Numerical Solutions for Fractional Bagley–Torvik Equation with Integral Boundary Conditions

Abstract

The Bagley–Torvik equation (BTE) is an important model in mathematical physics and mechanics, but obtaining its analytical solution remains challenging. For its numerical treatment, the presence of composite functions in the generalized BTE poses additional difficulties, and efficient approaches for handling nonlinear terms are still lacking in the literature. This study proposes an improved numerical method for the fractional BTE with integral boundary conditions. By employing an integration technique, the original problem is transformed into a weakly singular Fredholm–Hammerstein (F–H) integral equation of the second kind. To address the nonlinear terms, an enhanced piecewise Taylor expansion scheme is developed to construct the discrete form, while the uniqueness of the solution is proven using the contraction mapping theorem in Banach spaces. The convergence and error analyses are rigorously carried out, and numerical experiments confirm the accuracy and efficiency of the proposed approach.

1. Introduction

Fractional differential equations are deeply rooted in theoretical physics and offer rich insights. Over the past few decades, the theory of fractional calculus has found widespread use in a variety of fields, such as engineering, mechanics, chemistry, biology, and many others [1,2,3,4,5,6,7,8].

In fact, fractional differential equations offer a more general and effective modeling framework than classical differential equations, especially for systems with memory and hereditary properties. The BTE, originally proposed by Bagley and Torvik, is given by [9]

$$A_1{u}^{″}\left(t\right)+A_2{D}^{\alpha }u\left(t\right)+A_{3}u\left(t\right)=g\left(t\right),(0<\alpha <2).$$

(1)

The constants $A_1,A_2$, and $A_3$ depend on several parameters, including the mass and geometric area of the plate, the spring stiffness, and the density and viscosity of the surrounding fluid. The function $g\left(t\right)$ represents a predefined external force acting on the system. The displacement of the plate over time, denoted as $u\left(t\right)$, is the principal unknown to be determined. The BTE plays a significant role in mathematical modeling and applied mechanics. It has been extensively employed to describe a wide range of physical phenomena, including network dynamics, the motion of Newtonian fluids, and the behavior of viscoelastic damping materials [10,11,12,13,14].

Owing to its significance, many analytical and numerical methods have been developed to investigate the BTE. For initial value problems, several analytical and numerical approaches have been developed. Fractional integral formulations have been used to derive explicit and unique solutions [15]; collocation methods with cubic B-splines and the Caputo–Fabrizio derivative have been proposed [16]; and spectral tau techniques based on Schröder polynomials have been implemented [17]. Moreover, a direct piecewise polynomial collocation method has been applied to the numerical solution of the BTE, and both its convergence and superconvergence properties have been established [18]. In boundary value settings, several approaches have been explored for the BTE [19,20,21,22,23]. The solvability of Riemann–Stieltjes integral boundary value problems under resonance conditions has been investigated by combining the Laplace transform with Mawhin’s coincidence degree theory [24]. For models with variable coefficients, the BTE has been reformulated in the form of a Volterra integral form, in which the Caputo derivative and Robin boundary conditions are efficiently handled using spline approximation and finite difference schemes [25]. Other numerical strategies include the collocation–shooting method for generalized fractional orders [26], finite element methods for problems with $0<\alpha <2$ [27], and Chebyshev collocation schemes that automatically incorporate boundary conditions and require only a few basis functions to achieve high accuracy [28]. Furthermore, the generalized BTE with variable coefficients subject to nonlocal or multipoint boundary conditions has also been investigated [29].

For fractional-order equations (including the BTE) [11,26,30,31], the existence and uniqueness of solutions are typically investigated using fixed-point theorems and related analytical techniques. However, studies involving variable coefficients and integral boundary conditions remain relatively limited, and their computational procedures are often quite complex. To address these limitations, the present study investigates a nonlinear BTE with variable coefficients subject to a two-point integral boundary condition, expressed as

$$y^{″}\left(x\right)+\gamma \left(x\right)D^{\beta }y\left(x\right)+\phi \left(x\right)y\left(x\right)=F(x,y\left(x\right)),0<\beta <2,x\in \left[l_1,l_2\right],$$

(2)

with

$$y\left(l_1\right)={\int }_{l_1}^{l_2}{g}_0\left(s\right)y\left(s\right)\mathrm{d}s,y\left(l_2\right)={\int }_{l_1}^{l_2}{g}_1\left(s\right)y\left(s\right)\mathrm{d}s.$$

(3)

Here, $\gamma \left(x\right),\phi \left(x\right)$ and $F(x,y(x\left)\right)$ are prescribed functions, while $y\left(x\right)$ denotes the unknown solution. This formulation generalizes the classical boundary conditions and captures the dependence of coefficients on physical parameters such as fluid density and viscosity. For convenience and brevity, we denote the fractional derivative ${}_{t_0}D_t^{\nu }$ simply by $D^{\nu }$. The Riemann–Liouville fractional derivative is adopted and defined as

$${}_{t_0}D_t^{\nu }y\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }(m-\nu )}}{\displaystyle \frac{d^m}{d{t}^m}}{\int }_{t_0}^t{\displaystyle \frac{y\left(\tau \right)}{{(t-\tau )}^{\nu -m+1}}}\mathrm{d}\tau ,m-1<\nu <m.$$

Here, $\mathsf{\Gamma }$ denotes the Gamma function, which is defined as

$$\mathsf{\Gamma }\left(\mu \right)={\int }_0^{\infty }{e}^{-t}{t}^{\mu -1}\mathrm{d}t,\mu >0.$$

(4)

Building upon previous applications of Taylor series methods to the BTE [29,32], the present study introduces a novel segmented expansion algorithm specifically formulated for systems governed by Equations (2) and (3). The proposed scheme naturally accommodates integral boundary conditions, substantially enhances computational efficiency, and attains high accuracy for variable-coefficient problems, even under minimal smoothness constraints.

The structure of the paper is organized as follows. Section 2 discusses the application of the integration method to transform the integral boundary problems Equations (2) and (3) into an F–H integral equation. In Section 3, sufficient conditions for ensuring the uniqueness of the solution to the specified F–H integral equation are presented in $\mathbf{C}[l_1,l_2]$, with the application of Banach’s fixed-point theorem. In Section 4, F–H integral equations are solved using an improved numerical approach accompanied by a convergence analysis and error bounds for the approximate solution. In Section 5, several numerical examples are presented to illustrate the effectiveness of the suggested methods. Lastly, key conclusions are drawn in Section 6.

2. Fredholm–Hammerstein Integral Equations

For clarity and ease of reference, specific shorthand notations are adopted in this discussion:

$$W(t,x)=\left\{\begin{array}{c}{W}_1(t,x),0<\beta <1,\hfill \\ W_2(t,x),1\le \beta <2,\hfill \end{array}\right.$$

$$A(t,x)=\left\{\begin{array}{c}{A}_1(t,x),0<\beta <1,\hfill \\ A_2(t,x),1\le \beta <2,\hfill \end{array}\right.$$

where

$$W_1(t,x)=(x-t)\phi \left(t\right)+{\displaystyle \frac{1}{\mathsf{\Gamma }(1-\beta )}}{\int }_t^x{\displaystyle \frac{\gamma \left(s\right)-(x-s){\gamma }^{\prime }\left(s\right)}{{(s-t)}^{\beta }}}\mathrm{d}s,$$

$$A_1(t,x)={\displaystyle \frac{x-l_1}{l_2-l_1}}{W}_1(t,l_2)+{\displaystyle \frac{l_2-x}{l_2-l_1}}{g}_0\left(t\right)+{\displaystyle \frac{x-l_1}{l_2-l_1}}{g}_1\left(t\right),$$

and

$$W_2(t,x)=(x-t)\phi \left(t\right)+{\displaystyle \frac{1}{\mathsf{\Gamma }(2-\beta )}}\left[{\displaystyle \frac{\gamma \left(x\right)}{{(t-x)}^{1-\beta }}}-{\int }_t^x{\displaystyle \frac{2{\gamma }^{\prime }\left(s\right)-(s-x){\gamma }^{″}\left(s\right)}{{(t-s)}^{1-\beta }}}\mathrm{d}s\right],$$

$$A_2(t,x)={\displaystyle \frac{x-l_1}{l_2-l_1}}{W}_2(t,l_2)+{\displaystyle \frac{l_2-x}{l_2-l_1}}{g}_0\left(t\right)+{\displaystyle \frac{x-l_1}{l_2-l_1}}{g}_1\left(t\right).$$

In this section, we transform the integral boundary problems Equations (2) and (3) into F–H integral equations of the second kind. Subsequently, we derive the following theorems as a consequence of this transformation.

Theorem 1.

Assuming $0<\beta <2$, $\gamma \left(x\right)\in {\mathbf{C}}^2[l_1,l_2]$, $\phi \left(x\right)\in {\mathbf{L}}^1[l_1,l_2]$ and $F(x,y(x\left)\right)\in$ ${\mathbf{L}}^1[l_1,l_2;-\infty ,+\infty ]$, then we consider the integral boundary problem

$$\left\{\begin{array}{c}{y}^{″}\left(x\right)+\gamma \left(x\right)D^{\beta }y\left(x\right)+\phi \left(x\right)y\left(x\right)=F(x,y\left(x\right)),x\in \left[l_1,l_2\right],\hfill \\ y\left(l_1\right)={\int }_{l_1}^{l_2}{g}_0\left(s\right)y\left(s\right)\mathrm{d}s,y\left(l_2\right)={\int }_{l_1}^{l_2}{g}_1\left(s\right)y\left(s\right)\mathrm{d}s,\hfill \end{array}\right.$$

(5)

which can be reformulated as an F–H integral equation of the second kind

$$y\left(x\right)+{\int }_{l_1}^{l_2}\overline{K}(t,x)y\left(t\right)\mathrm{d}t={\int }_{l_1}^{l_2}\overline{G}(t,x)F(t,y\left(t\right))\mathrm{d}t,$$

(6)

where

$$\begin{array}{c}\hfill \overline{K}(t,x)=\left\{\begin{array}{c}{\displaystyle W_2(t,x)-A_2(t,x),l_1\le t\le x\le l_2,}\hfill \\ {\displaystyle -A_2(t,x),l_1\le x\le t\le l_2,}\hfill \end{array}\right.\end{array}$$

$$\begin{array}{c}\overline{G}(t,x)=\left\{\begin{array}{c}{\displaystyle (x-t)-\frac{x-l_1}{l_2-l_1}(l_2-t),l_1\le t\le x\le l_2,}\hfill \\ \frac{l_1-x}{l_2-l_1}(l_2-t),l_1\le x\le t\le l_2.\hfill \end{array}\right.\hfill \end{array}$$

Proof. 

