Existence, Stability, and Numerical Methods for Multi-Fractional Integro-Differential Equations with Singular Kernel

Abstract

This work investigates the solutions of fractional integro-differential equations (FIDEs) using a unique kernel operator within the Caputo framework. The problem is addressed using both analytical and numerical techniques. First, the two-step Adomian decomposition method (TSADM) is applied to obtain an exact solution (if it exists). In the second part, numerical methods are used to generate approximate solutions, complementing the analytical approach based on the Adomian decomposition method (ADM), which is further extended using the Sumudu and Shehu transform techniques in cases where TSADM fails to yield an exact solution. Additionally, we establish the existence and uniqueness of the solution via fixed-point theorems. Furthermore, the Ulam–Hyers stability of the solution is analyzed. A detailed error analysis is performed to assess the precision and performance of the developed approaches. The results are demonstrated through validated examples, supported by comparative graphs and detailed error norm tables ($L_{\infty }$, $L_2$, and $L_1$). The graphical and tabular comparisons indicate that the Sumudu-Adomian decomposition method (Sumudu-ADM) and the Shehu-Adomian decomposition method (Shehu-ADM) approaches provide highly accurate approximations, with Shehu-ADM often delivering enhanced performance due to its weighted formulation. The suggested approach is simple and effective, often producing accurate estimates in a few iterations. Compared to conventional numerical and analytical techniques, the presented methods are computationally less intensive and more adaptable to a broad class of fractional-order differential equations encountered in scientific applications. The adopted methods offer high accuracy, low computational cost, and strong adaptability, with potential for extension to variable-order fractional models. They are suitable for a wide range of complex systems exhibiting evolving memory behavior.

1. Introduction

Fractional calculus (FC) is a well-established mathematical theory; nonetheless, it went largely unexamined for many years due to a scarcity of practical applications. At the onset of the 20th century, advances in the natural sciences and increased technical requirements prompted a revival of interest in fuel cells. Since then, it has been utilized in various research domains [1,2,3], which include the examination of non-linear fluctuations in solitary gravity waves, gas transport, lithium-ion battery management, and dissipative acoustic equations. In the field of control systems, fractional-order controllers have emerged as a prominent research focus, with such controllers being designed for particular fractional-order systems.

Differential equations (DEs) [4] are essential to model real-world problems in diverse disciplines such as biology, physics, engineering, and chemistry [5,6,7,8]. DEs can accurately characterize many of these applications. Recently, there has been an increasing trend in using fractional differential equations (FDEs) for modeling applications in various fields, including fluid mechanics, biology, chemistry, viscoelasticity, engineering, finance, and physics. Moreover, recent research has investigated fractional-type operators with singular kernels, providing novel insights. A comprehensive body of literature delineates several features and significant findings in fractional calculus. In recent decades, FDEs have gained prominence in simulating intricate physical, biological, and engineering systems. Their non-local characteristics frequently render them more effective than classical local operators in capturing certain dynamics [9,10].

A fractional integro-differential equation (FIDE) [11] emerges in the modeling of processes within applied disciplines, including physics, engineering, finance, and biology [12,13]. Numerous issues are discussed in the fields of viscoelasticity, acoustics, electromagnetics, hydrology, and other domains [14]. Progression of scientific knowledge in applied sciences is notable in spheres like financial mathematics, mechanical engineering, reactor dynamics, and control systems, along with natural occurrences such as climate change and the growth of biological populations [15]. Thanks to their non-local and history-dependent nature, fractional-order derivatives have emerged as powerful tools in modeling systems with memory and hereditary properties. In engineering, they are widely used in modeling viscoelastic materials and designing robust control systems with superior performance [16]. In biology, fractional models have been successfully applied to describe anomalous transport processes in cells and tissues and memory-influenced neural dynamics. In physics, FC helps capture anomalous diffusion and complex quantum phenomena. Meanwhile, in finance, these models are used to represent long-range dependence and volatility clustering in financial time series. These diverse applications demonstrate the versatility and effectiveness of fractional calculus in modeling real-world systems [17]. This modification has substituted deterministic equations with stochastic equations to model physical systems, including differential, integral, and stochastic integro-differential equations [18]. Various real-world phenomena are better captured using stochastic fractional integro-differential equations (SFIDEs), as they consider the memory and randomness inherent in natural processes [19,20]. Thus, examining random effects and processes in fractional order investigations and formulating suitable models are crucial to advancing theoretical and applied research.

The class of fractional integro-differential equations, FIDEs, studied in this work is anomalous diffusion in viscoelastic media. In such systems, the transport of particles deviates from classical Fickian behavior because of memory effects and long-range temporal correlations in the medium’s response. This behavior is commonly modeled using fractional derivatives, which inherently capture non-local time dependencies [21].

In particular, the stress–strain relationship in viscoelastic materials is often governed by convolution-type integrals involving weakly singular kernels. These kernels naturally arise from the hereditary nature of the material, as described by the fractional Zener model or the fractional Maxwell model. When governing equations are derived from first principles with conservation laws and constitutive relations, the resulting equations include fractional Caputo derivatives and Riemann–Liouville integrals with singular kernels [22]. Therefore, the appearance of singular kernels in the FIDEs is not merely a mathematical artifact, but is physically motivated by the underlying structure of complex, time-dependent materials. Our analysis and methods aim to address precisely these challenges that arise in modeling such real-world phenomena [23].

In successive sections, we consider the class of fractional integro-differential equations (FIDEs) with initial conditions [1,2], described as

$$\begin{array}{c}\hfill {\lambda }^{″}\left(x\right)+{}_{0+}{}^{C}D_x^{{\zeta }_1}\lambda \left(x\right)+{}_{0+}{}^{RL}D_x^{-{\zeta }_2}\lambda \left(x\right)=\mathcal{H}\left(x\right),x\in [0,T]=\mathcal{I},0<{\zeta }_1,{\zeta }_2<2,\end{array}$$

(1)

with initial conditions $\lambda \left(0\right)={\eta }_1$ and ${\lambda }^{\prime }\left(0\right)={\eta }_2$, where ${}_{0+}{}^{C}D_x^{{\zeta }_1}$ and ${}_{0+}{}^{RL}D_x^{-{\zeta }_2}$ denote the Caputo fractional derivative and Riemann–Liouville fractional integral operators, respectively, ${\eta }_1$, ${\eta }_2$ are real constants, and $\mathcal{H}\left(x\right)$ is a known continuous function. Our interest lies in obtaining results on existence, uniqueness, and stability using fixed-point theory. Specifically, for stability analysis, we utilize the Ulam–Hyers stability concept. Such results are essential in validating mathematical models of real phenomena. For example, in [24], the authors analyzed non-linear FDEs with boundary conditions on the order $2<\alpha \le 3$. Wanassi et al. [25] investigated non-linear fractional-order differential equations using modified Caputo derivatives [26] and solved test cases using the Adomian decomposition method (ADM). Studies in [27,28] explored Ulam–Hyers (U-H) stability in the context of Caputo–Fabrizio operators and integral transforms.

It is clear that many analytical and numerical methods have been developed for FDE. Here, it is important to distinguish problems with singular and non-singular kernels. Many analytical and semi-analytical methods have been developed to study FIDEs, especially those involving singular kernels. Among the most prominent are the ADM, Laplace Adomian decomposition method (LADM) [29], Sumudu ADM, Shahu ADM, the difference method, the homotopy perturbation method (HPM), and the differential transform method (DTM).

So, FIDEs can be transformed into a recursive algebraic form by combining the Laplace transform with the LADM, making their analysis easier. However, as shown in the literature, it may not be able to handle complicated singular kernels or strong non-linearity. On the other hand, LADM has shown promise in solving linear and weakly non-linear situations. In [30], Lassa fever, a viral infection prevalent in West Africa, was discussed using fractional derivatives and solved analytically with the help of LADM. The provided solution is expressed as an infinite series with rapid convergence to the exact value. The study showed that varying the fractional-order parameter $\alpha$ significantly affects the dynamics of the system. Furthermore, sensitivity analysis revealed that controlling the contact rate in exposed, infected, and isolated compartments is crucial to managing disease spread, with graphic illustrations to validate the results. The authors of [31] examined the danger that the COVID-19 outbreak posed to worldwide public health, focusing on modeling its dissemination and control in Nigeria. The authors established a solution for the dynamics of the disease using the LADM and the fractional-order Caputo derivative.

The difference method (DM) is a powerful numerical technique that discretizes the domain and approximates both derivatives and integrals. DM is especially suitable for initial value problems, but it can suffer from instability and loss of accuracy when applied to fractional orders or when singularities are present [32]. Zhang and Tang [33] studied the q-fractional differential equation using the Caputo derivative:

$${}^{c}D_q^{\alpha }x\left(t\right)=f(t,x\left(t\right)),0<\alpha ,q<1.$$

The authors of [32] utilized a piecewise linear interpolation approximation to derive a difference formula to discretize the Caputo-type q-fractional derivative ${}^{c}D_q^{\alpha }x\left(t\right)$ and established the truncation error bound. This difference formula was then used to solve the initial value problem associated with the q-fractional differential equation. Moreover, in [34], an efficient numerical technique for a graded time mesh is proposed to address singular behavior at the initial time in time-fractional reaction–subdiffusion equations. The method employs the Caputo fractional derivative and combines a $L_1$ scheme in time with a compact finite difference scheme of high-order ADI in space. The rate of convergence of this method has been shown to depend on the grade exponent, and optimal accuracy is achieved, particularly on graded meshes.

The homotopy perturbation method, introduced by He [35], combines perturbation techniques with homotopy theory. It constructs a homotopy between a complex problem and a simpler problem and has been effectively applied to non-linear differential equations. Despite its flexibility, the convergence and accuracy of HPM can heavily depend on the initial assumption and the nature of the kernel involved. The authors in [36] addressed a non-linear oscillator with damping terms by modifying HPM using three effective expansions. They provided illustrative examples, including the Duffing equation with linear damping, to demonstrate the simplicity and effectiveness of the solution process. The study claims that the proposed scheme yields more accurate solutions for non-linear oscillators. Additional studies have explored and extended the method. In [37], a comparison between the homotopy analysis method (HAM) and HPM and their respective performance based on observed results was presented. In [38], the authors applied the modified homotopy perturbation method (MHPM) to solve a simple fractional problem and developed an error comparison table to demonstrate the method’s effectiveness.

The differential transform method (DTM) provides an approximate solution in a rapidly converging series. It is computationally efficient, but generally restricted to problems with analytic solutions and may not handle singular kernels effectively without significant modification [39]. In contrast to these methods, our work handles the complexity produced in the multi-fractional orders and singular kernels by establishing rigorous existence and stability results, along with a robust numerical scheme tailored to the complexities of the underlying operators. Ali et al. [40] provided details based on the suitability of the differential transform method for solving self-similar channel flow problems. In this work, they briefly presented that the quasi-linearization method is better than the DTM, as the latter becomes inappropriate for increasing the parameter values. These are beneficial details before considering this approach for fractional differential equations. In [41], authors used the DTM for integrating the Rössler system with fractional order and showed that, compared to the RK4 scheme and the DTM results, the DTM is more resistant to changes in the fractionality of the system. They identified the limitation of this method as its unsuitability for more than three-dimensional space, which subsequently made it more challenging for theoretical analyses and numerical experiments. We need to better assess which of the two integrating approaches of the fractional system gives results closer to accurate solutions.

Although existing methods such as the Legendre-based least squares method (LSM) and piecewise polynomial collocation offer good accuracy to solve fractional differential equations, they suffer from several limitations [1,2], including high computational cost, sensitivity to parameter choices (e.g., polynomial degree or collocation points), and complex implementation. Our proposed approaches, which combine the Sumudu and Shahu transform with ADM, yield more effective results than those presented in [1,2]. In addition, we solve the problems using two-step ADM, which provides analytical solutions with less computational effort. It effectively handles non-linearity, avoids discretization or collocation, and maintains high accuracy with a more straightforward implementation. Numerical comparisons confirm that our method offers a more stable and efficient alternative.

In 2022, a comprehensive framework for non-local fractional derivatives with singular kernels was established, enabling researchers to select derivatives most appropriate for specific physical models. These developments have improved the analytical and numerical tools available for studying the qualitative and quantitative behaviors of FDEs [42,43,44]. In [45,46], the authors proposed efficient numerical methods to solve fractional differential equations for terminal value problems of fractional linear systems. The discussion provides valuable insight and is worth further study in this area. However, a fundamental aspect of interest is the search for an analytical solution to FIDEs. It is often impossible to obtain one because of its complexity and time consumption. In this study, we provide analytical solutions to the problems considered. However, in some cases, we cannot derive an analytical solution. In that case, we resort to numerical techniques that yield approximate analytical solutions that can be considered exact for practical purposes. Our primary goal is to explore numerical methods that produce results comparable to those obtained by analytical methods. In some cases, applying numerical techniques is helpful, but not always. To identify the best approach for solving the FIDE mentioned, we found that combining transformation methods with numerical solutions yields better results than using numerical methods alone. Some of the studies that we present suggest combining transformations and numerical approaches.

In [47], an approach was proposed to solve KdV-type equations for homogeneous and non-homogeneous dispersive cases, known as the Shehu-ADM. The authors were impressed with the approximate solutions obtained in series form. They demonstrated that combining transformation methods and ADM offers a reliable and robust technique for solving various physical problems in applications. They introduced the combination of the Shehu transform and ADM to solve the Newell–Whitehead–Segel (NWS) equations in closed form. In addition, the convergence of the series solution and the absolute error analysis were presented in [48]. The above study combines the Shehu transformation and the numerical method, namely ADM, to solve different physical problems. For further results and to observe the efficiency of this powerful combination, we suggest referring to [49,50].

The authors in [51] discussed the combination of the Sumudu-ADM and the ADM for obtaining approximate solutions to non-linear systems of ordinary differential equations. In ADM, non-linear terms are handled using a special set of Adomian polynomials, which allow the non-linear part to be expressed as a series. This approach yields accurate solutions with a fast rate of convergence. In [52], the Sumudu-ADM was applied to obtain approximate solutions of a non-linear system of partial differential equations. The authors presented several examples and concluded that the method produced results closely aligned with known approximate solutions, aided by the Sumudu transform’s transformation properties and the ADM structure [53,54].

