^{\infty }e^{-\xi }|v(\xi )|^{2}\text {d}\xi \right) ^{\frac{1}{2}}\le \Vert v\Vert _{\infty }. \end{aligned}$$
(2.12)

Now for any \(v \in \mathbb {X}^{+}\) and \(j=0, 1, 2 \ldots \), we have

$$\begin{aligned} \Vert (\mathcal {K}v)^{(j)}\Vert _{\omega }\le&\Vert (\mathcal {K}v)^{(j)}\Vert _{\infty }\left( \int _{0}^{\infty }e^{-\xi }\text {d}\xi \right) ^{\frac{1}{2}}\nonumber \\ =&\Vert (\mathcal {K}v)^{(j)}\Vert _{\infty }\nonumber \\ =&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\left| (\mathcal {K}v)^{(j)}(\zeta )\right| \nonumber \\ =&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\left| \int _{0}^{\infty }\frac{\partial ^{j}}{\partial \zeta ^{j}}k(\zeta , \xi )v(\xi )\text {d}\xi \right| \nonumber \\ =&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\left| \int _{0}^{\infty }e^{-\xi }e^{\xi }\frac{\partial ^{j}}{\partial \zeta ^{j}}k(\zeta , \xi )v(\xi )\text {d}\xi \right| \nonumber \\ =&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\left| \int _{0}^{\infty }e^{-\xi }l_{j}(\zeta , \xi )v(\xi )\text {d}\xi \right| \nonumber \\ \le&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\Vert l_{j}(\zeta , .)\Vert _{\omega }\Vert v\Vert _{\omega }\nonumber \\ \le&\ \,c_{2}\Vert v\Vert _{\omega } \end{aligned}$$
(2.13)
$$\begin{aligned} \le&\, c_{2}\Vert v\Vert _{\infty }. \end{aligned}$$
(2.14)

Lemma 2

Let \(\pi _{n}^{\omega }:L^{2}_{\omega }(0, \infty )\rightarrow \mathbb {X}_{n}\) be projection given by (2.5) and the linear integral operator \(\mathcal {K}\) defined by (2.2), then the following hold:

$$\begin{aligned} {\text {(i)}}\,\Vert (\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}\Vert _{\omega }=\,&\mathcal {O}(n^{-\frac{r}{2}}), \end{aligned}$$
(2.15)
$$\begin{aligned} {\text {(ii)}}\,~\Vert \mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })\Vert _{\infty } =\,&\mathcal {O}(n^{-\frac{r}{2}}), \end{aligned}$$
(2.16)

and for \(v \in A^{r}(0, \infty ), r\ge 1,\)

$$\begin{aligned} {\text {(iii)}}~\Vert \mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })v\Vert _{\infty } =\,&\mathcal {O}(n^{-r}). \end{aligned}$$
(2.17)

Proof

Note that

$$\begin{aligned} \int _{0}^{\infty }\xi ^{r}e^{-\xi }\text {d}\xi =\varGamma (r+1),~ r>0. \end{aligned}$$
(2.18)

For any \(v \in L^{2}_{\omega }(0, \infty ),\) using (2.13) and (2.18), we have

$$\begin{aligned} \Vert (\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}v\Vert _{\omega }&= \Vert (\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}v\Vert _{A^{0}(0, \infty )}\nonumber \\&\le cn^{-\frac{r}{2}}|\mathcal {K}v|_{A^{r}(0, \infty )}\nonumber \\&=cn^{-\frac{r}{2}}\Vert (\mathcal {K}v)^{(r)}\Vert _{\omega _{r}}\nonumber \\&=cn^{-\frac{r}{2}}\left( \int _{0}^{\infty }\xi ^{r}e^{-\xi }|(\mathcal {K}v)^{(r)}(\xi )|^{2}\text {d}\xi \right) ^{\frac{1}{2}}\nonumber \\&\le cn^{-\frac{r}{2}} \Vert (\mathcal {K}v)^{(r)}\Vert _{\infty }\left( \int _{0}^{\infty }\xi ^{r}e^{-\xi }\text {d}\xi \right) ^{\frac{1}{2}}\nonumber \\&=c(\varGamma (r+1))^{\frac{1}{2}}n^{-\frac{r}{2}}\Vert (\mathcal {K}v)^{(r)}\Vert _{\infty }\end{aligned}$$
(2.19)
$$\begin{aligned}&\le cc_{2}(\varGamma (r+1))^{\frac{1}{2}}n^{-\frac{r}{2}}\Vert v\Vert _{\omega } \nonumber \\&=c_{4}n^{-\frac{r}{2}}\Vert v\Vert _{\omega }, \end{aligned}$$
(2.20)

where \(c_{4}=cc_{2}(\varGamma (r+1))^{\frac{1}{2}}.\) Hence, we obtain \(\Vert (\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}\Vert _{\omega }=\mathcal {O}(n^{-\frac{r}{2}}),\) which proves (2.15).