Given $1\le \beta <2$, the differential equation in Equation (2) is integrated twice over $(l_1,x)$, resulting in

$$\begin{array}{cc}\hfill & y\left(x\right)+{\int }_{l_1}^x\{(x-t)\phi \left(t\right)+{\displaystyle \frac{1}{\mathsf{\Gamma }(2-\beta )}}[{\displaystyle \frac{\gamma \left(x\right)}{{(t-x)}^{1-\beta }}}\hfill \\ \hfill -& {\int }_t^x{\displaystyle \frac{2{\gamma }^{\prime }\left(s\right)-(x-s){\gamma }^{″}\left(s\right)}{{(t-s)}^{1-\beta }}}\mathrm{d}s\left]\right\}y\left(t\right)\mathrm{d}t={\int }_{l_1}^x(x-t)F(t,y\left(t\right))\mathrm{d}t\hfill \\ \hfill +& y\left(l_1\right)+\left[{\displaystyle \frac{\gamma \left(l_1\right)}{\mathsf{\Gamma }(2-\beta )}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\int }_{l_1}^t{\displaystyle \frac{y\left(s\right)}{{(s-t)}^{1-\beta }}}\mathrm{d}s{|}_{t=l_1}+y^{\prime }\left(l_1\right)\right](x-l_1).\hfill \end{array}$$

(7)

By setting $x=l_2$ in Equation (7), we have

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\gamma \left(l_1\right)}{\mathsf{\Gamma }(2-\beta )}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\int }_{l_1}^t{\displaystyle \frac{y\left(s\right)}{{(s-t)}^{1-\beta }}}\mathrm{d}s{|}_{t=l_1}+y^{\prime }\left(l_1\right)\hfill \\ \hfill =& {\displaystyle \frac{1}{(l_2-l_1)}}\left[{\int }_{l_1}^{l_2}{W}_2(t,l_2)y\left(t\right)\mathrm{d}t-{\int }_{l_1}^{l_2}(l_2-t)F(t,y\left(t\right))\mathrm{d}t+y\left(l_2\right)-y\left(l_1\right)\right].\hfill \end{array}$$

(8)

Substituting Equation (8) into Equation (7) leads to

$$\begin{array}{cc}\hfill & y\left(x\right)+{\int }_{l_1}^x{W}_2(t,x)y\left(t\right)\mathrm{d}t-{\int }_{l_1}^{l_2}{\displaystyle \frac{x-l_1}{l_2-l_1}}{W}_2(t,l_2)y\left(t\right)\mathrm{d}t\hfill \\ \hfill =& {\int }_{l_1}^x(x-t)F(t,y\left(t\right))\mathrm{d}t-{\int }_{l_1}^{l_2}{\displaystyle \frac{x-l_1}{l_2-l_1}}(l_2-t)F(t,y\left(t\right))\mathrm{d}t+{\displaystyle \frac{l_2-x}{l_2-l_1}}y\left(l_1\right)+{\displaystyle \frac{x-l_1}{l_2-l_1}}y\left(l_2\right).\hfill \end{array}$$

(9)

By substituting the integral boundary problem from Equation (5) into Equation (9), we obtain

$$\begin{array}{cc}\hfill & y\left(x\right)+{\int }_{l_1}^x{W}_2(t,x)y\left(t\right)\mathrm{d}t-{\int }_{l_1}^{l_2}{\displaystyle \frac{x-l_1}{l_2-l_1}}{W}_2(t,l_2)y\left(t\right)\mathrm{d}t={\int }_{l_1}^x(x-t)F(t,y\left(t\right))\mathrm{d}t\hfill \\ \hfill -& {\int }_{l_1}^{l_2}{\displaystyle \frac{x-l_1}{l_2-l_1}}(l_2-t)F(t,y\left(t\right))\mathrm{d}t+{\displaystyle \frac{l_2-x}{l_2-l_1}}{\int }_{l_1}^{l_2}{g}_0\left(s\right)y\left(s\right)\mathrm{d}s+{\displaystyle \frac{x-l_1}{l_2-l_1}}{\int }_{l_1}^{l_2}{g}_1\left(s\right)y\left(s\right)\mathrm{d}s,\hfill \end{array}$$

(10)

or

$$\begin{array}{cc}\hfill & y\left(x\right)+{\int }_{l_1}^x{W}_2(t,x)y\left(t\right)\mathrm{d}t-{\int }_{l_1}^{l_2}{A}_2(t,x)y\left(t\right)\mathrm{d}t\hfill \\ \hfill =& {\int }_{l_1}^x(x-t)F(t,y\left(t\right))\mathrm{d}t-{\int }_{l_1}^{l_2}{\displaystyle \frac{x-l_1}{l_2-l_1}}(l_2-t)F(t,y\left(t\right))\mathrm{d}t.\hfill \end{array}$$

(11)

By considering different ranges of x and t, the following expressions are obtained:

Case 1: $l_1\le t\le x\le l_2$

$$\begin{array}{c}\hfill y\left(x\right)+{\int }_{l_1}^x\{W_2(t,x)-A_2(t,x)\}y\left(t\right)\mathrm{d}t={\int }_{l_1}^x\{(x-t)-{\displaystyle \frac{x-l_1}{l_2-l_1}}(l_2-t)\}F(t,y\left(t\right))\mathrm{d}t.\end{array}$$

(12)

Case 2: $l_1\le x\le t\le l_2$

$$\begin{array}{c}\hfill y\left(x\right)-{\int }_x^{l_2}{A}_2(t,x)y\left(t\right)\mathrm{d}t=-{\int }_x^{l_2}{\displaystyle \frac{x-l_1}{l_2-l_1}}(l_2-t)F(t,y\left(t\right))\mathrm{d}t.\end{array}$$

(13)

Hence, all the foregoing derivations confirm that the original fractional boundary value problem has been successfully reduced to the weakly singular F–H integral equation of the second kind given by Equation (6). □

In a similar fashion, we note that when for symmetrical interval $0<\beta <1$ the integral boundary problem in Equation (5) can be reformulated as a second-kind F–H integral equation. To conserve space, the detailed steps for this transformation are not presented here.

According to Theorem 1, we have successfully converted the integral boundary issues related to fractional BTE into a second-kind F–H integral equation. It is important to highlight that for $0<\beta \le 1$ the condition $\gamma \left(x\right)\in {\mathbf{C}}^2[l_1,l_2]$ can be relaxed to $\gamma \left(x\right)\in {\mathbf{C}}^1[l_1,l_2]$. Notably, the kernel $\overline{K}(x,t)$ exhibits weak singularity when the fractional order $\beta$ lies within the intervals $(0,1)\cup (1,2)$. Furthermore, when $\beta =1$ this aligns with the integral boundary problems arising from second-order ordinary differential equations. Specifically, the integration technique can be applied broadly to transform both linear and nonlinear multipoint boundary value issues of various differential equations into an integral equation format.

3. Uniqueness of the Solution

The subsequent sections will introduce the theorem concerning the uniqueness of solutions for the weakly singular F–H integral equation of the second-kind Equation (6), utilizing within continuous function spaces a theorem on contraction operators.

Theorem 2.

If $0<\beta <2$, $\gamma \left(x\right)\in {\mathbf{C}}^2[l_1,l_2]$, $\phi \left(x\right)\in {\mathbf{L}}^1[l_1,l_2]$, and $F(x,y(x\left)\right)$ satisfies the Lipschitz condition

$$\parallel F(x,y_1\left(x\right))-F(x,y_2\left(x\right)){\parallel }_{\infty }\le L{\parallel y_1\left(x\right)-y_2\left(x\right)\parallel }_{\infty },$$

where $L>0$ is the Lipschitz constant. Moreover, one has the constraint conditions as

$$\parallel {\int }_{l_1}^{l_2}\overline{G}(t,x)F(t,y\left(t\right))\mathrm{d}t{\parallel }_{\infty }\le P{\parallel y\parallel }_{\infty },$$

where $P>0$ is constant. If

$$\underset{l_1\le x\le l_2}{max}{\int }_{l_1}^{l_2}\left(|\overline{K}(t,x)|+L|\overline{G}(t,x)|\right)\mathrm{d}t<1$$

then the weakly singular F–H integral equation Equation (6) has a unique solution in $\mathbf{C}[l_1,l_2].$

Proof. 

For ease of reference, we rewrite Equation (6) as

$$\begin{array}{c}\hfill y=\overline{\mathbf{G}}y-\overline{\mathbf{K}}y=\mathbf{T}y,\end{array}$$

with

$$\begin{array}{c}\hfill \left(\overline{\mathbf{G}}y\right)\left(x\right)={\int }_{l_1}^{l_2}\overline{G}(t,x)F(t,y\left(t\right))\mathrm{d}t,\left(\overline{\mathbf{K}}y\right)\left(x\right)={\int }_{l_1}^{l_2}\overline{K}(t,x)y\left(t\right)\mathrm{d}t.\end{array}$$

It follows that the operators $\overline{\mathbf{G}}:\mathbf{C}[l_1,l_2]\to \mathbf{C}[l_1,l_2]$ and $\overline{\mathbf{K}}:\mathbf{C}[l_1,l_2]\to \mathbf{C}[l_1,l_2]$ are both linear and bounded. The function $\gamma \left(x\right)\in {\mathbf{C}}^2[l_1,l_2]$, $\phi \left(x\right)\in {\mathbf{L}}^1[l_1,l_2]$, $(x-t)\in \mathbf{C}[l_1,l_2]$, $1/{(x-t)}^{\beta -1}(1\le \beta <2)\in {\mathbf{L}}^1[l_1,l_2]$ for finite $l_1$ and $l_2$. It follows that $W_2(t,x),$ $A_2(t,x)\in$ ${\mathbf{L}}^1[l_1,l_2;l_1,l_2]$. Hence, we conclude that $\overline{K}(t,x)\in {\mathbf{L}}^1[l_1,l_2;l_1,l_2]$. By using

$$\parallel {\int }_{l_1}^{l_2}\overline{G}(t,x)F(t,y\left(t\right))\mathrm{d}t{\parallel }_{\infty }\le P{\parallel y\parallel }_{\infty },$$

and the kernel $\overline{G}(t,x)$ being a polynomial function with respect to x and $t,$ it is seen that $\overline{G}(t,x)\in$ ${\mathbf{L}}^1[l_1,l_2;l_1,l_2]$ and $\overline{\mathbf{G}}$ is a bounded operator from $\mathbf{C}[l_1,l_2]$ to $\mathbf{C}[l_1,l_2]$. Furthermore, we let

$$\begin{array}{c}\hfill \underset{l_1\le x\le l_2}{max}{\int }_{l_1}^{l_2}\left(|\overline{K}(t,x)|+L|\overline{G}(t,x)|\right)\mathrm{d}t=c_1,c_1\ge 0.\end{array}$$