This study aims to develop accurate and efficient analytical–numerical methods for solving FIDEs. To address this problem, we employ the ADM, along with two enhanced, transform-based variants aimed at improving solution accuracy and flexibility:

• Sumudu–Adomian decomposition method (Sumudu-ADM)
• Shehu–Adomian decomposition method (Shehu-ADM)

Using the Caputo and Riemann–Liouville operators, we develop a class of multi-fractional integro-differential equations with singular kernels in this paper. The main findings and contributions of this study include the following:

• We present the Two-Step ADM (TSADM), a modified form of the ADM that can yield exact solutions in a single iteration.
• We introduce two transform-based ADM variations, the Sumudu-ADM and Shehu- ADM techniques, which improve accuracy and convergence while preserving computational simplicity.
• These transform-based methods significantly improve the performance of ADM without requiring initial guesses or assumptions about the solution.
• Using fixed-point theorems [55,56], we rigorously establish the existence and uniqueness of solutions.
• The U-H stability [57] of the solutions is analyzed in depth, reinforcing the reliability of the proposed methods.
• We perform detailed error and convergence analysis and validate the methods using illustrative examples supported by comparative plots and error norm tables.

The proposed techniques are flexible, computationally efficient, and well-suited for a broad class of FIDEs encountered in practical applications, including systems with unequal fractional orders. In particular, the Shehu-ADM method exhibits enhanced performance due to its adaptive weighted formulation.

To validate the proposed approaches, several illustrative examples are presented in which the comparative graphs and error norm tables that involve the $L_{\infty }$, $L_2$, and $L_1$ norms are studied. Moreover, the results demonstrate the extreme accuracy of the methods, with Shehu-ADM often outperforming others due to its weighted formulation.

In our considered problem, the standard ADM and the LADM do not perform well. The solution tends to diverge because of the noise terms introduced in each iteration. However, when we apply the Shehu-ADM and Sumudu-ADM methods, both yield solutions that show good convergence and appear promising. We now compare the Shehu-ADM and Sumudu-ADM methods to determine which performs better. Since both methods often provide similarly strong results, we plan to use different norms to distinguish them more clearly. This comparison is the main objective of our current study.

The solutions are obtained in detail by applying Sumudu-ADM and Shehu-ADM, incorporating a weight function within the framework. The weight function, used in Shehu-ADM, is defined as

$$w({\zeta }_1,{\zeta }_2)={\displaystyle \frac{1+{\zeta }_1+{\zeta }_2}{3}},$$

with a significant impact of the weight function on the fractional integral and fractional derivative terms during iteration. It offers an adaptive balance between fractional orders ${\zeta }_1$ and ${\zeta }_2$; its function is essential when these orders diverge. This improves the accuracy, stability, and convergence rate of the Shehu-ADM approximation. On the other hand, Sumudu-ADM uses a transform to handle fractional terms directly without making modifications. Although this method simplifies the calculation, it might not adequately represent the complex behavior of fractional operators, particularly when working with unequal fractional orders. The study’s comparison findings show that, for a range of $\zeta$ values, the adaptive weighting mechanism in Shehu-ADM consistently produces better error norms and lower absolute errors. The efficiency of the weight function in improving the method’s overall performance is demonstrated here, making Shehu-ADM a more reliable and accurate method for solving FIDEs.

This paper is organized as follows. Section 2 outlines the essential definitions and preliminary results from fractional calculus required for the development of the study. Section 3 introduces the analytical techniques for solving fractional FIDEs. Section 4 presents the numerical methods designed to approximate the solutions. Section 5 is dedicated to obtaining error bounds and analysis of the convergence of the solution. Section 6 provides results for the existence and uniqueness under some conditions. In addition, we examine the stability of the solution using the Ulam–Hyers stability approach. Section 8 offers a comprehensive example to illustrate the effectiveness of both analytical and numerical methods. In Section 9, we include and discuss previous results, comparing them with our findings. Lastly, Section 10 summarizes the conclusions drawn from this work and discusses future work decisions.

2. Preliminaries

This section presents key ideas from fractional calculus in our study. We concentrate on the Riemann–Liouville and Caputo operators, which are essential for simulating memory-dependent systems, especially those with initial value formulations. We also include fixed-point concepts and transformation tools within the analytical framework, such as the Sumudu and Shehu transforms. These tools make it easier to handle non-local terms and help create trustworthy approximation methods (see [5,58,59,60,61,62]).

Definition 1.

([63]). A real function $g\left(y\right):{\mathbb{R}}^{+}\to \mathbb{R}$ is said to be in the space $C_{\theta }$, if $\theta \in \mathbb{R}$, ∃ a real number $v(>\theta )$, such that $g\left(y\right)=y^v{g}_1\left(y\right)$, where $g_1\left(y\right)\in C[0,\infty )$ and said to be in the space $C_{\theta }^n$ if $g^{\left(n\right)}\in C_{\theta }$, $n\in \mathbb{N}\cup \left\{0\right\}.$

Definition 2.

([63]). The Riemann–Liouville (RL) fractional integral operator of order $\sigma \ge 0$, of a function $g\in C_{\theta }$, $\theta \ge -1$ is defined as

$$\begin{array}{cc}\hfill {}_{0+}{}^{RL}D_y^{-\sigma }g\left(y\right)={}_{0+}{}^{RL}I_y^{\sigma }g\left(y\right)& ={\displaystyle \frac{1}{\Gamma \left(\sigma \right)}}{\int }_0^y{(y-\xi )}^{\sigma -1}g\left(\xi \right)d\xi ,\sigma >0,y>0,\hfill \\ \hfill {}_{0+}{}^{RL}I_y^{0}g\left(y\right)& =g\left(y\right).\hfill \end{array}$$

(2)

Remark 1.

The operator ${}_{0+}{}^{RL}I_y^{\sigma }:C^m[0,T]\to C^m[0,T]$ for any $m\in {\mathbb{N}}_0$ and is linear and bounded on this space [64].

The following properties of the operator ${{}_{0+}{}^{RL}I_y}^{\sigma }$ can be found in [59,60]. For $g\in C_{\theta }$, $\theta \ge -1$, $\sigma ,\omega \ge 0$ and $\zeta >-1$,

$$\left(i\right){}_{0+}{}^{RL}I_y^{\sigma }{}_{0+}{}^{RL}I_y^{\omega }g\left(y\right)={}_{0+}{}^{RL}I_y^{\sigma +\omega }g\left(y\right).$$

(3)

$$\left(ii\right){}_{0+}{}^{RL}I_y^{\sigma }{}_{0+}{}^{RL}I_y^{\omega }g\left(y\right)={}_{0+}{}^{RL}I_y^{\omega }{}_{0+}{}^{RL}I_y^{\sigma }g\left(y\right).$$

(4)

$$\left(iii\right){}_{0+}{}^{RL}I_y^{\sigma }{y}^{\zeta }={\displaystyle \frac{\Gamma (\zeta +1)}{\Gamma (\sigma +\zeta +1)}}{y}^{\sigma +\zeta }.$$

(5)

Definition 3.

The fractional derivative of a function $g\left(y\right)$ in the Caputo sense is defined as

$${}_{0+}{}^{C}D_y^{\sigma }g\left(y\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\Gamma (n-\sigma )}{\int }_0^y{(y-\xi )}^{n-\sigma -1}{g}^{\left(n\right)}\left(\xi \right)d\xi ,}\hfill & \mathit{if}\sigma <n,\hfill \\ g^{\left(n\right)}\left(y\right),\hfill & \mathit{if}\sigma =n,\hfill \end{array}\right.$$

(6)

for $n-1<\sigma \le n$, $n\in \mathbb{N}$, $y>0$, and $g\in C_{-1}^n$.

Moreover, for any constant $A\in \mathbb{R}$,

$${}_{0+}{}^{C}D_y^{\sigma }A=0.$$

(7)

The Caputo’s fractional derivatives are linear operators, as it is evident that

$${}_{0+}{}^{C}D_y^{\sigma }\left(As\left(y\right)+Bh\left(y\right)\right)=A{}_{0+}{}^{C}D_y^{\sigma }s\left(y\right)+B{}_{0+}{}^{C}D_y^{\sigma }h\left(y\right),$$

(8)

where A and B are constants.

Definition 4.

For every smallest integer n which exceeds σ, we can define the Caputo time-fractional derivative operator of order $\sigma >0$ as

$${}_{0+}{}^{C}D{}_{\xi }{}^{\sigma }w(y,\xi )={\displaystyle \frac{\partial {}^{\sigma }w(y,\xi )}{\partial {\xi }^{\sigma }}}=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\Gamma (n-\sigma )}}{\int }_0^{\xi }{(\xi -\tau )}^{n-\sigma -1}{\displaystyle \frac{{\partial }^{n}w(y,\tau )}{\partial {\tau }^n}}d\tau ,\hfill & \hfill n-1<\sigma <n,\\ \\ {\displaystyle \frac{\partial {}^{n}w(y,\xi )}{\partial {\xi }^n}},\hfill & \hfill \sigma =n\in \mathbb{N}.\end{array}\right.$$

(9)

Lemma 1.

If $n-1<\sigma \le n$, $n\in \mathbb{N}$ and $g\in C_{\theta }^n$, $\theta \ge -1$, then

$$\begin{array}{cc}\hfill {}_{0+}{}^{C}D_y{}^{\sigma }{}_{0+}{}^{RL}I_y^{\sigma }g\left(y\right)& =g\left(y\right).\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill {}_{0+}{}^{RL}I_y^{\sigma }{}_{0+}{}^{C}D{}_y{}^{\sigma }g\left(y\right)& =g\left(y\right)-\sum _{q=0}^{n-1}{g}^q\left(0^{+}\right){\displaystyle \frac{y^q}{q!}},y>0.\hfill \end{array}$$

(11)

In the following, we present the definitions of two transformations used to obtain the solution of the model.

Definition 5.

Let $f:{\mathbb{R}}^{+}\cup \left\{0\right\}\to \mathbb{R}$ be a function. The Sumudu transform [51,52] of f, denoted by $\mathcal{S}\left[f\right(t\left)\right]$, is defined as

$$\mathcal{S}\left[f\left(t\right)\right]\left(s\right)={\int }_0^{\infty }f\left(t\right)e^{-t/s}dt,s>0,$$

(12)

provided the integral exists. The Sumudu transform is particularly useful for solving differential equations, especially when the solution must remain in the same time domain.

Definition 6.

The Shehu transform [47,48] of $f\left(t\right)$, denoted by $\mathbb{H}\left[f\right(t\left)\right]$, is defined as

$$\mathbb{H}\left[f\left(t\right)\right]\left(v\right)={\int }_0^{\infty }{e}^{-v{t}^x}f\left(t\right)dt,v>0,x>0,$$

(13)

where x is a real order parameter. This transform is well-suited for solving fractional differential and integro-differential equations involving Caputo derivatives and singular kernels.

Remark 2.

$${}_{0+}{}^{RL}I_y^{\alpha }:C\left([0,T]\right)\to C\left([0,T]\right)$$

means that the Riemann–Liouville fractional integral operator ${}_{0+}{}^{RL}I_y^{\alpha }$ maps the Banach space of continuous functions on the interval $[0,T]$ into itself. That is, for any function $f\in C\left(\right[0,T\left]\right)$, the fractional integral

$$\left({}_{0+}{}^{RL}I_y^{\alpha }f\right)\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-\tau )}^{\alpha -1}f\left(\tau \right)d\tau$$

is also a continuous function on $[0,T]$. Therefore, ${}_{0+}{}^{RL}I_y^{\alpha }f\in C\left([0,T]\right)$, and the operator ${}_{0+}{}^{RL}I_y^{\alpha }$ is bounded on $C\left(\right[0,T\left]\right)$ (see [58]).

In the subsequent section, we present the analytical approach based on the TSADM, followed by the development of numerical schemes, particularly the Sumudu-ADM and Shehu-ADM methods, designed to approximate solutions of the proposed fractional integro-differential Equation (1).

3. Analytical Method

We now apply the TSADM to derive a solution to the problem formulated in Equation (1).

The algorithm consists of five steps:

Step 1. Apply the inverse operator ${}_{0+}{}^{RL}I_x^{{\zeta }_1}$ of ${}_{0+}{}^{C}D_x^{{\zeta }_1}$ to both sides of Equation (1). This yields

$$\lambda \left(x\right)={\eta }_1+x{\eta }_2+{}_{0+}{}^{RL}I_x^{{\zeta }_1}\left[\mathcal{H}\left(x\right)-{\lambda }^{″}\left(x\right)-{}_{0+}{}^{C}D_x^{-{\zeta }_2}\lambda \left(x\right)\right].$$

(14)

Step 2. Formulate the recursion relation for TSADM based on Equation (14) as

$${\lambda }^{\left(0\right)}\left(x\right)={\eta }_1+x{\eta }_2,$$

(15)

$${\lambda }^{(n+1)}\left(x\right)={}_{0+}{}^{RL}I_x^{{\zeta }_1}\left[\mathcal{H}\left(x\right)-{\left({\lambda }^{\left(n\right)}\right)}^{″}\left(x\right)-{}_{0+}{}^{C}D_x^{-{\zeta }_2}{\lambda }^{\left(n\right)}\left(x\right)\right],n=0,1,2,\cdots$$

(16)

Step 3. The initial iteration (zeroth term) from Equation (15) can be decomposed into multiple components:

$${\lambda }^{\left(0\right)}\left(x\right)=\mathcal{M}={\mathcal{M}}_1+{\mathcal{M}}_2+{\mathcal{M}}_3+\cdots +{\mathcal{M}}_M,$$

(17)

where ${\mathcal{M}}_1,{\mathcal{M}}_2,\cdots ,{\mathcal{M}}_M$ are the terms derived from the integration of the source term $\lambda \left(x\right)$ along with the corresponding initial and boundary conditions.

Step 4. Identify the first component of ${\lambda }^{\left(0\right)}$, denoted as ${\mathcal{M}}_1$. If ${\mathcal{M}}_1$ satisfies Equation (1) along with the initial conditions, then ${\mathcal{M}}_1$ is the exact solution. If it does not satisfy either of these, the next tests are performed, and the process is repeated. If any component ${\lambda }^{\left(0\right)}$ satisfies both Equation (1) and the initial conditions, it is taken as the exact solution. If none of the components satisfy the criteria, proceed to Step 5.

