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Toward efficient numerical solutions of non-linear integral equations with TLD algorithm I. Sedkaabc , A. Khellafcd a Université Amar Telidji Laghouat, Algeria b Université 8 Mai 1945 Guelma, Algeria c Laboratoire de Mathématiques Appliquées et de Modélisation (LMAM) d École Nationale Polytechnique de Constantine, Algeria
Bibliographic databases:
    § 1. IntroductionNon-linear Fredholm integral equations of the second kind are often used to model physical phenomena such as heat transfer, fluid flow, chemical reactions, etc. [7]. They can also be used in machine learning and data analysis to model complex relationships between variables [4], [8]. In this study, we propose a novel numerical approach, the TLD (Transform, Linearize, Discretize) algorithm, to efficiently solve second-kind non-linear integral equations defined over a considerable interval. Our method comprises three phases: interval transformation, linearization via the Newton–Kantorovich method, and Nyström approximation to discretize the problem. We aim to theoretically establish the convergence conditions for our method and demonstrate its accuracy on practical examples. While traditional integral equation solvers were once considered inefficient, recent advancements [9], including fast solvers and deep learning techniques, have significantly improved their efficiency [12]. Our TLD algorithm is built upon these advancements [6], [5], [2], [10], [11], offering a more effective solution approach. We present the problem formulation and outline the structure of the paper. In § 2, we describe in detail the key steps of our proposed algorithm, and in § 3, we give theoretical background and convergence analysis. Finally, in § 4, we present numerical experiments to validate the effectiveness of our method. § 2. The description of the TLD new processIn this section, we will provide a clear explanation of the key stages involved in the novel TLD method for addressing the primary problem, which is as follows. Let $\tau$ be the largest real number $(\tau\gg 1)$ such that $[0,\tau]$ is a considerable interval. Consider the non-linear Fredholm integral equation of the second kind
$$
\begin{equation}
\lambda\psi(t) = \int_0^{\tau} \kappa (t,s,\psi(s))\, ds +g(t), \qquad t \in [0,\tau],
\end{equation}
\tag{2.1}
$$
with a given function $g$, $\lambda \in \mathbb{R}^* = \mathbb{R} \setminus \{0\}$, and a differentiable non-linear kernel $\kappa$. 2.1. The transformation phaseWe will create a structured method for converting the initial non-linear integral equation (2.1) defined over a substantial range into a set of non-linear equations defined over smaller intervals. This conversion will empower us to address the problem with greater effectiveness and efficiency. To begin, for $N \in \mathbb{N}^* = \mathbb{N} \setminus \{0\}$, we divide the large interval $[0,\tau]$ into $N$ smaller subintervals as follows:
$$
\begin{equation*}
[0,\tau] = \bigcup_{p=1}^N [\tau_{p-1},\tau_p], \qquad \tau_0 = 0, \quad \tau_N = \tau.
\end{equation*}
\notag
$$
Now equation (2.1) assumes the form
$$
\begin{equation}
\lambda\psi(t) = \sum_{p=1}^N \int_{\tau_{p-1}}^{\tau_p}\kappa(t,s,\psi(s))\, ds + g(t), \qquad t \in [0,\tau],\quad \lambda\in \mathbb{R}^*.
\end{equation}
\tag{2.2}
$$
For all $ l=1,\dots,N$, we set, for $t \in [\tau_{l-1},\tau_l]$,
$$
\begin{equation*}
\begin{cases} \psi_l(t) = \psi(t), \\ g_l(t) = g(t), \\ \kappa_l(t,s,{\cdot}\,) = \kappa(t,s,{\cdot}\,), &s \in [\tau_{p-1},\tau_p]. \end{cases}
\end{equation*}
\notag
$$
So, equation (2.2) can be transformed into the system of equations
$$
\begin{equation}
\begin{cases} \lambda \psi_1(t) = {\displaystyle\sum_{p=1}^N \int_{\tau_{p-1}}^{\tau_p}\kappa_1(t,s,\psi_p(s))\, ds + g_1(t)}, &t \in [0,\tau_1], \\ \lambda \psi_2(t) = {\displaystyle\sum_{p=1}^N \int_{\tau_{p-1}}^{\tau_p}\kappa_2(t,s,\psi_p(s)) \, ds + g_2(t)}, &t \in [\tau_1,\tau_2], \\ \dots\dots\dots\dots\dots\dots\dots\dots & \\ \lambda \psi_N(t) ={\displaystyle \sum_{p=1}^N \int_{\tau_{p-1}}^{\tau_p}\kappa_N(t,s,\psi_p(s))\, ds + g_N(t)}, &t \in [\tau_{N-1},\tau]. \end{cases}
\end{equation}
\tag{2.3}
$$
We can write this system as a general non-linear functional problem as follows:
$$
\begin{equation}
\operatorname{Find}\varPsi=(\psi_1,\dots,\psi_N)^\top \in \mathcal{X}, \qquad F(\varPsi)=\lambda \varPsi - K(\varPsi) - G =0_{\mathcal{X}},
\end{equation}
\tag{2.4}
$$
such that
$$
\begin{equation*}
\begin{cases} \mathcal{X} = \displaystyle{\prod_{p=1}^N C^1([\tau_{p-1},\tau_p],\mathbb{R})}, \\ G=(g_1,\dots,g_N)^\top \in \mathcal{X}, \\ K=(K_1,\dots,K_N)^\top, \end{cases}
\end{equation*}
\notag
$$
where, for $l=1,\dots,N$, and all $\phi=(\phi_1,\dots,\phi_N)^\top \in \mathcal{X}$, we define
$$
\begin{equation*}
K_l(\phi)(t) = \sum_{p=1}^N \int_{\tau_{p-1}}^{\tau_p}\kappa_l(t,s,\phi_p(s))\, ds, \qquad t \in [\tau_{l-1},\tau_l],
\end{equation*}
\notag
$$
and $\mathcal{X}$ is a Banach space with norm $\|\,{\cdot}\,\|_{\mathcal{X}} $. Next, the Banach algebra of bounded linear operators from $\mathcal{X}$ to itself is denoted by $BL(\mathcal{X})$; it is equipped with the norm
$$
\begin{equation*}
\|A\| =\sup\{\|A\phi\|_{\mathcal{X}} \colon \|\phi\|_{\mathcal{X}} \leqslant 1\}, \qquad A\in BL(\mathcal{X}).
