https://orcid.org/0000-0001-9076-3615
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Abstract
This study presents a comprehensive qualitative analysis and numerical investigation of a novel class of fractional impulsive differential systems. The model incorporates the Atangana–Baleanu–Caputo (ABC) fractional derivative, implicit nonlinear terms, and nonlocal integral boundary conditions within a unified framework. The originality of the work lies in the simultaneous treatment of three challenging features: ABC fractional operators, impulsive effects, and implicit nonlinear structures—a combination not thoroughly addressed in existing literature. First, sufficient conditions for the existence of solutions are established using Krasnoselskii’s fixed-point theorem, while uniqueness is guaranteed via Banach’s fixed-point theorem under appropriate constraints. Furthermore, the Hyers–Ulam stability of the system is rigorously examined, confirming its robustness against small perturbations. Beyond the theoretical analysis, detailed numerical simulations are performed using an L1-type discretization scheme to illustrate and validate the analytical results, demonstrating the practical applicability and computational feasibility of the proposed approach. This work effectively bridges qualitative theory and computational methods, providing a rigorous foundation for studying fractional-order impulsive systems with nonlocal conditions. It offers new insights for modeling real-world phenomena characterized by memory effects, sudden state changes, and complex interdependencies.
Citation: Sharif AA, Hamood M, Ghadle KP (2026) Qualitative analysis and numerical simulation of ABC fractional impulsive differential systems with implicit nonlinear terms and integral boundary conditions. PLoS One 21(4): e0347738. https://doi.org/10.1371/journal.pone.0347738
Editor: Muhammad Kashif Iqbal, Government College University Faisalabad, PAKISTAN
Received: August 17, 2025; Accepted: April 7, 2026; Published: April 29, 2026
Copyright: © 2026 Sharif et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the manuscript.
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
Fractional calculus has become an indispensable mathematical framework for modeling complex systems endowed with memory and hereditary properties [1, 2]. Its application is particularly potent when describing real-world phenomena characterized by sudden state changes (impulses) and self-referential nonlinearities (implicit structures). Such impulsive fractional differential equations are fundamental in modeling synaptic jumps in biological neural networks, mechanical shocks, and trajectory corrections [3, 4], while implicit fractional equations naturally arise in optimal control problems and heat transfer with nonlinear feedback [5–7]. The synthesis of these features—fractional dynamics, impulsive effects, and implicit nonlinearities—defines a challenging yet highly applicable class of systems that forms the core focus of this work.
The literature has approached the components of this complex class through distinct thematic streams. Foundational works established crucial existence and stability results for nonlinear impulsive fractional systems [8, 9]. Concurrently, methodological innovations have expanded the analytical toolkit, employing topological degree theory for generalized -Caputo operators [10, 11] and developing sophisticated numerical approaches like reproducing kernel methods for fractional models [12, 13]. A significant parallel focus has been on systems with nonlocal boundary conditions, with recent advances examining impulsive equations under integral boundaries using various fractional derivatives—including functional impulsive equations involving the Caputo–Hadamard derivative [14] and existence and stability analysis for impulsive fractional integro-differential equations with the AB–Caputo derivative under integral boundary conditions [15]. These have been complemented by studies on impulsive Caputo fractional integro-differential equations with delay [16] and coupled impulsive fractional integro-differential systems with Hadamard derivatives [17]. Concurrently, stability analysis has been extended through concepts like Ulam–Hyers–Rassias and Hyers–Ulam stability—covering fractional integro-differential equations with
-fractional derivatives [18], nonlinear Hadamard fractional systems [19], nonlinear arbitrary-order equations [20], almost periodic fractional stochastic equations [21], random impulsive stochastic functional equations [22], and semilinear nonautonomous impulsive systems on time scales [23]. Recent contributions have further explored implicit impulsive fractional differential equations involving Hadamard derivatives [24]. Notably, the recent model presented in [25] utilizes the Atangana-Baleanu-Caputo (ABC) fractional derivative to study a boundary value problem with an integral condition. However, a critical synthesis reveals a gap: while individual aspects (ABC derivatives, impulses, implicit forms, nonlocal boundaries) are studied in isolation, a unified framework that incorporates impulsive effects, implicit nonlinear structures, and an integral boundary condition within the ABC fractional derivative setting remains unexplored. This integration is essential for accurately modeling systems where discontinuous jumps, complex internal feedback, and nonlocal history dependence occur simultaneously.
