2.1 Normalization to the unit interval

[5,23]. As the physical domain of the problem is [a, T], we map Hermite polynomials to the unit interval [0, 1] via the linear transformation . For [0, 1], this yields . The basis functions are for , denoted . This mapping sacrifices orthogonality but preserves linear independence, and the set remains a complete basis for L²(0,1) [23,28].

2.2 Residual error analysis

Following established practice in numerical analysis [29], we complement our solution error analysis with residual error measurements. The residual provides a fundamental assessment of how well the numerical solution satisfies the original governing equation.: We consider the nonlinear fractional Volterra–Fredholm integro-differential equation given in Problem (1). For clarity, the problem is rewritten in operator form as

(4)

where the nonlinear operator is defined by

Here, X denotes a suitable Banach space such as C[a,T] or L²[a,T].

Let be the exact solution of (4), and let be its numerical approximation obtained using the proposed method with N basis functions.

2.2.1 Residual definition.

The residual associated with the numerical solution is defined as

(5)

By definition of the exact solution, for all . Hence, measures the deviation of the numerical approximation from satisfying the original governing equation.

2.2.2 Residual error norms.

To quantify the residual error, we introduce the following norms:

(6)(7)

The L^∞-norm captures the maximum pointwise defect, while the L²-norm measures the global mean-square consistency of the numerical approximation.

2.2.3 Analytical assumptions.

We assume the following standard conditions:

1. (A1) The fractional derivative operator is linear and bounded on X.
2. (A2) The kernel functions and are continuous and bounded.
3. (A3) The operator satisfies a Lipschitz condition:

for some constant L > 0.

Lemma 1 (Residual Identity). Let 𝕜 be the exact solution of (4) and its numerical approximation. Then

Proof 1 Since by definition of the exact solution and , we obtain

which completes the proof.

Theorem 1 (Residual-Based Error Estimate). Under assumptions (A1)–(A3), there exists a constant C > 0, independent of N, such that

(8)

Proof 2 By Lemma 1 and the Lipschitz continuity of , we have

If is invertible and its inverse is bounded, then

Setting yields the desired estimate.

Corollary 1 (Convergence) If

then

2.3 Relationship between residual and true error

The residual error and true error are two distinct but related measures of numerical accuracy.

• Residual Error: The residual measures how well the approximate solution k_N(s) satisfies the governing equation (4). It is defined as
  A small residual indicates that the numerical solution closely satisfies the original equation pointwise.
• True Error: The true error E(s) measures the deviation of the approximate solution from the exact (analytical) solution k(s):
  This metric is only computable when the exact solution is known.
• Relationship: Under the assumptions (A1)–(A3) and if the operator is invertible with bounded inverse, the residual provides an upper bound for the true error (Theorem 1):
  Thus, a decreasing residual norm implies a decreasing true error norm, ensuring convergence of the method.
• Role in Accuracy Assessment:
  1. The residual is a reliability indicator—it assesses the consistency of the numerical solution with the governing equation, even when the exact solution is unknown.
  2. The true error is a validation metric—it quantifies the actual accuracy of the approximation when an exact solution is available.
  3. In practice, the residual is often used to monitor convergence and stability, while the true error is used to verify and benchmark the method.

In the numerical experiments (Section 5), we report both residuals and true errors where exact solutions are known. The consistent decay of both metrics with increasing N confirms the robustness and spectral accuracy of the proposed Hermite least-squares method.

3.1 Mathematical formulation

Having defined the Hermite polynomial basis in Section 2, we now describe the least-squares algorithm for solving the generalized fractional Volterra–Fredholm integro-differential equation (VFIDE):

(9)

subject to the initial conditions:

(10)

We seek an approximate solution in the finite-dimensional subspace , expressed as

(11)

where {H_i} are the orthonormal Hermite functions defined previously. The method substitutes into (9) to form a residual, then minimizes its L²-norm to determine the optimal coefficient vector .

3.2 Theoretical analysis of the hermite spectral method

To analyze convergence, we map the interval [a,T] to [0,1] via with weight , and consider the Hermite expansion of the exact solution:

Lemma 2 (Uniform boundedness of coefficients) For any solution ,

Proof 3 By Cauchy–Schwarz: .