For all $y_1,y_2\in \mathbf{C}[l_1,l_2]$, one has

$$\begin{array}{ccc}& & \parallel \mathbf{T}{y}_1-\mathbf{T}{y}_2{\parallel }_{\infty }={\parallel \overline{\mathbf{G}}(y_1-y_2)-\overline{\mathbf{K}}(y_1-y_2)\parallel }_{\infty }\hfill \\ & & \le {∥{\int }_{l_1}^{l_2}\overline{G}(t,x)[F(t,y_1\left(t\right))-F(t,y_2\left(t\right))]\mathrm{d}t∥}_{\infty }+{∥{\int }_{l_1}^{l_2}\overline{K}(t,x)[y_1\left(t\right)-y_2\left(t\right)]\mathrm{d}t∥}_{\infty }\hfill \\ & & \le \underset{l_1\le x\le l_2}{max}{\int }_{l_1}^{l_2}\left(|\overline{K}(t,x)|+L|\overline{G}(t,x)|\right)\mathrm{d}t·\parallel y_1-y_2{\parallel }_{\infty }=c_1{\parallel y_1-y_2\parallel }_{\infty }.\hfill \end{array}$$

By the Banach fixed-point theorem, when $c_1<1$ the weakly singular F–H integral equation given in Equation (6) admits a unique solution in $\mathbf{C}[l_1,l_2]$. □

Theorem 2 establishes the conditions ensuring the existence and uniqueness of the solution in $\mathbf{C}[l_1,l_2]$. It is important to highlight that for $0<\beta \le 1$ the requirement that $\gamma \left(x\right)\in$ ${\mathbf{C}}^2[l_1,l_2]$ can be relaxed to $\gamma \left(x\right)\in {\mathbf{C}}^1[l_1,l_2]$. This distinction arises because the kernel in Equation (6) is not the same as the conventional weakly singular kernel ${|x-t|}^{-1}$. These methods follow a similar style:

$$\begin{array}{c}\hfill \overline{K}(t,x)={\displaystyle \frac{A(t,x)}{{(x-t)}^{{\beta }_0}}}+{\displaystyle \frac{B(t,x)}{{(l_2-t)}^{{\beta }_0}}}.\end{array}$$

Given that $0<{\beta }_0<1$ and that $A(t,x)$ and $B(t,x)$ are piecewise continuous functions, an improved and efficient numerical approach, based on an existing method, is employed to approximate the solution following the establishment of its uniqueness. The details of this method will be presented in the subsequent section.

4. A New Numerical Method

We will now focus on finding numerical solutions to Equation (6). A recent approach that employs Taylor-series expansion has been introduced for addressing F–H integral equations of the second kind. Nonetheless, the kernels found in Equation (6) may demonstrate weak singularity, making the method outlined in [21] unsuitable for direct implementation. Therefore, we will modify the numerical technique from [21] to make it appropriate for tackling F–H integral equations of the second kind featuring weakly singular kernels. Additionally, it should also be noted that the existence of a solution in $\mathbf{C}[l_1,l_2]$ is guaranteed only under the conditions established in Theorem 2. To effectively use the piecewise Taylor series expansion method it is crucial that the solution $y\left(x\right)$ belongs to ${\mathbf{C}}^{n+1}[l_1,l_2]$ $(n\ge 0)$ in practical scenarios.

4.1. Formulation of the Approximate Solution

Typically, an F–H integral equation of the second kind can be expressed as

$$\begin{array}{c}\hfill y\left(x\right)+{\int }_{l_1}^{l_2}\overline{K}(t,x)y\left(t\right)\mathrm{d}t={\int }_{l_1}^{l_2}\overline{G}(t,x)F(t,y\left(t\right))\mathrm{d}t,x\in [l_1,l_2],\end{array}$$

(14)

where $y\left(x\right)$ is the unknown function to be determined, and where the kernel $\overline{K}(t,x)$ exhibits weak singularity. Unlike the differential approach presented in [21], we perform integration on both sides of Equation (14), a total of $l(l=1,2,\dots ,k)$ iterations, to obtain

$$\begin{array}{cc}\hfill & {\int }_{l_1}^{x}y\left(t\right){\displaystyle \frac{{(x-t)}^{l-1}}{(l-1)!}}\mathrm{d}t+{\int }_{l_1}^x\left[{\int }_{l_1}^{l_2}\overline{K}(\tau ,t)y\left(t\right)\mathrm{d}t\right]{\displaystyle \frac{{(x-\tau )}^{l-1}}{(l-1)!}}\mathrm{d}\tau \hfill \\ \hfill =& {\int }_{l_1}^x\left[{\int }_{l_1}^{l_2}\overline{G}(\tau ,t)F(t,y\left(t\right))\mathrm{d}t\right]{\displaystyle \frac{{(x-\tau )}^{l-1}}{(l-1)!}}\mathrm{d}\tau .\hfill \end{array}$$

(15)

Equation (14) corresponds to the special case of $l=0$. By exchanging the order of integration in Equation (15), we obtain

$$\begin{array}{cc}& {\int }_{l_1}^{x}y\left(t\right){\displaystyle \frac{{(x-t)}^{l-1}}{(l-1)!}}\mathrm{d}t+{\int }_{l_1}^{l_2}\left[{\int }_{l_1}^x{\displaystyle \frac{{(x-\tau )}^{l-1}}{(l-1)!}}\overline{K}(\tau ,t)\mathrm{d}\tau \right]y\left(t\right)\mathrm{d}t\hfill \\ \hfill =& {\int }_{l_1}^{l_2}\left[{\int }_{l_1}^x{\displaystyle \frac{{(x-\tau )}^{l-1}}{(l-1)!}}\overline{G}(\tau ,t)\mathrm{d}\tau \right]F(t,y\left(t\right))\mathrm{d}t.\hfill \end{array}$$

(16)

Following the idea of piecewise approximation, we first define a set of integration nodes as

$$l_1=x_0<x_1<\dots <x_m=l_2(m\ge 1),$$

where $c=(l_2-l_1)/m$ and $x_s=l_1+sc(s=0,1,\dots ,m)$. By setting $x=x_i(i=1,2,\dots ,m)$, Equation (16) can be rewritten in the following form:

$$\begin{array}{cc}\hfill & \sum _{s=0}^{i-1}{\int }_{x_s}^{x_{s+1}}{\displaystyle \frac{{(x_i-t)}^{l-1}}{(l-1)!}}y\left(t\right)\mathrm{d}t+\sum _{h=0}^{i-1}\sum _{s=0}^{m-1}{\int }_{x_s}^{x_{s+1}}\left[{\int }_{x_h}^{x_{h+1}}{\displaystyle \frac{{(x_i-\tau )}^{l-1}}{(l-1)!}}\overline{K}(\tau ,t)\mathrm{d}\tau \right]y\left(t\right)\mathrm{d}t\hfill \\ \hfill =& \sum _{h=0}^{i-1}\sum _{s=0}^{m-1}{\int }_{x_s}^{x_{s+1}}\left[{\int }_{x_h}^{x_{h+1}}{\displaystyle \frac{{(x_i-\tau )}^{l-1}}{(l-1)!}}\overline{G}(\tau ,t)\mathrm{d}\tau \right]F(t,y\left(t\right))\mathrm{d}t.\hfill \end{array}$$

(17)

It is now assumed that $y\left(t\right)$ and $F(t,y(t\left)\right)$ can be represented by a Taylor series expansion with a Lagrange remainder. Specifically, we have

$$\begin{array}{c}\hfill y(x_s+c\eta )=\sum _{n=0}^k{\displaystyle \frac{y^{\left(n\right)}\left(x_s\right)}{n!}}{\left(c\eta \right)}^n+{\displaystyle \frac{y^{(k+1)}\left({\theta }_s\right)}{(k+1)!}}{\left(c\eta \right)}^{k+1},x_s\le {\theta }_s\le x_{s+1}\end{array}$$

(18)

$$\begin{array}{c}\hfill F(x_s+c\eta ,y(x_s+c\eta ))=\sum _{n=0}^k{\displaystyle \frac{{\left(c\eta \right)}^n}{n!}}{\displaystyle \frac{{\mathrm{d}}^{n}F(u,y\left(u\right))}{\mathrm{d}{u}^n}}{|}_{u=x_s}\\ \hfill +{\displaystyle \frac{{\left(c\eta \right)}^{k+1}}{(k+1)!}}{\displaystyle \frac{{\mathrm{d}}^{k+1}F(u,y\left(u\right))}{\mathrm{d}{u}^{k+1}}}{|}_{u={\vartheta }_s},x_s\le {\vartheta }_s\le x_s+c\eta \end{array}$$

(19)

We define $\tau =x_h+c\xi$ and $t=x_s+c\eta$. By substituting Equations (18) and (19) into Equation (17), we obtain the following expression:

$$\begin{array}{cc}\hfill & \sum _{s=0}^{i-1}{\displaystyle \frac{c^l}{(l-1)!}}{\int }_0^1{(i-s-\eta )}^{l-1}\left[\sum _{n=0}^k{\displaystyle \frac{y^{\left(n\right)}\left(x_s\right)}{n!}}{\left(c\eta \right)}^n+{\displaystyle \frac{y^{(k+1)}\left({\theta }_s\right)}{(k+1)!}}{\left(c\eta \right)}^{k+1}\right]\mathrm{d}\eta \hfill \\ \hfill & +\sum _{h=0}^{i-1}\sum _{s=0}^{m-1}{\displaystyle \frac{c^{l+1}}{(l-1)!}}{\int }_0^1{\int }_0^1{(i-h-\xi )}^{l-1}\overline{K}(x_h+c\xi ,x_s+c\eta )\hfill \\ \hfill & ·\left[\sum _{n=0}^k{\displaystyle \frac{y^{\left(n\right)}\left(x_s\right)}{n!}}{\left(c\eta \right)}^n+{\displaystyle \frac{y^{(k+1)}\left({\theta }_s\right)}{(k+1)!}}{\left(c\eta \right)}^{k+1}\right]\mathrm{d}\xi \mathrm{d}\eta \hfill \\ \hfill & =\sum _{h=0}^{i-1}\sum _{s=0}^{m-1}{\displaystyle \frac{c^{l+1}}{(l-1)!}}{\int }_0^1{\int }_0^1{(i-h-\xi )}^{l-1}\overline{G}(x_h+c\xi ,x_s+c\eta )\hfill \\ \hfill & ·\left[\sum _{n=0}^k{\displaystyle \frac{{\left(c\eta \right)}^n}{n!}}{\displaystyle \frac{{\mathrm{d}}^{n}F(u,y\left(u\right))}{\mathrm{d}{u}^n}}{|}_{u=x_s}+{\displaystyle \frac{{\left(c\eta \right)}^{k+1}}{(k+1)!}}{\displaystyle \frac{{\mathrm{d}}^{k+1}F(u,y\left(u\right))}{\mathrm{d}{u}^{k+1}}}{|}_{u={\vartheta }_s}\right]\mathrm{d}\xi \mathrm{d}\eta .\hfill \end{array}$$