Next for any \(v \in \mathbb {X}^{+}\), consider

$$\begin{aligned} \Vert \mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })v\Vert _{\infty }=&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\left| \mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })v(\zeta )\right| \nonumber \\ =&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\left| \int _{0}^{\infty }k(\zeta , \xi )(\mathcal {I}-\pi _{n}^{\omega })v(\xi )\text {d}\xi \right| \nonumber \\ =&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\left| \int _{0}^{\infty }e^{-\xi }e^{\xi }k(\zeta , \xi )(\mathcal {I}-\pi _{n}^{\omega })v(\xi )\text {d}\xi \right| \nonumber \\ =&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\left| (l_{0}(\zeta , .), (\mathcal {I}-\pi _{n}^{\omega })v(.))_{\omega }\right| \end{aligned}$$
(2.21)
$$\begin{aligned} =&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\left| ((\mathcal {I}-\pi _{n}^{\omega })l_{0}(\zeta , .), v(.))_{\omega }\right| \nonumber \\ \le&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\Vert (\mathcal {I}-\pi _{n}^{\omega })l_{0}(\zeta , .)\Vert _{\omega }\Vert v\Vert _{\omega }. \end{aligned}$$
(2.22)

Thus, using (2.6) and (2.12) in (2.22), we have

$$\begin{aligned} \Vert \mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })v\Vert _{\infty }&\le \underset{\zeta \in \mathbb {R}^{+}}{\sup }n^{-\frac{r}{2}}|l_{0}(\zeta , .)|_{A^{r}(0, \infty )}\Vert v\Vert _{\omega }\\&\le \underset{\zeta \in \mathbb {R}^{+}}{\sup }n^{-\frac{r}{2}}\Vert (l_{0}(\zeta , .))^{(r)}\Vert _{\omega _{r}}\Vert v\Vert _{\infty }\\&\le n^{-\frac{r}{2}}c_{3}(\varGamma (r+1))^{\frac{1}{2}}\Vert v\Vert _{\infty }\\&= c_{5}n^{-\frac{r}{2}}\Vert v\Vert _{\infty }, \end{aligned}$$

where \(c_{5}=c_{3}(\varGamma (r+1))^{\frac{1}{2}}\), this proves (2.16).

Next using (2.21) and orthogonality of \(\pi _{n}^{\omega },\) for any \(v \in A^{r}(0, \infty ), r\ge 1\), we have

$$\begin{aligned} \Vert \mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })v\Vert _{\infty }=&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\left| (l_{0}(\zeta , .), (\mathcal {I}-\pi _{n}^{\omega })v(.))_{\omega }\right| \\ =&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\left| ((\mathcal {I}-\pi _{n}^{\omega })l_{0}(\zeta , .), (\mathcal {I}-\pi _{n}^{\omega })v(.))_{\omega }\right| \\ \le&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\Vert (\mathcal {I}-\pi _{n}^{\omega })l_{0}(\zeta , .)\Vert _{\omega }\Vert (\mathcal {I}-\pi _{n}^{\omega })v\Vert _{\omega }\\ =&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\Vert (\mathcal {I}-\pi _{n}^{\omega })l_{0}(\zeta , .)\Vert _{A^{0}(0, \infty )}\Vert (\mathcal {I}-\pi _{n}^{\omega })v\Vert _{A^{0}(0, \infty )}\\ \le&\,n^{-\frac{r}{2}}|l_{0}(\zeta , .)|_{A^{r}(0, \infty )}n^{-\frac{r}{2}}|v|_{A^{r}(0, \infty )}\\ =\,&n^{-r}\underset{\zeta \in \mathbb {R}^{+}}{\sup }|l_{0}(\zeta , .)|_{A^{r}(0, \infty )}|v|_{A^{r}(0, \infty )}\\ \le&\,n^{-r}c_{3}(\varGamma (r+1))^{\frac{1}{2}}|v|_{A^{r}(0, \infty )}\\ =\,&\mathcal {O}(n^{-r}). \end{aligned}$$

\(\square \)

Lemma 3

Let \(\pi _{n}^{\omega }:L^{2}_{\omega }(0, \infty )\rightarrow \mathbb {X}_{n}\) be the projection defined by (2.5) and the operator \(\mathcal {K}\) given by (2.2) such that \((\mathcal {I}-\mathcal {K})^{-1} \in B(L^{2}_{\omega }(0, \infty ))\), then there hold

$$\begin{aligned}&(i)~\Vert (\mathcal {I}-\pi _{n}^{\omega }\mathcal {K})^{-1}\Vert _{\omega }\le C_{1}<\infty , \end{aligned}$$
(2.23)
$$\begin{aligned}&(ii)~\Vert (\mathcal {I}-\mathcal {K}\pi _{n}^{\omega })^{-1}\Vert _{\omega }\le C_{2}<\infty \end{aligned}$$
(2.24)

for large n.

Proof

Since \(\Vert \mathcal {K}-\pi _{n}^{\omega }\mathcal {K}\Vert _{\omega }\rightarrow 0\), by applying Lemma 1, (2.23) holds.