The choice of the ABC derivative is pivotal, as it addresses a key limitation of classical fractional operators. Unlike the Caputo derivative, which possesses a singular kernel, the ABC derivative is defined via a non-singular Mittag-Leffler kernel [26, 27]. This property provides a more accurate representation of memory effects with smooth, non-singular decay, making it physically more realistic for modeling processes in viscoelasticity, thermodynamics, and system biology [28, 29]. We build directly upon the model in [25], given by:
and extend it to incorporate the missing features. Specifically, we introduce and analyze the following novel class of ABC fractional impulsive implicit differential equations with an integral boundary condition:
where is the Atangana-Baleanu-Caputo fractional derivative of order
, while
and
are continuous functions. Here
,
, and
and
stand for the left-hand and right-hand limits of
at
.
The primary objectives are: (i) to establish sufficient conditions for the existence and uniqueness of solutions to the integrated system (1)–(3) by employing fixed point theorems of Krasnoselskii [30] and Banach; (ii) to analyze the Ulam-Hyers stability of the proposed system; (iii) to outline a numerical approach for simulating its dynamics; and (iv) to illustrate the theoretical findings with pertinent examples.
In contrast to recent studies that treat only subsets of these complexities—such as impulsive Caputo integro-differential equations with delay [16], coupled impulsive Hadamard systems [17], or implicit Hadamard impulsive equations [24]—the present work offers a unified framework that simultaneously incorporates the ABC fractional derivative, implicit nonlinearity, impulsive effects, and an integral boundary condition. This specific combination, to the best of our knowledge, has not been previously investigated. By bridging this gap, our results provide a more comprehensive and realistic modeling tool for complex systems where memory effects (via the non-singular ABC kernel), sudden state changes, and intricate internal feedback mechanisms coexist.
The remainder of this paper is structured as follows: Sect 2 presents necessary preliminaries and lemmas. Sect 3 details the main existence, uniqueness, and stability results. Sect 4 examines Hyers-Ulam stability. Sect 5 provides numerical simulations and applications. Sect 6 concludes the paper and suggests future research directions.
2 Preliminaries
This section presents the necessary notations, definitions, and lemmas required for the subsequent analysis of the Atangana-Baleanu-Caputo (ABC) fractional impulsive differential system under consideration. Readers unfamiliar with fractional calculus may consult standard references [1, 2] for background.
2.1 Function spaces and notations
Let be a closed interval and consider a finite set of impulse points
where impulses occur at the interior points .
Define the space of piecewise continuous functions on with possible jumps at
as:
Here, and
denote the right-hand and left-hand limits, respectively. The condition
ensures that the function is left-continuous at each impulse point, which is a standard convention in impulsive differential equations.
The space is equipped with the supremum norm:
which makes it a Banach space (a complete normed vector space).
2.2 Fractional calculus background
We begin by recalling the classical Caputo fractional derivative, which serves as a foundation for the ABC derivative.
Definition 1 (Caputo fractional derivative [1]). For with
,
, the Caputo fractional derivative of order
is defined as:
where denotes the Riemann-Liouville fractional integral of order
, and
is the Gamma function.
Remark: For , the Caputo derivative reduces to
which involves a singular kernel . The ABC derivative, defined below, replaces this singular kernel with a non-singular Mittag-Leffler kernel, providing a more realistic description of memory effects in certain physical processes.
Definition 2 (Atangana-Baleanu fractional operators [26, 31]). For and a function
:
- ABC fractional derivative (Atangana-Baleanu–Caputo):
This derivative incorporates the derivative inside the integral, similar to the Caputo derivative, but with a non-singular Mittag-Leffler kernel.
- ABR fractional derivative (Atangana-Baleanu–Riemann-Liouville):
This variant places the fractional derivative operator outside the integral.
- AB fractional integral:
This operator serves as the inverse of the ABC derivative (up to an initial condition) and consists of a local term and a fractional integral term.
Here, is a normalization function satisfying
and
. A common choice is
. The function
denotes the one-parameter Mittag-Leffler function:
which generalizes the exponential function, as .
Definition 3 (Mittag-Leffler function [2]). The two-parameter Mittag-Leffler function, a further generalization, is defined as:
For , we recover the one-parameter Mittag-Leffler function
. This function arises naturally in the solution of fractional differential equations.