Step 5. Use ADM to derive the solution by setting ${\lambda }^{\left(0\right)}\left(x\right)=\mathcal{M}$ and iterating using Equation (16) from Step 2.

Approximate the solution $\lambda \left(x\right)$ by truncating the series

$${\lambda }^{\left(n\right)}=\sum _{m=0}^{N-1}{\lambda }^{\left(m\right)}\left(x\right),$$

(18)

$$\underset{N\to \infty }{lim}{\lambda }^{\left(n\right)}=\lambda \left(x\right).$$

(19)

Thus, $\lambda \left(x\right)$ is the solution to (1).

4. Numerical Methods

In this section, we develop two numerical strategies for approximating the solution of the FIDEs given in (1).

From Equation (14), we have

$$\lambda \left(x\right)={\eta }_1+x{\eta }_2+{}_{0+}{}^{RL}I_x^{{\zeta }_1}\left[\mathcal{H}\left(x\right)-{\lambda }^{″}\left(x\right)-{}_{0+}{}^{RL}D_x^{-{\zeta }_2}\lambda \left(x\right)\right],$$

(20)

where ${}_{0+}{}^{RL}I_x^{{\zeta }_1}$ denotes the RL-type fractional integral of order ${\zeta }_1$ and ${}_{0+}{}^{RL}D_x^{-{\zeta }_2}$ denotes the RL-type fractional integral of order ${\zeta }_2$. The function $\mathcal{H}\left(x\right)$ is assumed to be known, while ${\eta }_1$ and ${\eta }_2$ are constants.

4.1. Adomian Decomposition Method (ADM)

The solution $\lambda \left(x\right)$ is expressed as an infinite series for the FIDEs given in Equation (1), namely

$$\lambda \left(x\right)=\sum _{n=0}^{\infty }{\lambda }^{\left(n\right)}\left(x\right).$$

(21)

Substituting this into Equation (20) gives

$$\begin{array}{c}\hfill \sum _{n=0}^{\infty }{\lambda }^{\left(n\right)}\left(x\right)={\eta }_1+x{\eta }_2+{}_{0+}{}^{RL}I_x^{{\zeta }_1}\left[\mathcal{H}\left(x\right)-\sum _{n=0}^{\infty }{\left({\lambda }^{\left(n\right)}\right)}^{″}\left(x\right)-{}_{0+}{}^{RL}D_x^{-{\zeta }_2}\left(\sum _{n=0}^{\infty }{\lambda }^{\left(n\right)}\left(x\right)\right)\right].\end{array}$$

The iterative scheme begins with

$${\lambda }^{\left(0\right)}\left(x\right)={\eta }_1+x{\eta }_2,$$

(22)

and proceeds with the recursive relation

$${\lambda }^{(n+1)}\left(x\right)={}_{0+}{}^{RL}I_x^{{\zeta }_1}\left[{\mathcal{H}}_n\left(x\right)-{\left({\lambda }^{\left(n\right)}\right)}^{″}\left(x\right)-{}_{0+}{}^{RL}D_x^{-{\zeta }_2}{\lambda }^{\left(n\right)}\left(x\right)\right],n\ge 0.$$

(23)

4.2. Sumudu–Adomian Decomposition Method (Sumudu-ADM)

The Sumudu-ADM [54] employs the Sumudu transform, denoted by $\mathcal{S}$, which is particularly effective in solving linear and non-linear differential equations, including those involving fractional operators. We let w be the transform variable associated with the Sumudu transform.

Applying the Sumudu transform [65] to both sides of Equation (20), and using the known operational properties of the Sumudu transform for fractional operators, we obtain

$$\begin{array}{cc}\hfill \mathcal{S}\left[{\lambda }^{″}\left(x\right)\right]& =w^2\mathcal{S}\left[\lambda \left(x\right)\right]-w\lambda \left(0\right)-{\lambda }^{\prime }\left(0\right),\hfill \\ \hfill \mathcal{S}\left[{}_{0+}{}^{RL}D_x^{-{\zeta }_2}\lambda \left(x\right)\right]& =w^{-{\zeta }_2}\mathcal{S}\left[\lambda \left(x\right)\right],\hfill \\ \hfill \mathcal{S}\left[{}_{0+}{}^{C}D_x^{{\zeta }_1}\lambda \left(x\right)\right]& =w^{{\zeta }_1}\mathcal{S}\left[\lambda \left(x\right)\right]-\sum _{k=0}^1{w}^{{\zeta }_1-k-1}{\lambda }^{\left(k\right)}\left(0\right).\hfill \end{array}$$

Substituting these into the transformed form of Equation (20) gives

$$\mathcal{S}\left[\lambda \left(x\right)\right]=\mathcal{S}\left[{\eta }_1\right]+\mathcal{S}\left[x{\eta }_2\right]+w^{{\zeta }_1}\left[\mathcal{S}\left[\mathcal{H}\left(x\right)\right]-w^2\mathcal{S}\left[\lambda \left(x\right)\right]-w^{-{\zeta }_2}\mathcal{S}\left[\lambda \left(x\right)\right]\right].$$

Rewriting and solving for $\mathcal{S}\left[\lambda \right(x\left)\right]$, we then apply the inverse Sumudu transform ${\mathcal{S}}^{-1}$ to construct the recursive formula for ${\lambda }^{(n+1)}\left(x\right)$. The result is

$${\lambda }^{(n+1)}\left(x\right)={\mathcal{S}}^{-1}\left\{{\displaystyle \frac{\mathcal{S}\left[{\mathcal{H}}_n\left(x\right)\right]-w^{{\zeta }_1+2}\mathcal{S}\left[{\lambda }^{\left(n\right)}\left(x\right)\right]-w^{{\zeta }_1-{\zeta }_2}\mathcal{S}\left[{\lambda }^{\left(n\right)}\left(x\right)\right]}{w^2}}\right\},n\ge 0.$$

(24)

In this formulation,

• The term $w^{{\zeta }_1+2}$ corresponds to the Sumudu image of the second derivative composed of the Caputo integral;
• The term $w^{{\zeta }_1-{\zeta }_2}$ arises from the image of the nested RL integral term;
• The function ${\mathcal{H}}_n\left(x\right)$ can represent a known function or a non-linear function expressed as a series of decompositions.

The initial term is taken as

$${\lambda }^{\left(0\right)}\left(x\right)={\eta }_1+x{\eta }_2,$$

(25)

which satisfies the non-homogeneous initial structure of Equation (20). Then, each subsequent term is computed iteratively using Equation (24).

This transform-based ADM framework is especially useful when dealing with convolution-type fractional operators, as it converts complex integral-differential operations into algebraic manipulations in the transform domain. Using only a few terms in the decomposition often provides highly accurate approximate solutions with low computational effort.

4.3. Shehu–Adomian Decomposition Method (Shehu-ADM)

The Shehu-ADM [66] is based on the Shehu transform, which is similar in spirit to the Laplace and Sumudu transforms but provides better convergence. We let $\mathbb{H}$ denote the Shehu transform and $\omega$ its associated transform variable.

Applying the Shehu transform [67,68] to both sides of Equation (20) and incorporating the Shehu operational properties for non-zero initial conditions, we obtain the following relations:

$$\begin{array}{cc}\hfill \mathbb{H}\left[{\lambda }^{″}\left(x\right)\right]& ={\omega }^2\mathbb{H}\left[\lambda \left(x\right)\right]-\omega \lambda \left(0\right)-{\lambda }^{\prime }\left(0\right),\hfill \\ \hfill \mathbb{H}\left[{}_{0+}{}^{RL}D_x^{-{\zeta }_2}\lambda \left(x\right)\right]& ={\omega }^{-{\zeta }_2}\mathbb{H}\left[\lambda \left(x\right)\right],\hfill \\ \hfill \mathbb{H}\left[{}_{0+}{}^{C}D_x^{{\zeta }_1}\lambda \left(x\right)\right]& ={\omega }^{{\zeta }_1}\mathbb{H}\left[\lambda \left(x\right)\right]-\sum _{k=0}^1{\omega }^{{\zeta }_1-k-1}{\lambda }^{\left(k\right)}\left(0\right).\hfill \end{array}$$

Substituting these properties into the Shehu-transformed form of Equation (20), we get the following:

$$\mathbb{H}\left[\lambda \left(x\right)\right]=\mathbb{H}\left[{\eta }_1\right]+\mathbb{H}\left[x{\eta }_2\right]+{\omega }^{-{\zeta }_1}\left(\mathbb{H}\left[\mathcal{H}\left(x\right)\right]-{\omega }^2\mathbb{H}\left[\lambda \left(x\right)\right]+\omega \lambda \left(0\right)+{\lambda }^{\prime }\left(0\right)-{\omega }^{-{\zeta }_2}\mathbb{H}\left[\lambda \left(x\right)\right]\right).$$

Simplifying and solving for $\mathcal{H}\left[\lambda \right(x\left)\right]$ allows us to construct the iterative recursive formula via the inverse Shehu transform.

We obtain

$$\mathbb{H}\left[\lambda \left(x\right)\right]=\mathbb{H}\left[{\eta }_1\right]+\mathbb{H}\left[x{\eta }_2\right]+{\omega }^{{\zeta }_1}\left[\mathbb{H}\left[\mathcal{H}\left(x\right)\right]-{\omega }^2\mathbb{H}\left[\lambda \left(x\right)\right]-{\omega }^{-{\zeta }_2}\mathbb{H}\left[\lambda \left(x\right)\right]\right].$$

Now, solving for $\mathcal{H}\left[\lambda \right(x\left)\right]$ and applying the inverse Shehu transform, we derive the iterative formula:

$${\displaystyle {\lambda }^{(n+1)}\left(x\right)={\mathbb{H}}^{-1}\left\{\frac{\mathbb{H}\left[{\mathcal{H}}_n\left(x\right)\right]-{\omega }^{{\zeta }_1}{\omega }^2\mathbb{H}\left[{\lambda }^{\left(n\right)}\left(x\right)\right]-{\omega }^{{\zeta }_1}{\omega }^{-{\zeta }_2}\mathbb{H}\left[{\lambda }^{\left(n\right)}\left(x\right)\right]}{{\omega }^2}\right\},n\ge 0.}$$

(26)

For simplification, we combine exponents to get the final recursive form:

$${\lambda }^{(n+1)}\left(x\right)={\mathbb{H}}^{-1}\left\{{\displaystyle \frac{\mathbb{H}\left[{\mathcal{H}}_n\left(x\right)\right]-{\omega }^{{\zeta }_1+2}\mathbb{H}\left[{\lambda }^{\left(n\right)}\left(x\right)\right]-{\omega }^{{\zeta }_1-{\zeta }_2}\mathbb{H}\left[{\lambda }^{\left(n\right)}\left(x\right)\right]}{{\omega }^2}}\right\},n\ge 0.$$

(27)

This formulation leads to a recursive computation where

• ${\lambda }^{\left(0\right)}\left(x\right)={\eta }_1+x{\eta }_2$ is the initial approximation,
• ${\lambda }^{\left(1\right)}\left(x\right)$ incorporates the effect of $\mathcal{H}\left(x\right)$, the second derivative, and the fractional integral of ${\lambda }^{\left(0\right)}\left(x\right)$,
• higher-order terms are obtained iteratively using Equation (27).

The Shehu-ADM is beneficial when the convergence of the ADM series is slow using other transforms. It allows efficient algebraic manipulation in the transform domain and ensures better handling of convolution-type integrals in fractional models.

5. Error Estimate and Convergence Analysis

This section examines the convergence of ADM associated with TSADM, Sumudu-ADM, and Shehu-ADM. We show the necessary conditions for the convergence of the approach and the error estimate for Equations (1) and (14).

Theorem 1.

Let ${\lambda }^{\left(n\right)}\left(x\right)$ and $\lambda \left(x\right)$ be defined in the Banach space $(C^2\left(\mathcal{I}\right),\parallel ·{\parallel }_{\infty })$. Suppose that there exists a constant $\gamma \in (0,1)$ such that

$$\parallel {\lambda }^{\left(n\right)}\left(x\right){\parallel }_{\infty }\le \gamma \parallel {\lambda }^{(n-1)}\left(x\right){\parallel }_{\infty },\forall n\ge 1,x\in \mathcal{I}.$$

Then the series solution ${\left\{{\lambda }^{\left(n\right)}\left(x\right)\right\}}_{n=0}^{\infty }$ defined by

$$\lambda \left(x\right)=\sum _{n=0}^{\infty }{\lambda }^{\left(n\right)}\left(x\right),$$

(28)

converges uniformly to the solution $\lambda \left(x\right)$ in $C^2\left(\mathcal{I}\right)$.

Proof. 