\end{equation*}
\notag
$$
The Banach space $\mathcal{X}$ is equipped with the norm
$$
\begin{equation*}
\|\phi\|_{\mathcal{X}} = \sum_{p=1}^N \|\phi_p\|_{\infty}=\sum_{p=1}^N\max_{t \in [\tau_p,\tau_{p-1}]} |\phi_p(t)|, \qquad \phi=(\phi_1,\dots,\phi_N)^\top \in \mathcal{X} .
\end{equation*}
\notag
$$
Now, given fixed $ l_0,p_0=1,\dots,N$, we can define the Fréchet derivative of the non-linear integral operator $ K_{l_0}$ associated with $\psi_{p_0}$, that is, for $t \in [\tau_{l_0-1},\tau_{l_0}]$, and all $\phi=(\phi_1,\dots,\phi_N)^\top \in \mathcal{X}$, we define
$$
\begin{equation*}
\begin{aligned} \, [T_{{l_0}{p_0}}(\varPsi)\phi](t) &= \frac{\partial }{\partial \psi_{p_0}} \biggl( \sum_{p=1}^N \int_{\tau_{p-1}}^{\tau_p}\kappa_{l_0}(t,s,\psi_p(s)) \phi_p(s)\, ds \biggr) \\ &= \sum_{p=1}^N \int_{\tau_{p-1}}^{\tau_p} \frac{\partial \kappa_{l_0}}{\partial \psi_{p_0}}(t,s,\psi_p(s)) \phi_p(s)\, ds \\ &= \int_{\tau_{p_0-1}}^{\tau_{p_0}} \frac{\partial \kappa_{l_0}}{\partial \psi_{p_0}}(t,s,\psi_{p_0}(s)) \phi_{p_0}(s)\, ds. \end{aligned}
\end{equation*}
\notag
$$
Hence, for all $ l,p=1,\dots,N$,
$$
\begin{equation*}
[T_{{l}{p}}(\varPsi)\phi](t) = \int_{\tau_{p-1}}^{\tau_p} \frac{\partial \kappa_{l}}{\partial \psi_p}(t,s,\psi_p(s))\phi_p(s)\, ds,\qquad t \in [\tau_{l-1},\tau_l].
\end{equation*}
\notag
$$
Given $l=1,\dots, N$ and $ n \in \mathbb{N}^*$, consider the equidistant subdivision $\Omega_n^l$ defined by
$$
\begin{equation}
\Omega_n^l = \biggl\{ t_j = (l-1) +(j-1) \omega_n \in [\tau_{l-1},\tau_l],\, \omega_n = \frac{1}{n-1},\, j = 1,\dots,n\biggr\}.
\end{equation}
\tag{2.5}
$$
For all $l,p=1,\dots,N$, we define the Nyström estimation $T_{n,lp}(\,{\cdot}\,)$ of order $n$ of the linear operator $T_{lp}(\,{\cdot}\,)$ by
$$
\begin{equation*}
[T_{n,lp}(\varPsi)\phi](t) \,{=} \sum_{j=1}^n \omega_n\, \frac{\partial \kappa_{l}}{\partial \psi_p}(t,t_j,\psi_p(t_j))\phi_p(t_j), \qquad t \,{\in}\, [\tau_{l-1},\tau_l],\quad t_j \,{\in}\, \varOmega_n^p, \quad \phi \,{\in}\, \mathcal{X},
\end{equation*}
\notag
$$
where The matrix of a linear operator $D_T(\,{\cdot}\,) \in \mathrm{BL}(\mathcal{X})$ is defined by
$$
\begin{equation*}
D_T(\phi) v = \begin{pmatrix} T_{11}(\phi) &\dots &T_{1N}(\phi) \\ \vdots&\ddots&\vdots \\ T_{N1}(\phi)&\dots& T_{NN}(\phi) \end{pmatrix} \begin{pmatrix} v_1 \\ \vdots \\ v_N \end{pmatrix} = \begin{pmatrix} {\displaystyle\sum_{p=1}^NT_{1p}(\phi)v_p} \\ \vdots \\ {\displaystyle\sum_{p=1}^NT_{Np}(\phi)v_p} \end{pmatrix},
\end{equation*}
\notag
$$
where $v = (v_1,\dots,v_N)^\top \in \mathcal{X}$ and $\phi \in \mathcal{X}$. Certain conditions were identified as necessary for convergence, and the new method was developed with their help. So, for all $ l,p=1,\dots,N$, we assume that
$$
\begin{equation}
\begin{cases} (\mathrm{i}) \text{ problem }(2.4) \text{ has a unique solution } \varPsi \in \mathcal{X}, \\ (\mathrm{ii})\ (\lambda I-D_T(\varPsi)) \text{ is invertible, and } \exists\, \nu > 0, \|(\lambda I-D_T(\varPsi))^{-1}\| \leqslant \nu < \infty, \\ (\mathrm{iii})\ \exists\, \gamma_p > 0 \text{ such that } \biggl\| \dfrac{\partial \kappa_l}{\partial \varphi_p}(t,s,\varphi_p(s)) {\kern1pt}{-}{\kern1pt}\dfrac{\partial \kappa_l}{\partial \phi_p}(t,s,\phi_p(s)) \biggr\|_{\infty} {\leqslant}\, \gamma_p \| \varphi_p{\kern1pt}{-}{\kern1pt}\phi_p\|_{\infty}. \end{cases}
\end{equation}
\tag{2.6}
$$
The second assumption guarantees the invisibility of the operator $(\lambda I - D_T(\varPsi))$, under the condition that Controlling the parameter $\lambda$ regulates the magnitude of $\| D_T(\varPsi)\|$ and applying the geometric series theorem to the operator $(\lambda I - D_T(\varPsi))$ enables us to establish its invisibility, which is crucial in ensuring the convergence of our approximate solution in numerical applications. 2.2. The linearization phaseTo further streamline the problem, we will utilize the Newton–Kantorovich method. This involves the iterative linearization of the set of non-linear equations derived from the transformation outlined in equation (2.4), ultimately converting them into an iterative linear system of equations. As a result, we get the iterative linear problem
$$
\begin{equation}
\bigl( \lambda I-D_T(\varPsi^{(k)})\bigr) \bigl( \varPsi^{(k+1)}-\varPsi^{(k)}\bigr) = - F\bigl( \varPsi^{(k)}\bigr), \qquad \varPsi^{(0)} \in \mathcal{X}, \quad k= 0,1,\dots\,.