(20)

Based on Equation (20), a discrete formulation of the nonlinear F–H integral equation can be formulated as follows:

$$\begin{array}{cc}\hfill & \sum _{s=0}^{i-1}\sum _{n=0}^k{\displaystyle \frac{y_s^{\left(n\right)}}{n!}}{\displaystyle \frac{c^{l+n}}{(l-1)!}}{\int }_0^1{(i-s-\eta )}^{l-1}{\eta }^n\mathrm{d}\eta +\sum _{s=0}^{m-1}\sum _{h=0}^{i-1}\sum _{n=0}^k{\displaystyle \frac{y_s^{\left(n\right)}}{n!}}{\displaystyle \frac{c^{l+n+1}}{(l-1)!}}\hfill \\ \hfill & ·{\int }_0^1{\int }_0^1{(i-h-\xi )}^{l-1}\overline{K}(x_h+c\xi ,x_s+c\eta ){\eta }^n\mathrm{d}\xi \mathrm{d}\eta =\sum _{h=0}^{i-1}\sum _{s=0}^{m-1}\sum _{n=0}^k{\displaystyle \frac{c^{l+n+1}}{(l-1)!n!}}\hfill \\ \hfill & ·{\displaystyle \frac{{\mathrm{d}}^{n}F}{\mathrm{d}{u}^n}}{|}_{(x_s,y_s^{\left(n\right)})}{\int }_0^1{\int }_0^1{(i-h-\xi )}^{l-1}\overline{G}(x_h+c\xi ,x_s+c\eta ){\eta }^n\mathrm{d}\xi \mathrm{d}\eta ,\hfill \end{array}$$

(21)

where $l=0,1,2,\dots ,k$ and $i=1,2,\dots ,m$. It follows that we can determine the numerical solutions $y_s^{\left(n\right)}$ of $y^{\left(n\right)}\left(x_s\right)$ for $s=0,1,\dots ,m-1$ and $n=0,1,\dots ,k$. By substituting Equations (18) and (19) into Equation (14), the latter can be approximated by the following discrete formulation:

$$\begin{array}{cc}\hfill & y_{i-1}^{\left(0\right)}+\sum _{j=0}^k\sum _{s=0}^{m-1}{\displaystyle \frac{y_s^{\left(n\right)}{c}^{n+1}}{n!}}{\int }_0^1\overline{K}(x_{i-1},x_s+c\eta ){\eta }^n\mathrm{d}\eta \hfill \\ \hfill & =\sum _{n=0}^k\sum _{s=0}^{m-1}{\displaystyle \frac{c^{n+1}}{n!}}{\displaystyle \frac{{\mathrm{d}}^{n}F}{\mathrm{d}{u}^n}}{|}_{(x_s,y_s^{\left(n\right)})}{\int }_0^1\overline{G}(x_{i-1},x_s+c\eta ){\eta }^n\mathrm{d}\eta .\hfill \end{array}$$

(22)

After solving the nonlinear system defined by Equations (21) and (22), the approximate solution of $y\left(x\right)$ in the interval $l_1\le x\le l_2$ can be expressed as

$$\begin{array}{cc}\hfill y_{m,k}\left(x\right)=& c\sum _{j=0}^k\sum _{s=0}^{m-1}{\displaystyle \frac{c^n}{n!}}{\displaystyle \frac{{\mathrm{d}}^{n}F}{\mathrm{d}{u}^n}}{|}_{(x_s,y_s^{\left(n\right)})}{\int }_0^1\overline{G}(x,x_s+c\eta ){\eta }^n\mathrm{d}\eta \hfill \\ \hfill & -\sum _{n=0}^k\sum _{s=0}^{m-1}{\displaystyle \frac{y_s^{\left(n\right)}{c}^{n+1}}{n!}}{\int }_0^1\overline{K}(x,x_s+c\eta ){\eta }^n\mathrm{d}\eta .\hfill \end{array}$$

(23)

It can be observed from Equation (20) that the approximation technique utilized in this study involves two parameters, where the discretization points $x_s$ are used in the Taylor series expansion.

4.2. Convergence Analysis and Error Estimation

The convergence behavior and the corresponding error bounds of the approximate solution $y_{m,k}\left(x\right)$ are now examined. To begin with, Equations (21) and (22) are reformulated as follows:

$$(C+D)\tilde{\mathsf{\Phi }}=E\left(\tilde{\mathsf{\Phi }}\right),$$

(24)

where

$$\begin{array}{ccc}& & \tilde{\mathsf{\Phi }}={\left[y_s^{\left(n\right)}\right]}_{m(k+1)\times 1}={\left[y_0^{\left(0\right)},y_1^{\left(0\right)},\dots ,y_{m-1}^{\left(k\right)}\right]}^T,\hfill \\ & & C={\left[C_{pt}\right]}_{m(k+1)\times m(k+1)},\hfill \\ & & D={\left[d_{pt}\right]}_{m(k+1)\times m(k+1)},\hfill \\ & & E\left(\tilde{\mathsf{\Phi }}\right)={\left[w_p\left(\tilde{\mathsf{\Phi }}\right)\right]}_{m(k+1)\times 1},\hfill \end{array}$$

with

$$\begin{array}{ccc}& & C_{11}=1,\hfill \\ & & C_{12}=0,\hfill \\ & & \dots \dots \dots \dots \hfill \\ & & C_{m(k+1),m(k+1)}={\displaystyle \frac{c^{2k}}{(k-1)!k!}}{\int }_0^1{(1-\eta )}^{k-1}{\eta }^k\mathrm{d}\eta ,\hfill \\ & & d_{11}=c{\int }_0^1\overline{K}(x_0,x_0+c\eta )\mathrm{d}\eta ,\hfill \\ & & d_{12}=c{\int }_0^1\overline{K}(x_0,x_1+c\eta )\mathrm{d}\eta ,\hfill \\ & & \dots \dots \dots \dots \hfill \\ & & d_{m(k+1),m(k+1)}=\sum _{h=0}^{m-1}{\displaystyle \frac{c^{2k+1}}{(k-1)!k!}}{\int }_0^1{\int }_0^1{(m-h-\xi )}^{k-1}\overline{K}(x_h+c\xi ,x_{m-1}+c\eta ){\eta }^k\mathrm{d}\xi \mathrm{d}\eta ,\hfill \\ & & w_1\left(\tilde{\mathsf{\Phi }}\right)=\sum _{q=0}^{m-1}\sum _{n=0}^k{\displaystyle \frac{c^{n+1}}{n!}}{\displaystyle \frac{{\mathrm{d}}^{n}F}{\mathrm{d}{u}^n}}{|}_{(x_s,y_s^{\left(n\right)})}{\int }_0^1\overline{G}(x_0,x_s+c\eta ){\eta }^n\mathrm{d}\eta ,\hfill \\ & & w_2\left(\tilde{\mathsf{\Phi }}\right)=\sum _{s=0}^{m-1}\sum _{n=0}^k{\displaystyle \frac{c^{n+1}}{n!}}{\displaystyle \frac{{\mathrm{d}}^{n}F}{\mathrm{d}{u}^n}}{|}_{(x_s,y_s^{\left(n\right)})}{\int }_0^1\overline{G}(x_1,x_s+c\eta ){\eta }^n\mathrm{d}\eta ,\hfill \\ & & \dots \dots \dots \dots \hfill \\ & & w_{m(k+1)}\left(\tilde{\mathsf{\Phi }}\right)=\sum _{h=0}^{m-1}\sum _{s=0}^{m-1}\sum _{n=0}^k{\displaystyle \frac{c^{k+n+1}}{(k-1)!n!}}{\displaystyle \frac{{\mathrm{d}}^{n}F}{\mathrm{d}{u}^n}}{|}_{(x_s,y_s^{\left(n\right)})}\hfill \\ & & ·{\int }_0^1\left[{\int }_0^1{(m-h-\xi )}^{k-1}\overline{G}(x_h+c\xi ,x_s+c\eta )\mathrm{d}\xi \right]{\eta }^n\mathrm{d}\eta .\hfill \end{array}$$

By taking into account the Lagrange remainder, we deduce that

$$(C+D)\mathsf{\Phi }-E\left(\mathsf{\Phi }\right)=R,$$

(25)

where

$$\begin{array}{ccc}& & R={\left[r_p\right]}_{m(k+1)\times 1},\hfill \\ & & \mathsf{\Phi }={\left[y^{\left(n\right)}\left(x_s\right)\right]}_{m(k+1)\times 1}={[y^{\left(0\right)}\left(x_0\right),y^{\left(0\right)}\left(x_1\right),\dots ,y^{\left(k\right)}\left(x_{m-1}\right)]}^T,\hfill \\ & & E\left(\mathsf{\Phi }\right)={\left[w_p\left(\mathsf{\Phi }\right)\right]}_{m(k+1)\times 1},\hfill \end{array}$$