Next using the identity

$$\begin{aligned} \pi _{n}^{\omega }(\mathcal {I}-\mathcal {K}\pi _{n}^{\omega })^{-1} =(\mathcal {I}-\pi _{n}^{\omega }\mathcal {K})^{-1}\pi _{n}^{\omega }, \end{aligned}$$

and the property that \(\pi _{n}^{\omega }\) is bounded in weighted norm, it follows that \((\mathcal {I}-\mathcal {K}\pi _{n}^{\omega })^{-1} \in B(L^{2}_{\omega }(0, \infty ))\) and \(\Vert (\mathcal {I}-\mathcal {K}\pi _{n}^{\omega })^{-1}\Vert _{\omega }\le C_{2}.\) \(\square \)

Lemma 4

Let \(\pi _{n}^{\omega }:L^{2}_{\omega }(0, \infty )\rightarrow \mathbb {X}_{n}\) be projection defined by (2.5) and the operator \(\mathcal {K}\) defined by (2.2) such that \((\mathcal {I}-\mathcal {K})^{-1} \in B(\mathbb {X}^{+})\), then the following holds:

$$\begin{aligned} \Vert (\mathcal {I}-\mathcal {K}\pi _{n}^{\omega })^{-1}\Vert _{\infty }\le C_{3}<\infty . \end{aligned}$$
(2.25)

Proof

Using Lemma 2, we have

$$\begin{aligned} \Vert \mathcal {K}-\mathcal {K}\pi _{n}^{\omega }\Vert _{\infty }= \mathcal {O}(n^{-\frac{r}{2}})\rightarrow 0\,~ \ \hbox {as}~ n\rightarrow \infty . \end{aligned}$$

Hence using Lemma 1, results follows. \(\square \)

Next we give a detail discussion about the convergence results for the Galerkin method and its iterated version.

Theorem 5

Let \(u \in A^{r}(0, \infty ), r\ge 1\) be the unique solution of Eq. (2.3) and let \(u_{n}\) and \(\widetilde{u}_{n}\) be the approximate solutions of u defined as in (2.7) and (2.8), respectively. Then the following convergence rates hold

$$\begin{aligned} {(i)}&~\Vert u-u_{n}\Vert _{\omega }=\mathcal {O}(n^{-\frac{r}{2}}),\\ {(ii)}&~\Vert u-\widetilde{u}_{n}\Vert _{\omega }=\mathcal {O}(n^{-r}),\\ {(iii)}&~\Vert u-\widetilde{u}_{n}\Vert _{\infty }=\mathcal {O}(n^{-r}), \end{aligned}$$

where n is sufficiently large.

Proof

Since

$$\begin{aligned} u-u_{n}&=(\mathcal {I}-\mathcal {K})^{-1}y-(\mathcal {I}- \pi _{n}^{\omega }\mathcal {K})^{-1}\pi _{n}^{\omega }y\\&=(\mathcal {I}- \pi _{n}^{\omega }\mathcal {K})^{-1}(\mathcal {I}-\pi _{n}^{\omega })u, \end{aligned}$$

hence using Lemma 3, we have

$$\begin{aligned} \Vert u-u_{n}\Vert _{\omega }&\le \Vert (\mathcal {I}-\pi _{n}^{\omega }\mathcal {K})^{-1}(\mathcal {I}-\pi _{n}^{\omega })u\Vert _{\omega }\\&\le \Vert (\mathcal {I}-\pi _{n}^{\omega }\mathcal {K})^{-1}\Vert _{\omega }\Vert u-\pi _{n}^{\omega }u\Vert _{\omega }\\&\le C_{1}\Vert u-\pi _{n}^{\omega }u\Vert _{\omega }\\&= C_{1}\Vert u-\pi _{n}^{\omega }u\Vert _{A^{0}(0, \infty )}\\&\le C_{1}n^{-\frac{r}{2}}|u|_{A^{r}(0, \infty )}\\&=\mathcal {O}(n^{-\frac{r}{2}}). \end{aligned}$$

Again, since

$$\begin{aligned} u-\widetilde{u}_{n}=\,&\mathcal {K}u-\mathcal {K}\pi _{n}^{\omega } \widetilde{u}_{n}\\ =\,&\mathcal {K}u-\mathcal {K}\pi _{n}^{\omega }u+\mathcal {K}\pi _{n}^{\omega }u- \mathcal {K}\pi _{n}^{\omega }\widetilde{u}_{n}, \end{aligned}$$

we have

$$\begin{aligned} u-\widetilde{u}_{n}=(\mathcal {I} -\mathcal {K}\pi _{n}^{\omega })^{-1}\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u. \end{aligned}$$

This implies

$$\begin{aligned} \Vert u-\widetilde{u}_{n}\Vert _{\omega }\le&\,\Vert (\mathcal {I}-\mathcal {K}\pi _{n}^{\omega })^{-1}\Vert _{\omega }\Vert \mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u\Vert _{\omega }\\ \le&\,C_{2}\Vert \mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u\Vert _{\infty }\\ =\,&\mathcal {O}(n^{-r}). \end{aligned}$$