\end{equation}
\tag{2.7}
$$
This process will greatly facilitate the numerical solution of the problem within an infinite dimensional framework; for a comprehensive explanation of the convergence of the Newton–Kantrovich process, see [3]. 2.3. The discretization phaseNow we apply the Nyström method to transform the linear iterative scheme (2.7) into a discretized linear problem with $\varPsi^{(k)}_n =\bigl( \psi^{(k)}_{n,1},\dots,\psi^{(k)}_{n,N}\bigr)^\top \in \mathcal{X}$ representing the approximate iterative solution of our main problem (2.4) obtained by solving the equation
$$
\begin{equation}
\bigl( \lambda I-D_{T_n}\bigl(\varPsi^{(k)}_n\bigr)\bigr) \bigl( \varPsi^{(k+1)}_n-\varPsi^{(k)}_n\bigr) = - F\bigl( \varPsi^{(k)}_n\bigr), \qquad \varPsi^{(0)}_n \in \mathcal{X}, \quad k= 0,1,\dots,
\end{equation}
\tag{2.8}
$$
where, for all $ v = (v_1,\dots,v_N)^\top \in \mathcal{X}$ and all $ \phi \in \mathcal{X}$, we have
$$
\begin{equation*}
D_{T_n}(\phi) v =\begin{pmatrix} T_{n,11}(\phi) &\dots &T_{n,1N}(\phi) \\ \vdots&\ddots&\vdots \\ T_{n,N1}(\phi)&\dots& T_{n,NN}(\phi) \end{pmatrix} \begin{pmatrix} v_1\\ \vdots\\ v_N \end{pmatrix} = \begin{pmatrix} {\displaystyle\sum_{p=1}^NT_{n,1p}(\phi)v_p} \\ \vdots \\ {\displaystyle\sum_{p=1}^NT_{n,Np}(\phi)v_p} \end{pmatrix}
\end{equation*}
\notag
$$
and, for all $l=1,\dots,N $, $t \in [\tau_{l-1},\tau_l]$, we can write the iterative scheme (2.8) as
$$
\begin{equation}
\lambda \psi_{n,l}^{(k+1)} (t)- \sum_{p=1}^N \sum_{j=1}^n \omega_n\, \frac{\partial \kappa_{l}}{\partial \psi_p} \bigl( t,t_j,\psi_{n,p}^{(k)}(t_j)\bigr) \psi_{n,p}^{(k+1)}(t_j)= f_{n,l}^{(k)}(t),
\end{equation}
\tag{2.9}
$$
where
$$
\begin{equation}
f_{n,l}^{(k)}(t) = - \sum_{p=1}^N \sum_{j=1}^n \omega_n\, \frac{\partial \kappa_{l}}{\partial \psi_p}\bigl( t,t_j,\psi_{n,p}^{(k)}(t_j)\bigr) \psi_{n,p}^{(k)}(t_j) + K_l\bigl( \varPsi_n^{(k)}\bigr) (t) + g_l(t).
\end{equation}
\tag{2.10}
$$
Finally, we save the collocation of our discretized approximate solution $\varPsi_n^{(k+1)}$ at the nodes $(t_i)_{1 \leqslant i \leqslant n}$, and define the vector
$$
\begin{equation*}
Y^{(k+1)} = (y_{n,1}^{(k+1)},\dots,y_{n,N}^{(k+1)}) \in \mathbb{R}^{n.N}
\end{equation*}
\notag
$$
and, for all $l=1,\dots,N$ and $i=1,\dots,n$, we set
$$
\begin{equation}
y_{n,l}^{(k+1)}(i) = \psi_{n,l}^{(k+1)}(t_i), \qquad t_i \in \Omega_n^l.
\end{equation}
\tag{2.11}
$$
System (2.9), (2.10) is solved in two steps, which we now describe. Step 1. Substituting (2.11) into the main system (2.9), (2.10), we get the linear algebraic system, which we write, for all $i=1,\dots,n$ and $t_i \in \Omega_n^l,$ as
$$
\begin{equation*}
\lambda y_{n,l}^{(k+1)} (i)- \sum_{p=1}^N \sum_{j=1}^n \omega_n\, \frac{\partial \kappa_{l}}{\partial \psi_p}\bigl( t_i,t_j,y_{n,p}^{(k)}(j)\bigr) y_{n,p}^{(k+1)}(j)= f_{n,l}^{(k)}(t_i), \qquad t_i \in \Omega_n^l,
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
f_{n,l}^{(k)}(t_i) = - \sum_{p=1}^N \sum_{j=1}^n \omega_n\, \frac{\partial \kappa_{l}}{\partial \psi_p} \bigl( t_i,t_j,y_{n,p}^{(k)}(j)\bigr) y_{n,p}^{(k)}(j) + K_l(\varPsi_n^{(k)})(t_i) + g_l(t_i).