with

$$\begin{array}{ccc}& & w_1\left(\mathsf{\Phi }\right)=\sum _{n=0}^k\sum _{s=0}^{m-1}{\displaystyle \frac{c^{n+1}}{n!}}{\displaystyle \frac{{\mathrm{d}}^{n}F(u,y\left(u\right))}{d{u}^n}}{|}_{u=x_s}{\int }_0^1\overline{G}(x_0,x_s+c\eta ){\eta }^n\mathrm{d}\eta ,\hfill \\ & & w_2\left(\mathsf{\Phi }\right)=\sum _{n=0}^k\sum _{s=0}^{m-1}{\displaystyle \frac{c^{n+1}}{n!}}{\displaystyle \frac{{\mathrm{d}}^{n}F(u,y\left(u\right))}{\mathrm{d}{u}^n}}{|}_{u=x_s}{\int }_0^1\overline{G}(x_1,x_s+c\eta ){\eta }^n\mathrm{d}\eta ,\hfill \\ & & \dots \dots \dots \dots \hfill \\ & & w_{m(k+1)}\left(\mathsf{\Phi }\right)=\sum _{h=0}^{m-1}\sum _{s=0}^{m-1}\sum _{n=0}^k{\displaystyle \frac{c^{k+n+1}}{(k-1)!n!}}{\displaystyle \frac{{\mathrm{d}}^{n}F(u,y\left(u\right))}{d{u}^n}}{|}_{u=x_s}\hfill \\ & & ·{\int }_0^1\left[{\int }_0^1{(m-h-\xi )}^{k-1}\overline{G}(x_h+c\xi ,x_s+c\eta )d\xi \right]{\eta }^n\mathrm{d}\eta ,\hfill \\ & & r_1=\sum _{s=0}^{m-1}{\displaystyle \frac{c^{k+2}}{(k+1)!}}{\int }_0^1\overline{K}(x_0,x_s+c\eta )y^{(k+1)}\left({\theta }_s\right){\eta }^{k+1}\mathrm{d}\eta \hfill \\ & & -\sum _{s=0}^{m-1}{\displaystyle \frac{c^{k+2}}{(k+1)!}}{\int }_0^1\overline{G}(x_0,x_s+c\eta ){\displaystyle \frac{d^{k+1}F(u,y\left(u\right))}{d{u}^{k+1}}}{|}_{u={\vartheta }_s}{\eta }^{k+1}\mathrm{d}\eta ,\hfill \\ & & r_2=\sum _{s=0}^{m-1}{\displaystyle \frac{c^{k+2}}{(k+1)!}}{\int }_0^1\overline{K}(x_1,x_s+c\eta )y^{(k+1)}\left({\theta }_s\right){\eta }^{k+1}\mathrm{d}\eta \hfill \\ & & -\sum _{s=0}^{m-1}{\displaystyle \frac{c^{k+2}}{(k+1)!}}{\int }_0^1\overline{G}(x_1,x_s+c\eta ){\displaystyle \frac{{\mathrm{d}}^{k+1}F(u,y\left(u\right))}{\mathrm{d}{u}^{k+1}}}{|}_{u={\vartheta }_s}{\eta }^{k+1}\mathrm{d}\eta ,\hfill \\ & & \dots \dots \dots \dots \hfill \\ & & r_{m(k+1)}=\sum _{s=0}^{m-1}{\displaystyle \frac{c^{2k+1}}{(k-1)!(k+1)!}}{\int }_0^1{(m-s-\eta )}^{k-1}{y}^{(k+1)}\left({\theta }_s\right){\eta }^{k+1}\mathrm{d}\eta \hfill \\ & & +\sum _{h=0}^{m-1}\sum _{s=0}^{m-1}{\displaystyle \frac{c^{2k+2}}{(k-1)!(k+1)!}}{\int }_0^1{\int }_0^1{(m-h-\xi )}^{k-1}\hfill \\ & & ·\overline{K}(x_h+c\xi ,x_s+c\eta )y^{(k+1)}\left({\theta }_s\right){\eta }^{k+1}d\xi \mathrm{d}\eta \hfill \\ & & -\sum _{h=0}^{m-1}\sum _{q=0}^{m-1}{\displaystyle \frac{c^{2k+2}}{(k-1)!(k+1)!}}{\displaystyle \frac{{\mathrm{d}}^{k+1}F(u,y\left(u\right))}{\mathrm{d}{u}^{k+1}}}{|}_{u={\vartheta }_s}\hfill \\ & & ·{\int }_0^1\left[{\int }_0^1{(m-h-\xi )}^{k-1}\overline{G}(x_h+c\xi ,x_s+c\eta )\mathrm{d}\xi \right]{\eta }^{k+1}\mathrm{d}\eta .\hfill \end{array}$$

It is observed that the matrix C is non-singular.

Let $x\in {\mathcal{R}}^{m(k+1)}$, and suppose that

$$\overline{\mathcal{C}}:{\mathcal{R}}^{m(k+1)}\to {\mathcal{R}}^{m(k+1)}$$

is defined. We write this as

$$\overline{\mathcal{C}}x=Cx.$$

It can be rigorously verified that $\overline{\mathcal{C}}$ is a linear and invertible operator. The related computations are given by

$$\begin{array}{cc}\hfill {\parallel C\parallel }_{\infty }& =max\left\{1,\underset{1\le i\le m,1\le l\le k}{max}\sum _{s=0}^{i-1}\sum _{n=0}^k{\displaystyle \frac{c^{l+n}}{(l-1)!n!}}\left|{\int }_0^1{(i-s-\eta )}^{l-1}{\eta }^n\mathrm{d}\eta \right|\right\}\hfill \\ \hfill & =max\left\{1,\underset{1\le i\le m,1\le l\le k}{max}{\displaystyle \frac{c^l}{l!}}\sum _{s=0}^{i-1}[{(i-s)}^l-{(i-s-1)}^l·\sum _{n=0}^k{\displaystyle \frac{c^n}{n!}}]\right\}\hfill \\ \hfill & \le max\left\{1,e^c·\underset{1\le l\le k}{max}{\displaystyle \frac{{(l_2-l_1)}^l}{l!}}\right\}\hfill \\ \hfill & \le max\left\{1,e^{c+l_2-l_1}\right\}\le e^{c+l_2-l_1}<+\infty .\hfill \end{array}$$

(26)

Therefore, we conclude that $\overline{\mathcal{C}}$ is a bounded operator on the space ${\mathcal{R}}^{m(k+1)}$. By the inverse mapping theorem in Banach spaces, it follows that ${\overline{\mathcal{C}}}^{-1}$ is also continuous and bounded; hence, the norm $\parallel {\overline{\mathcal{C}}}^{-1}{\parallel }_{\infty }$ is finite.

Theorem 3.

We let $\parallel {\overline{\mathcal{C}}}^{-1}{\parallel }_{\infty }\le H$, and we assume that the function $d^{n}F(u,y\left(u\right))/d{u}^n$ satisfies the Lipschitz condition

$$\begin{array}{c}\hfill {∥{\displaystyle \frac{{\mathrm{d}}^{n}F(u,y_1\left(u\right))}{\mathrm{d}{u}^n}}-{\displaystyle \frac{{\mathrm{d}}^{n}F(u,y_2\left(u\right))}{\mathrm{d}{u}^n}}∥}_{\infty }\le \sum _{v=0}^n{L}_{n,v}{\Vert y_1^{\left(v\right)}\left(u\right)-y_2^{\left(v\right)}\left(u\right)\Vert }_{\infty },\end{array}$$

where the Lipschitz constants $L_{n,v}>0$ and $n=0,1,···,k$. Moreover, suppose the following boundedness conditions hold:

$$\begin{array}{c}\hfill \parallel \overline{G}{(t,x)\parallel }_{\infty }=\underset{l_1\le x,t\le l_2}{max}\left|\overline{G}(t,x)\right|=E_1<+\infty ,\end{array}$$

$$\begin{array}{c}\hfill \Vert y^{(k+1)}{\left(x\right)\Vert }_{\infty }=\underset{l_1\le x\le l_2}{max}\left|y^{(k+1)}\left(x\right)\right|=N_1<+\infty ,\end{array}$$

$$\begin{array}{c}\hfill {∥{\displaystyle \frac{{\mathrm{d}}^{k+1}}{\mathrm{d}{u}^{k+1}}}F(u,y\left(u\right))∥}_{\infty }=\underset{l_1\le u\le l_2}{max}\left|{\displaystyle \frac{{\mathrm{d}}^{k+1}}{\mathrm{d}{u}^{k+1}}}F(u,y\left(u\right))\right|=Q<+\infty ,\end{array}$$

$$\begin{array}{c}\hfill \underset{s=0,1,...,m-1}{max}\underset{l_1\le x\le l_2}{max}{\int }_0^1\left|\overline{K}(x,x_s+c\eta )\right|\mathrm{d}\eta =M_1<{\displaystyle \frac{1}{(l_2-l_1)H{e}^{c+l_2-l_1}}}-E_1\overline{L}<+\infty ,\end{array}$$

$$\overline{L}=\underset{0\le v\le n\le k}{max}{L}_{n,v}$$

Then, as shown in Equation (24), the approximate solution $y_{m,k}\left(x\right)$ converges uniformly to the exact solution $y\left(x\right)$:

$$\underset{k\to +\infty }{lim}{\parallel y_{m,k}\left(x\right)-y\left(x\right)\parallel }_{\infty }=0,$$

and

$$\underset{m\to +\infty (i.e.,h\to 0)}{lim}{\parallel y_{m,k}\left(x\right)-y\left(x\right)\parallel }_{\infty }=0.$$

Furthermore, the following error estimate holds:

$$\begin{array}{c}\hfill \parallel y_{m,k}{\left(x\right)-y\left(x\right)\parallel }_{\infty }\le {\displaystyle \frac{(l_2-l_1)[(M_1+E_1\overline{L})N_{1}H{e}^{c+(l_2-l_1)}+M_1{N}_1+E_{1}Q]}{(l_2-l_1)(M_1+E_1\overline{L})H{e}^{c+(l_2-l_1)}}}·{\displaystyle \frac{c^{k+1}}{(k+1)!}}.\end{array}$$

Proof. 