Similarly for infinity norm, we have

$$\begin{aligned} \Vert u-\widetilde{u}_{n}\Vert _{\infty }\le&\,\Vert (\mathcal {I}-\mathcal {K}\pi _{n}^{\omega })^{-1}\Vert _{\infty }\Vert \mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u\Vert _{\infty }\\ \le&\,C_{3}\Vert \mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u\Vert _{\infty }\\ =\,&\mathcal {O}(n^{-r}). \end{aligned}$$

\(\square \)

3 Multi-Galerkin and iterated multi-Galerkin methods for linear Fredholm integral equation on half-line

In this section, we discuss the multi-projection method and define the multi-projection operator (Kulkarni 2003, Kulkarni and Nelakanti 2004, Chen et al. 2007, Das and Nelakanti 2017, Mandal and Nelakanti 2017, Nahid et al. 2019) as follows

$$\begin{aligned} \mathcal {K}_{n}^{M}=\pi _{n}^{\omega }\mathcal {K}\pi _{n}^{\omega }+ (\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}\pi _{n}^{\omega }+ \pi _{n}^{\omega }\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega }). \end{aligned}$$
(3.1)

The multi-Galerkin method for integral Eq. (2.3) is defined as find \(u_{n}^{M} \in L^{2}_{\omega }(0, \infty )\) such that

$$\begin{aligned} u_{n}^{M}-\mathcal {K}_{n}^{M}(u_{n}^{M})=y. \end{aligned}$$
(3.2)

The solution \(u_{n}^{M}\) is called multi-Galerkin solution. The iterated solution corresponding to (3.2) is defined by

$$\begin{aligned} \widetilde{u}_{n}^{M}=\mathcal {K}(u_{n}^{M})+y, \end{aligned}$$
(3.3)

where \(\widetilde{u}_{n}^{M}\) is known as iterated multi-Galerkin approximate solution. Now to solve integral Eq. (2.3) by multi-projection method, we follow the following procedure.

Applying \(\pi _{n}^{\omega }\) and \((\mathcal {I}-\pi _{n}^{\omega })\) to Eq. (3.2), we have

$$\begin{aligned} \pi _{n}^{\omega }u_{n}^{M}-\pi _{n}^{\omega }\mathcal {K}\pi _{n}^{\omega }u_{n}^{M} -\pi _{n}^{\omega }\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u_{n}^{M}=\pi _{n}^{\omega }y, \end{aligned}$$
(3.4)

and

$$\begin{aligned} (\mathcal {I}-\pi _{n}^{\omega })u_{n}^{M}=(\mathcal {I}- \pi _{n}^{\omega })\mathcal {K}\pi _{n}^{\omega }u_{n}^{M}+(\mathcal {I}-\pi _{n}^{\omega })y. \end{aligned}$$
(3.5)

Substituting (3.5) in (3.4), we obtain

$$\begin{aligned} \pi _{n}^{\omega }u_{n}^{M}-(\pi _{n}^{\omega }\mathcal {K}+ \pi _{n}^{\omega }\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}) \pi _{n}^{\omega }u_{n}^{M}=\pi _{n}^{\omega }y+\pi _{n}^{\omega }\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })y. \end{aligned}$$

This implies that, we seek \(u_{n, 1}^{M}=\pi _{n}^{\omega }u_{n}^{M} \in \mathbb {X}_{n}\) from the equation

$$\begin{aligned} {[}\mathcal {I}-Q_{n}\mathcal {K}]u_{n, 1}^{M}=Q_{n}y, \end{aligned}$$
(3.6)

where \(Q_{n}=\pi _{n}^{\omega }+\pi _{n}^{\omega }\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })\) and then we obtain

$$\begin{aligned} u_{n}^{M}=u_{n, 1}^{M}+u_{n, 2}^{M}, \end{aligned}$$

where

$$\begin{aligned} u_{n, 2}^{M}=(\mathcal {I}-\pi _{n}^{\omega })u_{n}^{M}=(\mathcal {I}-\pi _{n}^{\omega })(\mathcal {K}u_{n, 1}^{M}+y). \end{aligned}$$

Remark 1

From (2.7) and (3.6), we observe that in Galerkin and multi-Galerkin methods the size of the systems of equations to be solved remains same as the dimension of the space \(\mathbb {X}_{n}\). However, to solve the systems of equations in multi-projection method the extra computation required are to evaluate the integrals of the form \(\pi _{n}^{\omega }\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega }\mathcal {K}) \mathcal {L}_{i}\) and \(\pi _{n}^{\omega }\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })y,\) where \(\mathcal {L}_{i} \in \mathbb {X}_{n}.\) Below we show that this extra computation is compensated by superconvergence result in multi-projection method as we establish that the iterated multi-Galerkin method converges with order \(\mathcal {O}(n^{-2r}),\) which is twice of the iterated Galerkin method and four times of the Galerkin method in weighted \(L^{2}-\) norm.