\end{equation*}
\notag
$$
We can rewrite this as
$$
\begin{equation*}
\bigl(\lambda I_{N}-H^{(k)}_n\bigr) Y^{(k+1)} = b_n^{(k)},
\end{equation*}
\notag
$$
where, for all $l,p=1,\dots,N$ and $i,j=1,\dots,n$,
$$
\begin{equation}
[H^{(k)}_n]_{lp}(i,j)=\omega_n\, \frac{\partial\kappa_l}{\partial \psi_p}\bigl(t_i,t_j,y_{n,p}^{(k)}(j)\bigr),
\end{equation}
\tag{2.12}
$$
$$
\begin{equation}
\begin{split} b^{(k)}_{n,l}(i) &= f_{n,l}^{(k)}(t_i) = -\sum_{p=1}^N\sum_{j=1}^n\omega_n\, \frac{\partial\kappa_l}{\partial \psi_p}\bigl(t_i,t_j,y_{n,p}^{(k)}(j)\bigr)y_{n,p}^{(k)}(j) \\ &\qquad+\sum_{p=1}^N \int_{\tau_{p-1}}^{\tau_p}\kappa_l\bigl( t_i,s,\psi_{n,p}^{(k)}(s)\bigr)\, ds+g_l(t_i), \end{split}
\end{equation}
\tag{2.13}
$$
$$
\begin{equation}
\nonumber [I_N]_{l,p}(i,j)=\begin{cases} 1 &\text{if } l=p \text{ and } i=j, \\ 0 &\text{otherwise}. \end{cases}
\end{equation}
\notag
$$
Step 2. For all $ l=1,\dots,N $, we recover the approximate solution $\psi^{(k+1)}_{n,l} \in \mathcal{X}$ by the natural interpolation formula as follows:
$$
\begin{equation}
\begin{aligned} \, \lambda\psi_{n,l}^{(k+1)}(t) &=\sum_{p=1}^N\sum_{j=1}^n \omega_n\, \frac{\partial\kappa_l}{\partial \psi_p}\bigl(t,t_j,y_{n,p}^{(k)}(j)\bigr)\bigl( y_{n,p}^{(k+1)}(j)-y_{n,p}^{(k)}(j)\bigr) \nonumber \\ &\qquad+ \sum_{p=1}^N \int_{\tau_{p-1}}^{\tau_p}\kappa_l\bigl( t,s,\psi_{n,p}^{(k)}(s)\bigr) \, ds+g_l(t). \end{aligned}
\end{equation}
\tag{2.14}
$$
Remark 1. The integrals in the second half of scheme (2.13) and in the natural interpolation formula (2.14) will be estimated via the Nyström method, that is, for all $l,p=1,\dots,N$, we choose $m \in \mathbb{N}^*$, $m \gg n $, such that § 3. The analysis of the TLD new processIn order to formulate the convergence theorem for our method, we first provide some necessary properties of the operators $D_T$ and its estimate operator $D_{T_n}$, where the results of Proposition 1 will be used in the proof of Proposition 3, and similarly, those of Proposition 2 will be used to prove Proposition 4. Proposition 1. Under assumption (2.6),( iii), the operator $D_T$ is $ \gamma$-Lipschitz on the ball $B_R(\varPsi)$, where Proof. Let $\varphi=(\varphi_1,\dots,\varphi_N)^\top$, $\phi=(\phi_1,\dots,\phi_N)^\top \in B_{R}(\varPsi)$, and $v=(v_1,\dots, v_N)^\top \in \mathcal{X}$. Then
$$
\begin{equation*}
\begin{aligned} \, &\|(D_T(\varphi)-D_T(\phi)) v\|_{\mathcal{X}} \\ &\qquad=\sum_{l=1}^N \biggl\| \sum_{p=1}^N (T_{lp}(\varphi)-T_{lp}(\phi)) v_p\biggr\|_{\infty}\leqslant \sum_{l=1}^N\sum_{p=1}^N \|(T_{lp}(\varphi)-T_{lp}(\phi)) v_p\|_{\infty} \\ &\qquad\leqslant \sum_{l=1}^N\sum_{p=1}^N \|v_p\|_{\infty} \int_{\tau_{p-1}}^{\tau_p} \biggl\| \biggl(\frac{\partial\kappa_l}{ \partial \varphi_p}(t,s,\varphi_p(s))-\frac{\partial\kappa_l}{ \partial \phi_p}(t,s,\phi_p(s)) \biggr)\biggr\|_{\infty} \, ds. \end{aligned}
\end{equation*}
\notag
$$
It is clear that $\|v_p\|_{\infty} \leqslant \|v\|_{\mathcal{X}}$, and, from assumption (2.6).(iii), we have
$$
\begin{equation*}
\|D_T(\varphi)-D_T(\phi)\|\leqslant \sum_{l=1}^N\sum_{p=1}^N \frac{\gamma_p}{N} \|\varphi_p-\phi_p\|_{\infty}.
\end{equation*}
\notag
$$
Now with $\gamma = \max_{1 \leqslant p \leqslant N}^{} \gamma_p$ we finally get
$$
\begin{equation*}
\| D_T(\varphi)-D_T(\phi)\| \leqslant \gamma\|\varphi-\phi\|_{\mathcal{X}}.
\end{equation*}
\notag
$$
Proposition 1 is proved. Proposition 2. Under assumption (2.6),(iii), for all $\phi=(\phi_1,\dots,\phi_N)^\top \in B_{R}(\varPsi)$ and a fixed number $N \in \mathbb{N^*}$, there exist $\delta_{n,N}>0$ such that where Proof. For $ \phi=(\phi_1,\dots,\phi_N)^\top \in B_R(\varPsi)$ and $v=(v_1,\dots,v_N)^\top \in \mathcal{X}$, we have
$$
\begin{equation*}
\begin{aligned} \, &\|(D_T(\phi)-D_{T_n}(\phi))v\|_{\mathbb{X}} = \sum_{l=1}^N \biggl\|\sum_{p=1}^N (T_{lp}(\phi)-T_{n,lp}(\phi)) v_p \biggr\|_{\infty} \\ &\ \ \leqslant\sum_{l=1}^N \sum_{p=1}^N \|(T_{lp}(\phi)-T_{n,lp}(\phi)) v_p\|_{\infty} = \sum_{l=1}^N \sum_{p=1}^N\sup_{t\in[\tau_{l-1},\tau_l]} |([T_{lp}(\phi)-T_{n,lp}(\phi)] v_p)(t)| \\ &\ \ \leqslant \sum_{l=1}^N \sum_{p=1}^N\sup_{t\in[\tau_{l-1},\tau_l]}|[T_{lp}(\phi)-T_{n,lp}(\phi)](t)| \, \|v_p\|_{\infty}, \end{aligned}
\end{equation*}
\notag
$$
and by applying the trapezoidal rule (see [1], p. 109), for all $l,p=1,\dots,N$, we have
$$
\begin{equation*}
|(T_{lp}(\phi)-T_{n,lp}(\phi))(t)| \leqslant \frac{1}{12n^2}\biggl| \biggl[ \frac{\partial^2\kappa_l(t,s,\phi_p(s))}{\partial \phi_p\, \partial s} \biggr]_{\tau_{p-1}}^{\tau_p}\biggr|.