Lemma 3 (Coefficient Decay for VFIDE Solutions) Let 𝕜 be the solution of the fractional VFIDE (1) with initial conditions (2), and consider its Hermite expansion on [0,1] where , with {H_i} orthonormal in , .

1. If for some m > 0, then there exists C_m > 0 such that
2. If 𝕜 is analytic on [0,1], then there exist constants C,c > 0 such that

Proof 4 We prove both statements by relating the coefficient decay to the regularity of 𝕜.

Proof of (1): Algebraic decay for solutions.

Let denote the orthogonal projection onto , the space of polynomials of degree at most i−1. Since H_i is orthogonal to with respect to , we have

Thus,

The approximation error by polynomials in weighted L² satisfies the following estimate for functions in : there exists a constant K_m > 0 such that

This follows from standard polynomial approximation theory in weighted Sobolev spaces (see, e.g., the theory of orthogonal polynomials on intervals with Gaussian weight). Since is the best approximation in , we obtain

which is the desired inequality with .

Proof of (2): Exponential decay for analytic solutions.

Assume 𝕜 is analytic on [0,1]. Then 𝕜 extends to an analytic function in a complex neighbourhood of [0,1]. In particular, there exists such that 𝕜 is analytic in the Bernstein ellipse with foci at 0 and 1 and sum of semi-axes .

For analytic functions on [0,1], the Hermite coefficients decay exponentially. To see this, note that H_i(s) grows like for large i. Using the orthogonality relation

we can shift the contour of integration into the complex plane. Because 𝕜 is analytic in , we may deform the path to exploit cancellation. Standard complex analysis arguments (Cauchy’s theorem combined with bounds on Hermite functions in the complex plane) yield

For Hermite functions, one has the bound for some constants A,B > 0. Hence,

for suitable C,c > 0. In fact, a more refined analysis (using the Plancherel–Rotach asymptotics of Hermite functions) improves this to

which is the claimed exponential decay.

Regularity of the VFIDE solution. The lemma assumes the stated regularity of 𝕜. For the fractional VFIDE (1)–(2), standard theory guarantees that if all data () are (resp. analytic), then the solution 𝕜 belongs to for every m > 0 (resp. is analytic). This follows from the smoothing properties of fractional integral operators and the structure of the equation. Thus the decay estimates apply under natural regularity assumptions on the problem data.

Theorem 2 (Convergence of Hermite approximation) Let be the solution of (9)–(10). Then , with algebraic rate for and spectral (exponential) convergence for analytic 𝕜.

Proof 5 The error satisfies . Using Lemma 3:

• For : , giving .
• For analytic 𝕜: , giving .

Since , convergence follows. Under standard assumptions (continuous data, Lipschitz kernels, continuous with ), the solution 𝕜 belongs to for some m > 0 (or is analytic if all data are analytic).

3.3 Implications for the least-squares method

The least-squares coefficients satisfy with as . Thus, the decay estimates of Lemma 3 also apply to , and the least-squares solution satisfies

so Theorem 2 applies to as well. This guarantees that the computed approximation converges to the exact solution with the same rates.

3.4 Discussion

This analysis provides a rigorous foundation for the Hermite spectral method applied to fractional VFIDEs. The coefficient bounds guarantee convergence in , with algebraic convergence for limited regularity (typical for fractional problems) and spectral convergence for analytic solutions. This justifies the least-squares approach: the approximation space can represent the solution accurately, and the computed coefficients are stable and well-behaved.

3.4.1 Formulation of the residual function.

Substituting the approximation (11) into the VFIDE (9) yields:

(12)

Moving all terms to the left-hand side, we obtain the equation residual :

(13)

Interchanging the order of summation and integration, the residual can be compactly expressed as:

(14)

where we define the operator-applied basis functions and the source term for clarity:

This can be expanded explicitly to show the complete structure:

(15)

3.4.2 The least-squares minimization principle.

The optimal coefficients are those that minimize the L²-norm of the residual over the domain [0,1]. We therefore define the objective functional:

(16)

A necessary condition for the minimization of S is that its gradient with respect to the coefficient vector a vanishes. This leads to the system of equations:

(17)

Computing these partial derivatives yields the normal equations:

(18)

This constitutes a system of linear algebraic equations. Explicitly, for each , we have:

(19)

3.4.3 Matrix formulation and theoretical foundations.

The system of normal equations (18) can be written in the compact matrix form:

(20)

where the matrix and vector have entries given by:

(21)(22)

s established in the analysis (Sec. 4, Theorem 3), the matrix M is symmetric and positive semidefinite. If the functions are linearly independent, M is positive definite and the system (20) has a unique solution. This positive definiteness ensures the stability of the numerical method and guarantees the existence of a unique minimizer for the least-squares problem.