3.1 Superconvergence results

Here in this, we give the superconvergence results for the approximation \(u_{n}^{M}\) and \(\widetilde{u}_{n}^{M}\) to the solution of integral Eq. (2.3). For this, we give the following Lemmas and Theorems.

Lemma 6

Let \(u \in A^{r}(0, \infty ), r\ge 1\) be the unique solution of (2.3) and \(\mathcal {K}_{n}^{M}\) be the operator given by (3.1), then the following holds

$$\begin{aligned} \Vert (\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u\Vert _{\omega } =\mathcal {O}\left( n^{-\frac{3r}{2}}\right) . \end{aligned}$$
(3.7)

Proof

Using equations (2.3) and (3.1), we have

$$\begin{aligned} \Vert (\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u\Vert _{\omega } =\,&\Vert (\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u\Vert _{A^{0}(0, \infty )}\nonumber \\ \le&\, n^{-\frac{r}{2}}|\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u|_{A^{r}(0, \infty )}\nonumber \\ =\,&n^{-\frac{r}{2}}\Vert (\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u)^{(r)}\Vert _{\omega _{r}}\nonumber \\ =\,&n^{-\frac{r}{2}}\left( \int _{0}^{\infty }\xi ^{r}e^{-\xi } |(\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u)^{(r)}(\xi )|^{2}\text {d}\xi \right) ^{\frac{1}{2}}\nonumber \\ \le&\,n^{-\frac{r}{2}}(\varGamma (r+1))^{\frac{1}{2}} \Vert (\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u)^{(r)}\Vert _{\infty }. \end{aligned}$$
(3.8)

Now using Cauchy–Schwarz inequality, we have

$$\begin{aligned} \Vert (\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u)^{(r)}\Vert _{\infty }=&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\left| (\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u)^{(r)}(\zeta )\right| \nonumber \\ =&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\left| \int _{0}^{\infty }\frac{\partial ^{r}}{\partial \zeta ^{r}}k(\zeta , \xi )(\mathcal {I}-\pi _{n}^{\omega })u(\xi )\text {d}\xi \right| \nonumber \\ =&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\left| \int _{0}^{\infty }e^{-\xi }e^{\xi }\frac{\partial ^{r}}{\partial \zeta ^{r}}k(\zeta , \xi )(\mathcal {I}-\pi _{n}^{\omega })u(\xi )\text {d}\xi \right| \nonumber \\ =&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\left| \int _{0}^{\infty }e^{-\xi }l_{r}(\zeta , \xi )(\mathcal {I}-\pi _{n}^{\omega })u(\xi )\text {d}\xi \right| \nonumber \\ =&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\left| (l_{r}(\zeta , .), (\mathcal {I}-\pi _{n}^{\omega })u(.))_{\omega }\right| \end{aligned}$$
(3.9)
$$\begin{aligned} =&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\left| ((\mathcal {I}-\pi _{n}^{\omega })l_{r}(\zeta , .), (\mathcal {I}-\pi _{n}^{\omega })u(.))_{\omega }\right| \nonumber \\ \le&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\Vert (\mathcal {I}-\pi _{n}^{\omega })l_{r}(\zeta , .)\Vert _{\omega }\Vert (\mathcal {I}-\pi _{n}^{\omega })u\Vert _{\omega }\nonumber \\ =&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\Vert (\mathcal {I}-\pi _{n}^{\omega })l_{r}(\zeta , .)\Vert _{A^{0}(0, \infty )}\Vert (\mathcal {I}-\pi _{n}^{\omega })u\Vert _{A^{0}(0, \infty )}\nonumber \\ \le&\underset{\zeta \in \mathbb {R}^{+}}{\sup }n^{-\frac{r}{2}}|l_{r}(\zeta , .)|_{A^{r}(0, \infty )}n^{-\frac{r}{2}}|u|_{A^{r}(0, \infty )}\nonumber \\ =&\underset{\zeta \in \mathbb {R}^{+}}{\sup }n^{-r}\Vert l_{r}^{(r)}(\zeta , .)\Vert _{\omega _{r}}|u|_{A^{r}(0, \infty )}\nonumber \\ \le&\underset{\zeta \in \mathbb {R}^{+}}{\sup }n^{-r} c_{3}(\varGamma (r+1))^{\frac{1}{2}}|u|_{A^{r}(0, \infty )}. \end{aligned}$$
(3.10)

From (3.8) and (3.10), we have

$$\begin{aligned} \Vert (\mathcal {I}-\pi _{n}^{\omega })\mathcal {K} (\mathcal {I}-\pi _{n}^{\omega })u\Vert _{\omega }=\mathcal {O}(n^{-\frac{3r}{2}}). \end{aligned}$$

\(\square \)

Theorem 7

Let \(\pi _{n}^{\omega }:L^{2}_{\omega }(0, \infty )\rightarrow \mathbb {X}_{n}\) be projection given by (2.5) and the operator \(\mathcal {K}\) defined by (2.2), then \(\Vert (\mathcal {I}-\mathcal {K}_{n}^{M})^{-1}\Vert _{\omega }\le C_{4}<\infty ,\) for large n.

