1. Introduction
While classical integer-order derivatives remain highly effective for modeling many phenomena, their locality operators may not fully capture certain complex phenomena in applied mathematics that depend on a system’s past states, particularly those with non-local characteristics. Fractional (non-local) derivatives can offer an alternative to the classical framework by offering additional flexibility for describing processes with memory and hereditary effects, including those in epidemiology [
1,
2], viscoelastic materials [
3], and gas-film dynamics [
4]. Nevertheless, it is challenging to solve most fractional differential equations analytically. This situation has led to the development of numerous numerical techniques for solving fractional differential equations. Numerous studies have been conducted on integro-differential equations. The topic of applying various methods to solve integral equations with fractional derivatives has been the subject of multiple prior studies. For instance, fractional integro-differential equations of weakly singular kernels have been solved using the
-collocation method [
5], Galerkin spectral and finite difference methods [
6], the mesh-free methods [
7,
8], the Haar wavelet method [
9], double Laplace transform [
10], parallel-in-time (PinT) algorithm [
11,
12], and high-order finite difference [
13]. Moreover, a
-Galerkin approach to solving the fourth-order partial integro-differential equation with a weakly singular kernel is proposed in [
14].
In this work, we examine the
-collocation and the iterative Laplace methods to solve a time-fractional partial integro-differential equation (FPIDE) with a weakly singular kernel. Let
be a differentiable function. Then, the Caputo-fractional derivative of order
is defined as follows [
15]:
where
is the Riemann–Liouville fractional integral operator is given by the following:
We adopt the Caputo definition of the fractional derivative here because it is well-suited to modeling physical processes with classical initial conditions.
Consider FPIDE with a weakly singular kernel:
where
,
is the Caputo-fractional derivative,
, and
is a given function.
To guarantee the existence of a unique solution of Equation (
2) in
, we assume that
, the source term
, and the kernel
is completely monotone. A comprehensive review of the relevant frameworks and detailed proofs can be found in [
16]. In the rest of the manuscript, we assume that the solution
of (
2) is unique and sufficiently smooth in both time and space, to consider the second-order weighted and shifted Grünwald difference formula to discretize the Caputo derivative in Equation (
2), see, e.g., [
17].
The parameter
is called the order of the memory kernel, and it represents the order of singularity in the kernel
when
. This kind of singularity leads to a convergent integral; hence, Equation (
2) is said to have a weakly singular kernel. In this work, we aim to solve Equation (
2) and discuss the influence of the orders of the fractional derivative
and memory kernel
on the robustness of the proposed methods. This kernel models a secondary relaxation mechanism that decays, allowing Equation (
2) to capture both instantaneous and retarded diffusion. Applications include anomalous tracer transport in fractured media [
18], thermal waves in polymers [
19], and charge migration in amorphous semiconductors [
20]. Recent analytical and numerical works study related initial–boundary value problems [
21,
22], underscoring the current interest in mixed Caputo–Volterra models. In [
5], the authors studied the solution of Equation (
2) when
. However, varying
in
will allow testing the accuracy of our proposed methods across a range of kernel singularities as the kernel becomes sharper when
or milder when
[
23].
The paper is organized as follows:
Section 2 presents the spatial discretization of
in Equation (
2) using the
-collocation method, along with related convergence results. In
Section 3, the time-fractional derivative and integral term in Equation (
2) are approximated using the weighted and shifted Grünwald operator and a quadrature formula based on the product trapezoidal integration rule, respectively.
Section 4 establishes the convergence results of the proposed method. An overview of the iterative Laplace transform method is provided in
Section 5.
Section 6 presents two numerical examples that illustrate the implementation of the proposed method and its comparison with the iterative Laplace transform method. Finally,
Section 7 discusses the obtained results.
2. The -Collocation Method for Spatial Discretization
In this section, we provide notations and definitions and review the main results for the
-collocation method [
24,
25]. Throughout this manuscript, we denote the set of all integers, the real line, and the complex plane by
,
, and
, respectively.
First, we begin by defining the
function, which is a function defined on the entire set of real numbers
by
For the
-collocation method, we derive its basis from the Whittaker cardinal functions
for
and any step size
. Note that the function
is a shifted and scaled version of the function
, that is, it is symmetric about
. More precisely, the parameter
j introduces a horizontal shift, moving the center of symmetry of
from
to
, while
scales the function accordingly, see
Figure 1.