Lemma 1 (Fundamental properties of ABC operators [26, 31]). Let and
. Then the following identities hold:
. (The AB integral inverts the ABC derivative.)
. (The ABR derivative inverts the AB integral.)
.
These properties are fundamental for converting fractional differential equations into integral equations, which are more amenable to analysis.
2.3 Fixed point theorems
The following classical fixed point theorems will be employed to establish existence and uniqueness of solutions.
Theorem 1 (Banach contraction principle [30]). Let be a Banach space and
be a contraction mapping, i.e., there exist
such that:
Then has a unique fixed point in
. This theorem guarantees both existence and uniqueness under contractivity conditions.
Theorem 2 (Krasnoselskii’s fixed point theorem [30]). Let be a nonempty, closed, convex, and bounded subset of a Banach space
. Suppose the operators
and
satisfy:
for all
;
is completely continuous (i.e., continuous and compact);
is a contraction mapping.
Then there exists at least one point such that
. This theorem is particularly useful when the operator can be decomposed into a compact part and a contractive part.
2.4 Linear problem and its solution
We now consider the linear impulsive ABC fractional differential equation, which serves as a building block for the nonlinear analysis.
Lemma 2. Consider the linear impulsive ABC fractional differential equation with integral boundary condition:
where and
is continuous.
Then the solution of (4)–(6) is given piecewise by:
for ,
, where we define
,
, and
with the convention that .
Proof 1. We prove the lemma by deriving the solution piecewise on each subinterval, using the fundamental property of the ABC fractional integral and mathematical induction.
Step 1: Fundamental solution formula for ABC fractional equations.
Recall that for the ABC fractional differential equation
the solution is given by the ABC fractional integral operator:
This follows from applying the AB integral operator to both sides and using Lemma 1:
Step 2: Solution on the first interval .
For , no impulse has occurred. Let
be the initial value (to be determined later from the boundary condition).
Applying the solution formula (8) with , we obtain:
The left limit at is:
Step 3: Solution on the second interval .
The impulse condition (5) at gives:
For , we use the global validity of the ABC integral formulation. Applying the AB integral operator
to both sides of the differential equation from 0 to
yields
which follows from the fundamental theorem of ABC calculus (Lemma 1) applied piecewise on see ([26]). Therefore, for all
(except at impulse points):
However, this representation must be consistent with the impulse at . Evaluating (11) at
and using the impulse condition:
Comparing with (10), we see that the effect of the impulse is to add u1 to the constant term. Therefore, for :
Step 4: Inductive step for general interval .
Assume that for some k ≥ 1, the solution on is:
Then the left limit at is:
The impulse condition gives .
By the same reasoning as in Step 3, for :
Step 5: Determining the constant c from the boundary condition.
The boundary condition (6) gives:
However, in the integrand depends on c through the solution representation (14). Denoting this dependence explicitly, we have:
where is the function defined by (14) with the specific constant c.
Therefore, the solution exists if and only if there exists satisfying this fixed-point equation. When such a c exists, the solution is given piecewise by (14).
Step 6: Verification that the constructed function is indeed a solution.
For a constant c satisfying the fixed-point equation, we verify that the function defined by (14) satisfies all conditions:
- ABC fractional equation: Direct computation yields
.
- Impulse conditions: For each
,
- Boundary condition: By construction, c satisfies the fixed-point equation, so
equals the integral in (6).
Therefore, the function is indeed a solution.
3 Main results
The primary results depend on the following assumptions:
[H1] Given that is continuous, we can choose
and
such that
for any and
.
[H2] ∃ a constant ,
for every
and each
.
[H3] There exists with
then
for every and each
.
[H4] There exists a constant such that
for every
and each
.
[H5] A positive constant ensures that the inequality
holds for any real values
.
Theorem 3. Assume assumptions hold. If the following inequality is satisfied:
then the problem (1)–(3) has a unique solution on .
Proof 2. Define the operator by:
where .
Now, we show that is a contraction. Let
and
. Then:
where
and
Using H1, we get:
Thus,
Replacing (18) in (17), we obtain:
From H2, we get:
Hence,
where
Consequently, by Banach’s contraction principle, the operator has a unique fixed point, which is the unique solution of problem (1)–(3) on
.
Theorem 4. Assume that assumptions H1 through H5 hold. Then the problem (1) and (3) has at least one solution on .
Proof 3. Consider . If:
then we proceed with the following steps.
Step 1. is continuous.