Proof

Note that

$$\begin{aligned} \mathcal {K}-\mathcal {K}_{n}^{M}=(\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega }). \end{aligned}$$

Hence, using Lemma 2, we have

$$\begin{aligned} \Vert \mathcal {K}-\mathcal {K}_{n}^{M}\Vert _{\omega }&=\Vert (\mathcal {I}-\pi _{n}^{\omega }) \mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })\Vert _{\omega }\\&\le \Vert (\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}\Vert _{\omega }\Vert (\mathcal {I}-\pi _{n}^{\omega })\Vert _{\omega }\\&\le \Vert (\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}\Vert _{\omega }(1+\Vert \pi _{n}^{\omega }\Vert _{\omega })\\&\le (1+c)\Vert (\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}\Vert _{\omega }\\&=\mathcal {O}(n^{-\frac{r}{2}})\rightarrow 0~ ~~\hbox {as}~ n\rightarrow \infty . \end{aligned}$$

Thus, applying Lemma 1, we obtain \(\Vert (\mathcal {I}-\mathcal {K}_{n}^{M})^{-1}\Vert _{\omega }\le C_{4}< \infty .\) \(\square \)

Theorem 8

Let \(u \in A^{r}(0, \infty ), r\ge 1\) be the unique solution of Eq. (2.3) and let \(u_{n}^{M}\) be the multi-Galerkin approximation of u, then the following holds

$$\begin{aligned} \Vert u-u_{n}^{M}\Vert _{\omega }=\mathcal {O}\left( n^{-\frac{3r}{2}}\right) , \end{aligned}$$

where n is sufficiently large.

Proof

Since

$$\begin{aligned} u-u_{n}^{M}&=\mathcal {K}u-\mathcal {K}_{n}^{M}u_{n}^{M}\\&=\mathcal {K}u-\mathcal {K}_{n}^{M}u+\mathcal {K}_{n}^{M}u- \mathcal {K}_{n}^{M}u_{n}^{M}, \end{aligned}$$

we have

$$\begin{aligned} u-u_{n}^{M}=(\mathcal {I}-\mathcal {K}_{n}^{M})^{-1}(\mathcal {K}- \mathcal {K}_{n}^{M})u. \end{aligned}$$

Hence applying Lemma 6 and Theorem 7, we obtain

$$\begin{aligned} \Vert u-u_{n}^{M}\Vert _{\omega }\le&\, \Vert (\mathcal {I}-\mathcal {K}_{n}^{M})^{-1}\Vert _{\omega }\Vert (\mathcal {K}- \mathcal {K}_{n}^{M})u\Vert _{\omega }\\ \le&\, C_{4}\Vert (\mathcal {K}-\mathcal {K}_{n}^{M})u\Vert _{\omega }\\ =\,&C_{4}\Vert (\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u\Vert _{\omega }\\ =\,&\mathcal {O}(n^{-\frac{3r}{2}}). \end{aligned}$$

\(\square \)

Theorem 9

Let \(u \in A^{r}(0, \infty ), r\ge 1\) be the unique solution of Eq. (2.3) and let \(\widetilde{u}_{n}^{M}\) be the iterated multi-Galerkin solution of u, then the following holds:

$$\begin{aligned} \Vert u-\widetilde{u}_{n}^{M}\Vert _{\omega }=\mathcal {O}(n^{-2r}), \end{aligned}$$

where n is sufficiently large.

Proof

Using equations (2.3) and (3.3), we have

$$\begin{aligned} \Vert u-\widetilde{u}_{n}^{M}\Vert _{\omega }=&\Vert \mathcal {K}(u-u_{n}^{M})\Vert _{\omega }\nonumber \\ =&\Vert \mathcal {K}(\mathcal {I}-\mathcal {K}_{n}^{M})^{-1}(\mathcal {K}-\mathcal {K}_{n}^{M})u\Vert _{\omega }\nonumber \\ =&\Vert \mathcal {K}\left( \mathcal {I}+(\mathcal {I}- \mathcal {K}_{n}^{M})^{-1}\mathcal {K}_{n}^{M}\right) (\mathcal {K}-\mathcal {K}_{n}^{M})u\Vert _{\omega }\nonumber \\ \le&\Vert \mathcal {K}(\mathcal {K}-\mathcal {K}_{n}^{M})u\Vert _{\omega }+ \Vert \mathcal {K}(\mathcal {I}-\mathcal {K}_{n}^{M})^{-1}\mathcal {K}_{n}^{M}(\mathcal {K} -\mathcal {K}_{n}^{M})u\Vert _{\omega }. \end{aligned}$$
(3.11)

