A computational method for fractional equations using Legendre polynomials and Mittag-Leffler kernels

1. Introduction

In recent years, the Atangana-Baleanu-Caputo (ABC) fractional derivative has attracted growing interest due to its non-singular and non-local kernel, which better captures memory effects in various physical and biological processes. Unlike classical fractional derivatives with singular kernels, the ABC derivative incorporates a generalized three-parameter Mittag-Leffler function, offering a more realistic modeling framework for systems with fading memory. This makes the ABC derivative particularly suitable for modeling complex dynamical systems, where traditional integer-order or singular fractional models fall short. In this work, we focus on developing a numerical method tailored to fractional equations involving the ABC derivative, with an emphasis on achieving high accuracy and stability. Fractional calculus generalizes classical calculus by allowing derivatives and integrals to be taken to arbitrary (including non-integer) orders. Although its conceptual foundations trace back over three centuries-initiating with the correspondence between Leibniz and L’Hôpital on half-integer derivatives. It is only in recent years that fractional calculus has truly flourished (Petras, 2011).

With the growing availability of computational tools and increased awareness of memory and hereditary effects in real-world systems, fractional derivatives have been increasingly used to model complex phenomena in physics, biology, finance, and engineering (Abu-Shady & Kaabar, 2021). Major breakthroughs in fractional calculus revolve around the definitions and applications of the Riemann–Liouville and Caputo fractional derivatives (Atangana & Secer, 2013). While both have been widely used to express a broad class of fractional differential equations, it is specifically the Caputo derivative that allows for initial conditions analogous to those in classical integer-order calculus, making it more suitable for physical and engineering applications. However, additional formulations, such as the Grünwald-Letnikov and the more recent ABC fractional derivatives, further expand the scope and adaptability of fractional modeling by offering different kernels, boundary conditions, or regularity properties (Podlubny, 1998).

Among these newer definitions, the ABC fractional derivative is especially noteworthy for incorporating a generalized Mittag-Leffler kernel with three parameters. This construction provides additional flexibility in modeling complex dynamical behaviors, including anomalous diffusion, viscoelasticity, and processes with long-range temporal correlations. The accompanying AB fractional integral likewise preserves these generalized attributes. Together, they form a compelling framework for numerical analysis, enabling finer control over the behavior of the integral and derivative operators based on the parameter choices.

Despite these advances, a perennial challenge lies in devising stable, accurate, and computationally efficient numerical schemes for fractional differential equations. Polynomial approximation methods, particularly those based on orthogonal functions such as Legendre polynomials, have proven indispensable. Traditional approaches typically assume that the solution itself can be approximated as a polynomial series expansion, reducing the fractional differential equation to a system of algebraic equations in terms of the series coefficients. This method has produced reliable results in many instances, yet there remains potential for improving convergence rates and error reduction. Accordingly, the present work introduces a new perspective, approximating the ABC fractional derivative of the unknown function directly using shifted Legendre polynomials.

A similar approach has been explored in the literature using alternative polynomial bases. For example, the Bernstein polynomials were employed to construct the numerical scheme (Tamimi, 2023). In contrast, the present study adopts shifted Legendre polynomials, which offer favorable orthogonality properties and improved spectral convergence behavior for the class of problems considered. This technique not only preserves the polynomial framework but also tends to yield more compact, accurate formulations compared to directly approximating the solution. Consequently, it opens a pathway to faster convergence, improved precision, and less computational overhead.

Several recent works further support the use of spectral and orthogonal polynomial-based techniques in the fractional setting. For instance, in a paper (Ali, Haq, Hussain, Nisar, & Arifeen, 2024), a spectral collocation method based on Lucas and Fibonacci polynomials was applied to a two-dimensional nonlinear fractional diffusion equation. The authors provided an error analysis and demonstrated high computational accuracy at a low cost. In (Ghafoor, Hussain, Ahmad, & Arifeen, 2024), the Haar wavelet approximations were used to solve the fractional Benjamin–Bona–Mahony–Burgers’ equations, showcasing stability and efficiency in both one- and two-dimensional cases. Similarly, (Ali, Arifeen, Hussain, & Idris, 2025) presented an integrated approach using the Lucas and Fibonacci polynomials for time-fractional diffusion models, validated through comparisons with other numerical techniques. These studies reinforce the trend of employing orthogonal polynomial bases to tackle complex fractional systems and offer valuable points of comparison for the present work.