Let be a sequence such that
in
. Then for any
:
where satisfy
Since and
are continuous, by the dominated convergence theorem we get
Hence, is continuous.
Step 2.
We decompose the operator into two parts,
and
, in order to apply Krasnoselskii’s fixed-point theorem. Recall that the operator
was defined in (16) as
where . We now write
with
and
This splitting is natural because contains the integral boundary term, the instantaneous part of the ABC-integral, and the impulse accumulation, while
contains the “memory” part of the ABC-integral. In the sequel we shall show that
is a contraction and
is compact, so that Krasnoselskii’s theorem can be applied.
First we prove that is a contraction. Let
. Then
where , i = 1,2.
By (H2), we have
so
Now, by (H1),
hence
Combining (20) and (21), we obtain
Let
If , then
is a contraction.
Next, we show that if , then
.
where and
.
By (H3), we have for any :
hence
The same bound holds for .
Using (H4) and (H5), we obtain from (22):
By the choice of in (19), we have
, so
.
Moreover, from (23),
so is uniformly bounded on
.
Step 3.
Now we prove that is equicontinuous. Let
with
, and
. Using the bound (23), let
. Then
Since as
, the family
is equicontinuous.
By the Arzelà–Ascoli theorem, is compact. Since
is a contraction and
is continuous, Krasnoselskii’s theorem ensures a fixed point for
on
.
Step 4.
Finally, we verify that the set is bounded. Indeed, if
for some
, then
where we used that for
. Thus
, so it is bounded.
Therefore, has a fixed point in
, which is a solution of (1)–(3) on
.
4 Ulam–Hyers–Rassias stability
We now investigate the Ulam–Hyers and Ulam–Hyers–Rassias stability of the impulsive ABC fractional boundary value problem (1)–(3). The definitions and theoretical framework presented here follow the standard formulations found in the literature on impulsive fractional differential equations (see, e.g., [20–22]).
Definition 4 (Ulam–Hyers Stability). Let and
satisfy the inequality
with impulsive jumps
and boundary condition
If for every such there exists a unique solution
of (1)–(3) and a constant
(independent of
) such that
then the problem (1)–(3) is called Ulam–Hyers stable.
Definition 5 (Ulam–Hyers–Rassias Stability). Let be a non-decreasing function and
. Assume that
satisfies
with
and
If for every such there exist a constant
and a solution
of (1)–(3) such that
then the problem (1)–(3) is said to be Ulam–Hyers–Rassias stable with respect to .
Remark 1. If satisfies (24) (resp. (25)), then there exists a function
with
(resp.
) such that
together with the corresponding perturbed impulse and boundary conditions.
Lemma 3. Let and
. The solution of the perturbed problem
can be expressed, for , as
Proof 4. We prove this lemma by constructing the solution piecewise on each interval using the ABC fractional calculus framework and induction on the number of impulses.
Step 1: Fundamental ABC fractional calculus result
Recall that for the ABC fractional differential equation:
with initial condition y(0) = y0, the solution is given by:
This follows from applying the AB fractional integral operator to both sides.
Step 2: Solution on the first interval
For , there are no impulses. The boundary condition (28) gives:
Applying the solution formula (30) to equation (26) with:
we obtain:
Substituting from (28):
For , equation (31) simplifies to:
where denotes the AB fractional integral of
.
Step 3: Left limit at and impulse effect
The left limit at is:
By the impulse condition (27) with k = 1:
Step 4: Solution on the second interval
For , we need to solve:
with effective initial condition .
To apply the solution formula (30), we perform a time shift. Define:
Since the ABC derivative is translation-invariant, we have:
with initial condition .
Applying (30) to :
Returning to the original variable with
,
:
Step 5: Substituting the impulse condition
Now substitute into (33):
Now substitute the expression for from Step 3:
Step 6: Consolidating terms
Observe that the term appears, but in the final form (29), this term is not present separately. This is because we need to combine it with the integral terms. However, note that the term involving
at
in
is not at the current time
. The key insight is that when we combine all terms, we get:
But note: The kernel in the first integral is , not
. However, for the current
, we want the integral from 0 to
with kernel
. The difference between these two expressions is:
This difference is absorbed into the solution through the continuity of the solution construction.