Consider the first term of the estimate (3.11)

$$\begin{aligned} \Vert \mathcal {K}(\mathcal {K}-\mathcal {K}_{n}^{M})u\Vert _{\omega }=&\Vert \mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u\Vert _{\omega }\nonumber \\ =&\left( \int _{0}^{\infty }e^{-\xi }|\mathcal {K}(\mathcal {I}- \pi _{n}^{\omega })\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u(\xi )|^{2}\text {d}\xi \right) ^{\frac{1}{2}}\nonumber \\ \le&\Vert \mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u\Vert _{\infty }\nonumber \\ =&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\left| \mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}(\mathcal {I}- \pi _{n}^{\omega })u(\zeta )\right| \nonumber \\ =&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\left| \int _{0}^{\infty }k(\zeta , \xi )(\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u(\xi )\text {d}\xi \right| \nonumber \\ =&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\left| \int _{0}^{\infty }e^{-\xi }e^{\xi }k(\zeta , \xi )(\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u(\xi )\text {d}\xi \right| \nonumber \\ =&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\left| \int _{0}^{\infty }e^{-\xi }l_{0}(\zeta , \xi )(\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u(\xi )\text {d}\xi \right| . \end{aligned}$$
(3.12)

Hence using Cauchy–Schwarz inequality and the orthogonality of \(\pi _{n}^{\omega },\) estimate (3.12) becomes

$$\begin{aligned} \Vert \mathcal {K}(\mathcal {K}-\mathcal {K}_{n}^{M})u\Vert _{\omega }\le&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\left| (l_{0}(\zeta , .), (\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u(.))_{\omega }\right| \end{aligned}$$
(3.13)
$$\begin{aligned} =&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\left| ((\mathcal {I}-\pi _{n}^{\omega })l_{0}(\zeta , .), (\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u(.))_{\omega }\right| \nonumber \\ \le&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\Vert (\mathcal {I}-\pi _{n}^{\omega })l_{0}(\zeta , .)\Vert _{\omega }\Vert (\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u\Vert _{\omega }\nonumber \\ =&\underset{\zeta \in \mathbb {R}^{+}}{\sup }\Vert (\mathcal {I}-\pi _{n}^{\omega })l_{0}(\zeta , .)\Vert _{A^{0}(0, \infty )}\Vert (\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u\Vert _{\omega }\nonumber \\ \le&\underset{\zeta \in \mathbb {R}^{+}}{\sup }n^{-\frac{r}{2}}|l_{0}(\zeta , .)|_{A^{r}(0, \infty )}\Vert (\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u\Vert _{\omega }\nonumber \\ =&\underset{\zeta \in \mathbb {R}^{+}}{\sup }n^{-\frac{r}{2}}\Vert (l_{0}(\zeta , .))^{(r)}\Vert _{\omega _{r}}\Vert (\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u\Vert _{\omega }\nonumber \\ \le&n^{-\frac{r}{2}}c_{3}(\varGamma (r+1))^{\frac{1}{2}} \Vert (\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u\Vert _{\omega }. \end{aligned}$$
(3.14)

Now using Lemma 6 in (3.14), we obtain

$$\begin{aligned} \Vert \mathcal {K}(\mathcal {K}-\mathcal {K}_{n}^{M})u\Vert _{\omega } =&\mathcal {O}(n^{-2r}). \end{aligned}$$
(3.15)

Note that

$$\begin{aligned} \mathcal {K}(\mathcal {I}-\mathcal {K}_{n}^{M})^{-1}\mathcal {K}_{n}^{M}(\mathcal {K}-\mathcal {K}_{n}^{M}) =\mathcal {K}(\mathcal {I}-\mathcal {K}_{n}^{M})^{-1}\pi _{n}^{\omega }\mathcal {K} (\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega }). \end{aligned}$$

Hence for the second term of the estimate (3.11), using Theorem 7 and estimates (2.13), (3.15), we have

$$\begin{aligned} \Vert \mathcal {K}(\mathcal {I}&-\mathcal {K}_{n}^{M})^{-1}\mathcal {K}_{n}^{M} (\mathcal {K}-\mathcal {K}_{n}^{M})u\Vert _{\omega }\nonumber \\ =\,&\Vert \mathcal {K}(\mathcal {I}-\mathcal {K}_{n}^{M})^{-1}\pi _{n}^{\omega }\mathcal {K} (\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u\Vert _{\omega }\nonumber \\ \le&\,\Vert \mathcal {K}(\mathcal {I}-\mathcal {K}_{n}^{M})^{-1}\pi _{n}^{\omega }\Vert _{\omega }\Vert \mathcal {K} (\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u\Vert _{\omega }\nonumber \\ \le&\,\Vert \mathcal {K}\Vert _{\omega }\Vert (\mathcal {I}-\mathcal {K}_{n}^{M})^{-1}\Vert _{\omega } \Vert \pi _{n}^{\omega }\Vert _{\omega }\Vert \mathcal {K} (\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u\Vert _{\omega }\nonumber \\ \le&\, cc_{2}C_{4}\Vert \mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })\mathcal {K}(\mathcal {I}-\pi _{n}^{\omega })u\Vert _{\omega }\nonumber \\ =\,&\mathcal {O}(n^{-2r}). \end{aligned}$$
(3.16)

Now combining estimates (3.15), (3.16), we have

$$\begin{aligned} \Vert u-\widetilde{u}_{n}^{M}\Vert _{\omega }=\mathcal {O}(n^{-2r}). \end{aligned}$$

\(\square \)

Remark 2

(i) From Theorem 5, we see that for the iterated Galerkin solution, we get the optimal convergence rates in both infinity and weighted \(L^{2}-\)norm which improves over the Galerkin method.