Let us $(C^2\left(\mathcal{I}\right),\parallel ·{\parallel }_{\infty })$ denote the Banach space comprising all continuous functions defined in $\mathcal{I}$, equipped with the norm.

$${∥\lambda \left(x\right)∥}_{\infty }=\underset{x\in \mathcal{I}}{sup}\left|\lambda \left(x\right)\right|.$$

(29)

We define $\left\{{\mathcal{S}}_n\right\}$ as the sequence of partial sums of the series ${\sum }_{n=0}^{\infty }{\lambda }^{\left(n\right)}\left(x\right)$ by

$$\begin{array}{cc}\hfill {\mathcal{PS}}_0& ={\lambda }^{\left(0\right)}\left(x\right),\hfill \\ \hfill {\mathcal{PS}}_1& ={\lambda }^{\left(0\right)}\left(x\right)+{\lambda }^{\left(1\right)}\left(x\right),\hfill \\ \hfill {\mathcal{PS}}_2& ={\lambda }^{\left(0\right)}\left(x\right)+{\lambda }^{\left(1\right)}\left(x\right)+{\lambda }^{\left(2\right)}\left(x\right),\hfill \\ \hfill ⋮\\ \hfill {\mathcal{PS}}_n& ={\lambda }^{\left(0\right)}\left(x\right)+{\lambda }^{\left(1\right)}\left(x\right)+{\lambda }^{\left(2\right)}\left(x\right)+\cdots +{\lambda }^{\left(n\right)}\left(x\right).\hfill \end{array}$$

(30)

It is necessary to demonstrate that the sequence ${\left\{{\mathcal{PS}}_n\right\}}_{n=0}^{\infty }$ constitutes a Cauchy sequence within the Banach space $(C^2\left(\mathcal{I}\right),\parallel ·{\parallel }_{\infty })$. To this end, we examine

$$\begin{array}{cc}\hfill \parallel {\mathcal{PS}}_{n+1}-{\mathcal{PS}}_n{\parallel }_{\infty }& =\parallel {\lambda }^{\left(n\right)}\left(x\right){\parallel }_{\infty }\le \gamma {\parallel {\lambda }^{(n-1)}\left(x\right)\parallel }_{\infty }\hfill \\ & \le {\gamma }^2\parallel {\lambda }^{(n-2)}\left(x\right){\parallel }_{\infty }\le \cdots \le {\gamma }^{n+1}{\parallel {\lambda }^{\left(0\right)}\left(x\right)\parallel }_{\infty }.\hfill \end{array}$$

(31)

For every, $n,m\in \mathbb{N},n\ge m,$ form (31), we have

$$\begin{array}{cc}\hfill \parallel {\mathcal{PS}}_n-{\mathcal{PS}}_m{\parallel }_{\infty }& =\parallel ({\mathcal{PS}}_n-{\mathcal{PS}}_{n-1})+({\mathcal{PS}}_{n-1}-{\mathcal{PS}}_{n-2})+\cdots +({\mathcal{PS}}_{m+1}-{\mathcal{PS}}_m){\parallel }_{\infty }\hfill \\ \hfill & \le \parallel {\mathcal{PS}}_n-{\mathcal{PS}}_{n-1}{\parallel }_{\infty }+\parallel {\mathcal{PS}}_{n-1}-{\mathcal{PS}}_{n-2}{\parallel }_{\infty }+\cdots +\parallel {\mathcal{PS}}_{m+1}-{\mathcal{PS}}_m{)\parallel }_{\infty }\hfill \\ \hfill & \le {\gamma }^n\parallel {\lambda }^{\left(0\right)}\left(x\right){\parallel }_{\infty }+{\gamma }^{n-1}\parallel {\lambda }^{\left(0\right)}\left(x\right){\parallel }_{\infty }+\cdots +{\gamma }^{m+1}{\parallel {\lambda }^{\left(0\right)}\left(x\right)\parallel }_{\infty }\hfill \\ \hfill & ={\displaystyle \frac{1-{\gamma }^{n-m}}{1-\gamma }}{\gamma }^{m+1}{\parallel {\lambda }^{\left(0\right)}\left(x\right)\parallel }_{\infty }.\hfill \end{array}$$

(32)

Since $0<\gamma <1,$ we have $1-{\gamma }^{n-m}<1.$ Thus

$$\parallel {\mathcal{PS}}_n-{\mathcal{PS}}_m{\parallel }_{\infty }\le {\displaystyle \frac{1-{\gamma }^{m+1}}{1-\gamma }}\parallel {\lambda }^{\left(0\right)}\left(x\right){\parallel }_{\infty }.$$

(33)

Again, ${\lambda }^{\left(0\right)}\left(x\right)$ is bounded, which gives

$$\underset{n,m\to 0}{lim}\parallel {\mathcal{PS}}_n-{\mathcal{PS}}_m{\parallel }_{\infty }=0.$$

Therefore, it ${\left\{{\mathcal{PS}}_n\right\}}_{n=0}^{\infty }$ is a Cauchy sequence in the Banach space, $(C^2\left(\mathcal{I}\right),\parallel ·{\parallel }_{\infty }),$ so the series solution of the series $\sum {\lambda }^{\left(n\right)}$ converges. □

Theorem 2.

The maximum absolute truncation error of the series solution given in Equation (28), corresponding to the problem formulated in Equation (1), is approximated as

$$|\lambda \left(x\right)-\sum _{b=0}^m{\lambda }^{\left(b\right)}\left(x\right)|\le {\displaystyle \frac{1-{\gamma }^{m+1}}{1-\gamma }}\parallel {\lambda }^{\left(0\right)}\left(x\right){\parallel }_{\infty }.$$

(34)

Proof. 

From Theorem 1 and (32), we have

$$\parallel {\mathcal{PS}}_n-{\mathcal{PS}}_m{\parallel }_{\infty }\le {\displaystyle \frac{1-{\gamma }^{n-m}}{1-\gamma }}{\gamma }^{m+1}\parallel {\lambda }^{\left(0\right)}\left(x\right){\parallel }_{\infty },$$

(35)

for $n\ge m.$

If $n\to \infty ,$ then ${\mathcal{PS}}_n\to \lambda \left(x\right).$ So,

$$\parallel \lambda \left(x\right)-{\mathcal{PS}}_m{\parallel }_{\infty }\le {\displaystyle \frac{1-{\gamma }^{m+1}}{1-\gamma }}\parallel {\lambda }^{\left(0\right)}\left(x\right){\parallel }_{\infty }.$$

(36)

Since $0<\gamma <1,$$1-{\gamma }^{n-m}<1,$ the above inequality becomes

$$\parallel \lambda \left(x\right)-\sum _{b=0}^m{\lambda }^{\left(b\right)}\left(x\right){\parallel }_{\infty }\le {\displaystyle \frac{1-{\gamma }^{m+1}}{1-\gamma }}\parallel {\lambda }^{\left(0\right)}\left(x\right){\parallel }_{\infty }.$$

(37)

Finally, we have

$$\parallel \lambda \left(x\right)-\sum _{b=0}^m{\lambda }^{\left(b\right)}\left(x\right){\parallel }_{\infty }=\mathcal{O}\left({\gamma }^m\right).$$

(38)

The inequality above (38) indicates the exponential convergence of the series in relation to the number of iterations m. Although the method does not rely on a mesh size or time step, its approximation order in fractional differential equations aligns with the classical understanding that Caputo-type solutions exhibit limited smoothness. According to Podlubny [69], the fractional integral operator of order ${\zeta }_1\in (0,1)$ introduces regularity consistent with a first-order convergence behavior when interpreted through the lens of series expansion. Thus, the proposed decomposition methods provide exponential convergence in m and are effectively second-order accurate in fractional regularity. □

Lemma 2.

Let $\mathcal{H}\in C\left(\right[0,T\left]\right)$. Then the solution of the FIDEs (1) is given by

$$\lambda \left(x\right)={\eta }_1+x{\eta }_2+{}_{0+}{}^{\mathrm{RL}}I_x^{{\zeta }_1}\left[\mathcal{H}\left(x\right)-{\lambda }^{″}\left(x\right)-{}_{0+}{}^{\mathrm{RL}}D_x^{-{\zeta }_2}\lambda \left(x\right)\right].$$

(39)

6. Existence and Uniqueness of Solutions

This section is devoted to proving the existence and uniqueness of solutions to Equation (1), subject to well-defined assumptions.

Lemma 3.

There exist constants ${\mathcal{M}}_1,{\mathcal{M}}_2>0$ such that, for all $x\in [0,T]$,

(a)

$\left|{}_{0+}{}^{C}D_x^{{\zeta }_2}\lambda \left(x\right)\right|\le {\mathcal{M}}_1$,

(b)

$\left|{}_{0+}{}^{RL}I_x^{{\zeta }_1}\lambda \left(x\right)\right|\le {\mathcal{M}}_2$,

provided that $\lambda \in C^2\left([0,T]\right)$ and ${\lambda }^{\prime }\in L^{\infty }\left([0,T]\right)$.

Proof. 

(a)

For the Caputo fractional derivative

$$\begin{array}{cc}\hfill \left|{}_{0+}{}^{C}D_x^{{\zeta }_2}\lambda \left(x\right)\right|& =\left|{\displaystyle \frac{1}{\Gamma (1-{\zeta }_2)}}{\int }_0^x{\displaystyle \frac{{\lambda }^{\prime }\left(\xi \right)}{{(x-\xi )}^{{\zeta }_2}}}d\xi \right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma (1-{\zeta }_2)}}{\int }_0^x{\displaystyle \frac{|{\lambda }^{\prime }\left(\xi \right)|}{{(x-\xi )}^{{\zeta }_2}}}d\xi \hfill \\ \hfill & \le {\displaystyle \frac{sup|{\lambda }^{\prime }|}{\Gamma (1-{\zeta }_2)}}{\int }_0^x{(x-\xi )}^{-{\zeta }_2}d\xi \hfill \\ \hfill & ={\displaystyle \frac{\parallel {\lambda }^{\prime }{\parallel }_{\infty }}{\Gamma (1-{\zeta }_2)}}·{\displaystyle \frac{T^{1-{\zeta }_2}}{1-{\zeta }_2}}\hfill \\ \hfill & \le {\mathcal{M}}_1,\hfill \end{array}$$

where ${\mathcal{M}}_1={\displaystyle \frac{\parallel {\lambda }^{\prime }{\parallel }_{\infty }{T}^{1-{\zeta }_2}}{(1-{\zeta }_2)\Gamma (1-{\zeta }_2)}}$, which bounds the Caputo derivative on $[0,T]$. Suppose $\parallel {\lambda }^{\prime }{\parallel }_{\infty }\le {\mathcal{P}}^{\prime }{\parallel \lambda \parallel }_{\infty }$; then ${\mathcal{M}}_1={\displaystyle \frac{{\mathcal{P}}^{\prime }{\parallel \lambda \parallel }_{\infty }{T}^{1-{\zeta }_2}}{\Gamma (2-{\zeta }_2)}}.$

(b)

For the RL fractional integral:

$$\begin{array}{cc}\hfill \left|{}_{0+}{}^{RL}I_x^{{\zeta }_1}\lambda \left(x\right)\right|& =\left|{\displaystyle \frac{1}{\Gamma \left({\zeta }_1\right)}}{\int }_0^x{(x-\xi )}^{{\zeta }_1-1}\lambda \left(\xi \right)d\xi \right|\hfill \\ \hfill & \le {\displaystyle \frac{sup\left|\lambda \right|}{\Gamma \left({\zeta }_1\right)}}{\int }_0^x{(x-\xi )}^{{\zeta }_1-1}d\xi \hfill \\ \hfill & \le {\displaystyle \frac{{\parallel \lambda \parallel }_{\infty }}{\Gamma \left({\zeta }_1\right)}}·{\displaystyle \frac{T^{{\zeta }_1}}{{\zeta }_1}}\hfill \\ \hfill & \le {\displaystyle \frac{{\parallel \lambda \parallel }_{\infty }{T}^{{\zeta }_1}}{{\zeta }_1\Gamma \left({\zeta }_1\right)}}\hfill \\ \hfill & \le {\mathcal{M}}_2,\hfill \end{array}$$

where ${\mathcal{M}}_2={\displaystyle \frac{{\parallel \lambda \parallel }_{\infty }{T}^{{\zeta }_1}}{\Gamma ({\zeta }_1+1)}}$, which bounds the RL integral on $[0,T]$.

  □

To proceed further, we assume that

Assumption 1

([A1]). There exist constants $L_1,L_2>0$ such that

$$\left|{\left({\lambda }^{\left(1\right)}\right)}^{″}\left(x\right)-{\left({\lambda }^{\left(2\right)}\right)}^{″}\left(x\right)\right|\le L_1\left|{\lambda }^{\left(1\right)}\left(x\right)-{\lambda }^{\left(2\right)}\left(x\right)\right|,\forall x\in [0,T].$$

This assumption is justified under the regularity condition ${\lambda }^{\left(n\right)}\in C^2\left([0,T]\right)$.

$$\left|{}_{0+}{}^{RL}D_x^{-{\zeta }_2}{\lambda }^{\left(1\right)}\left(x\right)-{}_{0+}{}^{RL}D_x^{-{\zeta }_2}{\lambda }^{\left(2\right)}\left(x\right)\right|\le L_2\left|{\lambda }^{\left(1\right)}\left(x\right)-{\lambda }^{\left(2\right)}\left(x\right)\right|,\forall x\in [0,T].$$

We let $\mathcal{X}=C^2\left(\mathcal{I}\right)$, where $\mathcal{I}=[0,T]$. Then $\mathcal{X}$ is a Banach space with the norm

$${\parallel \lambda \parallel }_{\infty }=\underset{x\in \mathcal{I}}{sup}\left|\lambda \left(x\right)\right|.$$

Remark 3.

Since the Caputo derivative of order ${\zeta }_1\in (0,2)$ appears in the problem formulation, we assume that each iterative function ${\lambda }^{\left(n\right)}\left(x\right)\in C^2\left([0,T]\right)$. This ensures that the Caputo derivative is well-defined and allows us to work with second-order derivatives in the convergence analysis.

Theorem 3.

Under hypothesis $\left[A1\right]$, if condition $\left[{\displaystyle \frac{(L_1+L_2)T^{{\zeta }_1}}{\Gamma ({\zeta }_1+1)}}\right]<1$ holds, then the proposed problem (1) has a unique solution.

Proof. 