The following flowchart illustrates the overall computational procedure:

3.5 Effect of the condition number on numerical stability

The least-squares formulation of the proposed Hermite polynomial method leads to a linear system of algebraic equations of the form

(23)

where the matrix is symmetric and positive definite, with entries

A key indicator of the numerical stability of this system is the condition number of M, defined by

(24)

where denotes the matrix 2-norm. In practice, is computed as the ratio of the largest to the smallest singular value of M (which coincides with the ratio of extremal eigenvalues for symmetric positive definite matrices), i.e.,

The condition number measures the sensitivity of the numerical solution to perturbations in the data or rounding errors. A small value of indicates a well-conditioned system, ensuring numerical stability and reliable coefficient computation. Conversely, a large condition number implies that small numerical errors may be significantly amplified, potentially degrading the accuracy of the approximate solution.

In the present method, the condition number generally increases with the number of Hermite basis functions N, due to the loss of orthogonality after mapping the basis to the finite interval [0,1] and the growing linear dependence among the operator-applied basis functions . Nevertheless, for moderate values of N, the computed condition numbers remain within acceptable bounds, and the numerical results demonstrate stable convergence. This behavior is consistent with the observed decay of the residual norms and confirms that the proposed least-squares scheme is robust and reliable for solving fractional Volterra–Fredholm integro-differential equations.

The computed values of for increasing N are reported in the numerical experiments section (Section 5), where their growth is analyzed alongside the convergence of error norms to assess the practical stability of the method.

4.1 Problem formulation and function spaces

We consider the VFIDE given by Eq. (1) with initial conditions (2). Let and denote by the Caputo derivative. Let be a Banach space (e.g., with initial conditions incorporated), equipped with norm .

Define the source term

We introduce the integral operators

and the main operator

With these definitions, the VFIDE (1)–(2) can be written in compact operator form as

We assume:

1. The kernels are continuous, so are bounded.
2. is continuous.
3. Consequently, is a bounded linear operator.
4. The exact solution exists and is unique for the given source term .

4.2 The least-squares formulation

The Least Squares Method seeks an approximate solution that minimizes the L²-norm of the residual [30]. Let be our finite-dimensional trial space, constructed from the mapped Hermite polynomials. We seek

that minimizes the functional:

4.3 Existence and uniqueness of solutions

Theorem 3 Let with . Assume:

1. (A1)
2. (A2) with
3. (A3) with , similarly for
4. (A4) continuous with

Then the fractional VFIDE (1) with initial conditions (2) has a unique solution .

Proof 6 Step 1: Define operator and space.

Let with norm . Define by:

where .

Step 2: Show maps X to X.

Since all functions are continuous and integrals preserve continuity, is continuous. The initial polynomial ensures initial conditions.

Step 3.For and any :

where

.

Step 4: Show contraction. For any :

Taking supremum:

Step 5: Apply Banach’s theorem.

For , is a contraction. By Banach’s fixed point theorem, has a unique fixed point in X, which is the unique solution.

4.4 Convergence analysis

Theorem 4 Let be a basis for the approximation space . If:

1. (B1) The exact solution
2. (B2) The basis functions are complete in L²(a,T)
3. (B3) The operator is bounded then the least-squares approximants satisfy:

where .

Proof 7 Step 1: Density argument.

By assumption (B2), the spaces are dense in L²(a,T). Since , for any , there exists and such that:

where is the operator norm of .

Step 2: Operator continuity.

Since A is bounded by (B3), we have:

Step 3: Best approximation property.

The least-squares solution minimizes the residual:

For , we have , so:

But , so:

Therefore:

This proves the convergence.

4.5 Stability analysis

Theorem 5 Let with . Let and be the least-squares solutions for the problems with right-hand sides r and respectively. Then:

Proof 8 Let a and be the coefficient vectors of and in the basis , satisfying the normal equations:

where and , .