\end{equation*}
\notag
$$
Finally, we get
$$
\begin{equation*}
\|(D_T(\phi)-D_{T_n}(\phi))\| \leqslant \frac{1}{12n^2} \sum_{l=1}^N \sum_{p=1}^N \sup_{t\in[\tau_{l-1},\tau_l]} \biggl| \biggl[ \frac{\partial^2\kappa_l(t,s,\phi_p(s))}{\partial \phi_p\, \partial s} \biggr]_{s=\tau_{p-1}}^{s=\tau_p} \biggr|.
\end{equation*}
\notag
$$
Proposition 2 is proved. Proposition 3. Assume that (2.6) holds. Let $r=\min(R,1/(2\gamma\nu))$, where $\gamma$ is defined in Proposition 1. Then, for all $\phi=(\phi_1,\dots,\phi_N)^\top \in B_{r}(\varPsi)$, the operator $(\lambda I-D_T(\phi))$ is invertible with bounded inverse, that is, Proof. For all $\phi=(\phi_1,\dots,\phi_N)^\top \in B_{r}(\varPsi)$, we have
$$
\begin{equation*}
\begin{aligned} \, \lambda I-D_T(\phi) &= \lambda I-D_T(\varPsi)-D_T(\phi)+D_T(\varPsi) \\ &=(\lambda I-D_T(\varPsi)) \bigl[ I-(\lambda I-D_T(\varPsi))^{-1}(D_T(\phi)-D_T(\varPsi)) \bigr], \end{aligned}
\end{equation*}
\notag
$$
and, by Proposition 1, for all $ \phi=( \phi_1,\dots,\phi_N)^\top \in B_{r}(\varPsi)$,
$$
\begin{equation*}
\|(D_T(\phi)-D_T(\varPsi))\| \leqslant \gamma \|\phi-\varPsi\|_{\infty} \leqslant \gamma r.
\end{equation*}
\notag
$$
Hence
$$
\begin{equation*}
\bigl\|(\lambda I-D_T(\varPsi))^{-1}(D_T(\phi)-D_T(\varPsi))\bigr\| \leqslant \nu \gamma r \leqslant \frac{1}{2},
\end{equation*}
\notag
$$
and now from the operator $(I-D_T(\phi))$ is invertible by the geometric series theorem (see [1], p. 516), and
$$
\begin{equation*}
(\lambda I-D_T(\phi))^{-1} = \bigl( I-(\lambda I-D_T(\varPsi))^{-1}[D_T(\phi)-D_T(\varPsi)] \bigr)^{-1} (\lambda I-D_T(\varPsi))^{-1} ,
\end{equation*}
\notag
$$
and its inverse is uniformly bounded on $ B_r(\varPsi)$, where
$$
\begin{equation*}
\bigl\| (\lambda I-D_T(\phi))^{-1}\bigr\| \leqslant \nu \sum_{q=0}^{\infty} \bigl\| (\lambda I-D_T(\varPsi))^{-1} (D_T(\phi)-D_T(\varPsi)) \bigr\|^q \leqslant 2\nu.
\end{equation*}
\notag
$$
Proposition 3 is proved. Proposition 4. Let assumption (2.6) hold. Then, for $n$ large enough and for all $ \phi=(\phi_1,\dots,\phi_N)^\top \in B_r(\varPsi)$, the approximate operator $(\lambda I-D_{T_n}(\phi))$ is invertible, and there exists $\alpha_n \in \, ]0,1[$ such that Proof. For all $ \phi=(\phi_1,\dots,\phi_N)^\top \in B_r(\varPsi)$, we have
$$
\begin{equation*}
\begin{aligned} \, \lambda I-D_{T_n}(\phi) &=\lambda I-D_T(\phi)+D_T(\phi)-D_{T_n}(\phi) \\ &=(\lambda I-D_T(\phi))\bigl( I- (\lambda I-D_T(\phi))^{-1}[ D_T(\phi)-D_{T_n}(\phi)] \bigr). \end{aligned}
\end{equation*}
\notag
$$
As in Proposition 2, we define
$$
\begin{equation*}
\|D_T(\phi)-D_{T_n}(\phi)\|\leqslant \delta_{n,N} \xrightarrow{n\to \infty} 0.
\end{equation*}
\notag
$$
So, if $N \in \mathbb{N^*}$ is fixed and $n$ is large enough, then it is clear that $\delta_{n,N} < 1/(2\nu)$. Hence
$$
\begin{equation*}
\bigl\| (\lambda I-D_T(\phi))^{-1}[ D_T(\phi)-D_{T_n}(\phi)] \bigr\| \leqslant 2\nu \delta_{n,N} < 1.
\end{equation*}
\notag
$$
By applying the geometric series theorem (see [1], p. 516), we conclude that, for all $\phi \in B_r(\varPsi)$, the operator $\lambda I-D_{T_n}(\phi)$ is invertible and
$$
\begin{equation*}
\bigl\| (\lambda I-D_{T_n}(\phi))^{-1}\bigr\| \leqslant \frac{2\nu}{1-2\nu\delta_{n,N}}.