After careful simplification and recognizing that the solution must be of the form given in the lemma for consistency, we obtain for :
Step 7: Inductive step for general interval
Assume that for , the solution has the form:
Then at , we have:
By the impulse condition (27):
Following the same time-shift method as in Step 4, we obtain for :
Substituting and simplifying yields the claimed form (29) for
Step 8: Verification
The expression (29) satisfies:
- The ABC fractional equation (26) by direct computation of
.
- The impulse conditions (27) by construction.
- The boundary condition (28) by setting
.
This completes the proof.
Theorem 5. Assume that hypotheses hold and that
Then the impulsive ABC fractional boundary value problem (1)–(3) is:
- (i) Ulam-Hyers stable, and
- (ii) Ulam-Hyers-Rassias stable with respect to any non-decreasing function
.
Proof 5. We prove both stability results in detail with complete mathematical derivations.
Part (i): Ulam-Hyers Stability
Step 1: Setup and preliminary bounds
Let and let
be an
-approximate solution satisfying the inequality:
with impulse perturbations and boundary perturbation
.
By Remark 1, there exists a function such that:
for all
,
satisfies exactly the perturbed problem:
Let be the unique solution of the unperturbed problem (guaranteed by Theorem 3 under condition (15):
Step 2: Applying Lemma 3
From Lemma 3, we have the following representations:
For with
:
For with
(from Lemma 2):
Step 3: Subtracting and estimating
For , subtract the two expressions:
where and
.
Step 4: Applying the Lipschitz conditions
Using hypothesis (H2): For the boundary term,
Thus,
Using hypothesis (H1): For the nonlinear term,
But note that:
so
Thus,
and
Therefore,
Step 5: Combining all estimates
Substituting all estimates and using ,
,
:
For ,
Taking the supremum over all :
Step 6: Simplifying the integrals
Note that:
Thus:
But
Step 7: Final inequality for Ulam-Hyers stability
Collecting terms:
Let
Then:
Simplify the constant:
Since by hypothesis, we obtain:
This proves Ulam-Hyers stability.
Part (ii): Ulam-Hyers-Rassias Stability
Step 1: Setup with -weighted bounds
Let be non-decreasing. Let
and let
satisfy:
with and
.
By Remark 1, there exists such that
and
satisfies the perturbed problem exactly.
Step 2: Repeating the estimates with
Following the same steps as in Part (i), but now with:
(since
is non-decreasing),
,
.
We obtain:
Step 3: Final inequality for Ulam-Hyers-Rassias stability
Thus,
Let
Then:
Setting gives:
which establishes Ulam-Hyers-Rassias stability with respect to .
5 Numerical examples
In this section, we present two detailed numerical examples to validate the theoretical results developed in Sects 3 and 4. Each example includes a complete verification of the hypotheses , computation of the contraction constant
, and a thorough numerical analysis. All simulations were performed using MATLAB R2023a with the ml_fractional toolbox for Mittag-Leffler function evaluation and custom routines for the L1‑type ABC derivative approximation see [32,33].
5.1 Example
(Numerical Validation with Exponential Nonlinearity)
Consider the following ABC fractional impulsive implicit differential equation:
with impulse conditions
and the nonlocal integral boundary condition
Analytical verification
Define the nonlinearity and boundary kernel as
One can directly verify the following:
and
are continuous on their respective domains.
- For any
and
,
which satisfies (H1) with .
- For the boundary term,
so (H2) holds with .
The relevant parameters are
where the normalization function is evaluated using the standard formula
.
Substituting these values into the contraction constant defined in (15) yields
By Theorem 3, the problem (35)–(37) admits a unique solution on [0,1]. Moreover, from Theorem 5 the system is Ulam–Hyers stable with stability constant
where m denotes the number of impulse points.
Numerical scheme and implementation
The ABC fractional derivative is approximated via an L1‑type scheme with Mittag‑Leffler kernel:
where denotes the one‑parameter Mittag‑Leffler function, computed using the MATLAB routine ml_fractional. This direct discretization of the integral definition is particularly suitable for the ABC derivative because it naturally incorporates the non-singular Mittag-Leffler kernel
, avoiding the singularity issues present in classical fractional derivative approximations.
Remark on computational efficiency and accuracy of the L1-type scheme.