The comparative numerical experiments span multiple example classes, including third-order nonlinear fractional differential equations (Blasius boundary value problem and nonlinear fractional differential equations (population growth in a closed system). In each case, emphasis is placed on analyzing both residual errors of the obtained solutions. We observe that the proposed method typically achieves lower errors across a range of parameter settings, including the important regime where $\alpha$ and $\mu$ approximate integers. The increase of interest in fractional differential and integral equations has attracted the attention of researchers, particularly those involving generalized fractional operators with non-singular kernels such as the Atangana-Baleanu operator in the Caputo sense. These models are essential in capturing memory and hereditary properties of complex systems, which are inadequately described by classical integer-order models. Despite recent progress, efficient numerical methods that combine spectral accuracy with the ability to handle such operators remain limited. This work addresses the growing need for accurate and efficient numerical methods to solve fractional differential and integral equations that arise in modeling complex phenomena in science and engineering.

In particular, fractional models have gained prominence due to their ability to capture memory effects and anomalous dynamics, which are prevalent in fluid mechanics and biological systems. This paper addresses two important applications: the classical Blasius boundary layer problem and a fractional integral equation describing population growth in a closed system. The Blasius problem, a cornerstone in fluid dynamics, is extended here to fractional order to incorporate memory effects that classical models overlook. Similarly, the fractional integral equation provides a realistic framework for modeling population dynamics with non-local temporal interactions.

The Blasius boundary problem has been solved by several methods such as the Homotopy perturbation method (Miansari, Miansari, Barari, & Domairry, 2010), the numerical solution of the Blasius equation found by spectral method based on the Chybeshev polynomials (Shoukat, et al., 2024), the variational iteration method (Liu & Kurra, 2011), the Homotopy analysis method (Allan & Syam, 2005) and so on. Also, the second example (the population growth model) has been solved by methods such as the Homotopy analysis method (Vosughi, Shivanian, & Abbasbandy, 2011), the sinc-Galerkin method (Stenger, 1993), and the Adomain decomposition method (Al-Khaled K. , 2005). To the best of our knowledge, the use of shifted Legendre polynomials has not been explored for either of these problems. This underscores the novelty and contribution of the present work.

The novelty of our approach lies in employing the shifted Legendre polynomials combined with the AB fractional integral operator featuring the Mittag-Leffler kernel to develop a spectral method that achieves high accuracy on finite intervals. We rigorously analyze convergence and stability within Sobolev spaces and demonstrate the method’s effectiveness in approximating both solutions and their fractional derivatives. To the best of our knowledge, this unified spectral framework applied to these classical and fractional models is new and offers a powerful computational tool for a broad class of fractional problems.

This paper is structured as follows: In Section 0, we present the theoretical background of the ABC fractional derivative and the AB fractional integral, summarizing their key properties and pertinent definitions. In Section 0, we establish the existence of an approximate solution for the proposed method. In Section 0, we recall the construction of the shifted Legendre polynomials with their properties and explain how these polynomials can be employed to approximate fractional derivatives efficiently. We provide in Section 0 a detailed analysis of the convergence and stability of the method. Section 0 is devoted to giving details of the numerical formulation of the applied technique and providing extensive examples to demonstrate its effectiveness. Finally, Sections 0 and 0 discuss the performance implications and summarize the main conclusions of the study.

2. Preliminary and Illustrations

In this section, we present some definitions and theorems that are related to the left generalized ABC fractional derivative and left AB fractional integral. In classical definitions of fractional derivatives, the conventional singular power-law kernels are frequently substituted with the non-singular, smooth Mittag-Leffler kernel. The ABC fractional derivative formulation relies on it, and it is defined in terms of the generalized Mittag-Leffler function and denoted by $E_{{\alpha }_1, \mu }^{\gamma }\left(\lambda , x\right)$.