We define the fixed-point operator $\Sigma :\mathcal{X}\to \mathcal{X}$ by using Equation (1) as

$$\Sigma \lambda \left(x\right)={}_{0+}{}^{RL}I_x^{{\zeta }_1}\left[\mathcal{H}\left(x\right)-{\lambda }^{″}\left(x\right)-{}_{0+}{}^{RL}D_x^{-{\zeta }_2}\lambda \left(x\right)\right].$$

(40)

Then, for any ${\lambda }^{\left(1\right)},{\lambda }^{\left(2\right)}\in X$, by applying Equation (40), we obtain

$$\begin{array}{cc}\hfill |\Sigma {\lambda }^{\left(1\right)}-\Sigma {\lambda }^{\left(2\right)}|& =|\Sigma {\lambda }^{\left(1\right)}\left(x\right)-\Sigma {\lambda }^{\left(2\right)}\left(x\right)|\hfill \\ \hfill & \le \left|{}_{0+}{}^{RL}I_x^{{\zeta }_1}\right[\mathcal{H}\left(x\right)-{\left({\lambda }^{\left(1\right)}\right)}^{″}\left(x\right)-{}_{0+}{}^{RL}D_x^{-{\zeta }_2}{\lambda }^{\left(1\right)}\left(x\right)]\hfill \\ & -{}_{0+}{}^{RL}I_x^{{\zeta }_1}[\mathcal{H}\left(x\right)-{\left({\lambda }^{\left(2\right)}\right)}^{″}\left(x\right)-{}_{0+}{}^{RL}D_x^{-{\zeta }_2}{\lambda }^{\left(2\right)}\left(x\right)]|\hfill \\ & \le \left|{}_{0+}{}^{RL}I_x^{{\zeta }_1}\left({\left({\lambda }^{\left(1\right)}\right)}^{″}\left(x\right)-{\left({\lambda }^{\left(2\right)}\right)}^{″}\left(x\right)\right)\right|\hfill \\ & +\left|{}_{0+}{}^{RL}I_x^{{\zeta }_1}\left({}_{0+}{}^{RL}D_x^{-{\zeta }_2}{\lambda }^{\left(1\right)}\left(x\right)-{}_{0+}{}^{RL}D_x^{-{\zeta }_2}{\lambda }^{\left(2\right)}\left(x\right)\right)\right|\hfill \\ \hfill & \le L_1{}_{0+}{}^{RL}I_x^{{\zeta }_1}|{\lambda }^{\left(1\right)}\left(x\right)-{\lambda }^{\left(2\right)}\left(x\right)|+L_2{}_{0+}{}^{RL}I_x^{{\zeta }_1}|{\lambda }^{\left(1\right)}\left(x\right)-{\lambda }^{\left(2\right)}\left(x\right)|\hfill \\ \hfill & \le (L_1+L_2){}_{0+}{}^{RL}I_x^{{\zeta }_1}|{\lambda }^{\left(1\right)}\left(x\right)-{\lambda }^{\left(2\right)}\left(x\right)|\hfill \\ \hfill & \le {\displaystyle \frac{(L_1+L_2)T^{{\zeta }_1}}{\Gamma ({\zeta }_1+1)}}sup|{\lambda }^{\left(1\right)}-{\lambda }^{\left(2\right)}|\hfill \\ \hfill & \le {\displaystyle \frac{(L_1+L_2)T^{{\zeta }_1}}{\Gamma ({\zeta }_1+1)}}{\parallel {\lambda }^{\left(1\right)}-{\lambda }^{\left(2\right)}\parallel }_{\infty }.\hfill \end{array}$$

(41)

Finally,

$$\parallel \Sigma {\lambda }^{\left(1\right)}-\Sigma {\lambda }^{\left(2\right)}{\parallel }_{\infty }\le {\displaystyle \frac{(L_1+L_2)T^{{\zeta }_1}}{\Gamma ({\zeta }_1+1)}}\parallel {\lambda }^{\left(1\right)}-{\lambda }^{\left(2\right)}{\parallel }_{\infty },$$

(42)

where $\left[{\displaystyle \frac{(L_1+L_2)T^{{\zeta }_1}}{\Gamma ({\zeta }_1+1)}}\right]<1.$ Hence, $\Sigma$ is a contraction [70,71]; therefore, it has a unique fixed point. Consequently, the corresponding problem (1) has a unique solution in $C^2\left(\mathcal{I}\right)$. □

Lemma 4.

Let $L=max\{L_1,L_2\}$, where $L_1$ and $L_2$ are Lipschitz constants associated with the classical derivative and fractional integral terms, respectively. Assume $\lambda \in C^2\left([0,T]\right)$. Then the associated operator Σ is a contraction on $C^2\left([0,T]\right)$ if the time domain T satisfies

$$\left[{\displaystyle \frac{2L{T}^{{\zeta }_1}}{\Gamma ({\zeta }_1+1)}}\right]<1.$$

Under Assumption [A1] and for any ${\lambda }^{\left(1\right)},{\lambda }^{\left(2\right)}\in C^2\left([0,T]\right)$, we obtain

$$\parallel \Sigma {\lambda }^{\left(1\right)}-\Sigma {\lambda }^{\left(2\right)}{\parallel }_{\infty }\le {\displaystyle \frac{2L{T}^{{\zeta }_1}}{\Gamma ({\zeta }_1+1)}}\parallel {\lambda }^{\left(1\right)}-{\lambda }^{\left(2\right)}{\parallel }_{\infty }.$$

(43)

Clearly, inequality (43) is obtained from (41). It holds that $\Sigma$ is a contraction mapping in $C^2\left([0,T]\right)$ if the time domain T satisfies

$$\left[{\displaystyle \frac{2L{T}^{{\zeta }_1}}{\Gamma ({\zeta }_1+1)}}\right]<1.$$

(44)

Hence, by Banach’s fixed point theorem [59], the operator (40) admits a unique fixed point in $C^2\left([0,T]\right)$, which corresponds to the unique solution of the problem (1).

Error Propagation

This subsection utilizes the previously stated Theorem 3 to analyze the propagation of errors of the proposed iterative methods: ADM, Shehu-ADM, and Sumudu-ADM.

Theorem 4.

Let $\lambda \left(x\right)$ be the exact solution of a linear FIDEs, and let ${\lambda }^{\left(k\right)}\left(x\right)$ denote the kth approximation obtained using ADM, Sumudu-ADM, or Shehu-ADM. We define the error in iteration k as

$$e_k={\parallel {\lambda }^{\left(k\right)}-\lambda \parallel }_{\infty },$$

where ${\lambda }^{\left(k\right)}$ is the kth approximation and λ is the exact solution.

Assume that the iterative process satisfies the contraction condition

$$e_{k+1}\le \kappa e_k,\kappa \in (0,1).$$

Then the error decays geometrically:

$$e_k\le {\kappa }^k{e}_0,k\in \mathbb{N}.$$

Proof. 

Let us define the error at iteration k by

$$e_k:={\parallel {\lambda }^{\left(k\right)}-\lambda \parallel }_{\infty }.$$

The recursive terms ${\lambda }_k$ in the decomposition are generated by a linear bounded operator $\Sigma$, i.e.,

$${\lambda }^{(k+1)}=\Sigma {\lambda }^{\left(k\right)}.$$

We know that from Theorem 3, the operator $\Sigma$ is a contraction mapping in the Banach space $(C^2\left(\mathcal{I}\right),\parallel ·{\parallel }_{\infty })$; we obtain

$$e_{k+1}=\parallel {\lambda }^{(k+1)}{-\lambda \parallel }_{\infty }=\parallel \Sigma {\lambda }^{\left(k\right)}{-\Sigma \lambda \parallel }_{\infty }\le |\Sigma |\parallel {\lambda }^{\left(k\right)}{-\lambda \parallel }_{\infty }=\kappa e_k,$$

where $\kappa =|\Sigma |<1$. Thus, the ADM sequence converges geometrically:

$$e_k\le {\kappa }^k{e}_0.$$

The contraction constant $\kappa$ depends on the properties of the transformation used in the method. For Sumudu-ADM and Shehu-ADM, $\kappa$ is influenced by the scaling properties of the transform. Under these conditions, the convergence of the ADM is guaranteed with geometric decay. □

Theorem 5.

Let $\eta \subset X$ be a closed, convex and bounded subset of the real Banach space, X, and let ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$ be operators on η satisfying the following conditions [72]:

(i) 

${\mathit{Q}}_1\left({\eta }^{\prime }\right)+{\mathit{Q}}_2\left({\eta }^{″}\right)\in \eta ,\forall {\eta }^{\prime },{\eta }^{″}\in \eta ,$

(ii) 

${\mathit{Q}}_1$ is a strict contraction on η, that is, there exists a $r\in [0,1)$ such that $|{\mathit{Q}}_1\left(x_1\right)-{\mathit{Q}}_1\left(x_2\right)|\le r|x_1-x_2|,\forall x_1,x_2\in \eta ,$

(iii) 

${\mathit{Q}}_2$ is continuous on η and ${\mathit{Q}}_2$ is a relatively compact subset of $X.$

Then there exists at least one solution ${\eta }^{\prime }\in \eta$ such that ${\mathit{Q}}_1\left({\eta }^{\prime }\right)+{\mathit{Q}}_2\left({\eta }^{\prime }\right)={\eta }^{\prime }.$

For further results, let the given hypothesis hold:

Assumption 2

($\left[A2\right]$). There exists a constant $A_{\mathcal{H}}>0$ such that

$$\left|\mathcal{H}\left(x\right)\right|\le A_{\mathcal{H}}\mathit{for}\mathit{all}x\in [0,T].$$

Assumption 3

($\left[A3\right]$). There exists a constant $\mathcal{B}>0$ such that

$$|{\lambda }^{″}\left(x\right)|\le \mathcal{B}\mathit{for}\mathit{all}x\in [0,T].$$

Assumption 4

($\left[A4\right]$). There exists a constant $\mathcal{C}>0$ such that

$$\left|{}_{0+}{}^{RL}D_x^{-{\zeta }_2}\lambda \left(x\right)\right|\le \mathcal{C}\mathit{for}\mathit{all}x\in [0,T].$$

Assumption 5

($\left[A5\right]$). There exists a constant ${\mathcal{A}}_1>0$ such that

$$|{\lambda }^{″}\left(x_1\right)-{\lambda }^{″}\left(x_2\right)|\le {\mathcal{A}}_1|x_1-x_2|\mathit{for}\mathit{all}{x}_1,x_2\in [0,T].$$

Assumption 6

($\left[A6\right]$). There exists a constant ${\mathcal{B}}_1>0$ such that

$$\left|{}_{0+}{}^{RL}D_{x_1}^{-{\zeta }_2}\lambda \left(x_1\right)-{}_{0+}{}^{RL}D_{x_2}^{-{\zeta }_2}\lambda \left(x_2\right)\right|\le {\mathcal{B}}_1|x_1-x_2|\mathit{for}\mathit{all}{x}_1,x_2\in [0,T].$$

Theorem 6.

Under Assumptions $\left[A2\right]$–$\left[A6\right]$, if

$$\left[{\displaystyle \frac{(L_1+L_2)T^{{\zeta }_1}}{\Gamma ({\zeta }_1+1)}}\right]<1,$$

then the problem considered in (1) has at least one solution in $C^2\left(\mathcal{I}\right)$.

Proof. 

Let us define two operators from Equation (1) as

$${\Sigma }_1\lambda \left(x\right)={\eta }_1\left(x\right)+x{\eta }_2\left(x\right),$$

(45)

and

$${\Sigma }_2\lambda \left(x\right)={}_{0+}{}^{RL}I_x^{{\zeta }_1}\left[\mathcal{H}\left(x\right)-{\lambda }^{″}\left(x\right)-{}_{0+}{}^{RL}D_x^{-{\zeta }_2}\lambda \left(x\right)\right].$$

(46)

We let $\mathcal{E}=\{\lambda \in \mathcal{X}:\parallel \lambda {\parallel }_{\infty }\le r\}$. Since $\mathcal{H}$ is continuous, it follows that ${\Sigma }_1$ is also continuous. Moreover, for any ${\lambda }^{\left(1\right)},{\lambda }^{\left(2\right)}\in \mathcal{E}$, we have from (45)

$$\parallel {\Sigma }_1{\lambda }^{\left(1\right)}-{\Sigma }_1{\lambda }^{\left(2\right)}{\parallel }_{\infty }=0.$$

(47)

Therefore, ${\Sigma }_1$ is a constant operator and hence trivially a contraction.

Next, we demonstrate that ${\Sigma }_2$ is continuous and compact. For each $\lambda \in \mathcal{E}$, from (46) we get

$$\begin{array}{cc}\hfill \parallel {\Sigma }_2\lambda {\parallel }_{\infty }& =\underset{x\in [0,T]}{sup}\left|{\Sigma }_2\lambda \left(x\right)\right|\hfill \\ \hfill & \le {\displaystyle \frac{A_{\mathcal{H}}+\mathcal{B}+\mathcal{C}}{\Gamma ({\zeta }_1+1)}}=\mathcal{P}.\hfill \end{array}$$

(48)

This shows that ${\Sigma }_2$ is bounded.

Now, for $x_1,x_2\in [0,T]$ with $x_1<x_2$, we estimate

$$\begin{array}{cc}\hfill & |{\Sigma }_2\lambda \left(x_1\right)-{\Sigma }_2\lambda \left(x_2\right)|\hfill \\ \hfill & =|{}_{0+}{}^{RL}I_{x_1}^{{\zeta }_1}\left[\mathcal{H}\left(x_1\right)-{\lambda }^{″}\left(x_1\right)-{}_{0+}{}^{RL}D_{x_1}^{-{\zeta }_2}\lambda \left(x_1\right)\right]\hfill \\ \hfill & -{}_{0+}{}^{RL}I_{x_2}^{{\zeta }_1}\left[\mathcal{H}\left(x_2\right)-{\lambda }^{″}\left(x_2\right)-{}_{0+}{}^{RL}D_{x_2}^{-{\zeta }_2}\lambda \left(x_2\right)\right]|\hfill \\ \hfill & \le {\displaystyle \frac{T^{{\zeta }_1}({\mathcal{A}}_1+{\mathcal{B}}_1)}{\Gamma ({\zeta }_1+1)}}\underset{x_1,x_2\in \mathcal{I}}{sup}|x_1-x_2|.\hfill \end{array}$$

(49)

Thus,

$$\parallel {\Sigma }_2\lambda \left(x_1\right)-{\Sigma }_2\lambda \left(x_2\right){\parallel }_{\infty }\le {\displaystyle \frac{T^{{\zeta }_1}({\mathcal{A}}_1+{\mathcal{B}}_1)}{\Gamma ({\zeta }_1+1)}}{\parallel x_1-x_2\parallel }_{\infty },$$

(50)

which implies that ${\Sigma }_2\lambda$ is equicontinuous. Since ${\Sigma }_2$ maps bounded sets into relatively compact sets of continuous functions, it follows from the Arzelà–Ascoli theorem that ${\Sigma }_2$ is a compact operator.

Therefore, the operator $\Sigma ={\Sigma }_1+{\Sigma }_2$ maps the closed convex bounded subset $\mathcal{E}\subset \mathcal{X}$ to itself. Since ${\Sigma }_1$ is a contraction and ${\Sigma }_2$ is compact and continuous, it follows from the Krasnoselskii fixed point theorem that $\Sigma$ has at least one fixed point.

Hence, the problem (1) has at least one solution in $C^2\left(\mathcal{I}\right)$. □

7. Stability

Here, we establish the U–H stability of Equation (1). Before proceeding with the stability analysis, we define U–H stability for the proposed model (1).