It can be shown that

$$\begin{array}{cc}\hfill {\parallel D\parallel }_{\infty }=& max\{\underset{1\le i\le m}{max}\sum _{s=0}^{m-1}\sum _{n=0}^k{\displaystyle \frac{c^{n+1}}{n!}}\left|{\int }_0^1\overline{K}(x_{i-1},x_s+c\eta ){\eta }^n\mathrm{d}\eta \right|,\underset{1\le i\le m,1\le l\le k}{max}\hfill \\ \hfill & \sum _{s=0}^{m-1}\sum _{n=0}^k\sum _{h=0}^{i-1}{\displaystyle \frac{c^{l+n+1}}{(l-1)!n!}}\left|{\int }_0^1{\int }_0^1\overline{K}(x_h+c\xi ,x_s+c\eta ){\eta }^n{(i-h-\xi )}^{l-1}\mathrm{d}\xi \mathrm{d}\eta \right|\}\hfill \\ \hfill \le & max\left\{cm{M}_1{e}^c,cm{M}_1\sum _{n=0}^k{\displaystyle \frac{c^n}{n!}}\underset{1\le i\le m,1\le l\le k}{max}{\displaystyle \frac{c^l}{(l-1)!}}\sum _{h=0}^{i-1}{\int }_0^1{(i-h-\xi )}^{l-1}\mathrm{d}\xi \right\}\hfill \\ \hfill \le & cm{M}_1{e}^c·\underset{0\le l\le k}{max}{\displaystyle \frac{{(l_2-l_1)}^l}{l!}}\hfill \\ \hfill \le & (l_2-l_1)M_1{e}^{c+(l_2-l_1)},\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill {\parallel R\parallel }_{\infty }=& max\left\{\underset{1\le i\le m}{max}\right|\sum _{s=0}^{m-1}{\displaystyle \frac{c^{k+2}}{(k+1)!}}{\int }_0^1\overline{K}(x_{i-1},x_s+c\eta )y^{(k+1)}\left({\theta }_s\right){\eta }^{k+1}\mathrm{d}\eta \hfill \\ \hfill & -\sum _{s=0}^{m-1}{\displaystyle \frac{c^{k+2}}{(k+1)!}}{\int }_0^1\overline{G}(x_{i-1},x_s+c\eta ){\displaystyle \frac{{\mathrm{d}}^{k+1}F(u,y\left(u\right))}{\mathrm{d}{u}^{k+1}}}{|}_{u={\vartheta }_s}·{\eta }^{k+1}\mathrm{d}\eta |,\hfill \\ \hfill & \underset{1\le i\le m,1\le l\le k}{max}|\sum _{s=0}^{i-1}{\displaystyle \frac{c^{k+l+1}}{(l-1)!(k+1)!}}{\int }_0^1{y}^{(k+1)}\left({\theta }_s\right){\eta }^{k+1}{(i-s-\eta )}^{l-1}\mathrm{d}\eta \hfill \\ \hfill & +\sum _{s=0}^{m-1}\sum _{h=0}^{i-1}{\displaystyle \frac{c^{k+l+2}}{(l-1)!(k+1)!}}{\int }_0^1{\int }_0^1\overline{K}(x_h+c\xi ,x_s+c\eta ){(i-h-\xi )}^{l-1}\hfill \\ \hfill & ·y^{(k+1)}\left({\theta }_s\right){\eta }^{k+1}\mathrm{d}\xi \mathrm{d}\eta -\sum _{s=0}^{m-1}\sum _{h=0}^{i-1}{\displaystyle \frac{c^{k+l+2}}{(l-1)!(k+1)!}}{\int }_0^1{\int }_0^1{(i-h-\xi )}^{l-1}\hfill \\ \hfill & ·\overline{G}(x_h+c\xi ,x_s+c\eta ){\displaystyle \frac{{\mathrm{d}}^{k+1}F(u,y\left(u\right))}{\mathrm{d}{u}^{k+1}}}{|}_{u={\vartheta }_s}·{\eta }^{k+1}\mathrm{d}\xi \mathrm{d}\eta \left|\right\}\hfill \\ \hfill & \le max\{{\displaystyle \frac{(l_2-l_1)(M_1{N}_1+E_{1}Q)c^{k+1}}{(k+1)!}},{\displaystyle \frac{N_1{e}^{l_2-l_1}{c}^{k+1}}{(k+1)!}}\hfill \\ \hfill & +{\displaystyle \frac{(l_2-l_1)(M_1{N}_1+E_{1}Q)c^{k+1}}{(k+1)!}}·\underset{1\le l\le k}{max}{\displaystyle \frac{{(l_2-l_1)}^l}{l!}}\}\hfill \\ \hfill & \le {\displaystyle \frac{[N_1+(l_2-l_1)(M_1{N}_1+E_{1}Q)]e^{l_2-l_1}{c}^{k+1}}{(k+1)!}},\hfill \end{array}$$

(28)

and

$$\begin{array}{cc}\hfill & \parallel E\left(\mathsf{\Phi }\right)-E\left(\tilde{\mathsf{\Phi }}\right){\parallel }_{\infty }\hfill \\ \hfill =& max\left\{\underset{1\le i\le m}{max}\right|\sum _{n=0}^k\sum _{s=0}^{m-1}{\displaystyle \frac{c^{n+1}}{n!}}{\displaystyle \frac{{\mathrm{d}}^{n}F(u,y\left(u\right))}{\mathrm{d}{u}^n}}{|}_{u=x_s}{\int }_0^1\overline{G}(x_{i-1},x_s+c\eta ){\eta }^n\mathrm{d}\eta \hfill \\ \hfill & -\sum _{n=0}^k\sum _{s=0}^{m-1}{\displaystyle \frac{c^{n+1}}{n!}}{\displaystyle \frac{{\mathrm{d}}^{n}F}{\mathrm{d}{u}^n}}{|}_{(x_s,y_s^{\left(n\right)})}{\int }_0^1\overline{G}(x_{i-1},x_s+c\eta ){\eta }^{n}d\eta |,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \underset{1\le i\le m,1\le l\le k}{max}|\sum _{h=0}^{i-1}\sum _{s=0}^{m-1}\sum _{n=0}^k{\displaystyle \frac{{\mathrm{d}}^{n}F(u,y\left(u\right))}{\mathrm{d}{u}^n}}{|}_{u=x_s}{\displaystyle \frac{c^{l+n+1}}{(l-1)!n!}}\hfill \\ \hfill & ·{\int }_0^1{\int }_0^1\overline{G}(x_h+c\xi ,x_s+c\eta ){\eta }^n{(i-h-\xi )}^{l-1}\mathrm{d}\xi \mathrm{d}\eta \hfill \\ \hfill & -\sum _{h=0}^{i-1}\sum _{s=0}^{m-1}\sum _{n=0}^k{\displaystyle \frac{{\mathrm{d}}^{n}F}{\mathrm{d}{u}^n}}{|}_{(x_s,y_s^{\left(n\right)})}{\displaystyle \frac{c^{l+n+1}}{(l-1)!n!}}\hfill \\ \hfill & ·{\int }_0^1{\int }_0^1\overline{G}(x_h+c\xi ,x_s+c\eta ){\eta }^n{(i-h-\xi )}^{l-1}\mathrm{d}\xi \mathrm{d}\eta \left|\right\}\hfill \\ \hfill & \le max\{E_1\sum _{s=0}^{m-1}\sum _{n=0}^k{\displaystyle \frac{c^{n+1}}{(n+1)!}}·{∥{\displaystyle \frac{{\mathrm{d}}^{n}F(u,y\left(u\right))}{\mathrm{d}{u}^n}}{|}_{u=x_s}-{\displaystyle \frac{{\mathrm{d}}^{n}F}{\mathrm{d}{u}^n}}{|}_{(x_s,y_s^{\left(n\right)})}∥}_{\infty },\hfill \\ \hfill & E_1\sum _{s=0}^{m-1}\sum _{n=0}^k{\displaystyle \frac{c^{n+1}}{(n+1)!}}·\underset{1\le i\le m,1\le i\le k}{max}{\displaystyle \frac{c^l}{(l-1)!}}\sum _{h=0}^{i-1}{\int }_0^1{(i-h-\xi )}^{l-1}\mathrm{d}\xi \hfill \\ \hfill & \times {∥{\displaystyle \frac{{\mathrm{d}}^{n}F(u,y\left(u\right))}{\mathrm{d}{u}^n}}{|}_{u=x_s}-{\displaystyle \frac{{\mathrm{d}}^{n}F}{\mathrm{d}{u}^n}}{|}_{(x_s,y_s^{\left(n\right)})}∥}_{\infty }\}\hfill \\ \hfill & \le E_1{e}^{l_2-l_1}\sum _{s=0}^{m-1}\sum _{n=0}^k{\displaystyle \frac{c^{n+1}}{(n+1)!}}·\sum _{v=0}^n{L}_{n,v}{\Vert y^{\left(v\right)}\left(x_s\right)-y_s^{\left(v\right)}\Vert }_{\infty }\hfill \\ \hfill & \le E_1\overline{L}{e}^{l_2-l_1}\sum _{s=0}^{m-1}\sum _{n=0}^k{\displaystyle \frac{c^{n+1}}{n!}}·{\Vert \mathsf{\Phi }-\tilde{\mathsf{\Phi }}\Vert }_{\infty }\hfill \\ \hfill & \le (l_2-l_1)E_1\overline{L}{e}^{c+(l_2-l_1)}{\Vert \mathsf{\Phi }-\tilde{\mathsf{\Phi }}\Vert }_{\infty }.\hfill \end{array}$$

(29)

Now let us assume that

$$\begin{array}{c}\hfill 0<M_1+E_1\overline{L}<{\displaystyle \frac{1}{(l_2-l_1)H{e}^{c+l_2-l_1}}},\end{array}$$

(30)

Consequently, it follows that

$$\begin{array}{c}\hfill \parallel D{C}^{-1}{\parallel }_{\infty }\le {\parallel D\parallel }_{\infty }{\parallel C^{-1}\parallel }_{\infty }<1\end{array}$$

It is straightforward to verify that $D{C}^{-1}$ satisfies the condition of strict diagonal dominance. Therefore, we conclude that $D{C}^{-1}$ is a non-singular matrix. From Equations (24) and (25), it then follows that

$$\begin{array}{cc}\hfill & \Vert \mathsf{\Phi }-\tilde{\mathsf{\Phi }}{\Vert }_{\infty }={\Vert {\left[\left(D{C}^{-1}\right)C\right]}^{-1}[E\left(\mathsf{\Phi }\right)-E\left(\tilde{\mathsf{\Phi }}\right)+R]\Vert }_{\infty }\hfill \\ \hfill & \le \parallel C^{-1}{\parallel }_{\infty }\parallel {\left(D{C}^{-1}\right)}^{-1}{\parallel }_{\infty }[\parallel E\left(\mathsf{\Phi }\right)-E\left(\tilde{\mathsf{\Phi }}\right){\parallel }_{\infty }+{\parallel R\parallel }_{\infty }]\hfill \\ \hfill & \le {\displaystyle \frac{\Vert C^{-1}{\Vert }_{\infty }}{1-{\parallel D\parallel }_{\infty }{\parallel C^{-1}\parallel }_{\infty }}}[\parallel E\left(\mathsf{\Phi }\right)-E\left(\tilde{\mathsf{\Phi }}\right){\parallel }_{\infty }+{\parallel R\parallel }_{\infty }]\hfill \\ \hfill & \le {\displaystyle \frac{(l_2-l_1)H{E}_1\overline{L}{e}^{c+(l_2-l_1)}\parallel \mathsf{\Phi }-\tilde{\mathsf{\Phi }}{\parallel }_{\infty }+H{\parallel R\parallel }_{\infty }}{1-(l_2-l_1)H{M}_1{e}^{c+(l_2-l_1)}}}.\hfill \end{array}$$

(31)

Furthermore, one has

$$\begin{array}{cc}\hfill & \parallel \mathsf{\Phi }-\tilde{\mathsf{\Phi }}{\parallel }_{\infty }\le {\displaystyle \frac{H}{1-(l_2-l_1)(M_1+E_1\overline{L})H{e}^{c+(l_2-l_1)}}}{\parallel R\parallel }_{\infty }.\hfill \end{array}$$

(32)