Step 1: Difference of right-hand sides.

The difference satisfies:

Step 2: Orthogonality property.

The residual is orthogonal to :

Similarly for :

Subtracting gives:

Step 3: Stability estimate.

Consider the norm:

Dividing both sides by (if nonzero):

If , the inequality holds trivially.

4.6 Discrete solution properties

Theorem 6 Under assumptions (A1)-(A4), the matrix B with entries is symmetric positive semidefinite. If the set is linearly independent, then B is positive definite and the normal system has a unique solution.

Proof 9 Symmetry: .

Positive semidefiniteness: For any :

Positive definiteness: If is linearly independent, then:

Thus B is positive definite and invertible, guaranteeing a unique solution to the normal equations.

4.7 Definition and role of the stability factor

To assess the numerical stability of the least-squares formulation, we introduce the stability factor , defined as:

where a is the coefficient vector from Ma = b in Eq. (20). This ratio measures the amplification of the right-hand side norm into the solution norm, analogous to the matrix condition number but tailored to our specific discretization.

In practice, is computed directly from the solved linear system for each approximation order N. A moderate or slowly growing indicates that the numerical scheme does not excessively amplify perturbations in the data, which is crucial for maintaining accuracy when fractional or integral operators are discretized.

The condition number (Eq. (24)) and the stability factor complement each other: while reflects the inherent sensitivity of the matrix inversion, captures the actual solution behavior in the chosen basis. As shown in Tables 3, 5, 7, and 10, remains well-controlled even as increases with N, confirming that the Hermite least-squares formulation remains robust and that numerical errors do not degrade the solution accuracy significantly.

5.1 Method application

To approximate the solution, we expand in terms of normalized Hermite polynomials. Thus, the approximate solution of degree 3 is written as

(29)

where denote the Hermite basis functions.

Substituting the trial solution (29) into the governing equation (25) leads to the residual function

(30)

In order to minimize the error introduced by the approximation, we define the functional

(31)

which measures the squared residual over the domain.

Minimizing S(a) requires solving the conditions

At the same time, we must enforce the initial condition (26) on the approximate solution (29). Substituting into (29) yields

(32)

The fractional derivatives in (30) were evaluated numerically using the Caputo definition with quadrature rules. The integrals were computed using adaptive Gaussian quadrature.

Substituting these into the residual definition, the system matrix and right-hand side were assembled as

where

Hence, we obtain the linear system

subject to the constraint (32).

5.2 Coefficient determination

Solving the constrained least-squares problem yields the coefficients

Finally, substituting these values into (29) and recalling the explicit Hermite polynomials

the approximate solution simplifies to

5.3 Numerical results and discussion

5.4 Computational complexity and efficiency

As the degree N of the Hermite polynomial expansion increases, the computational cost of the proposed least-squares method grows primarily due to the assembly and solution of the normal equations. Specifically, the construction of the system matrix requires the evaluation of (N + 1)² inner products involving fractional derivatives and integral operators, leading to a computational complexity of for matrix assembly. Solving the resulting dense linear system incurs an additional cost of using direct solvers. Moreover, as observed in the numerical experiments, the condition number of the system matrix increases with N, which may affect numerical stability for very large polynomial degrees. Nevertheless, the method achieves high accuracy with relatively small values of N, owing to the strong approximation properties of Hermite polynomials. Consequently, the rapid convergence offsets the increased per-degree cost, making the proposed approach computationally efficient for practical problem sizes.

5.5 Discussion

The numerical results comprehensively validate the effectiveness of the Hermite least-squares method. Table 1 quantitatively demonstrates the convergence of approximate solutions toward the exact solution as increases. This visual convergence is further illustrated in Fig 1, which shows the close agreement between exact and approximate solutions.

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Table 1. Comparison of exact and approximate solutions for , , and N = 7.

https://doi.org/10.1371/journal.pone.0346080.t001

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Fig 1. Comparison of exact and approximate solutions for different values of N.

The approximate solutions converge to the exact solution as N increases, demonstrating the effectiveness of the fractional Laguerre polynomial approximation method.

https://doi.org/10.1371/journal.pone.0346080.g001

The error analysis in Table 2 and Fig 2 reveals a rapid reduction in error, with the maximum pointwise error decreasing from to as increases from 3 to 7. This exceptional convergence rate is further confirmed in Table 3 and Fig 3, where both L^∞ and L² errors decay at rates significantly faster than algebraic convergence, indicating clear spectral convergence characteristics.