Definition 2.1 (Srivastava, Alomari, Saad, & Hamanah, 2021) The left generalized ABC fractional derivative with kernel $E_{{\alpha }_1, \mu }^{\gamma }\left(\lambda , x\right)$ is defined as

where $M\left({\alpha }_1\right)$ is the normalization function such that

and $n<\alpha \le n+1$, $n\in {\mathbb{N}}_0�\mathbb{N}\cup \left(0\right)=\left(0,1,2,\mathrm{...}\right)$,

where

and

Definition 2.2 (Morita & Sato, 2013) The left Riemann-Liouville fractional integral of order $\alpha -n$ where $\alpha , a, t\in ℝ$ and $n-1<\alpha <n$ is defined as follows

Definition 2.3 (Abdeljawad, 2019) The left AB fractional integral for $0<\alpha \left(1, \mu \right)0$ and $\gamma \in ℝ$

where $\left({}_a{}^{RL}I{}^{\alpha i-\mu +1}f\right)\left(t\right)$ is the Riemann-Liouville fractional integral of order $\alpha i-\mu +1.$

The previous definition establishes the AB fractional integral for $0<\alpha <1$, we make the following definition and theorem to find the AB fractional integral and derivative for $n<\alpha <n+1$ which can be found in (Tamimi, 2023).

Definition 2.4 (Tamimi, 2023) The left AB fractional integral for $n<\alpha \left(n+1, \mu \right)0$, ${\alpha }_1=\alpha -n$ and $\gamma =1$ is given by

where $\left({}_a{}^{RL}I{}^{\alpha i-\mu +n+1}f\right)\left(t\right)$ is the left Riemann-Liouville fractional integral of order $\alpha i-\mu ++n+1.$

Theorem 2.1 (Tamimi, 2023) For $n<\alpha \left(n+1, \mu \right)0$ and $\gamma =1$, we conclude that

where $n\in \mathbb{N}$.

3. Existence of Solutions

The ABC derivative is involved in the fractional differential equation

with the initial conditions

where $n<\alpha <n+1$, and $f:\left(a\text{, }b\right)\times ℝ\to ℝ$ is a given function.

Theorem 3.1 (Existence of solution). Assume that

• (i)

  The function $f\left(t,y\right)$ is continuous on $\left(a\text{,} b\right)\times ℝ$.

• (ii)

  There exists a constant $L>0$ such that for all $t\in \left(a, b\right)$ and $y_1$, $y_2\in ℝ$,

• (iii)

  The initial conditions $y^{\left(k\right)}\left(a\right)=y_k$ are given and finite.

Then, at least one continuous function $y\in C\left(a\text{,} b\right)$ that satisfies the initial conditions and (7) exists.

Proof. The problem (7) should be rewritten as the integral equation

where the fractional integral operator Atangana-Baleanu is defined by, ${}_a{}^{AB}I{}^{\alpha ,\mu ,\gamma }g$,

with the Mittag-Leffler function $E_{\alpha }\left(\cdot \right)$ and a normalization constant $B\left(\alpha \right)$. The operator $\text{T }:C\left(a\text{,} b\right)\to C\left(a\text{,} b\right)$ is defined by

We aim to show that $\text{T}$ preserves a fixed point in an appropriate convex, bounded, and closed subset $S\subset C\left(a\text{,} b\right)$.

Step 1: $\text{T}$ maps $S$ into itself.

Choose $S=\left(y\in C\left(a\text{,} b\right) : y_{\infty }\le M\right)$ for some large $M>0$. According to the properties of the fractional integral operator and the continuity and boundedness of

$f$ on $\left(a\text{,} b\right)\times \left(-M\text{,} M\right)$, $\text{T} y$ is both continuous and bounded for sufficiently large

Step 2: $\text{T}$ is compact and continuous.

The continuity of $f$ and the integral operator implies continuity. Compactness results from the fractional integral operator’s smoothing property, which uses the Arzelà-Ascoli theorem to convert bounded sets in $C\left(a\text{,} b\right)$ into relatively compact subsets.

Step 3: Schauder Fixed Point Theorem application.

The Schauder Fixed Point Theorem ensures that at least one fixed point exists since

$S$ is nonempty, convex, closed, and bounded, and $\text{T}$ is continuous and compact with

which is the solution to the original problem.

4. Shifted Legendre Polynomials and their Fractional Integral

We use the shifted Legendre polynomials as basis functions to implement our numerical method because of their computational efficiency on finite intervals and orthogonality. To build the approximate solution and examine the behavior of the ABC fractional derivative, we give the formal definition of these polynomials and formulate their fractional integrals in this section.