Consequently, for any function
, we define the series
as the Whittaker cardinal expansion of
f, provided it converges [
26]. Fix
. For this fixed
d, we choose the infinite strip-shaped region
in the complex plane
to provide a domain for the Whittaker cardinal expansion that guarantees convergence. The choice of
d, such that
, is related to the growth constraints of
f of exponential type less than
, i.e.,
with
and
.
Let
, the set of all positive integers, and choose a step size
appropriately depending on
such that the Whittaker cardinal expansion
is convergent, then the function
f is approximated by (see
Figure 2)
Note that Equation (
2) is focused on the interval
. To transform the basis functions on the interval
, we consider a conformal map, which is a complex function that preserves angles locally, that is, it is analytic and its derivative never vanishes [
27]. We take the conformal map [
24,
25]
which transforms the simple connected region (see
Figure 3)
to the strip-shaped region
. In this case, the branch of the logarithm is chosen such that the argument of
is restricted to
, mapping the boundaries of the
z-domain to the lines
(see
Figure 3). It is easy to check that
Note that
, i.e., on the boundary of
, with
and
. Consequently, we set (see
Figure 3)
Let
and consider the uniformly distributed points
on the real line. Then, the corresponding points
are (see
Figure 3)
Hence, the basis functions on
are
Note that these
functions satisfy
To approximate the function
f by a finite series on
using
functions in (
4)
we provide the following conditions on
f to guarantee convergence.
Definition 1 ([
24]).
Let , and for setHere, . Then, we define the following families of functions and on : The following theorem provides an approximation of the
p-derivative of
[
25].
Theorem 1. If the conformal map ϕ is one-to-one and . Then, for all , we haveMoreover, when and , then there exists such thatfor . The constant depends on p, ϕ, d, κ, and f. Note that Theorem 1 provides an approximation of the
p-derivative of
with an exponential convergent, that is,
Recall that
. Hence, at
, we have
Let
. To help represent discrete systems, we define the column vectors
where
, and the following
matrices, the diagonal matrix
and using (
5), we have
The matrix
is the identity matrix,
is the skew symmetric Toeplitz matrix
and
is and the symmetric Toeplitz matrix
Hence, we can write Equation (
7) as
Consequently, we approximate
in (
2) by (
7) or (
8)
or
respectively, where
3. The Temporal Discretization
In this section, we approximate the time-fractional derivative and integral term in (
2). Let
and set
. Denote
,
, and
for
.
To approximate the time-fractional derivative in (
2), we use the weighted and shifted Grünwald difference (WSGD) operator. Suppose that
and
. Then, the shifted Grünwald difference operator is defined by [
28]
where
and the coefficients
can be evaluated recursively as
Moreover, we have
uniformly for
as
.
Theorem 2 ([
29])
. Assume , , and belong to the Lebesgue space , where denotes the Fourier transform. Then, the weighted and shifted Grünwald difference operator is defined bywhere m and r are integers such that . Then,uniformly for as . Furthermore, the m and r are symmetric, that is, . Theorem 2 implies that the discrete approximations for Riemann–Liouville fractional derivatives when
is simplified as [
30]:
where
and
For the integral term in (
2) involving the weakly singular kernel, we use a quadrature approximation based on a product trapezoidal integration rule [
31,
32]. For
, we have
where
and
is the order of the product trapezoidal integration rule [
32]. Consequently, we have
where
Substituting Equations (
11) and (
12) into Equation (
2) leads to a temporal semi-discrete form of Equation (
2) as follows:
where
. Dropping the error term
, we obtain
with
,
, and
. Suppose that the approximate solution to Equation (
13) using Equation (
9) is given by
At
, denote
for
. Now, substituting
in Equation (
13) leads to
with initial condition
. Multiplying Equation (
15) by
gives
Then, the matrix form of Equation (
16) is
Consequently, we write it as
where
Then, Equation (
17) can be written as the iteration form
with the initial condition
For each
, the iteration Equation (
19) forms a system of
linear equations and
variables. The coefficients of the approximate solution in Equation (
14) can be obtained by solving this system.
Note that due to its diagonal shape and strictly positive entries, the matrix
is invertible and positive definite. A discretized second derivative gives rise to the symmetric positive semi-definite matrix
. Since the coefficient
is small for sufficiently small time steps
, the matrix
is a minor perturbation of a positive definite matrix. Thus,
is, therefore, invertible, and hence, Equation (
19) has a unique solution
.