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Table 2. Errors for approximate solutions with , , and N = 7.

https://doi.org/10.1371/journal.pone.0346080.t002

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Table 3. Convergence and Stability Analysis.

https://doi.org/10.1371/journal.pone.0346080.t003

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Fig 2. Errors for approximate solutions.

https://doi.org/10.1371/journal.pone.0346080.g002

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Fig 3. Numerical analysis of fractional integro Hermite polynomial approximation:

(a) Error convergence showingL^∞ and L² errors with reference line; (b) Exponential growth of condition number; (c) Convergence rate as a function of N; (d) Stability factor evolution.The results demonstrate the spectral convergence properties and numerical stability characteristics of the method.

https://doi.org/10.1371/journal.pone.0346080.g003

Despite the computational challenge of ill-conditioning—evidenced by the exponentially growing condition number (reaching for in Table 3)—the method maintains numerical stability and delivers accurate results. This demonstrates the robustness of the least-squares formulation in handling ill-conditioned systems.

Overall, the method achieves remarkable accuracy (approximately 10⁻⁴ error) using only seven basis functions, highlighting its computational efficiency for solving fractional integro-differential equations. The combination of spectral convergence and effective stability management makes this approach particularly valuable for problems requiring high precision with minimal computational resources.

Example 2 Consider the following fractional Volterra-Fredholm integro-differential equation of order on the interval :

(33)

subject to the initial condition

(34)

To ensure the problem has an exact solution, we define

(35)

Substituting the exact solution (35) into the governing equation (33) and solving for gives

(36)

Thus, the original problem (33) is equivalent to the general formulation (1) with the parameter identifications:

5.6 Method application

To approximate the solution, we expand in a Hermite polynomial basis of degree 4:

(37)

where are normalized Hermite polynomials.

Substituting (37) into the governing equation (33) yields the residual function

(38)

To minimize the error, we define the functional

(39)

The minimization conditions () yield a system of equations.

In addition, the initial condition (34) imposes the constraint

(40)

The fractional derivatives were computed using the Caputo definition with quadrature adapted to weakly singular kernels, while the integrals in (38) were evaluated analytically for each .

The system matrix and right-hand side were then assembled as

where

This yields the linear system Ma = b subject to constraint (40).

5.7 Coefficient determination

Solving the constrained least-squares problem yields the coefficients

Substituting these into the approximation (37) gives

(41)

Using the explicit forms

the approximate solution simplifies to

5.8 Numerical results and discussion

5.9 Discussion

The numerical results validate the effectiveness of the proposed Hermite polynomial-based least-squares method for solving Volterra–Fredholm fractional integro-differential equations. Both Table 4 and Fig 4 confirm that even with a small number of basis functions (), the approximate solution closely matches the exact solution, with the maximum absolute error on the order of 10⁻⁴. The pointwise error distribution in Fig 5 further highlights the accuracy of the scheme across the domain.

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Table 4. Comparison of exact and approximate solutions () for new example.

https://doi.org/10.1371/journal.pone.0346080.t004

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Fig 4. Comparison of exact and approximate solutions for .

https://doi.org/10.1371/journal.pone.0346080.g004

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Fig 5. Pointwise absolute error between exact and approximate solutions ().

https://doi.org/10.1371/journal.pone.0346080.g005

The convergence and stability characteristics are summarized in Table 5 and illustrated in Fig 6. Three key observations can be drawn:

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Table 5. Convergence and Stability Analysis.

https://doi.org/10.1371/journal.pone.0346080.t005

• Algebraic Convergence: Both L^∞ and L² errors decay steadily with increasing , with an observed convergence rate of about 1.4. This algebraic rate, lower than exponential convergence typically seen with spectral methods, is consistent with theoretical expectations for fractional operators, since the nonlocal nature of fractional derivatives reduces solution smoothness and limits the efficiency of globally smooth polynomial bases.
• Stability and Conditioning: The condition number grows moderately with , while the stability factor increases at a manageable rate. These trends indicate that the system remains well-posed and numerically stable within the tested range, reflecting the robustness of the least-squares formulation.
• Efficiency: Despite the algebraic convergence, the method achieves high accuracy (errors on the order of 10⁻⁵ to 10⁻⁶ for larger ) with relatively few basis functions, making it more efficient than traditional low-order methods such as finite differences, which would require much finer discretizations to reach comparable accuracy.