Definition 4.1 (Çerdik Yaslan & Mutlu, 2019), (Liu, Ding, Saray, Juraev, & Elsayed, 2025). The recurrence relation of Legendre polynomials $L_i\left(t\right)$ on $\left(-1\text{,} 1\right)$ is defined as follows

with initial terms

The standard form (Rodrigue’s formula) of Legendre polynomials $L_i\left(t\right)$ on [−1,1] is

which are orthogonal on $\left(-1,1\right)$ with respect to the wight function $w\left(t\right)=1,$ and satisfy

As well as the recurrence relation of the shifted Legendre polynomials ${\tilde{L}}_i\left(t\right)$ on $\left(a,b\right)$ that is defined as

with

Also, we can obtain the Rodrigues formulas of shifted Legendre polynomials as

such that the orthogonality of shifted Legendre polynomials on $\left(a,b\right)$ are defined by

In this paper, we apply the proposed method by representing the ABC fractional derivative as a linear combination of shifted Legendre polynomials, formulated as follows

Then, we apply left AB fractional integral on both sides of Eq. (8) for $n<\alpha <n+1$ using Theorem 3.1, Definition 2.2 and Definition 2.4, so we have

where ${}_a{}^{RL}\text{I}{}^{\alpha s-\mu +n+1}$ is the left Riemann-Liouville fractional integral of order $\alpha s-\mu +n+1$ which is defined in Definition 2.2. Thus, by balancing equation (12), we obtain the solution in the following form

We employ the shifted Legendre polynomials as basis functions in our numerical scheme due to their orthogonality on finite intervals and favorable computational properties. These polynomials allow for efficient and accurate approximation of smooth functions on $\left(a\text{,} b\right)$, particularly when combined with fractional integral operators. The approximate solution is expressed as in equation (13), where ${\tilde{L}}_i\left(t\right)$ are the shifted Legendre polynomials and ${}_a{}^{AB}\text{I}{}^{\alpha ,\mu ,\gamma }$ denotes the Atangana-Baleanu fractional integral operator with Mittag-Leffler kernel. The Taylor polynomial ensures accurate enforcement of the initial conditions and captures the local behavior near $t=a$.

This elegant representation seamlessly integrates spectral accuracy with the nonlocal memory effect of fractional operators, offering a powerful and robust tool for solving a wide class of fractional differential equations.

5. Convergence Analysis

In this section, we prove the convergence of our approach, where the target class is the Sobolev Space $H^r\left(0\text{,} 1\right)$, see (Adams & Fournier, 2003).

Theorem 5.1 Let $y\in C^n\left(0\text{,} 1\right)$, and assume that the ABC fractional derivative of the remainder is given by

for some $r>0$, where

is the Taylor polynomial of degree $n$ centered at $a$. We define the approximate solution $y_N\left(t\right)$ by

where ${\tilde{L}}_i\left(t\right)$ are the shifted Legendre polynomials on $\left(0\text{,} 1\right)$, and ${}_a{}^{AB}I{}^{\alpha ,\mu ,\gamma }$ denotes the corresponding Atangana-Baleanu fractional integral operator.

Then there exists a constant $C>0$ such that

and hence,

Proof. Let us denote the residual function as

By the assumption of the theorem, we have

Now, consider the ABC fractional integral approximation of $R\left(t\right)$

where the coefficients $c_i$ are chosen such that $R_N$ approximates $R$ in the $L^2$-norm. Then the full approximate solution is given by

We want to estimate $y-y_N{}_{L^2\left(0\text{,} 1\right)}=R-R_N{}_{L^2\left(0\text{,} 1\right)}$. By the properties of the AB fractional integral operator and the orthogonality and completeness of shifted Legendre polynomials in $L^2\left(0,1\right)$, the projection of $R\left(t\right)$ onto the span of ${\left({}_a{}^{AB}I{}^{\alpha ,\mu ,\gamma }{\tilde{L}}_i\left(t\right)\right)}_{t=0}^N$ yields the best approximation in $L^2$-norm.

Moreover, by a standard result in spectral approximation theory (see (Canuto, Quarteroni, Hussaini, & Zang, 2007)), for any function $f\in H^r\left(0\text{,} 1\right)$, its Legendre expansion satisfies the following estimate

$f-\sum _{i=0}^N{\hat{f}}_i{\tilde{L}}_i{}_{L^2\left(0\text{, }1\right)}\le C{N}^{-r}{f}_{H^r\left(0\text{, }1\right)}\text{,}$

for some constant $C>0$ independent of N.