The U–H stability analysis ensures that the problem’s solution depends continuously on the initial data and the source term. This provides theoretical support for the reliability of both analytical and numerical solutions under small perturbations, which is especially important in modeling real-world systems.

Definition 7.

The problem Equation (1) is said to be U-H stable if, for every function $\mu \left(x\right)$ satisfying

$$|{\mu }^{″}\left(x\right)+{}_{0+}{}^{C}D_x^{{\zeta }_1}\mu \left(x\right)+{}_{0+}{}^{RL}D_x^{-{\zeta }_2}\mu \left(x\right)-\mathcal{H}\left(x\right)|\le ϵ,$$

(51)

there exists a solution $\lambda \left(x\right)$ of Equation (1) such that

$$|\mu \left(x\right)-\lambda \left(x\right)|\le h_fϵ,\mathit{where}{h}_f\in \mathbb{R}.$$

(52)

Theorem 7.

Suppose that all assumptions required for the existence of a solution to Equation (1) hold and that

$$\left[{\displaystyle \frac{(L_1+L_2)T^{{\zeta }_1}}{\Gamma ({\zeta }_1+1)}}\right]<1.$$

Then the problem (1) is U-H stable in $C^2\left(\mathcal{I}\right)$.

Proof. 

We let $\mu \left(x\right)$ satisfy the inequality Equation (51). Then,

$${\mu }^{″}\left(x\right)+{}_{0+}{}^{C}D_x^{{\zeta }_1}\mu \left(x\right)+{}_{0+}{}^{RL}D_x^{-{\zeta }_2}\mu \left(x\right)-\mathcal{H}\left(x\right)=\Phi \left(x\right),$$

(53)

where $|\Phi (x\left)\right|\le ϵ$.

Applying the operator ${}_{0+}{}^{RL}I_x^{{\zeta }_1}$ to both sides of Equation (53), we obtain

$$\begin{array}{cc}\hfill |\mu \left(x\right)-{}_{0+}{}^{RL}I_x^{{\zeta }_1}\left[\mathcal{H}\left(x\right)-{\mu }^{″}\left(x\right)-{}_{0+}{}^{RL}D_x^{-{\zeta }_2}\mu \left(x\right)\right]|& =\left|{}_{0+}{}^{RL}I_x^{{\zeta }_1}(\Phi \left(x\right))\right|\hfill \\ \hfill & \le {\displaystyle \frac{T^{{\zeta }_1}}{\Gamma ({\zeta }_1+1)}}·|\Phi \left(x\right)|\hfill \\ \hfill & \le ϵ·{\displaystyle \frac{T^{{\zeta }_1}}{\Gamma ({\zeta }_1+1)}}.\hfill \end{array}$$

(54)

Now, we let $\lambda \left(x\right)$ be the unique solution of Equation (1). Then

$$\begin{array}{cc}\hfill \left|\mu \right(x)-\lambda (x\left)\right|& =|\mu \left(x\right)-{}_{0+}{}^{RL}I_x^{{\zeta }_1}\left[\mathcal{H}\left(x\right)-{\mu }^{″}\left(x\right)-{}_{0+}{}^{RL}D_x^{-{\zeta }_2}\mu \left(x\right)\right]\hfill \\ \hfill & +{}_{0+}{}^{RL}I_x^{{\zeta }_1}\left[\mathcal{H}\left(x\right)-{\mu }^{″}\left(x\right)-{}_{0+}{}^{RL}D_x^{-{\zeta }_2}\mu \left(x\right)\right]\hfill \\ \hfill & -{}_{0+}{}^{RL}I_x^{{\zeta }_1}\left[\mathcal{H}\left(x\right)-{\lambda }^{″}\left(x\right)-{}_{0+}{}^{RL}D_x^{-{\zeta }_2}\lambda \left(x\right)\right]|\hfill \\ \hfill & \le |\mu \left(x\right)-{}_{0+}{}^{RL}I_x^{{\zeta }_1}\left[\mathcal{H}\left(x\right)-{\mu }^{″}\left(x\right)-{}_{0+}{}^{RL}D_x^{-{\zeta }_2}\mu \left(x\right)\right]|\hfill \\ \hfill & +{}_{0+}{}^{RL}I_x^{{\zeta }_1}|({\mu }^{″}\left(x\right)-{\lambda }^{″}\left(x\right))+\left({}_{0+}{}^{RL}D_x^{-{\zeta }_2}\mu \left(x\right)-{}_{0+}{}^{RL}D_x^{-{\zeta }_2}\lambda \left(x\right)\right)|\hfill \\ \hfill & \le ϵ·{\displaystyle \frac{T^{{\zeta }_1}}{\Gamma ({\zeta }_1+1)}}+{\displaystyle \frac{(L_1+L_2)T^{{\zeta }_1}}{\Gamma ({\zeta }_1+1)}}·|\mu \left(x\right)-\lambda \left(x\right)|.\hfill \end{array}$$

(55)

Rearranging the inequality in Equation (55), we get

$$|\mu \left(x\right)-\lambda \left(x\right)|\le {\displaystyle \frac{ϵ·\frac{T^{{\zeta }_1}}{\Gamma ({\zeta }_1+1)}}{1-\frac{(L_1+L_2)T^{{\zeta }_1}}{\Gamma ({\zeta }_1+1)}}}.$$

Hence, $|\mu \left(x\right)-\lambda \left(x\right)|\le h_fϵ$, where

$$h_f={\displaystyle \frac{\frac{T^{{\zeta }_1}}{\Gamma ({\zeta }_1+1)}}{1-\frac{(L_1+L_2)T^{{\zeta }_1}}{\Gamma ({\zeta }_1+1)}}}.$$

This proves the U–H stability of the problem Equation (1) in $C^2\left(\mathcal{I}\right)$. □

Mittag–Leffler Bound for Establishing Ulam–Hyers Stability

We recall the following asymptotic property of the Mittag–Leffler function, which is used in subsequent stability analysis. For our considered problem Equation (1) and ${ϵ}^{″}$ associated with the following inequality

$$|{\lambda }^{″}\left(x\right)+{}_{0+}{}^{C}D_x^{{\zeta }_1}\lambda \left(x\right)+{}_{0+}{}^{RL}D_x^{-{\zeta }_2}\lambda \left(x\right)-\mathcal{H}\left(x\right)|\le {ϵ}^{″},x\in [0,T],$$

(56)

$E_{\alpha }$ is the Mittag–Leffler function [73] defined by

$$E_{{\zeta }_1}\left(\lambda \right):=\sum _{q=0}^{\infty }{\displaystyle \frac{{\lambda }^q}{\Gamma (q{\zeta }_1+1)}},\lambda \in \mathbb{C},\mathbb{R}>0.$$

Definition 8.

The problem in Equation (1) is Ulam–Hyers–Mittag–Leffler-stable with respect to $E_{{\zeta }_1}\left(x^{{\zeta }_1}\right)$ if there exists ${\mathcal{C}}_{E_{{\zeta }_1}}>0$ such that for each ${ϵ}^{″}>0$ and for each solution $\lambda \in C^2([0,T],\mathbb{R})$ to Equation (56), there exists a solution ${\lambda }^{\left(1\right)}\in C^2([0,T],\mathbb{R})$ to Equation (1) with

$$|\lambda \left(x\right)-{\lambda }^{\left(1\right)}\left(x\right)|\le {\mathcal{C}}_{E_{{\zeta }_1}}{E}_{{\zeta }_1}\left(x^{{\zeta }_1}\right),x\in [0,T].$$

Remark 4.

A function $\lambda \in C^2(I,\mathbb{R})$ is a solution of the inequality Equation (56) if and only if there exists a function $\mathcal{O}\in C^2(I,\mathbb{R})$ (which depends on λ) such that

(i) 

$\left|\mathcal{O}\left(x\right)\right|\le {ϵ}^{″}{E}_{{\zeta }_1}\left(x^{{\zeta }_1}\right)$ for all $x\in [0,T]$,

(ii) 

${\lambda }^{″}\left(x\right)+{}_{0+}{}^{C}D_x^{{\zeta }_1}\lambda \left(x\right)+{}_{0+}{}^{RL}D_x^{-{\zeta }_2}\lambda \left(x\right)=\mathcal{H}\left(x\right)+\mathcal{O}\left(x\right)$ for all $x\in [0,T]$.

By Remark 4, we have the following Ulam–Hyers–Mittag–Leffler stable estimate.

Remark 5.

Let $\lambda \in C^2(I,\mathbb{R})$ be a solution of the inequality Equation (56). Then λ is a solution of the following integral inequality:

$$\begin{array}{cc}\hfill \left|\lambda \right(x)& -\lambda \left(0\right)-x{\lambda }^{\prime }\left(0\right)-{}_{0+}{}^{RL}I_x^{{\zeta }_1}\left(\mathcal{H}\left(x\right)-{\lambda }^{″}\left(x\right)-{}_{0+}{}^{RL}D_x^{-{\zeta }_2}\lambda \left(x\right)\right)|\hfill \\ \hfill & \le {\displaystyle \frac{{ϵ}^{″}}{\Gamma \left({\zeta }_1\right)}}{\int }_0^t{(x-s)}^{{\zeta }_1-1}{E}_{{\zeta }_1}\left(s^{{\zeta }_1}\right)ds\hfill \\ \hfill & \le {\displaystyle \frac{{ϵ}^{″}}{\Gamma \left({\zeta }_1\right)}}{\int }_0^t{(x-s)}^{{\zeta }_1-1}\sum _{q=0}^{\infty }{\displaystyle \frac{s^{q{\zeta }_1}}{\Gamma (q{\zeta }_1+1)}}ds\hfill \\ \hfill & ={\displaystyle \frac{{ϵ}^{″}}{\Gamma \left({\zeta }_1\right)}}\sum _{q=0}^{\infty }{\displaystyle \frac{1}{\Gamma (q{\zeta }_1+1)}}{\int }_0^t{(x-s)}^{{\zeta }_1-1}{s}^{q{\zeta }_1}ds\hfill \\ \hfill & ={\displaystyle \frac{{ϵ}^{″}}{\Gamma \left({\zeta }_1\right)}}\sum _{q=0}^{\infty }{\displaystyle \frac{x^{(q+1){\zeta }_1}}{\Gamma (q{\zeta }_1+1)}}{\int }_0^t{\displaystyle \frac{\Gamma \left({\zeta }_1\right)\Gamma (q{\zeta }_1+1)}{\Gamma ((q+1){\zeta }_1+1)}}\hfill \\ \hfill & ={ϵ}^{″}\sum _{q=0}^{\infty }{\displaystyle \frac{x^{(q+1){\zeta }_1}}{\Gamma ((q+1){\zeta }_1+1)}}\hfill \\ \hfill & \le {ϵ}^{″}\sum _{n=0}^{\infty }{\displaystyle \frac{x^{n{\zeta }_1}}{\Gamma (n{\zeta }_1+1)}}\hfill \\ \hfill & \le {ϵ}^{″}{E}_{{\zeta }_1}\left(x^{{\zeta }_1}\right).\hfill \end{array}$$

(57)

Inequality Equation (56) provides an upper bound for the Mittag–Leffler function, which is essential to establish Definition 8, that is, the stability of Mittag–Leffler–Ulam–Hyers. This result can be proven as in Theorem 7 by applying the asymptotic estimate in conjunction with fixed-point arguments or integral inequalities.

Remark 6.

Using the same parameters as previously, consider Equation (1) with the non-zero initial condition $\lambda \left(0\right)={\lambda }_0\ne 0$. According to the previously stated U–H stability framework, numerical studies verify that the solution maintains stability and shows a bounded deviation under perturbations [74].

8. Illustrative Examples and Discussion

Modeling systems where the current state depends on the past and present behavior is a key application of FIDEs in the form of Equation (1). Such equations are extensively used in various fields such as image processing, population dynamics, control systems, heat transfer, epidemiology, and viscoelasticity [75,76,77,78]. Their ability to capture memory and hereditary effects makes them essential in many real-world applications. The examples presented in this study demonstrate the effectiveness of the proposed methods in solving FIDE and highlight their practical relevance. This section provides solutions using the TSADM method and numerical techniques, as well as relevant remarks and numerical comparisons. In the upcoming examples, the proposed FIDEs, Equation (1), present both categories, incorporating local and non-local operators. These play a significant role in physical interpretations in complex media [79]. The classical second-order derivative ${\lambda }^{″}\left(x\right)$ represents standard elastic or diffusive behavior [80]. The term ${}_{0+}{}^{C}D_x^{{\zeta }_1}\lambda \left(x\right)$, involving the Caputo fractional derivative [81] of order $0<{\zeta }_1<2$, models memory-dependent processes and is commonly used in viscoelasticity to describe stress–strain relations in materials with hereditary properties. The presence of ${}_{0+}{}^{RL}D_x^{-{\zeta }_2}\lambda \left(x\right)$, the RL fractional integral [82], introduces a long-range temporal effect, characteristic of subdiffusive behavior or creep phenomena in viscoelastic systems [83]. Due to non-locality, these fractional operators capture short- and long-memory effects [84], making the model suitable for describing anomalous diffusion. In contrast, classical models fail to describe physical complexities more accurately [85].

Example 1.

Consider the fractional integro-differential equation (FIDE) with initial conditions as described in [1,2] given by

$$\begin{array}{cc}\hfill {\lambda }^{″}\left(x\right)+{}_{0+}{}^{C}D_x^{{\zeta }_1}\lambda \left(x\right)+{}_{0+}{}^{RL}D_x^{-{\zeta }_2}\lambda \left(x\right)& =\mathcal{H}\left(x\right),x\in [0,1],0<{\zeta }_1<2,\hfill \end{array}$$

(58)

with initial conditions

$$\begin{array}{cc}\hfill \lambda \left(0\right)& =0,\hfill \\ \hfill {\lambda }^{\prime }\left(0\right)& =0,\hfill \end{array}$$

where

$$\mathcal{H}\left(x\right)=5x^4-6x^5+{\displaystyle \frac{720}{6\Gamma (7-{\zeta }_1)}}{x}^{6-{\zeta }_1}-{\displaystyle \frac{5040}{7\Gamma (8-{\zeta }_1)}}{x}^{7-{\zeta }_1}+{\displaystyle \frac{720}{6\Gamma (7+{\zeta }_2)}}{x}^{6+{\zeta }_2}-{\displaystyle \frac{5040}{7\Gamma (8+{\zeta }_2)}}{x}^{7+{\zeta }_2}.$$

(59)

The exact solution is given by

$$\lambda \left(x\right)={\displaystyle \frac{1}{6}}{x}^6-{\displaystyle \frac{1}{7}}{x}^7,$$

(60)

using the technique of the TSADM.