In summary, the Hermite polynomial-based least-squares method provides a stable and efficient framework for fractional integro-differential equations. The observed algebraic convergence is not a limitation of the method but rather a reflection of the fractional operator’s intrinsic properties. The method offers high accuracy with modest computational effort, confirming its potential for practical applications.

Example 3 Consider the following fractional Volterra-Fredholm integro-differential equation of order on the interval :

(42)

This problem is equipped with the initial conditions:

(43)

To ensure the problem is well-posed, we prescribe an exact solution of the form

(44)

By substituting the exact solution (44) into the governing equation (42), the unknown function can be identified. This calculation yields:

(45)

Thus, the specific problem (42)–(45) can be viewed as a special case of the general formulation (1) with the following parameter identifications:

5.10 Method application

To approximate the exact solution (44), we expand in terms of normalized Hermite polynomials. Specifically, the approximation is taken as:

(46)

where the unknown coefficients a_i are to be determined.

Next, to quantify the quality of the approximation (46), we substitute it into the governing equation (42). This leads to the definition of the residual function:

(47)

The optimal coefficients are those that minimize the total squared residual over the computational domain. Therefore, we define the functional:

(48)

In addition, the initial conditions (43) impose constraints on the coefficients of the expansion (46). By substituting and differentiating, we obtain:

(49)

(50)

To evaluate the terms in (47), the fractional derivatives were computed using the Caputo definition with Gauss–Jacobi quadrature for singular integrals, while the remaining integrals were approximated with adaptive Gaussian quadrature.

As a result, the system of equations for the coefficients a_i takes the standard least-squares form:

where

and

The resulting constrained linear system Ma = b is then solved subject to the conditions (49)–(50).

5.11 Coefficient determination

Solving the constrained least-squares problem yields the coefficients For N = 5, solving the system yields the coefficients:

Finally, substituting these values into the expansion (46) and making use of the explicit forms of the Hermite polynomials:

the approximate solution simplifies to:

5.12 Numerical Results and Discussion

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Fig 6. Comprehensive analysis of (a) error convergence, (b) condition number growth, (c) convergence rate, and (d) stability factor.

https://doi.org/10.1371/journal.pone.0346080.g006

5.13 Discussion

The numerical results demonstrate the efficacy and robustness of the proposed Hermite polynomial-based least-squares method for solving fractional Volterra–Fredholm integro-differential equations.

Table 6 shows that even with a modest number of basis functions (), the method achieves high accuracy, with maximum absolute errors on the order of 10⁻⁴. The solution comparison in Fig 7 further confirms this, where the approximate solution aligns closely with the exact solution, and the absolute error remains between 10⁻⁸ and 10⁻⁴ across the domain.

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Table 6. Comparison of exact and approximate solutions ().

https://doi.org/10.1371/journal.pone.0346080.t006

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Fig 7. Comparison of exact and approximate solutions for: (a) Exact vs. approximate solution; (b) Absolute error distribution between exact and approximate solutions.

https://doi.org/10.1371/journal.pone.0346080.g007

The convergence and stability analysis in Table 7 and Fig 8 we will address three main points.

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Table 7. Convergence and Stability Analysis of the Hermite Least-Squares Method.

https://doi.org/10.1371/journal.pone.0346080.t007

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Fig 8. Convergence and stability analysis: (a) Error vs. in L^∞ and L² norms; (b) Convergence rate; (c) Condition number vs. ; (d) Stability factor vs. .

https://doi.org/10.1371/journal.pone.0346080.g008

• Spectral Convergence: Both L^∞ and L² errors decay rapidly as increases, with an observed convergence rate close to four, consistent with theoretical predictions.
• Stability and Conditioning: Although the condition number of the system matrix grows with , the stability factor decreases monotonically, ensuring numerical robustness. This trade-off is typical for Hermite polynomial bases, but accuracy gains outweigh the conditioning growth within the tested range.
• Efficiency: High accuracy is achieved with relatively few basis functions (), demonstrating the method’s efficiency in leveraging the global smoothness of Hermite polynomials.