It is easy to obtain

$$\begin{array}{cc}\hfill & \parallel y\left(x\right)-y_{m,k}{\left(x\right)\parallel }_{\infty }\hfill \\ \hfill \le & {∥\sum _{s=0}^{m-1}c{\int }_0^1\left[y(x_s+c\eta )-\sum _{n=0}^k{\displaystyle \frac{y_s^{\left(n\right)}}{n!}}{\left(c\eta \right)}^n\right]\overline{K}(x,x_s+c\eta )\mathrm{d}\eta ∥}_{\infty }\hfill \\ \hfill & +{∥\sum _{s=0}^{m-1}c{\int }_0^1\overline{G}(x,x_s+c\eta )\left[F(x_s+c\eta ,y(x_s+c\eta ))-\sum _{n=0}^k{\displaystyle \frac{{\mathrm{d}}^{n}F}{\mathrm{d}{u}^n}}{|}_{(x_s,y_s^{\left(j\right)})}{\displaystyle \frac{{\left(c\eta \right)}^n}{n!}}\right]\mathrm{d}\eta ∥}_{\infty }\hfill \\ \hfill \le & {∥\sum _{s=0}^{m-1}c{\int }_0^1\left[y(x_s+c\eta )-\sum _{n=0}^k{\displaystyle \frac{y^{\left(n\right)}\left(x_s\right)}{n!}}{\left(c\eta \right)}^n\right]\overline{K}(x,x_s+c\eta )\mathrm{d}\eta ∥}_{\infty }\hfill \\ \hfill & +{∥\sum _{s=0}^{m-1}c{\int }_0^1\left[\sum _{n=0}^k{\displaystyle \frac{y^{\left(n\right)}\left(x_s\right)}{n!}}{\left(c\eta \right)}^n-\sum _{n=0}^k{\displaystyle \frac{y_s^{\left(n\right)}}{n!}}{\left(c\eta \right)}^n\right]\overline{K}(x,x_s+c\eta )\mathrm{d}\eta ∥}_{\infty }\hfill \\ \hfill & +{∥\sum _{s=0}^{m-1}c{\int }_0^1\overline{G}(x,x_s+c\eta )\left[F(x_s+c\eta ,y(x_s+c\eta ))-\sum _{n=0}^k{\displaystyle \frac{{\left(c\eta \right)}^n}{n!}}{\displaystyle \frac{{\mathrm{d}}^{n}F}{\mathrm{d}{u}^n}}{|}_{u=x_s}\right]\mathrm{d}\eta ∥}_{\infty }\hfill \\ \hfill & +{∥\sum _{s=0}^{m-1}c{\int }_0^1\overline{G}(x,x_s+c\eta )\left[\sum _{n=0}^k{\displaystyle \frac{{\left(c\eta \right)}^n}{n!}}{\displaystyle \frac{{\mathrm{d}}^{n}F}{\mathrm{d}{u}^n}}{|}_{u=x_s}-\sum _{n=0}^k{\displaystyle \frac{{\left(c\eta \right)}^n}{n!}}{\displaystyle \frac{{\mathrm{d}}^{n}F}{\mathrm{d}{u}^n}}{|}_{(x_s,y_s^{\left(n\right)}))}\right]\mathrm{d}\eta ∥}_{\infty }\hfill \\ \hfill \le & {∥\sum _{s=0}^{m-1}{\displaystyle \frac{c^{k+2}}{(k+1)!}}{\int }_0^1\overline{K}(x,x_s+c\eta )y^{(k+1)}\left({\theta }_s\right){\eta }^{k+1}\mathrm{d}\eta ∥}_{\infty }\hfill \\ & +{∥\sum _{s=0}^{m-1}\sum _{n=0}^k{\displaystyle \frac{c^{n+1}}{n!}}{\int }_0^1\overline{K}(x,x_s+c\eta ){\eta }^n\mathrm{d}\eta \left[y^{\left(n\right)}\left(x_s\right)-y_s^{\left(n\right)}\right]∥}_{\infty }\hfill \\ \hfill & +{∥\sum _{s=0}^{m-1}{\displaystyle \frac{c^{k+2}}{(k+1)!}}{\int }_0^1\overline{G}(x,x_s+c\eta ){\displaystyle \frac{{\mathrm{d}}^{k+1}F(u,y\left(u\right))}{\mathrm{d}{u}^{k+1}}}{|}_{u={\vartheta }_s}{\eta }^{k+1}\mathrm{d}\eta ∥}_{\infty }\hfill \\ \hfill & +{∥\sum _{s=0}^{m-1}\sum _{n=0}^k{\displaystyle \frac{c^{n+1}}{n!}}{\int }_0^1\overline{G}(x,x_s+c\eta ){\eta }^n\mathrm{d}\eta \left[{\displaystyle \frac{{\mathrm{d}}^{n}F}{\mathrm{d}{u}^n}}{|}_{u=x_s}-{\displaystyle \frac{{\mathrm{d}}^{n}F}{\mathrm{d}{u}^n}}{|}_{(x_s,y_s^{\left(n\right)})}\right]∥}_{\infty }\hfill \\ \hfill \le & {\displaystyle \frac{(l_2-l_1)(M_1{N}_1+E_{1}Q)c^{k+1}}{(k+1)!}}+(l_2-l_1)(M_1+E_1\overline{L})e^c{\parallel \mathsf{\Phi }-\tilde{\mathsf{\Phi }}\parallel }_{\infty }\hfill \\ \hfill \le & {\displaystyle \frac{(l_2-l_1)[(M_1+E_1\overline{L})N_{1}H{e}^{c+(l_2-l_1)}+M_1{N}_1+E_{1}Q]}{1-(l_2-l_1)(M_1+E_1\overline{L})H{e}^{c+(l_2-l_1)}}}{\displaystyle \frac{c^{k+1}}{(k+1)!}}.\hfill \end{array}$$

Now we have the following limits:

$$\underset{k\to +\infty }{lim}{\parallel y_{m,k}\left(x\right)-y\left(x\right)\parallel }_{\infty }=0,$$

$$\underset{m\to +\infty (i.e.,h\to 0)}{lim}{\parallel y_{m,k}\left(x\right)-y\left(x\right)\parallel }_{\infty }=0.$$

□

As established in Theorem 3, two key parameters are introduced to characterize the convergence behavior and error estimates of the approximate solution. Furthermore, appropriate values of m and k can be selected to ensure an effective approximation to the exact solution. These observations will be further confirmed through numerical examples in the subsequent section.

5. Numerical Results

The effectiveness of the proposed methods is substantiated through a series of numerical experiments. In this section, we have carefully selected three examples to thoroughly elaborate on the proposed method. The following examples will examine how fractional parameters m and k influence the approximate solution. Furthermore, a comparative analysis will be carried out between the proposed approach and the traditional difference method commonly applied to such equations of fractional order.

Example 1.

First, let us analyze the case $\beta =0$ and provide the following example:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{y}^{″}\left(x\right)-(1+sinx)y\left(x\right)=-y\left(x\right)sinx,\hfill \\ y\left(0\right)=1,y\left(1\right)={\int }_0^{1}esy\left(s\right)ds,\hfill \end{array}\right.\end{array}$$

(33)

with $x\in \left[0,1\right]$ and the exact solution $y\left(x\right)=e^x$.

The numerical method proposed here is utilized to address the derived F–H integral equation. In line with the notation established in [29], representing the parameters m and k as a vector $(m,k)$ is practical. For the computations, we selected the following combinations of $(m,k):$ $(4,0),(4,1),(4,2),(8,0),(8,1),(8,2)$.

Table 1. The absolute errors of approximate and exact solutions for Example 1.

x	$(4,0)$	$(4,1)$	$(4,2)$	$(8,0)$	$(8,1)$	$(8,2)$
$0.1$	$8.4450\times {10}^{-4}$	$5.9990\times {10}^{-4}$	$5.9556\times {10}^{-5}$	$7.5716\times {10}^{-4}$	$5.9527\times {10}^{-5}$	$1.3671\times {10}^{-6}$
$0.2$	$4.5541\times {10}^{-4}$	$1.0074\times {10}^{-4}$	$4.0513\times {10}^{-5}$	$2.4250\times {10}^{-4}$	$4.0420\times {10}^{-6}$	$2.6842\times {10}^{-7}$
$0.3$	$1.1551\times {10}^{-4}$	$1.0014\times {10}^{-4}$	$1.4058\times {10}^{-5}$	$1.2421\times {10}^{-4}$	$1.4037\times {10}^{-5}$	$2.0640\times {10}^{-6}$
$0.4$	$2.1548\times {10}^{-3}$	$2.4020\times {10}^{-4}$	$2.4065\times {10}^{-5}$	$2.2417\times {10}^{-4}$	$2.4032\times {10}^{-5}$	$9.3411\times {10}^{-6}$
$0.5$	$3.1547\times {10}^{-3}$	$3.4027\times {10}^{-4}$	$3.4073\times {10}^{-5}$	$3.2413\times {10}^{-4}$	$3.4027\times {10}^{-5}$	$7.9005\times {10}^{-6}$
$0.6$	$4.4043\times {10}^{-3}$	$4.1538\times {10}^{-3}$	$4.4083\times {10}^{-4}$	$4.3671\times {10}^{-4}$	$4.4025\times {10}^{-5}$	$9.3941\times {10}^{-6}$
$0.7$	$5.4542\times {10}^{-3}$	$5.1049\times {10}^{-3}$	$5.4095\times {10}^{-4}$	$5.7431\times {10}^{-3}$	$5.3867\times {10}^{-5}$	$6.2793\times {10}^{-6}$
$0.8$	$6.1543\times {10}^{-3}$	$6.4065\times {10}^{-4}$	$6.4112\times {10}^{-5}$	$6.8235\times {10}^{-4}$	$6.2937\times {10}^{-6}$	$6.3974\times {10}^{-7}$
$0.9$	$7.1552\times {10}^{-3}$	$7.4086\times {10}^{-4}$	$7.4133\times {10}^{-5}$	$7.1584\times {10}^{-4}$	$7.4038\times {10}^{-5}$	$8.6745\times {10}^{-6}$
$1.0$	$8.1555\times {10}^{-3}$	$8.4110\times {10}^{-4}$	$8.4157\times {10}^{-5}$	$8.2431\times {10}^{-4}$	$8.2204\times {10}^{-6}$	$6.3716\times {10}^{-6}$

presents the absolute errors $|y\left(x\right)-y_{m,k}\left(x\right)|$ for different parameter combinations. As observed, increasing either m or k results in a significant reduction in the error, which confirms the convergence properties predicted by the theoretical analysis in Section 4. Specifically, for a fixed m, increasing k enhances the local approximation accuracy through higher-order Taylor expansions, whereas for a fixed k, enlarging m improves the global resolution by refining the discretization grid. Furthermore, Figure 1 illustrates the comparison between the numerical and exact solutions, as well as the corresponding absolute error, for $m=4$ and $k=2$. The results clearly demonstrate that the proposed piecewise expansion method achieves excellent numerical accuracy and stability.

Example 2.