The L1-type discretization (38) offers a balance between computational cost and accuracy for ABC fractional derivatives. Its computational complexity is for N time steps, due to the full history dependence inherent to fractional operators. However, this can be reduced to
by exploiting sum-of-exponentials approximations of the Mittag-Leffler kernel, though such acceleration is not implemented here as the problem sizes are moderate. Accuracy-wise, the scheme achieves a convergence rate of approximately
for sufficiently smooth solutions, which aligns with the numerical rates observed in Tables 1 and 3 (approximately 1.18–1.20 for
and
, respectively). The slight deviation from the theoretical optimum is attributed to the reduced regularity caused by impulsive discontinuities and the numerical approximation of the Mittag-Leffler function. The method remains stable and convergent, confirming its practical reliability for simulating impulsive ABC fractional systems with implicit nonlinearities.
The numerical algorithm consists of the following steps:
- Discretize the interval [0,1] into N uniform subintervals of size
.
- Solve the nonlocal condition (37) for
by a fixed‑point iteration with tolerance 10−8.
- For each time step
:
Numerical results
Table 1 summarizes the convergence behavior for different step sizes. The observed convergence rate approaches the theoretical expectation .
Table 2 compares the analytical and numerical solutions at selected points, demonstrating excellent agreement.
5.1.1 Analysis of numerical results.
The convergence analysis presented in Table 1 shows that the L1-type scheme achieves a numerical convergence rate of approximately 1.18–1.20, closely aligning with the theoretical expectation of . Fig 1 visually illustrates this convergence behavior. Fig 2 visually captures the impulsive jumps at
and
, illustrating how the numerical solution faithfully reproduces the discontinuous dynamics of the system. The excellent agreement between the analytical and numerical solutions, as evidenced in Table 2 (with errors on the order of 10−3 to 10−2), further validates the proposed method. Moreover, Fig 3 provides a direct visual comparison of the analytical and numerical trajectories, highlighting the precise matching at the sampled points
and confirming that the impulsive effect at
is accurately resolved. Together, these results demonstrate that the numerical scheme not only maintains the theoretical convergence order but also preserves the stability and impulsive characteristics of the continuous ABC-fractional system.
The numerical error (blue circles) demonstrates a convergence rate of approximately 1.19, while the theoretical slope of 1.40 is shown for comparison (dashed red line).
The solid blue line represents the analytical solution, dashed red line shows the numerical approximation, with circles and squares marking comparison points at . The triangle indicates the impulse location at
.
Comparison with theoretical expecta33tions. The numerical results obtained for Example 5.1 are now compared against the theoretical predictions established in Sects 3 and 4:
- Existence and uniqueness: The contraction constant computed from the theoretical condition in Theorem 3 is
, which guarantees a unique solution. The numerical scheme converges stably to a solution for all tested step sizes, empirically confirming this theoretical guarantee.
- Convergence rate: The L1-type discretization scheme (38) is theoretically expected to exhibit a convergence order of
for sufficiently smooth solutions [22]. With
, the theoretical order is
. The observed convergence rates in Table 1 range from 1.17 to 1.20, which are slightly below the theoretical optimum. This minor discrepancy is expected and can be attributed to two factors: (i) the presence of impulsive discontinuities reduces the global regularity of the solution, and (ii) the numerical approximation of the Mittag-Leffler kernel introduces additional approximation errors. The convergence rates remain consistent and approach the theoretical value as the mesh refines, confirming the expected asymptotic behavior.
- Ulam–Hyers stability: Theorem 5 provides an explicit stability constant
. In practice, the numerical solution remained well-behaved under the perturbations inherent in discretization (with tolerance 10−8 for the boundary condition and 10−6 for the Newton iteration). The errors observed in Table 2 remain bounded and do not exhibit amplification, consistent with the theoretical stability guarantee.
- Impulse handling: The piecewise solution representation derived in Lemma 2 forms the basis for the numerical implementation. The ability of the scheme to accurately capture the prescribed jumps
(as illustrated in Figs 2 and 3) validates the theoretical decomposition of the solution into a sum of accumulated impulses plus a continuous memory component.
- Error growth: Table 2 shows that the absolute error increases from 9.2 × 10−4 at
to 1.1 × 10−2 at
. This gradual error accumulation is typical for numerical methods applied to fractional differential equations, where the memory effect (via the fractional integral) propagates local errors forward. The observed error magnitude remains acceptable for practical purposes and decreases with mesh refinement, as shown in Table 1.
Overall, the numerical results for Example 5.1 are in strong agreement with the theoretical framework, confirming both the mathematical well-posedness of the problem and the reliability of the computational approach.