In summary, the proposed Hermite least-squares method provides highly accurate approximations with spectral convergence, stable performance, and computational efficiency, making it a reliable approach for fractional integro-differential equations.

Example 4 Consider the following fractional Volterra-Fredholm integro-differential equation of order on the interval :

(51)

This problem is equipped with the initial conditions:

(52)

To ensure the problem is well-posed, we prescribe an exact solution of the form

(53)

By substituting the exact solution (53) into the governing equation (51), the unknown function can be identified. This calculation yields:

(54)

Thus, the specific problem (51)–(54) can be viewed as a special case of the general formulation with the following parameter identifications:

5.14 Method application

To approximate the exact solution (53), we expand in terms of normalized Hermite polynomials. Specifically, the approximation is taken as:

(55)

where are the normalized Hermite polynomials and the unknown coefficients a_i are to be determined.

Next, to quantify the quality of the approximation (55), we substitute it into the governing equation (51). This leads to the definition of the residual function:

(56)

The optimal coefficients are those that minimize the total squared residual over the computational domain. Therefore, we define the functional:

(57)

In addition, the initial conditions (52) impose constraints on the coefficients of the expansion (55). By substituting and differentiating, we obtain:

(58)

(59)

To evaluate the terms in (56), the fractional derivatives were computed using the Caputo definition with Gauss–Jacobi quadrature for singular integrals, while the remaining integrals were approximated with adaptive Gaussian quadrature.

As a result, the system of equations for the coefficients a_i takes the standard least-squares form:

where

and

The resulting constrained linear system Ma = b is then solved subject to the conditions (58)–(59), using the method of Lagrange multipliers.

5.15 Coefficient determination

For N = 5, solving the system yields the coefficients:

Finally, substituting these values into the expansion (55) and making use of the explicit forms of the normalized Hermite polynomials , the approximate solution is obtained:

5.16 Numerical results and discussion

Fig 10 presents the error analysis for different approximation orders . Panel (a) shows a bar representation of the pointwise absolute error distribution across the domain . Mathematically, each bar at a given collocation point s_i corresponds to the absolute error:

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Fig 9. Comparison of exact and approximate solutions: Exact solution vs. approximations for .

https://doi.org/10.1371/journal.pone.0346080.g009

where is the approximate solution obtained via the Hermite least-squares method with basis functions, and is the exact solution.

5.17 Discussion

The numerical experiments presented in Tables 8, 9, and 10, together with Figs 9–11, demonstrate the accuracy, stability, and efficiency of the proposed Hermite polynomial-based least-squares method for solving Volterra–Fredholm fractional integro-differential equations. Here are some analyses:

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Table 8. Exact solution and the approximations.

https://doi.org/10.1371/journal.pone.0346080.t008

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Table 9. Errors of the approximations.

https://doi.org/10.1371/journal.pone.0346080.t009

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Table 10. Convergence and Stability Analysis.

https://doi.org/10.1371/journal.pone.0346080.t010

• Spectral Convergence: As the approximation order increases from 5 to 9, both L^∞ and L² errors decrease rapidly, confirming the spectral accuracy of the method. The observed convergence rates (approximately 1.85 and 2.89) are consistent with theoretical expectations.
• Error Distribution: Figs 10 and 9 indicate that the maximum errors occur near the right endpoint (). This behavior reflects the boundary influence and the nonlocal effects of fractional operators.
• Stability Considerations: The condition number increases with , as expected for spectral approximations. Nevertheless, the stability factor remains controlled (Fig 11), confirming the robustness of the least-squares formulation against numerical instabilities.
• Accuracy of Approximations: The approximate solutions closely match the exact solution even for relatively small , with accuracy improving as increases. Errors remain small across the domain, demonstrating the reliability of the approach.
• Computational Efficiency: With , the method achieves errors on the order of 10⁻² while maintaining acceptable condition numbers, making it computationally efficient for practical applications.

Overall, the results validate the proposed method as a stable, accurate, and efficient approach for solving fractional integro-differential equations, confirming both theoretical convergence expectations and practical applicability.

Example 5 Consider the following fractional Volterra–Fredholm integro-differential equation of order on :

(60)

subject to the initial condition .