(ii) From Theorems 8 and 9, we notice that in both multi-Galerkin and iterated multi-Galerkin methods, we get the superconvergence results. Also we observe that, iterated multi-Galerkin method provides finer accuracy than multi-Galerkin method in weighted \(L^{2}-\)norm.

Remark 3

The results obtained for the linear Fredholm integral equation of the second kind on the half-line can be extended to nonlinear Fredholm–Hammerstein integral equations on the half-line which we will explore in our future papers.

4 Numerical results

In this section, we present numerical results. Here the approximating space \(\mathbb {X}_{n}\) is the space of Laguerre polynomials, discussed in the section 2. We give the errors for the approximate solutions of Galerkin and multi-Galerkin methods and their iterated versions using Laguerre polynomials as basis functions in Tables 1, 2, 5 and 6.

Table 1 Galerkin and iterated-Galerkin methods using Laguerre polynomials
Full size table
Table 2 Multi-Galerkin and iterated Multi-Galerkin methods using Laguerre polynomials
Full size table

We compare our results with piecewise polynomial-based method for finite section integral operator for the example 1 below. For this consider the graded mesh of \([0, \infty )\) as

$$\begin{aligned} 0=t_{1}<t_{2}<\ldots<t_{n}=\beta <t_{n+1}=+\infty , \end{aligned}$$

where \(\{t_{i}\}\) are

$$\begin{aligned} t_{i}=\frac{4r}{\mu }\log \left( \frac{n}{1+n-i}\right) , ~~i=1, 2, \ldots n,~\mu >0, \end{aligned}$$

and we choose approximating space as the space of piecewise constant functions, i.e., \(r=1.\) The errors are presented in Tables 3 and 4.

The numerical results are attained using matlab(2013b) on a Computer Intel(R) Core (TM) i5-8400 CPU @ 2.80 GHz (6 CPUs) Processor, 16.00GB RAM and 64-bit OS.

Example 1

(Rahmoune (2013)) Consider the following integral equation

$$\begin{aligned} u(\zeta )-\int _{0}^{\infty }k(\zeta , \xi )u(\xi )d\xi =y(\zeta ), ~\zeta \in [0, \infty ), \end{aligned}$$

where

$$\begin{aligned} k(\zeta , \xi )=e^{-(1+\zeta )\xi },\,y(\zeta )=e^{-\zeta }\sin (\zeta )-\frac{1}{(2+\zeta )^{2}+1}. \end{aligned}$$

The exact solution of the above integral equation is \(u(\zeta )=e^{-\zeta }\sin (\zeta ),\) which is a smooth function and decays exponentially at infinity.

Table 3 Galerkin and iterated-Galerkin methods using piecewise polynomials
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Table 4 Multi-Galerkin and iterated Multi-Galerkin methods using piecewise polynomials
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Table 5 Galerkin and iterated-Galerkin methods using Laguerre polynomials
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Remark 4

From Tables 2 and 4 clearly we can observe that to obtain the error of order \(10^{-7}\) in iterated multi-Galerkin method using Laguerre polynomials, we need to solve a system of size \(5\times 5,\) whereas in piecewise polynomial-based method, we need to solve a system of size \(32\times 32.\) Similarly from Tables 1 and 3, we observe that to get the error of order \(10^{-5}\) in iterated Galerkin method using Laguerre polynomials, we need to solve a system of size \(8\times 8,\) whereas in piecewise polynomial-based method, we need to solve a system of size \(64\times 64.\) Thus, we solve a smaller linear system in Laguerre polynomials as basis functions in Galerkin and multi-Galerkin methods compare to piecewise polynomials as basis functions.

Example 2

(Long et al. 2008) Consider the following integral equation

$$\begin{aligned} u(\zeta )-\int _{0}^{\infty }k(\zeta , \xi )u(\xi )\text {d}\xi =y(\zeta ), ~\zeta \in [0, \infty ), \end{aligned}$$

where

$$\begin{aligned} k(\zeta , \xi )=e^{-\xi ^{2}-\zeta ^{2}},~~~~~y(\zeta )=\zeta ^{4}-\frac{3\sqrt{\pi }}{8}e^{-\zeta ^{2}}. \end{aligned}$$

The exact solution is \(u(\zeta )=\zeta ^{4}.\)

Table 6 Multi-Galerkin and iterated Multi-Galerkin methods using Laguerre polynomials
Full size table

Remark 5

From Tables 1, 2, 5 and 6, it is clear that iterated Galerkin method gives better accuracy than Galerkin method and iterated multi-Galerkin method provides finer result than multi-Galerkin method using Laguerre polynomials as basis functions.