Applying this to the function

we obtain

${}_a^{ABC}{D}^{\alpha ,\mu ,\gamma }R-\sum _{i=0}^N{\hat{f}}_i{\tilde{L}}_i{}_{L^2\left(0\text{, }1\right)}\le C{N}^{-r}{}_a^{ABC}{D}^{\alpha ,\mu ,\gamma }{R}_{H^r\left(0\text{, }1\right)}\text{.}$

Now, using the boundedness of the AB fractional integral operator ${}_a{}^{AB}I{}^{\alpha ,\mu ,\gamma }$ on $L^2\left(0\text{,} 1\right)$, we conclude

for some constant $C>0$. Hence,

This proves the desired estimate. Since $r>0$, it follows that

Remark 5.1 (On the Regularity Assumption) The Assumption $y\in C^n\left(0,1\right)$ ensures the validity of the Taylor expansion near $t=a$. However, since the problem involves the ABC fractional derivative, the appropriate setting for the analysis is a fractional Sobolev space. Specifically, we assume the remainder $\left(y-T_{n}y\right)$ satisfies ${}_{a }{}^{ABC}D{}^{\alpha , \mu , \gamma }\left(y-T_{n}y\right)\in H^r\left(0,1\right)$ for some $r>0$. This fractional regularity ensures convergence of the spectral approximation and is standard in the analysis of fractional differential equations.

6. Illustrative Applications

In this section, we show the fractional derivative of a solution expressed as a combination of shifted Legendre Polynomials in examples such as a third-order nonlinear differential equation (the Blasius boundary value problem) and a nonlinear fractional differential equation, as we shall explain below.

Example 1: Realistic modeling of biological systems

Consider the Blasius boundary value problem (Al-Khaled, Ajeel, Darweesh, & Al-Khalid, 2024) as follows

along with the initial-boundary conditions

where the parameter $a$ represents the Blasius constant, ${}_a{}^{ABC}D{}^{\alpha ,\mu ,\gamma }$ is the ABC fractional derivative of order $\alpha$, $y\left(t\right)$ is the dimensionless stream function and $t$ is the similarity variable.

The fractional Blasius model is suitable for describing fluid flow through heterogeneous, viscoelastic biological tissues, such as muscle, fat, and skin, where blood or medicinal fluids exhibit non-Newtonian and memory-dependent behavior. In particular, it can capture the nonlocal and inherited behavior in the velocity profile when $\alpha \in (2\text{,} 3)$. The use of a fractional derivative captures these nonlocal and history effects more accurately than classical models. For instance, the rate at which a drug diffuses and passes through tissue has a direct impact on dosage and efficacy in drug delivery. This shows how fractional-order modeling improves our capacity to depict fluid behavior in non-classical settings, like fluid motion in memory-sensitive smart materials or blood flow through tissue. We can fine-tune the model to more accurately represent real-world phenomena like anomalous diffusion and viscoelasticity by varying the fractional order $\alpha$. This directly affects drug delivery, biomedical engineering, and intelligent material design.

To this end, we estimate the value of $y^{″}\left(0\right)$ and make a transformation for shifted Legendre polynomials from $[0\text{,} 1]$ to $[0\text{,} b]$. Then, we apply the proposed method to our example, which assumes the fractional derivative as a combination of Legendre polynomials, and appeal to the AB fractional integral for both sides of the ABC fractional derivative in (9); hence, we get $y\left(t\right)$ as explained in (13). We thereafter substitute the fractional derivative and $y\left(t\right)$ into (14) to get

where $2<\alpha \le 3$, $\mu >0$, $\gamma \in N.$Thus, equation (13) becomes

By applying $m+4-\alpha$ points into equation (16), and from the initial-boundary conditions, we get a system of nonlinear equations; by solving it, we have the $c_i$ for $i=0, . . . ,m$. Finally, we substitute $c_i$’s into equation (15) and, hence, we get the approximate solution. In Fig. 1, we plot the approximate solutions of Example 1 when $a=2$ with different values of $\alpha$ and $\gamma$ and fixed values of μ for $m=10$. In Fig. 2, we plot the approximate solutions for Example 1 and their derivatives with different values of Blasius constant. In Fig. 3, we plot the approximate solutions of Example 1 with fixed value for $a$ when $\mu =1$, $\gamma =1$ and with different values for $\alpha$.