5.2 Example
(Numerical Validation with Logarithmic Nonlinearity)
Consider the ABC fractional impulsive implicit equation with a logarithmic nonlinearity:
subject to impulses
and the nonlocal condition
Analytical verification
Define the nonlinearity and boundary kernel as
One can directly verify the following:
and
are continuous on their respective domains.
- For any
and
,
which satisfies (H1) with and
.
- For the boundary term,
so (H2) holds with .
The relevant parameters are
where the normalization function is evaluated using the standard formula
.
Substituting these values into the contraction constant defined in (15) yields
By Theorem 3, the problem (39)–(41) admits a unique solution on [0,1]. Moreover, from Theorem 5 the system is Ulam–Hyers stable with stability constant
where m denotes the number of impulse points.
Numerical scheme and implementation
The same L1‑type scheme (38) is employed with :
where denotes the one‑parameter Mittag‑Leffler function, computed using the MATLAB routine ml_fractional. This direct discretization of the integral definition is particularly suitable for the ABC derivative because it naturally incorporates the non-singular Mittag-Leffler kernel
, avoiding the singularity issues present in classical fractional derivative approximations. The numerical algorithm follows the same steps as in Example 1:
- Discretize the interval [0,1] into N uniform subintervals of size
.
- Solve the nonlocal condition (41) for
by a fixed‑point iteration with tolerance 10−8.
- For each time step
:
- If
coincides with an impulse point
(set at
), apply the jump
.
- Compute the ABC derivative via the L1-type scheme.
- Solve the implicit equation (39) using Newton–Raphson iteration (tolerance 10−6).
- If
Numerical results
Table 3 summarizes the convergence behavior for different step sizes. The observed convergence rate approaches the theoretical expectation .
Table 4 compares the analytical and numerical solutions at selected points, demonstrating excellent agreement.
5.2.1 Analysis of numerical results.
The convergence analysis presented in Table 3 shows that the L1-type scheme achieves a numerical convergence rate of approximately 1.18–1.20, closely aligning with the theoretical expectation of . The excellent agreement between the analytical and numerical solutions, as evidenced in Table 4 (with errors on the order of 10−4), further validates the proposed method. Moreover, Fig 4 demonstrates that the numerical solution accurately captures the three impulsive discontinuities at
. Together, these results confirm that the numerical scheme maintains the theoretical convergence order and preserves the stability and impulsive characteristics of the continuous ABC-fractional system, even with logarithmic nonlinearity.
Comparison with theoretical expectations. The numerical results obtained for Example 5.2 are now compared against the theoretical predictions established in Sects 3 and 4:
- Existence and uniqueness: The contraction constant computed from the theoretical condition in Theorem 3 is
, which guarantees a unique solution. This is notably smaller than the value obtained in Example 1 (
), reflecting the more contractive nature of the logarithmic nonlinearity. The numerical scheme converges stably for all tested step sizes, empirically confirming this theoretical guarantee.
- Convergence rate: The L1-type discretization scheme (38) is theoretically expected to exhibit a convergence order of
for sufficiently smooth solutions [22]. With
, the theoretical order is
. The observed convergence rates in Table 3 range from 1.18 to 1.20, which are slightly below the theoretical optimum. This minor discrepancy is consistent with Example 1 and can be attributed to: (i) the reduced regularity caused by impulsive discontinuities at three distinct points (
), and (ii) the numerical approximation of the Mittag-Leffler kernel. The convergence rates remain stable across mesh refinements and approach the theoretical value as the step size decreases.
- Ulam–Hyers stability: Theorem 5 provides an explicit stability constant
for this example. The numerical solution remained well-behaved under the perturbations inherent in discretization (with tolerance 10−8 for the boundary condition and 10−6 for the Newton iteration). The errors observed in Table 4 are consistently small (on the order of 10−4) and do not exhibit amplification, providing empirical support for the theoretical stability guarantee. Notably, the errors in Example 2 are an order of magnitude smaller than those in Example 1, which can be attributed to the smaller contraction constant
and the smoother nature of the logarithmic nonlinearity.
- Impulse handling: The numerical scheme successfully handles three impulse points at
, 0.5, and 0.75 with varying jump magnitudes
, which take values 0, −1/8, and 0 respectively. Fig 4 clearly shows the solution jumping downward at
(where the impulse is non-zero) while remaining continuous at the other impulse points (where the jump is zero). This demonstrates that the numerical implementation correctly respects the piecewise solution representation derived in Lemma 2 and accurately captures impulses of varying magnitudes.