Applying the inverse operator ${}_{0+}{}^{RL}I_x^{{\zeta }_1}$ to Equation (58), we obtain

$$\lambda \left(x\right)=\lambda \left(0\right)+x{\lambda }^{\prime }\left(0\right)+{}_{0+}{}^{RL}I_x^{{\zeta }_1}\left(\mathcal{H}\left(x\right)-{\lambda }^{″}\left(x\right)-{}_{0+}{}^{RL}D_x^{-{\zeta }_2}\lambda \left(x\right)\right).$$

(61)

The recursion formula for iterations from Equation (61) works as follows:

$${\lambda }^{\left(0\right)}\left(x\right)={}_{0+}{}^{RL}I_x^{{\zeta }_1}\left(\mathcal{H}\left(x\right)\right),$$

(62)

and

$${\lambda }^{(n+1)}\left(x\right)=-{}_{0+}{}^{RL}I_x^{\zeta }\left({\displaystyle \frac{d^2}{d{x}^2}}{\lambda }^{\left(n\right)}\left(x\right)+{}_{0+}{}^{RL}D_x^{-{\zeta }_2}{\lambda }^{\left(n\right)}\left(x\right)\right),$$

(63)

where $n=1,2,\dots$.

The initial version of the TSADM may be divided into three components as shown in [59],

$${\lambda }^{\left(0\right)}={\mathcal{L}}_0+{\mathcal{L}}_1+{\mathcal{L}}_2,$$

(64)

where

$$\begin{array}{cc}\hfill {\mathcal{L}}_0& ={\displaystyle \frac{5\Gamma \left(5\right)}{\Gamma (5+{\zeta }_1)}}{x}^{4+{\zeta }_1}-{\displaystyle \frac{6\Gamma \left(6\right)}{\Gamma (6+{\zeta }_1)}}{x}^{5+{\zeta }_1},\hfill \\ \hfill {\mathcal{L}}_1& ={\displaystyle \frac{1}{6}}{x}^6-{\displaystyle \frac{1}{7}}{x}^7,\hfill \\ \hfill {\mathcal{L}}_2& ={\displaystyle \frac{720}{6\Gamma (7+{\zeta }_1+{\zeta }_2)}}{x}^{6+{\zeta }_2}-{\displaystyle \frac{5040}{\Gamma (78+{\zeta }_1+{\zeta }_2)}}{x}^{7+{\zeta }_2}-{\displaystyle \frac{5040}{\Gamma (7+{\zeta }_1+{\zeta }_2)}}{x}^7.\hfill \end{array}$$

(65)

Now, let us take ${\lambda }^{\left(0\right)}={\mathcal{L}}_1={\displaystyle \frac{1}{6}}{x}^6-{\displaystyle \frac{1}{7}}{x}^7$ and check whether this assumption ${\lambda }^{\left(0\right)}$ satisfies Equation (58) along with the initial and boundary conditions given. If this choice ${\lambda }^{\left(0\right)}$ is verified, then the selected term is the solution to the problem.

It can be verified that ${\mathcal{L}}_0$ and ${\mathcal{L}}_2$ satisfy the associated conditions but do not satisfy Equation (58). Therefore, for Example 1, we can conclude that the proposed TSADM method is more efficient in providing the exact solution with only one iteration compared to other existing numerical methods [59,60]. Furthermore, we include the plots for varying values of ${\zeta }_1$ and ${\zeta }_2$ to offer a clearer visualization of the behavior of the solution under different conditions specific to the values of ${\zeta }_1$ and ${\zeta }_2$. The series of solutions has been derived in terms of the second order to facilitate comparative analysis for simplicity and fairness in the evaluation. Although additional terms could be computed using the same approach, we limit the expansion here to align with the existing literature and allow a consistent comparison. Our proposed methods, Sumudu-ADM and the Shehu-ADM, are compared with traditional numerical techniques found in previous works, such as the LADM and the difference method (DM), particularly for fractional orders ${\zeta }_1,{\zeta }_2\in (0,2]$, which do not help solve these types of FIDE Equation (58).

The detailed plots and tabulated results demonstrate the accuracy of the proposed methods for equal and unequal fractional orders. The 2D graphs illustrate the behavior of the approximate and exact solutions, while the corresponding absolute error graphs clearly show the negligible deviation between them, especially as ${\zeta }_1$ and ${\zeta }_2$ approach a value of two. In particular, when ${\zeta }_1={\zeta }_2=2$, the approximate solutions obtained from Shehu-ADM and Sumudu-ADM coincide with the exact solution, as observed in the line and surface plots.

In addition, a detailed analysis of the error norms using the norms $L_{\infty }$, $L_2$, and $L_1$ is presented in tabular form. The tables indicate that, because of its weighted framework, the Shehu-ADM approach frequently produces marginally more precise results than Sumudu-ADM and markedly surpasses other traditional methods for accuracy and computing efficiency.

The comparison results confirm that the proposed approaches, especially Shehu-ADM, provide a superior and efficient alternative for solving FIDE, particularly where high precision and minimal iterations are required. The findings emphasize the robustness of our approach in handling varying fractional orders and demonstrate clear improvements over traditional methods in terms of absolute error, computational effort, and adaptability.

Overall Observations for Example 1:

Example 1 was first solved analytically using the TSADM, through which the exact solution was obtained successfully. However, in cases where obtaining an exact solution is not feasible, particularly for homogeneous FIDE with associated homogeneous conditions, numerical methods become essential.

To address such situations, Example 1 was also solved using two numerical approaches that integrate transformation techniques with ADM-based approximations: Sumudu-ADM and Shehu-ADM. The behavior of these methods was analyzed under two different scenarios: when the fractional orders are equal (${\zeta }_1={\zeta }_2$) and when they are different (${\zeta }_1\ne {\zeta }_2$). As shown in Figure 1 and Figure 2, both methods performed effectively in both cases, producing results very close to the exact solution.

To further evaluate the accuracy of the methods, the absolute error plots are presented in Figure 3 and Figure 4. These plots reveal that Sumudu-ADM generally outperforms Shehu-ADM in terms of accuracy for this example, although both methods exhibit nearly equivalent overall performance.

Furthermore, bar graphs of different error norms are included in Figure 5 and Figure 6 for better clarity and comparison. The plotted graphs show that the tabulated values in

Table 1. Comparison of error norms for equal fractional orders (${\zeta }_1={\zeta }_2$) using Sumudu-ADM and Shehu-ADM methods involving two iterations.

$({\mathit{\zeta }}_1={\mathit{\zeta }}_2)$	${\mathit{L}}_{\mathit{\infty }}^{\mathbf{Sumudu}}$	${\mathit{L}}_2^{\mathbf{Sumudu}}$	${\mathit{L}}_1^{\mathbf{Sumudu}}$	${\mathit{L}}_{\mathit{\infty }}^{\mathbf{Shehu}}$	${\mathit{L}}_2^{\mathbf{Shehu}}$	${\mathit{L}}_1^{\mathbf{Shehu}}$
0.5	0.000581	0.001810	0.010145	0.000541	0.001698	0.009503
0.7	0.000767	0.002435	0.013888	0.000732	0.002341	0.013386
0.9	0.001017	0.003276	0.019148	0.000987	0.003203	0.018734
1.0	0.001220	0.003992	0.023371	0.001199	0.003942	0.023015
1.2	0.001757	0.006017	0.035532	0.001761	0.006027	0.035609
1.4	0.002405	0.008461	0.051930	0.002498	0.008778	0.053857
1.6	0.003163	0.011340	0.072578	0.003379	0.012107	0.077822
1.8	0.004030	0.014652	0.097478	0.004379	0.015913	0.106326
2.0	0.005004	0.018395	0.126631	0.005569	0.020568	0.141598

and

Table 2. Extended error norm comparison for unequal fractional orders $({\zeta }_1,{\zeta }_2)$ using Sumudu-ADM and Shehu-ADM methods involving two iterations.

$({\mathit{\zeta }}_1,{\mathit{\zeta }}_2)$	${\mathit{L}}_{\mathit{\infty }}^{\mathbf{Sumudu}}$	${\mathit{L}}_2^{\mathbf{Sumudu}}$	${\mathit{L}}_1^{\mathbf{Sumudu}}$	${\mathit{L}}_{\mathit{\infty }}^{\mathbf{Shehu}}$	${\mathit{L}}_2^{\mathbf{Shehu}}$	${\mathit{L}}_1^{\mathbf{Shehu}}$
(0.5, 0.9)	0.000539	0.000644	0.001129	0.000438	0.000523	0.000916
(0.7, 1.2)	0.000733	0.000887	0.001580	0.000709	0.000858	0.001528
(0.9, 1.4)	0.001019	0.001245	0.002254	0.001119	0.001369	0.002477
(1.0, 1.6)	0.001199	0.001475	0.002690	0.001437	0.001768	0.003224
(1.2, 1.8)	0.001665	0.002070	0.003836	0.002218	0.002758	0.005112
(1.4, 2.0)	0.002297	0.002892	0.005444	0.003367	0.004239	0.007982
(0.3, 0.7)	0.000367	0.000523	0.001194	0.000366	0.000520	0.001183
(0.6, 1.1)	0.000625	0.000794	0.001527	0.000619	0.000780	0.001490
(0.8, 1.5)	0.000919	0.001176	0.002155	0.000989	0.001278	0.002359
(1.0, 1.8)	0.001291	0.001667	0.003035	0.001515	0.001982	0.003675
(1.3, 2.0)	0.002069	0.002747	0.005008	0.002901	0.003884	0.007263
(0.4, 1.0)	0.000447	0.000625	0.001348	0.000443	0.000612	0.001313
(0.9, 1.6)	0.001101	0.001372	0.002465	0.001304	0.001668	0.003102
(1.1, 1.7)	0.001456	0.001897	0.003422	0.001835	0.002421	0.004575

reinforce the effectiveness and reliability of the proposed numerical schemes.

Example 2.

Consider the FIDE with initial conditions as described in [1,2] given by

$$\begin{array}{cc}\hfill {\lambda }^{″}\left(x\right)+{}_{0+}{}^{C}D_x^{{\zeta }_1}\lambda \left(x\right)+{}_{0+}{}^{RL}D_x^{-{\zeta }_2}\lambda \left(x\right)& =\mathcal{H}\left(x\right),x\in [0,1],0<{\zeta }_1<2,\hfill \end{array}$$

(66)

with initial conditions

$$\begin{array}{cc}\hfill \lambda \left(0\right)& =0,\hfill \\ \hfill {\lambda }^{\prime }\left(0\right)& =0,\hfill \end{array}$$

where

$$\mathcal{H}\left(x\right)=x^5-x^6+{\displaystyle \frac{\Gamma \left(8\right)x^{7-{\zeta }_1}}{42\Gamma (8+{\zeta }_1)}}-{\displaystyle \frac{\Gamma \left(9\right)x^{8-{\zeta }_1}}{56\Gamma (9+{\zeta }_1)}}+{\displaystyle \frac{\Gamma \left(8\right)x^{7+{\zeta }_2}}{42\Gamma (8+{\zeta }_2)}}-{\displaystyle \frac{\Gamma \left(9\right)x^{8+{\zeta }_2}}{56\Gamma (9+{\zeta }_2)}}.$$

(67)

The exact solution is given by

$$\lambda \left(x\right)={\displaystyle \frac{1}{42}}{x}^7-{\displaystyle \frac{1}{56}}{x}^8,$$

(68)

using the technique of the TSADM.

By applying the inverse operator ${}_{0+}{}^{RL}I_x^{{\zeta }_1}$ to Equation (66), we obtain

$$\lambda \left(x\right)=\lambda \left(0\right)+x{\lambda }^{\prime }\left(0\right)+{}_{0+}{}^{RL}I_x^{{\zeta }_1}\left(\mathcal{H}\left(x\right)-{\lambda }^{″}\left(x\right)-{}_{0+}{}^{RL}D_x^{-{\zeta }_2}\lambda \left(x\right)\right).$$

(69)

The recursion formula for iterations from Equation (61) works as follows:

$${\lambda }^{\left(0\right)}\left(x\right)={}_{0+}{}^{RL}I_x^{{\zeta }_1}\left(\mathcal{H}\left(x\right)\right),$$

(70)

and

$${\lambda }^{(n+1)}\left(x\right)=-{}_{0+}{}^{RL}I_x^{\zeta }\left({\left({\lambda }^{\left(n\right)}\right)}^{″}\left(x\right)+{}_{0+}{}^{RL}D_x^{-{\zeta }_2}{\lambda }^{\left(n\right)}\left(x\right)\right),$$

(71)

where $n=1,2,\dots .$

The initial version of the TSADM may be divided into three components as shown in [59],

$${\lambda }^{\left(0\right)}={\mathcal{L}}_0+{\mathcal{L}}_1+{\mathcal{L}}_2,$$

(72)

where

$$\begin{array}{cc}\hfill {\mathcal{L}}_0& ={\displaystyle \frac{\Gamma \left(6\right)}{\Gamma (6+{\zeta }_1)}}{x}^{6+{\zeta }_1}-{\displaystyle \frac{\Gamma \left(7\right)}{\Gamma (7+{\zeta }_1)}}{x}^{7+{\zeta }_1},\hfill \\ \hfill {\mathcal{L}}_1& ={\displaystyle \frac{1}{42}}{x}^7-{\displaystyle \frac{1}{56}}{x}^8,\hfill \\ \hfill {\mathcal{L}}_2& ={\displaystyle \frac{\Gamma \left(8\right)}{42\Gamma (8+{\zeta }_2+{\zeta }_1)}}-{\displaystyle \frac{\Gamma \left(9\right)}{56\Gamma (9+{\zeta }_2+{\zeta }_1)}}.\hfill \end{array}$$

(73)

Now, let us take ${\lambda }^{\left(0\right)}={\mathcal{L}}_1={\displaystyle \frac{1}{42}}{x}^7-{\displaystyle \frac{1}{56}}{x}^8$ and check whether this assumption of ${\lambda }^{\left(0\right)}$ satisfies the Equation (66) along with the given initial or boundary conditions. If this choice of ${\lambda }^{\left(0\right)}$ is verified, then the selected term is the solution to the problem. It can be verified that ${\mathcal{L}}_0$ and ${\mathcal{L}}_2$ satisfy the associated conditions but do not satisfy Equation (66). Therefore, for Example 2, we can conclude that the proposed TSADM method is more efficient in providing the exact solution with fewer iterations than other existing numerical methods [59,60]. When the analytical approach is not applicable, we resort to numerical methods for solving FIDE.