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Fig. 1.

Graph of the approximate solutions of Example 1 with $a=2$ for $\mu =\gamma =1$ and different values of $\alpha$.

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Fig. 2.

Graph of the Approximate solutions and their derivatives for Example 1: (a) approximate solutions, (b) first derivatives, and (c) second derivatives, all plotted for different values of the Blasius constant.

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Fig. 3.

Graph of the approximate solutions of Example 1 with $a=2$ for $\mu =\gamma$ = 1 and different values of $\alpha$.

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It is worth noting that this problem has been solved by the Adomain Decomposition method, which contains tables that show the values of the approximate solution. In our method, we explain the approximate solution in figures for different values of the Blasius constant. The results demonstrate that the present method achieves excellent agreement with the ADM (Al-Khaled, Ajeel, Darweesh, & Al-Khalid, 2024), while they also offer improved computational efficiency, particularly in terms of faster convergence and the reduced number of basis functions required to achieve comparable accuracy.

Example 2: Population growth in a closed system

Consider the nonlinear fractional differential equation (Al-Khaled K. , 2005) for $0<\alpha <1$, $\mu >0$ and $\gamma \in \mathbb{N}$ as follows

where $a$, $b$, $c$ and $\beta$ are some constants. $y\left(t\right)$ is the population size at time $t$. The nonlinear term $cy\left(t\right){\int }_0^{t}y\left(\tau \right)d\tau$ which is the hereditary interaction—the population is affected not just by current density, but also by the cumulative population history, scaled by the current state. Also, this term models memory-driven reinforcement or self-interaction, where the current rate of change is affected by: The accumulated population presence over time (e.g., cumulative stress, chemical buildup, or environmental modification) multiplied by the current size, which can represent feedback intensity, $\alpha =1$, reduces to a classical first-order integro-differential equation, no memory—the system reacts instantly to current population levels. While when $\alpha <1$, the system evolves under an ABC fractional derivative, which introduces memory—the rate of change depends on the entire history of $y\left(t\right)$.

Case 1: $\alpha =0.9$ (Weak Fractional Effect) In this case, slight memory is added where the system still grows, but it is slightly slow, and the growth curve lags behind the classical case. Also, accumulated past values still push the solution upward via the integral term, but the overall growth is more restrained. Peak and saturation occur later in time.

Case 2: $\alpha =0.7$ (Moderate Memory). Here, memory has more weight, and the fractional derivative causes a delay in acceleration. While growth is smooth and slow, the initial population stays small for longer. But the integral feedback can eventually cause acceleration, possibly leading to overshoot or oscillation (depending on the strength of the integral term). Biological meaning: population is slow to respond but may later accelerate sharply due to accumulated environmental effects.

Case 3: $\alpha =0.4$ (Strong Memory). This case has a significant delay, and the system hesitates to grow. Also, early growth is very flat (like dormancy), then it slowly increases; even with the reinforcing integral term, the population may take a long time to noticeably grow and might even stagnate depending on the initial condition or parameter balance. The biological meaning indicates that the system is highly history-dependent, e.g., stress-dominated, diseased, or chemically suppressed environments.

In this example, we impose the ABC fractional derivative as a combination of shifted Legendre polynomials that is defined in (9). By applying the AB fractional integral, we have $y\left(t\right)$ in the form defined in (13) for $n<\alpha <n+1.$ Here, we have $0<\alpha <1$. Finally, by using this fractional derivative along with $y\left(t\right)$, we get equation (18) in the form

and the initial condition becomes

We make a transformation for the shifted Legendre polynomials from $[0\text{,} 1]$ to $[0\text{,} b]$ and then we substitute $m+2-\alpha$ points into equation (18) to get a system of nonlinear equations. By solving it, we get $c_i$’s.

We plot the approximate solutions of Example 2 once with different values of $\mu$ and $\alpha$ where $\kappa =1$ as shown in Fig. 4, and again we put $\mu =\gamma =1$ with two cases of $\kappa$ for different values of $\alpha$ as illustrated in Fig. 5.

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Fig. 4.

The approximate solutions of Example 2 with two cases where in (a) that is taken over different values of $\mu$, and in (b) we take different values of $\alpha$ where $\kappa =1$.