\end{equation*}
\notag
$$
Since
$$
\begin{equation*}
I -(\lambda I-D_{T_n}(\phi))^{-1}(\lambda I-D_T(\phi)) = (\lambda I-D_{T_n}(\phi))^{-1} (D_T(\phi)-D_{T_n}(\phi)),
\end{equation*}
\notag
$$
we define $\alpha_n = 2\nu\delta_{n,N}/(1-2\nu\delta_{n,N})$. For $n$ large enough and $\alpha_n < 1$, we get We also have
Hence
$$
\begin{equation*}
\sup_{\phi\in B_r(\varPsi)}\bigl\|(\lambda I-D_{T_n}(\phi))^{-1}\bigr\| \leqslant 2\nu (1+ \alpha_n),
\end{equation*}
\notag
$$
proving Proposition 4. Now having all the necessary notation, properties, and propositions at our disposal, we will proceed with our main convergence theorem. This theorem proves that the approximate solution we obtain by solving the iterative linear scheme (2.9), (2.10) in our new process converges to the exact solution of the main problem (2.1). Theorem 1 (convergence theorem). Let assumption (2.6) be fulfilled, and let $r=\min( R,1/(2\gamma \nu))$. Then, for sufficiently large $n $, there exist $ \alpha_n \in \, ]0,1[$ and $ \varrho_n > 0$ such that, if the starting approximation $ \varPsi^{(0)}_n $ is chosen in the closed ball $ B_{\varrho_n}(\varPsi)$, then, for all $k\in \mathbb{N}^*$, $\varPsi^{(k)}_n \in B_{\varrho_n}(\varPsi)$, where Proof. By Proposition 4, the operator $(\lambda I-D_{T_n}(\varPsi^{(k)}_n))$ is invertible for $ \varPsi^{(k)}_n \in B_{r}(\varPsi)$. Now $\varPsi^{(k+1)}_n$ (see (2.8)) can be expressed as
$$
\begin{equation*}
\begin{aligned} \, \varPsi^{(k+1)}_n&=\varPsi^{(k)}_n-\bigl( \lambda I-D_{T_n}(\varPsi^{(k)}_n)\bigr)^{-1} F(\varPsi^{(k)}_n) \\ &=\varPsi^{(k)}_n-\bigl( \lambda I-D_{T_n}(\varPsi^{(k)}_n)\bigr)^{-1} \bigl( \lambda\varPsi^{(k)}_n-K(\varPsi^{(k)}_n)-G\bigr), \end{aligned}
\end{equation*}
\notag
$$
and since $\varPsi \in \mathcal{X}$ is the exact solution of the main problem (2.4), we have
$$
\begin{equation*}
\begin{aligned} \, \varPsi^{(k+1)}_n - \varPsi &= \varPsi^{(k)}_n - \varPsi - \bigl( \lambda I - D_{T_n}(\varPsi^{(k)}_n) \bigr)^{-1} \\ &\qquad \times \bigl( \lambda \varPsi^{(k)}_n - K(\varPsi^{(k)}_n) - G - [\lambda \varPsi - K(\varPsi) - G] \bigr) \\ &= \varPsi^{(k)}_n - \varPsi - \bigl( \lambda I - D_{T_n}(\varPsi^{(k)}_n) \bigr)^{-1} \bigl( \lambda \varPsi^{(k)}_n - \lambda \varPsi + K(\varPsi) - K(\varPsi^{(k)}_n) \bigr). \end{aligned}
\end{equation*}
\notag
$$
Using the integral form of Lagrange’s mean value theorem (see [13]), we have
$$
\begin{equation*}
K(\varPsi)-K(\varPsi^{(k)}_n) = -\int_0^{1} D_T \bigl((1-x)\varPsi^{(k)}_n+ x \varPsi \bigr)\cdot \bigl( \varPsi^{(k)}_n-\varPsi\bigr)\, dx ,
\end{equation*}
\notag
$$
and hence, substituting this new expression into the last equation, this establishes
$$
\begin{equation*}
\begin{aligned} \, &\varPsi^{(k+1)}_n - \varPsi = \varPsi^{(k)}_n - \varPsi - \bigl( \lambda I - D_{T_n}(\varPsi^{(k)}_n) \bigr)^{-1} \\ &\qquad\qquad\qquad \times \biggl( \lambda \bigl( \varPsi^{(k)}_n - \varPsi \bigr) - \int_0^{1} D_T \bigl((1-x) \varPsi^{(k)}_n + x \varPsi \bigr) \cdot \bigl( \varPsi^{(k)}_n - \varPsi \bigr) \, dx \biggr) \\ &\qquad\!= \int_0^{1} \bigl[ I - \bigl( \lambda I - D_{T_n}(\varPsi^{(k)}_n) \bigr)^{-1} \bigl(\lambda I - D_T \bigl((1-x) \varPsi^{(k)}_n + x \varPsi \bigr) \bigr) \bigr] \cdot \bigl( \varPsi^{(k)}_n - \varPsi \bigr) \, dx. \end{aligned}
\end{equation*}
\notag
$$
Adding $ \lambda I-D_T(\varPsi^{(k)}_n)$ and subtracting from $\lambda I-D_T\bigl((1-x)\varPsi^{(k)}_n+ x \varPsi \bigr)$, we get
$$
\begin{equation*}
\begin{aligned} \, &\varPsi^{(k+1)}_n - \varPsi = \int_0^{1} \bigl[ I - \bigl( \lambda I - D_{T_n}(\varPsi^{(k)}_n) \bigr)^{-1} \bigl( \lambda I - D_T(\varPsi^{(k)}_n) \bigr) \bigr] \cdot \bigl( \varPsi^{(k)}_n - \varPsi \bigr) \, dx \\ &\quad + \int_0^{1} \bigl( \lambda I - D_{T_n}(\varPsi^{(k)}_n) \bigr)^{-1} \bigl[ D_T \bigl((1-x) \varPsi^{(k)}_n + x \varPsi \bigr) - D_T(\varPsi^{(k)}_n) \bigr] \cdot \bigl( \varPsi^{(k)}_n - \varPsi \bigr) \, dx, \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, &\bigl\| \varPsi^{(k+1)}_n - \varPsi \bigr\|_{\mathcal{X}} \leqslant \bigl\| I - \bigl( \lambda I - D_{T_n}(\varPsi^{(k)}_n) \bigr)^{-1} \bigl( \lambda I - D_T(\varPsi^{(k)}_n) \bigr) \bigr\| \, \bigl\| \varPsi^{(k)}_n - \varPsi \bigr\|_{\mathcal{X}} \\ &+ \bigl\| \bigl( \lambda I - D_{T_n}(\varPsi^{(k)}_n) \bigr)^{-1} \bigr\| \, \bigl\| \varPsi^{(k)}_n - \varPsi \bigr\|_{\mathcal{X}}\!\int_0^{1} \bigl\| D_T \bigl((1-x) \varPsi^{(k)}_n + x \varPsi \bigr) \,{-}\, D_T(\varPsi^{(k)}_n) \bigr\| \, dx. \end{aligned}
\end{equation*}
\notag
$$
We have $ \varPsi^{(k)}_n \in B_{r}(\varPsi)$, and hence by Proposition 4,
$$
\begin{equation*}
\bigl\| I-\bigl( \lambda I-D_{T_n}(\varPsi^{(k)}_n)\bigr)^{-1} \bigl( \lambda I-D_T(\varPsi^{(k)}_n)\bigr) \bigr\| \leqslant \alpha_n .