4. Stability and Convergence Analysis
Now, we analyze the convergence of the iteration Equation (
19) for the FPIDE Equation (
2). To this end, we rewrite, for simplicity, Equation (
13) as an ordinary differential equation
where
taking into account the boundary conditions
. Suppose that
is the exact solution of Equation (
20), which satisfies Equation (
2) at the
-th time step. Assume that
be the approximate solution of Equation (
20) using the
-collocation formula in Equation (
14). At
, the solution of Equation (
2) is computed by
In the following, we first determine a suitable upper bound for , then we establish an upper bound for .
The following result from [
33] provides the necessary upper bound for
.
Theorem 3. Denote by the conjugate transpose of the matrix in Equation (17). Let , then we havewhere Moreover, if the eigenvalues of the matrix are nonnegative, then there exists , which is independent of , such that for a sufficiently large , we have Note that Theorem 3 implies that the matrix is bounded as long as does not vanish and is sufficiently large. Consistently, we have the following result.
Theorem 4. The numerical scheme in Equation (19) is stable. Proof. Suppose that
, with solution
in Equation (
19), contains a small error
. Denote
and the corresponding solution
. Hence, Equation (
19) gives
Define the error
. Then,
Thus, the error in the perturbed solution is bounded, and hence, the numerical scheme in Equation (
19) is stable. □
Now, we find an upper bound for in the following result.
Theorem 5. Suppose (resp. ) is an approximate solution of Equation (13) (resp. Equation (20)). Then, there exists a constant , which is independent of , such thatwhere . Proof. Applying the Cauchy–Schwarz inequality to
gives
Since
where
is a constant independent of
. Denote
Using the iteration Equation (
19), we obtain
To find an bound for
, we denote
Then, Equation (
20) gives
Then, Theorem 1 implies that there exist two constants
and
, which are independent of
, such that
where
. Since,
it follows from Equation (
23) that
Now, using the upper bound of
in Theorem 3, we obtain
Hence,
where
□
In the following theorem, we establish an upper bound for .
Theorem 6. Suppose that be the exact solution of (20), which satisfies the Equation (2) at the -th time step. Assume that be the approximate solution of Equation (2) using the -collocation method in Equation (14). Then, there exists a constant , which is independent of , such thatwhere . Proof. Note that the triangular inequality implies that
Then, we have from Theorems 1 and 5 that there are constants
and
, which are independent of
, such that
and
respectively.
Consequently, we have
where
. □
5. Iterative Laplace Transform Method
For comparison purposes, and since the kernel of the integral equation under study is of the convolution type, we will take advantage of this property and find another approximate solution using an iterative method linked with the Laplace transform, called the Iterative Laplace Transform Method (ILTM). Because of its computational efficiency and ability to treat weakly singular kernels and convolution terms, ILTM has become a widely used technique for solving fractional integro-differential equations. The accuracy and convergence of the method have been examined in depth (see e.g., [
34,
35]). These studies report that ILTM typically attains a super-linear convergence rate and converges rapidly for linear problems with smooth initial and boundary conditions. In the following, we illustrate and describe the properties of this iterative method.
Definition 2 ([
36])
. The Laplace transform of the Caputo-fractional derivative of order α of the function is given bywhere is the order derivative of at . First, we introduce the method when there is a nonlinear term
in Equation (
2). Consider the nonlinear time-fractional integro-differential equation
subject to the same initial and boundary conditions in Equation (
2). We apply the Laplace–Adomian Decomposition Method (LADM) [
34] and solve Equation (
24). Taking the Laplace transform of both sides in Equation (
24) and using the convolution property of the Laplace transform gives
Consequently, we obtain from Equations (
25) that
Dividing by
and using the boundary conditions in Equations (
24) imply
Assume the nonlinear term can be written as
where
are the Adomian polynomials and given by
Then, using the iterative solution
implies
We now introduce the following recurrence relation [
34]:
Let
, we obtain an approximate solution in series form as
Assume that the initial and boundary conditions in Equation (
2) are smooth. Then, since there is no nonlinear term, the iterative Laplace transform method converges.
Note that when there is no nonlinear term
as in Equation (
2), the method is called
Iterative Laplace transform method [
35] and the recurrence relation Equation (
26) becomes
For instance, when
, we have
6. Numerical Results
In this section, we present two numerical examples to demonstrate the validity and accuracy of the proposed method. We set , , , and .