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Fig. 5.

Graph of the approximate solutions of Example 2 with $\mu =1$, $\gamma =1$ including two cases where in (a) we take $\kappa <1$ and in (b) $\kappa >1$ with different values of $\alpha$.

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This example has been solved using methods like the ADM and the sinc-Galerkin method, as illustrated in (Al-Khaled K., 2005). By a quick comparison between the results of the two methods with ours, we find a correspondence of results such that the approximate solution curve starts from an initial value, then increases, and finally decreases. We see that the tables and figures in the aforementioned methods agree with the outcomes of our method, as shown in Figs. 4 and ‎5.

7. Discussion

In this paper, we introduced a novel numerical technique for solving fractional differential and integral equations involving the ABC fractional derivative with a generalized Mittag-Leffler kernel. By directly approximating the fractional derivative using shifted Legendre polynomials, rather than approximating the solution itself, we developed a method that is both efficient and highly accurate. The theoretical foundation of the method was established through a rigorous convergence analysis, ensuring its mathematical soundness and practical reliability.

To evaluate the performance and versatility of the proposed approach, we applied it to two well-known problems. The first example involved solving the fractional Blasius boundary value problem, which models the steady laminar boundary layer over a flat plate in fluid dynamics. This problem, characterized by its nonlinear nature and boundary conditions at infinity, serves as a benchmark for the accuracy and stability of numerical solvers. Our results closely match those in the literature (Al-Khaled, Ajeel, Darweesh, & Al-Khalid, 2024), as illustrated in the accompanying figures, demonstrating the method’s capacity to accurately resolve nonlinear boundary layer behavior. The second example tackled a fractional integro-differential equation that models population growth in a closed system. This type of equation is typical in biological and ecological modeling, where memory effects and hereditary properties play a significant role. The numerical results obtained agree significantly well with those reported in (Al-Khaled K., 2005), and show the potential of the method in applications involving fractional models of real-world phenomena. Beyond these applications, the generality of the method suggests that it could be extended to a broader class of problems. Potential future work includes

• Extending the technique to systems of coupled fractional differential equations;

• Applying the method to time-dependent problems in multiple spatial dimensions;

• Investigating the method’s performance with other types of non-singular fractional derivatives, such as those with exponential or stretched Mittag-Leffler kernels;

• Incorporating adaptive mesh or polynomial degree refinement to further improve efficiency;

• Exploring real-world applications in finance, viscoelasticity, epidemiology, and other areas where fractional models naturally arise.

In summary, the proposed shifted Legendre polynomial-based method offers a powerful and accurate framework for solving a wide variety of fractional equations, and its success in the presented examples highlights its potential as a versatile tool in fractional calculus and applied modeling.

8. Conclusions

In this paper, we introduced a new numerical technique for solving fractional differential and integral equations involving the ABC derivative with a generalized three-parameter Mittag-Leffler kernel. By directly approximating the fractional derivative using shifted Legendre polynomials, we constructed highly accurate solutions with very low residual error. We rigorously proved the convergence of the proposed method, confirming its theoretical reliability. The effectiveness of the technique was demonstrated through two illustrative examples: the fractional Blasius boundary value problem and a fractional integro-differential equation modeling population dynamics in a closed system. In both cases, the approximate solutions aligned closely with previously reported results, reinforcing the validity and precision of our approach. This method provides a robust and efficient tool for tackling a wide class of fractional equations that arise in applied sciences. Several interesting directions for future research are as follows:

• Extension to problems with multiple dimensions: The suggested approach can be extended to solve fractional partial differential equations (PDEs) in two or three dimensions of space-time. These PDEs are frequently encountered in fluid dynamics and anomalous diffusion models.

• Variable-order fractional models or time-dependent models: Utilizing variable-order Atangana-Baleanu operators to investigate fractional equations where the order of differentiation changes over time or space.

• Coupled equations and nonlinear systems: Utilizing the approach for nonlinear fractional equation systems, particularly those that emerge in biological modeling, reaction-diffusion systems, and viscoelasticity.

• Stability and error estimates: Creating stability results and strict a priori error bounds under less strict regularity assumptions, perhaps in more general Banach or Sobolev spaces.

• Control and optimization applications: Applying the spectral fractional framework to fractional dynamics-governed optimal control problems.