We present the graphical and tabulated results based on different values of fractional orders ${\zeta }_1$ and ${\zeta }_2$. The following figures and tables illustrate the behavior of the solution for the considered Example 2. In this example, we solve FIDE (Example 2) with equal fractional orders, that is, ${\zeta }_1={\zeta }_2$ and ${\zeta }_1\ne {\zeta }_2$ using the proposed methods. The exact solution is compared with its approximations obtained via the Sumudu-ADM and Shehu-ADM methods through 2D plots and absolute error curves. The error norms $L_{\infty }$, $L_2$, and $L_1$ are computed and presented. These norms further confirm the accuracy of numerical approximations, with the Shehu-ADM method demonstrating superior performance in this particular case (a reduction in iteration complexity of more than 30% compared to ADM).

Remark 7.

(i) 

The suggested approach is universal and can be used for any interval $[0,T]$, $T>0$, even if the numerical examples are given on the interval $[0,1]$. Because fractional-order operators have a memory effect that can affect accuracy and computational cost, the method might need finer time discretization for larger T.

(ii) 

The exact solution, such as $\lambda \in C^n[0,T]$, where n depends on the order of the fractional derivative and the numerical approach, is assumed to be sufficiently smooth for the convergence analysis. This regularity assumption guarantees the validity of the observed convergence rates and the computed error estimates. The equation studied in Equation (1) is a problem in the literature due to its relevance in modeling systems with memory effects and its analytical tractability, allowing for comparison with exact solutions.

(iii) 

The convergence behavior established in this Section 5 is supported by the numerical results presented in Section 8. In particular, the error plots and tables based on the $L_1$, $L_2$, and $L_{\infty }$ norms (see Table 1 and Table 2 and Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6 for Example 1, and

Table 3. Error norms for Example 2 using Sumudu-ADM and Shehu-ADM involving two iterations for extended unequal $({\zeta }_1,{\zeta }_2)$ pairs.

$({\mathit{\zeta }}_1,{\mathit{\zeta }}_2)$	${\mathit{L}}_{\mathit{\infty }}^{\mathbf{Sumudu}}$	${\mathit{L}}_2^{\mathbf{Sumudu}}$	${\mathit{L}}_1^{\mathbf{Sumudu}}$	${\mathit{L}}_{\mathit{\infty }}^{\mathbf{Shehu}}$	${\mathit{L}}_2^{\mathbf{Shehu}}$	${\mathit{L}}_1^{\mathbf{Shehu}}$
(0.5, 0.9)	0.003715	0.006381	0.015866	0.003212	0.005497	0.013609
(0.6, 1.0)	0.003090	0.003751	0.006604	0.002811	0.003403	0.005972
(0.7, 1.2)	0.002419	0.004162	0.010406	0.002361	0.004059	0.010142
(0.8, 1.3)	0.002002	0.002437	0.004319	0.002051	0.002497	0.004429
(0.9, 1.4)	0.001654	0.002849	0.007158	0.001774	0.003061	0.007710
(1.0, 1.6)	0.001295	0.001584	0.002834	0.001493	0.001834	0.003294
(1.1, 1.7)	0.001011	0.001230	0.002189	0.001214	0.001485	0.002673
(1.2, 1.8)	0.000876	0.001068	0.001931	0.001087	0.001337	0.002356
(1.3, 1.9)	0.000679	0.000826	0.001493	0.000888	0.001102	0.001940
(1.4, 2.0)	0.000585	0.000721	0.001311	0.000797	0.000991	0.001819
(0.5, 1.5)	0.003414	0.004062	0.007204	0.002976	0.003540	0.006337
(0.6, 1.8)	0.002525	0.003105	0.005511	0.002384	0.002946	0.005152

and

Table 4. Error norms for Example 2 using Sumudu-ADM and Shehu-ADM involving two iterations for equal $({\zeta }_1={\zeta }_2)$ values.

$({\mathit{\zeta }}_1,{\mathit{\zeta }}_2)$	${\mathit{L}}_{\mathit{\infty }}^{\mathbf{Sumudu}}$	${\mathit{L}}_2^{\mathbf{Sumudu}}$	${\mathit{L}}_1^{\mathbf{Sumudu}}$	${\mathit{L}}_{\mathit{\infty }}^{\mathbf{Shehu}}$	${\mathit{L}}_2^{\mathbf{Shehu}}$	${\mathit{L}}_1^{\mathbf{Shehu}}$
(0.5, 0.5)	0.005030	0.006108	0.010714	0.004191	0.005066	0.008833
(0.7, 0.7)	0.003492	0.004221	0.007385	0.003143	0.003786	0.006592
(0.9, 0.9)	0.002401	0.002892	0.005053	0.002321	0.002792	0.004869
(1.0, 1.0)	0.001984	0.002387	0.004169	0.001984	0.002387	0.004169
(1.2, 1.2)	0.001346	0.001616	0.002825	0.001436	0.001729	0.003036
(1.4, 1.4)	0.000912	0.001096	0.001973	0.001084	0.001304	0.002348
(1.6, 1.6)	0.000639	0.000767	0.001392	0.000841	0.001013	0.001825
(1.8, 1.8)	0.000460	0.000551	0.001000	0.000671	0.000806	0.001470
(2.0, 2.0)	0.000344	0.000411	0.000745	0.000552	0.000663	0.001203

and Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 for Example 2) demonstrate a decreasing trend in the error values, consistent with the theoretical convergence expectations. These results confirm the practical effectiveness and reliability of the proposed method.

Results and Discussion

The numerical results demonstrate that the Shehu-ADM approach outperforms standard ADM and Sumudu-ADM in accuracy and convergence speed, especially for problems involving weakly singular kernels. Error norms confirm the exponential decay of the residuals across all test cases, and the Shehu transform’s weighted structure contributes to greater numerical stability. These findings validate the theoretical convergence analysis and illustrate the practical effectiveness of the proposed methods in solving the FIDE Equation (1).

The approach presented in this work is formulated for one-dimensional problems but can be extended to higher-dimensional systems. In such systems, fractional operators (e.g., Caputo, Riemann–Liouville) can be generalized using multi-dimensional kernels. The Sumudu/Shehu transform-based ADM remains applicable as long as separability in the variables is maintained, and the decomposition process can be applied iteratively along each spatial dimension. Future research will explore these extensions in two- and three-dimensional domains under relevant boundary conditions (see [20,60,71,86,87,88,89] for related works).

Several studies have addressed the treatment of FIDEs in multi-dimensional spaces. For example, Azarnavid et al. [20] presented an approach claiming an efficient numerical method to solve FIDEs with weakly singular kernels in both two- and three-dimensional domains. They introduced an unconditionally stable scheme using temporal discretization and handled regular and irregular spatial domains. Furthermore, the authors developed alternating direction implicit (ADI) difference approaches for distributed-order integro-differential equations in 2D and 3D. These methods combine spatial discretization with the weighted and shifted Grünwald–Letnikov expansion, while a second-order convolution quadrature (CQ) is employed for the distributed-order time-fractional derivative. The convergence of the method is rigorously established through numerical experiments [86]. Similar contributions can be found in [87,88,89].

The approach proposed in this work compares favorably with other numerical methods, as demonstrated in the numerical examples.

As per our concern for the numerical solution for proposed FIDEs, the standard ADM as well as the LADM were found to be unsuitable for the considered FIDEs, primarily due to convergence issues and incompatibility with the structure of the problem, including singular kernels and multi-fractional operators. However, the proposed approaches, such as the TSADM, Sumudu-ADM, and Shehu-ADM methods, were successfully applied and produced accurate results, as demonstrated in the numerical examples. These transform-based variants overcome the limitations of ADM and LADM while offering improved numerical stability and convergence behavior. A direct comparison of the error norms $L_{\infty }$, $L_2$, and $L_1$ confirms the effectiveness of the proposed methods.

ADM and LADM were tested on the problem but failed to converge or appropriately handle the FIDEs in our case; therefore, they are not included in the numerical comparison.

It is also important to note that other numerical techniques, such as HPM and the reduced differential transform method (RDTM), are conceptually related to ADM and inherit similar limitations. These methods are generally unsuited for problems involving singular kernels or multi-fractional dynamics. In our systems, HPM and RDTM did not yield reliable results for the current model, mainly due to their lack of robustness in handling singular behavior and structural complexity.

Consequently, direct comparisons are limited to the proposed methods, which are specifically adapted to address the singular and fractional nature of the problem. The success of these transform-based approaches underscores the need for problem-specific modifications when dealing with singular kernels and complex fractional structures.

9. Comparative Analysis

This section presents a comparative study between the analytical approach proposed in the literature and theoretical methods. The primary objective is to demonstrate the efficiency, accuracy, and computational benefits of the method developed in solving FIDEs [1,2]. By examining assumptions, iteration requirements, and solution accuracy, the study highlights the improvements introduced by the proposed techniques.

In [1], the authors suggested least-squares-based methods for similar problem classes. Despite their effectiveness, these techniques frequently depend on global basis functions and mesh-based discretization. Our proposed methods, on the other hand, circumvent these dependencies and provide more flexibility for non-linear, variable-order, and fractional extensions. The suggested technique is more flexible and efficient than the approach in [1], especially when resolving the memory effects present in fractional operators. The examples’ reduced errors provide evidence for this.

Our approach utilizes enhanced versions of the ADM, specifically the Sumudu-ADM [51,52] and the Shehu-ADM [47,48]. These transform-based methods significantly reduce computational complexity and yield highly accurate approximations, often in two iterations. Unlike ADM, which typically requires multiple recursive steps, Sumudu-ADM and Shehu-ADM avoid extensive iteration while maintaining solution precision [49,53,54].

The numerical comparisons are supported by error norm tables based on the norms $L_{\infty }$, $L_2$, and $L_1$, as well as graphical illustrations for various values of fractional orders ${\zeta }_1$ and ${\zeta }_2$. These results confirm the superior performance of Shehu-ADM in terms of lower error magnitude, which is attributed to its built-in weighted structure. Both proposed methods surpass conventional approaches by eliminating the need for discretization or linearization.

Furthermore, TSADM [59,60] has been explored and provides an exact solution more efficiently than traditional ADM. However, even TSADM typically involves one iteration to converge. In contrast, Sumudu-ADM and Shehu-ADM achieve the desired accuracy in a single iteration, making them more efficient alternatives.

In conclusion, the comparative study validates that the proposed methods, particularly Shehu-ADM, outperform traditional numerical schemes and ADM variants by offering faster convergence, fewer computational steps, and higher accuracy. These features make the proposed methodology attractive and reliable for solving FIDEs encountered in various scientific and engineering applications.

As shown in

Table 5. Comparison of computational complexity and convergence behavior.

Method	Transform Inversions per Iteration	Convergence Behavior
Standard ADM	-	Not applicable (fails to converge)
Laplace-ADM	-	Diverges or produces inaccurate results
Sumudu-ADM	2	Converges slowly
Shehu-ADM	2	Stable and accurate due to weighted formulation
Two-Step ADM (TSADM)	1	Yields exact results

, standard ADM fails to handle singular kernels due to divergence issues. In contrast, the proposed Shehu-ADM and TSADM methods maintain accuracy and stability, even for weakly singular problems, while requiring comparable or lower computational overhead.

10. Conclusions

The primary objective of this research is to obtain analytical solutions for fractional integro-differential equations using enhanced variants of the Adomian decomposition method. This study develops and applies the two-step ADM, Sumudu-ADM, and Shehu-ADM without requiring prior assumptions or initial approximations. These advanced methods exhibit excellent computational efficiency and symbolic accuracy. It is important to note that ADM and its transform-based variant, LADM, are generally ineffective when applied to the class of FIDEs considered in this study, particularly in the presence of singular kernels. These methods typically assume smooth kernel behavior and require the analytic inversion of Laplace transforms. However, when the kernel is weakly singular—such as in the form ${(t-\tau )}^{-\gamma }$ with $0<\gamma <1$—the resulting transformed expressions become analytically intractable or lead to divergent series in the iterative ADM framework; moreover, these approaches assume well-posed initial conditions and regular kernel properties, assumptions that are violated in the problems addressed in this work. As a result, ADM and LADM often fail to converge or yield inaccurate approximations when applied to such singular FIDEs. In particular, in this work, the TSADM is an enhanced version of ADM that successfully produces the exact solution in just one iteration. The Sumudu-ADM and Shehu-ADM methods also yield highly accurate approximate analytical solutions within just two iterations, demonstrating their reliability and robustness. Unlike traditional numerical methods, the proposed transform-based and two-step ADM approaches offer a symbolic framework that is easy to implement and highly accurate. Theoretical validation of the methods is established through fixed-point theorems to ensure the existence and uniqueness of the solution. In addition, U-H stability is employed to confirm the robustness and reliability of the solutions obtained. Several test examples are included to evaluate the performance of these methods. Each example is supplemented with graphical comparisons between exact and approximate solutions and tabulated error norms using the $L_{\infty }$, $L_2$, and $L_1$ norms. The discussed numerical results validate the effectiveness of numerical approaches, such as the analytical method, TSADM, which provided the exact solution in just one iteration. The Shehu-ADM, because of its weighted formulation, often yields slightly lower error values than the Sumudu-ADM, further reinforcing its practical utility. In summary, we have the following description of our main focus in this study. We propose and validate a highly accurate TSADM and improved transform-based ADM techniques for solving FIDEs and establish comprehensive theoretical support, including results on existence, uniqueness, and stability, thus offering a rigorous foundation for the proposed methods. For future work, these methods can be extended to different fractional operators, including non-singular fractional derivatives, and applied to relevant problems to analyze the outcomes. As an extension, the proposed method can be adapted to variable-order fractional models, allowing for greater flexibility in modeling systems with time- or space-dependent memory effects.