Example 1. Choose in Equation (2), and consider the FPIDE with a weakly singular kernel:with . Then, the exact solution is It is well established in fractional calculus theory that for fractional derivatives with orders
, the approximate solution converges continuously to the exact solution of the corresponding problem when
[
37].
Figure 4 and
Figure 5 illustrate that the exact solution corresponding to
is quite close to the approximate solution when
and
. We note that the error distribution aligns with the behavior of the exact solution, with higher errors where the solution has its lowest magnitude when
. Moreover, the behavior of the approximate solutions across different values of
is consistent with that of the exact solution when
. These observations underscore the efficiency and robustness of the approximate solution method. Moreover,
Figure 4 and
Figure 5 show certain patterns in the numerical results. The tendency for errors to become more pronounced near
and the natural accumulation and propagation of discretization errors, which may become more evident in regions with significant solution dynamics or evolving gradients as the simulation progresses.
Applying the recurrence relation for the iterative Laplace transform method in Equation (
26) leads to
and, in all cases,
, for all
. Hence, the approximate solution is
Thus, the approximate solution converges to the exact solution after two iterations. This is because the structure of the function
which consists only of integer powers of
x. As a result, the term
will disappear after a finite number of derivatives, ensuring the rapid convergence of the approximation.
Example 2. Consider the FPIDE with a weakly singular kernel:with . Then, the exact solution is In
Figure 6,
Figure 7,
Figure 8,
Figure 9,
Figure 10 and
Figure 11, we plot the exact and approximate solutions along with
-error and absolute error. The same qualitative behavior is observed as in Example 1 when
is decreased. We note that, for
, the exact solution remains close to the approximate one for values of
near 1. As
decreases further, the solution curve shifts upward, which results in larger error values; see
Figure 6,
Figure 8 and
Figure 10. Furthermore, the absolute error distribution aligns with the shape of the exact solution, reaching its largest values where the solution itself is largest—particularly at
, as illustrated in
Figure 7,
Figure 9 and
Figure 11. Additionally, we observe that the error decreases as
increases.
For comparison in this example, we apply the recurrence relation for the iterative Laplace transform method in Equation (
26) when
and
. Consequently, we have
Hence, the approximate solution is
Table 1 shows that both the
-collocation and iterative Laplace transform methods provide good approximations to the exact solution of the FPIDE in Example 2 at
when
and
. However, the iterative Laplace transform method outperforms the
-collocation method in terms of accuracy across all tested
points.
The temporal errors of the suggested method at the spatial point
for different time step sizes
with fixed parameters
,
, and spatial resolution
are shown in
Table 2. We can calculate the maximum norm of the error
at different time steps. From
Table 2, we observe that when the time step size
is halved, the corresponding error is reduced by approximately a factor of 40. This behavior indicates that the spatial error, which is of exponential order, consequently took center stage, hiding the actual temporal convergence and producing false results.
7. Conclusions
In this article, we have solved a time-fractional partial integro-differential equation (FPIDE) with a weakly singular kernel using the -collocation and iterative Laplace transform methods. The -collocation method has been employed to discretize the spatial domain, while a combination of numerical techniques has been utilized for temporal discretization. Consequently, a symmetric discrete system of equations has been developed. Subsequently, an upper bound on the error was determined, and a convergence analysis was conducted. Since the considered FPIDE has the convolution property, we have applied the iterative Laplace transform method to solve it. For comparison, we have considered two numerical examples. We have observed that the absolute error distribution of the -collocation method exhibits an almost perfect symmetry about the spatial domain’s midpoint due to the discrete system’s symmetrical property. Moreover, we have noted that the approximation solution with various fractional-order exhibits the same behavior as the exact solution for which . Within the theory of fractional calculus, it is evident that the approximate solution consistently tends to the exact solution of the problem when the fractional derivative tends to 1, Additionally, the parameter , the order of singularity inside the kernel in the FPIDE, has shown effects on the maximum error with different . Numerical experiments have demonstrated that both the -collocation method and the iterative Laplace transform approach yield accurate approximations to the exact solution.
Future research may include fractional integro-differential equations with nonlinear term and tempered kernels , to model systems with fading memory, where past influences decay exponentially over time. Due to its exponential damping, this kernel circumvents the infinite-memory problem of classical fractional models. It is used in applications such as viscoelastic materials, financial models with finite variance, and anomalous diffusion in heterogeneous media. Moreover, we may extend the theoretical analysis to two- and three-dimensional problems, as well as semi-linear formulations.