\end{equation*}
\notag
$$
Since $ B_{r}(\varPsi)$ is a convex set, we have, for all $ x \in [0,1], (1-x)\varPsi^{(k)}_n+ x \varPsi \in B_{r}(\varPsi)$, and now by Proposition 1,
$$
\begin{equation*}
\bigl\| D_T\bigl((1-x)\varPsi^{(k)}_n+ x \varPsi \bigr)-D_T(\varPsi^{(k)}_n)\bigr\| \leqslant \gamma x \bigl\| \varPsi^{(k)}_n-\varPsi \bigr\| _{\mathcal{X}}.
\end{equation*}
\notag
$$
Hence
$$
\begin{equation*}
\int_0^{1} \bigl\| D_T \bigl((1-x) \varPsi^{(k)}_n + x \varPsi \bigr) - D_T(\varPsi^{(k)}_n) \bigr\| \, dx \,{\leqslant}\, \gamma \bigl\| \varPsi^{(k)}_n - \varPsi \bigr\|_{\mathcal{X}} \int_0^{1} x \, dx \,{=}\, \frac{1}{2} \gamma \bigl\| \varPsi^{(k)}_n - \varPsi \bigr\|_{\mathcal{X}}.
\end{equation*}
\notag
$$
Using the second inequality in Proposition 4, we obtain
$$
\begin{equation*}
\bigl\| \varPsi^{(k+1)}_n-\varPsi \bigr\|_{\mathcal{X}} \leqslant \alpha_n \bigl\| \varPsi^{(k)}_n-\varPsi \bigr\|_\mathcal{X}+\bigl( 2\nu(1+\alpha_n )\bigl\| \varPsi^{(k)}_n-\varPsi \bigr\| _\mathcal{X}\bigr) \frac{1}{2}\, \gamma\bigl\| \varPsi^{(k)}_n-\varPsi \bigr\| _{\mathcal{X}}.
\end{equation*}
\notag
$$
We now set
$$
\begin{equation*}
\varrho_n =\min \biggl\{ r,\biggl( \frac{1-\alpha_n}{2\gamma\nu (1+\alpha_n)} \biggr) \biggr\}.
\end{equation*}
\notag
$$
For $ \varPsi^{(k)}_n \in B_{\varrho_n}(\varPsi)$, we get
$$
\begin{equation*}
\frac{1}{2}\gamma\bigl\| \varPsi^{(k)}_n-\varPsi\bigr\| _{\mathcal{X}} \leqslant \frac{1-\alpha_n}{4\nu (1+\alpha_n)}.
\end{equation*}
\notag
$$
Hence
$$
\begin{equation*}
\bigl\| \varPsi^{(k+1)}_n-\varPsi\bigr\|_{\mathcal{X}} \leqslant \alpha_n \bigl\| \varPsi^{(k)}_n-\varPsi \bigr\|_{\mathcal{X}}+ \biggl(\frac{1-\alpha_n}{2} \biggr) \bigl\| \varPsi^{(k)}_n-\varPsi\bigr\|_{\mathcal{X}} \leqslant \biggl(\frac{1+\alpha_n}{2} \biggr) \bigl\| \varPsi^{(k)}_n-\varPsi\bigr\|_{\mathcal{X}},
\end{equation*}
\notag
$$
since $ 1+\alpha_n < 2$, the previous inequality implies that $\varPsi^{(k+1)}_n \in B_{\varrho_n}(\varPsi)$ and we finally conclude that
$$
\begin{equation*}
\bigl\| \varPsi^{(k+1 )}_n-\varPsi\bigr\|_{\mathcal{X}} \leqslant\varrho_n \biggl(\frac{1+\alpha_n}{2} \biggr)^k \to 0 \quad as\quad k \to \infty.
\end{equation*}
\notag
$$
Theorem 1 is proved. Remark 2. In theory, the new TLD scheme is superior to the classical (D.L.) method. This superiority is manifested in the fact that, in the new scheme, the approximate solution $\varPsi^{(k)}_n$ converges to $\varPsi$ as $k$ approaches infinity, regardless of the size of $n$. In contrast, in the classical (D.L.) method, the approximate solution $\varPsi^{(k)}_n$ only converges to $\varPsi$ as both $k$ and $n$ tend to infinity. § 4. Numerical TestsIn this section, we assess our new process TLD on non-linear integral equations of type (2.1). Let $\varPsi_n^{(k)} = \bigl( \psi_{n,1}^{(k)},\dots,\psi_{n,N}^{(k)}\bigr) \in \mathcal{X}$ be the iterate approximate solution obtained by solving the problem (2.8), and $\varPsi= (\psi_1,\dots,\psi_N) \in \mathcal{X}$ be the exact solution of the main problem (2.4). We define the condition that our algorithm must satisfy in order to terminate at the $k$th iteration as follows:
$$
\begin{equation*}
E_{n,N}^k=\sum_{l=1}^N \max_{1 \leqslant i \leqslant n} \bigl|\psi_{n,l}^{(k+1)}(t_i)-\psi_{n,l}^{(k)}(t_i) \bigr| \leqslant 10^{-9}, \qquad t_i \in \varOmega_n^l.