- Error growth and accuracy: Table 4 shows that the absolute errors remain remarkably consistent across the domain, ranging from 3.0 × 10−4 to 4.0 × 10−4. Unlike Example 1, where error accumulated over time, the errors in Example 2 do not exhibit significant growth. This improved accuracy is consistent with the smaller contraction constant
and the more contractive nature of the nonlinearity, which tends to dampen error propagation.
- Nonlinearity effects: The logarithmic nonlinearity in
introduces a mild singularity at
due to the term
. Despite this, the numerical scheme maintains stable convergence and accuracy, demonstrating its robustness to non-smooth nonlinearities. This aligns with the theoretical assumptions (H1) and (H3), which require only Lipschitz continuity and linear growth conditions, both of which are satisfied by the chosen
.
Overall, the numerical results for Example 5.2 are in strong agreement with the theoretical framework. The smaller contraction constant yields improved accuracy and stability compared to Example 1, while the scheme successfully handles multiple impulses with varying magnitudes. These results further confirm that the proposed theoretical conditions are both sufficient and practically relevant for guaranteeing well-posedness and computational reliability.
5.3 Discussion
All computations were performed in MATLAB R2023a, using the ml_fractional toolbox for evaluating the Mittag-Leffler function and custom iterative routines for solving the implicit equations. Both examples validate the theoretical framework and demonstrate the reliability of the numerical implementation. The computed contraction constants guarantee uniqueness and existence, while the explicit stability constants (
for Example 5.1 and
for Example 5.2) quantify the robustness of the solutions to small perturbations.
The L1-type approximation for the ABC derivative, combined with Newton-type solvers for the implicit nonlinearity, proved to be stable and convergent, even in the presence of impulsive discontinuities. The slightly lower observed convergence rates (1.17–1.20 vs. theoretical 1.33–1.40) are attributed to the reduced regularity caused by the impulse jumps and the numerical approximation of the Mittag-Leffler kernel. Importantly, the convergence rates remained consistent across mesh refinements and did not degrade further with additional impulses (Example 5.2), indicating that the method scales well to problems with multiple discontinuities.
The close alignment between theoretical predictions and numerical observations—particularly in terms of the contraction condition, convergence rates, stability constants, and accurate impulse handling—confirms that ABC fractional impulsive systems with implicit nonlinearities and nonlocal boundary conditions are both mathematically well-posed and computationally tractable.
Note on application context: While these examples serve primarily to validate the theoretical results, the mathematical structure of Eqs (35)–(37) and (39)–(41) is motivated by models in systems with memory effects, sudden state changes, and nonlocal feedback. The exponential and logarithmic nonlinearities, combined with ABC fractional derivatives and impulsive effects, can describe various physical phenomena such as: (i) thermal systems with sudden heat injections, (ii) biological neuron models with spike-timing-dependent plasticity, and (iii) mechanical systems with impacts and hereditary properties. The nonlocal boundary conditions model situations where the system state depends on its historical behavior rather than just endpoint values, which occurs in many real-world applications including viscoelastic materials, population dynamics, and control systems with integral feedback.
6 Conclusion
In this work, we analyzed and numerically simulated ABC fractional impulsive differential systems with implicit nonlinear terms and integral boundary conditions. We established existence and stability results for the considered integral boundary value problems by employing a fixed-point framework, providing sufficient conditions for the existence of at least one solution and ensuring H-U stability. The proposed results were supported by numerical examples (Sect 5), which confirmed the theoretical findings and demonstrated the computational feasibility of the approach.
The framework developed here can be effectively applied to a variety of physical models, including those arising in thermal and hydrodynamic systems.
Future research directions include:
- Extension to nonlinear coupled systems with integral boundary conditions
- Investigation of Mittag-Leffler-type derivatives in more complex scenarios
- Development for implicit fractional integro-differential equations
- Application to broader classes of problems in physics and engineering
These extensions would significantly enhance the modeling capabilities for systems with memory effects, discontinuous behaviors, and complex boundary interactions. The current work provides a solid foundation for such future developments in fractional calculus and its applications.
Acknowledgments
This section is intended only for general acknowledgements and thanks. Any information related to funding, data availability, author contributions, etc. should be entered directly into their dedicated fields in the PLOS Editorial Manager submission system, which will then be incorporated into the appropriate section in your article during the production process.
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