Second, we investigate the case $0\le \beta <1$ and provide the following example:

$$\left\{\begin{array}{c}{y}^{″}\left(x\right)-5\sqrt{\pi }x{D}^{{\displaystyle \frac{1}{2}}}y\left(x\right)+(16x^{{\displaystyle \frac{1}{2}}}+x^3)y\left(x\right)=6x+y^2\left(x\right),\hfill \\ y\left(0\right)={\int }_0^{{\displaystyle \frac{1}{5}}}(s-{\displaystyle \frac{4}{25}})y\left(s\right)\mathrm{d}s,y\left({\displaystyle \frac{1}{5}}\right)={\int }_0^{{\displaystyle \frac{1}{5}}}125sy\left(s\right)\mathrm{d}s,\hfill \end{array}\right.$$

(34)

where $x\in \left[0,{\displaystyle \frac{1}{5}}\right]$, and where the exact solution is $y\left(x\right)=x^3$. We apply the proposed method to calculate the error with respect to the exact solution in

Table 2. The absolute errors of approximate and exact solutions for Example 2.

x	$(4,0)$	$(4,1)$	$(4,2)$	$(8,0)$	$(8,1)$	$(8,2)$
$0.02$	$9.8957\times {10}^{-4}$	$3.3150\times {10}^{-4}$	$0.3417\times {10}^{-4}$	$1.0783\times {10}^{-5}$	$1.1531\times {10}^{-4}$	$3.2971\times {10}^{-6}$
$0.04$	$1.5956\times {10}^{-3}$	$1.4750\times {10}^{-3}$	$2.3479\times {10}^{-4}$	$1.3152\times {10}^{-4}$	$1.8101\times {10}^{-5}$	$3.6974\times {10}^{-6}$
$0.06$	$2.2029\times {10}^{-3}$	$9.3883\times {10}^{-4}$	$6.4787\times {10}^{-5}$	$2.2565\times {10}^{-3}$	$2.4016\times {10}^{-4}$	$5.3241\times {10}^{-6}$
$0.08$	$2.8117\times {10}^{-3}$	$1.1800\times {10}^{-3}$	$8.3616\times {10}^{-4}$	$1.6555\times {10}^{-3}$	$2.8801\times {10}^{-4}$	$5.9642\times {10}^{-6}$
$0.10$	$3.4217\times {10}^{-3}$	$1.3221\times {10}^{-4}$	$0.3971\times {10}^{-4}$	$2.8657\times {10}^{-3}$	$3.1980\times {10}^{-4}$	$8.3852\times {10}^{-5}$
$0.12$	$4.0357\times {10}^{-3}$	$1.3116\times {10}^{-3}$	$0.8963\times {10}^{-4}$	$2.8402\times {10}^{-3}$	$3.3097\times {10}^{-4}$	$8.3974\times {10}^{-5}$
$0.14$	$4.6541\times {10}^{-3}$	$1.0775\times {10}^{-3}$	$1.0352\times {10}^{-4}$	$2.5318\times {10}^{-3}$	$3.1695\times {10}^{-4}$	$8.3741\times {10}^{-5}$
$0.16$	$5.2776\times {10}^{-3}$	$6.0459\times {10}^{-4}$	$3.0568\times {10}^{-4}$	$1.8938\times {10}^{-3}$	$2.7298\times {10}^{-4}$	$6.3147\times {10}^{-5}$
$0.18$	$5.9089\times {10}^{-3}$	$1.9551\times {10}^{-4}$	$5.0327\times {10}^{-4}$	$8.7931\times {10}^{-4}$	$1.9432\times {10}^{-4}$	$2.3679\times {10}^{-5}$
$0.20$	$6.5469\times {10}^{-3}$	$1.3438\times {10}^{-3}$	$7.0421\times {10}^{-4}$	$5.5859\times {10}^{-4}$	$7.6247\times {10}^{-5}$	$1.3479\times {10}^{-6}$

.

It can be noted that the suggested numerical approach is employed to address the derived second-kind F–H integral equation. In this context, we choose various pairs of $(m,k)$ for separate calculations, and Figure 2 presents the comparison between the numerical and exact solutions, as well as the absolute error, for $m=8$ and $k=2.$ A review of Table 1 and Table 2 indicates that we have successfully attained a satisfactory approximation of the precise solution for the integral boundary value problem involving fractional BTE with variable coefficients. As either m or k increases, the absolute error $|y\left(x\right)-y_{m,k}\left(x\right)|$ tends to diminish. These numerical findings are in excellent agreement with the theoretical predictions presented in Theorem 3, thereby confirming the validity and robustness of the developed approach.

Example 3.

Finally, let us examine the scenario where $1\le \beta <2$ and provide an illustration. In cases where an exact solution is unattainable, the effectiveness of the proposed approach is assessed by contrasting it with a standard difference scheme through the computation of the subsequent example:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{y}^{″}\left(x\right)+\sqrt{\pi }{x}^2{D}^{{\displaystyle \frac{3}{2}}}y\left(x\right)+(-4x^{{\displaystyle \frac{1}{2}}}+x^2)y\left(x\right)=y^2\left(x\right)+2,\hfill \\ y\left(0\right)=0,y\left({\displaystyle \frac{1}{10}}\right)={\int }_0^{{\displaystyle \frac{1}{10}}}400sy\left(s\right)\mathrm{d}s,\hfill \end{array}\right.\end{array}$$

(35)

with $x\in \left[0,{\displaystyle \frac{1}{10}}\right]$.

According to Theorem 1, Equation (35) can be transformed into an equivalent F–H integral equation of the second kind, as follows:

$$\begin{array}{c}\hfill y\left(x\right)+{\int }_{l_1}^{l_2}\overline{K}(t,x)y\left(t\right)\mathrm{d}t={\int }_{l_1}^{l_2}\overline{G}(t,x)[y^2\left(t\right)+2]\mathrm{d}t,\end{array}$$

(36)

where, for $l_1\le t\le x\le l_2$,

$$\begin{array}{ccc}& & y\left(x\right)+{\int }_{l_1}^x\{W_2(t,x)-A_2(t,x)\}y\left(t\right)\mathrm{d}t\hfill \\ & & ={\int }_{l_1}^x\{(x-t)-{\displaystyle \frac{x-l_1}{l_2-l_1}}(l_2-t)\}F(t,y\left(t\right))\mathrm{d}t.\hfill \end{array}$$

We will now examine the singularity of the solution presented in Example 3. The calculation shows that

$$\begin{array}{c}\hfill W_2(t,x)=(x-t)(-4\sqrt{t}+t^2)+{\displaystyle \frac{x^2}{\sqrt{x-t}}}+4x\sqrt{x-t}-4{(x-t)}^{{\displaystyle \frac{3}{2}}}-12t\sqrt{x-t}.\end{array}$$

We first prove the uniqueness of the solution and choose the Lipschitz constant as

$$\begin{array}{c}\hfill L={∥{\displaystyle \frac{F(x,y_1\left(x\right))-F(x,y_2\left(x\right))}{y_1\left(x\right)-y_2\left(x\right)}}∥}_{\infty }={\parallel y_1\left(x\right)+y_2\left(x\right)\parallel }_{\infty }\le 2.\end{array}$$

Now, we have applied the typical central difference format, as described in reference [33]:

$$\begin{array}{ccc}& & y^{″}\left(x\right)={\displaystyle \frac{y(x-c)-2y\left(x\right)+y(x+c)}{c^2}}-{\displaystyle \frac{c^2}{12}}{y}^{\left(4\right)}\left(\xi \right),x-c<\xi <x+c,\hfill \\ & & D^{\beta }y\left(x\right)=\underset{c\to 0}{lim}{c}^{-\beta }\sum _{n=0}^{\left[\right(x-a)/c]}{(-1)}^n\left(\begin{array}{c}\beta \\ n\end{array}\right)y(x-nc),\hfill \end{array}$$

where

$$\begin{array}{c}\hfill \left(\begin{array}{c}\beta \\ n\end{array}\right)={\displaystyle \frac{\beta (\beta -1)\dots (\beta -n+1)}{n!}}.\end{array}$$

Table 3. Contrasting the proposed method with the difference methods for Example 3.

	${\mathit{y}}_{\mathit{m},\mathit{k}}\left(\mathit{x}\right)$: The Present Method				${\mathit{y}}_{\mathit{c}}\left(\mathit{x}\right)$: The Difference Method
$\mathit{x}$	$(\mathbf{4},\mathbf{1})$	$(\mathbf{4},\mathbf{2})$	$(\mathbf{8},\mathbf{1})$	$(\mathbf{8},\mathbf{2})$	$\mathit{h}={\displaystyle \frac{\mathbf{1}}{\mathbf{20}}}$	$\mathit{h}={\displaystyle \frac{\mathbf{1}}{\mathbf{40}}}$
$0.01$	0.0006999	0.0001596	0.0001560	0.0001017	0.0001693	0.0001002
$0.02$	0.0005007	0.0004051	0.0004404	0.0004003	0.0004205	0.0004003
$0.03$	0.0010001	0.0009141	0.0009140	0.0009021	0.0009582	0.0009002
$0.04$	0.0018402	0.0016241	0.0016240	0.0016093	0.0016703	0.0016005
$0.05$	0.0028403	0.0025341	0.0025340	0.0025079	0.0025863	0.0025006
$0.06$	0.0077538	0.0040408	0.0036440	0.0036094	0.0036955	0.0036062
$0.07$	0.0100049	0.0054500	0.0049539	0.0049063	0.0049763	0.0049005
$0.08$	0.0070407	0.0064641	0.0064063	0.0064006	0.0064271	0.0064010
$0.09$	0.0088409	0.0081741	0.0081740	0.0081109	0.0081648	0.0081017
$0.10$	0.0108411	0.0100842	0.0100008	0.0100006	0.0102180	0.0100002

shows the approximate solutions $y_{m,k}\left(x\right)$ obtained by applying the method proposed in this article and the approximate solutions $y_c\left(x\right)$ obtained by applying the difference scheme. For any given x, it is evident that the values of $y_{m,k}\left(x\right)$ and $y_c\left(x\right)$ are quite similar. This information illustrates the effectiveness and practicality of the suggested approach.

6. Conclusions

Three classes of integral boundary value problems involving the fractional BTE with variable coefficients have been investigated. In contrast to traditional approaches that rely on constructing two linearly independent solutions, the integration technique was employed to reformulate the problems into second-kind F–H integral equations. The Banach fixed-point theorem was then applied to establish the existence and uniqueness of the corresponding solutions. Furthermore, an improved numerical approach, based on a modified piecewise Taylor expansion technique, was developed to effectively handle weakly singular second-kind F–H integral equations. This enhanced scheme provides approximate solutions together with convergence and error analyses. Distinct from conventional approaches, two parameters are adopted to evaluate the deviation between the approximate and exact solutions. Finally, a set of numerical experiments was presented to verify the effectiveness of the proposed method and to benchmark its performance against existing techniques.

In upcoming research, the approximate method will be applied to further investigate multipoint boundary value problems associated with different fractional differential equations.