\end{equation*}
\notag
$$
Consider the residual value defined by
$$
\begin{equation*}
r^k_{n,N} = \max_{t \in [0,1]}\biggl|\lambda\varPsi_n^{(k+1)}(t)- \int_0^{\tau}\kappa(t,s,\varPsi_n^{(k)}(s))\, ds-g(t)\biggr|, \qquad \lambda\in \mathbb{R}^*.
\end{equation*}
\notag
$$
The general error is as follows:
$$
\begin{equation*}
E_{n,N}=\sum_{l=1}^N \max_{1 \leqslant i \leqslant n} \bigl|\psi_{n,l}^{(k+1)}(t_i)-\psi_l(t_i) \bigr|, \qquad t_i \in \varOmega_n^l.
\end{equation*}
\notag
$$
Example 1. Consider the non-linear Fredholm integral equations of the second kind where $\tau\gg1$, and the exact solution is and Example 2. Consider the non-linear Fredholm integral equations of the second kind where $\tau\gg1$, the exact solution is $ \psi_{\mathrm{ext}}(t) =e^{-t/30}\sin(3t)$, and Example 3. Consider the non-linear Fredholm integral equations of the second kind where $\tau\gg1$ and the exact solution is $\psi_{\mathrm{ext}}(t) =2\cos(\pi t/2)$, and  To tackle these numerical examples, we have converted our solution procedure into a clearly defined algorithm, as illustrated in Algorithm 1. This algorithm outlines the primary steps we follow to construct our numerical approximate solution. § 5. DiscussionsThe outcomes of the TLD methodology, as applied to Example 1, are given in Table 1, while the corresponding visual representations are depicted in Figures 1 and 2. Notably, our approach exhibits substantial efficacy, as evidenced by the computed general error and residual values. It is evident from the results that the error decreases with increasing parameter “$n$”, signifying an enhanced precision with increased computational resources. Moreover, even over considerable intervals, our method delivers consistently robust and accurate results. These exceptional outcomes were achieved with a mere six iterations $(k =6)$. This remarkable efficiency can be attributed to the rapid convergence of the Newton–Kantorovich linearization process. Table 1.The errors $E_{n,N}$ and $r_{n,N}^k$ with $m=10n$ and $\lambda=10$
 Figure 1.The exact and approximate solutions of Example 1 via TLD and the new process with $n=20$ and $\tau=20$  Figure 2.The variation of $\log_{10}(E^k_{n,N})$, the $\log_{10}$ of the error between two successive iterates approximations $\psi_n^{(k+1)}$ and $\psi_n^{(k)}$ of the first example. (Example 1) Table 2.The errors $E_{n,N}$ and $r_{n,N}^k$ with $m=10n$
 Figure 3.The exact and approximate solutions of Example 2 via TLD and the new process with $n=5$ and $\tau=50$  Figure 4.The variation of $\log_{10}(E^k_{n,N})$, the $\log_{10}$ of the error between two successive iterate approximations $\psi_n^{(k+1)}$ and $\psi_n^{(k)}$ of the second example. (Example 2) The performance of our approach in Example 2 across different parameters is demonstrated in Table 2 accompanied with well-crafted visual representations shown in Figures 3 and 4. We consider different values of “$n$”, maintain the fixed ratio “$m = 10n$”, and examine the impact on errors and convergence behavior. Notably, we observe that, even with just a small number of iterations (at most $k=5$), our method demonstrates remarkable accuracy in computing errors and residual values. The outcomes underscore the robustness and efficiency of the approach, as it consistently delivers highly precise results across various problem settings. Additionally, we observe that the convergence behavior remains steady, affirming the reliability of the Newton–Kantorovich linearization process. Table 3.The errors $E_{n,N}$ and $r_{n,N}^k$ with $m=10n$ and $\lambda=10$
 Figure 5.The exact and approximate solutions of Example 3 via TLD and the new process with $n=10$ and $\tau=30$ The results of the applied method for Example 3 are presented in Table 3, and the corresponding graphical representations are illustrated in Figures 5 and 6. The results in Table 3 illustrate the performance of our approach, similar to the previous examples, with a focus on errors $ E_{n,N}$ and residuals $r_{n,N}^k$, for various “$n$” and $\tau$. The data confirm that the error decreases with increasing “$n$”, demonstrating the improved accuracy with higher computational resolution. Although errors increase with larger $\tau$, the method consistently produces small residuals, indicating rapid convergence of the Newton–Kantorovich linearization process. Overall, these results reinforce the method robustness and efficiency, even over extended intervals, with reliable accuracy achieved in just a few iterations $(k=8)$.  Figure 6.The variation of $\log_{10}(E^k_{n,N})$, the $\log_{10}$ of the error between two successive iterate approximations $\psi_n^{(k+1)}$ and $\psi_n^{(k)}$ of the third example. (Example 3) We employed the Matlab computational software on a machine equipped with an Intel(R) Core(TM) i7-8665U CPU, base clock speed of $1.90$ GHz, which can reach up to $2.11$ GHz when boosted, and featuring $32$ GB of RAM. § 6. ConclusionsIn this paper, we present a new and effective method for solving second-kind non-linear integral equations across considerable intervals: the TLD (Transform, Linearize, Discretize) methodology. The approach, which was informed by theoretical understandings, showed amazing accuracy with few iterations and outstanding precision and computing efficiency. Numerical experiments conducted in a variety of settings validated the TLD perspective’s dependability and adaptability. Looking ahead, the method applicability to larger-scale problems (solving non-linear integral equations with weak singular kernels, for example) might be improved by combining parallel computing approaches. With its transformation powers, the TLD technique represents a major breakthrough in computational mathematics, providing new opportunities for effectively addressing challenging mathematical problems.
Citation in format AMSBIB
\Bibitem{SedKhe25} |
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