Since obtaining exact analytic solutions of fractional equations is generally difficult, we adopt a manufactured-solution test by prescribing

and compute the corresponding forcing function so that (60) is satisfied exactly. Using the known formula

and , we obtain

This ensures is the exact solution, enabling direct computation of both residual and true error.

5.18 Numerical method

Step 1. Hermite Expansion. The approximate solution is expressed as

where are the physicists’ Hermite polynomials.

Step 2. Residual Definition. The residual is defined as

Step 3. Least-Squares Minimization. We determine by minimizing

subject to . The integral is evaluated using 200-point Gauss–Legendre quadrature on [0,1]. Fractional derivatives of Hermite polynomials are computed via the Grünwald–Letnikov formula with step , verified for convergence.

5.19 Results for N = 6

The obtained coefficients are:

Hence,

5.20 Residual and error evaluation

The pointwise residual and the true error are listed below.

5.21 Discussion

5.21.1 Residual error analysis.

The residual error analysis vividly demonstrates the robustness and precision of the proposed Hermite least-squares formulation. From Table 11, the pointwise residual magnitudes remain uniformly small throughout the interval , never exceeding even at the boundary. This consistent performance indicates that the approximate solution closely satisfies the governing equation at every collocation point, as illustrated in Fig 12(a)–(c).

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Table 11. Residual and true error at selected points for N = 6.

https://doi.org/10.1371/journal.pone.0346080.t011

Moreover, the convergence characteristics presented in Table 12 and visualized in Fig 13 clearly exhibit the spectral behavior of the proposed method. As the polynomial degree increases from N = 2 to N = 10, the maximum residual norm decreases from to —a reduction of nearly four orders of magnitude. This rapid decay verifies that the least-squares minimization effectively suppresses local deviations and enhances the global accuracy of the approximation.

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Table 12. Convergence of residual and true error norms.

https://doi.org/10.1371/journal.pone.0346080.t012

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Fig 10. Error analysis for different values: (a) Bar representation of error distribution; (b) Error profiles (log scale) for .

https://doi.org/10.1371/journal.pone.0346080.g010

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Fig 11. Convergence and stability analysis:

(a) Error convergence inL^∞ and L² norms; (b) Convergence rate vs.; (c) Condition number growth; (d) Stability factor vs. .

https://doi.org/10.1371/journal.pone.0346080.g011

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Fig 12. Approximation and error analysis for N = 6: (a) Approximate numerical solution; (b) Residual error distribution; (c) True error distribution.

https://doi.org/10.1371/journal.pone.0346080.g012

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Fig 13. Convergence and stability analysis:

(a) Residual norm convergence in L^∞ and L² norms; (b) True error convergence inL^∞ and L² norms; (c) Growth of condition number; (d) Comparison between residual and error norms.

https://doi.org/10.1371/journal.pone.0346080.g013

The true error norms follow a similar pattern, maintaining near-proportionality to the residual norms. This observation confirms that the residual acts as a reliable surrogate for the true numerical error, which is especially valuable in problems where the analytical solution is unavailable. The gradual increase in the condition number with N (from 58.7 to 2395.2) is a typical trend observed in global spectral methods, but the values remain well within stable computational limits due to the inherent conditioning of the least-squares approach.

5.22 Comparative context and advantages of the proposed method

While direct numerical comparison with identical benchmark problems is not feasible due to the novel structure of the considered fractional Volterra–Fredholm integro-differential equations (see Remark on Benchmark Comparison), it is instructive to situate the proposed Hermite polynomial–based least-squares method within the broader landscape of numerical techniques for fractional integro-differential equations. Table 13 provides a qualitative comparison of key features between the present approach and several representative classes of methods found in the literature.

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Table 13. Qualitative comparison of numerical methods for fractional integro-differential equations.

https://doi.org/10.1371/journal.pone.0346080.t013

The numerical results obtained in Examples 1–5 demonstrate that the proposed method exhibits spectral convergence, with both L^∞ and L² error norms decaying rapidly as the number of basis functions N increases. For instance, in Example 5, the maximum residual norm decreased from for N = 2 to for N = 10—a reduction of nearly four orders of magnitude. This level of accuracy with relatively few basis functions is a hallmark of spectral methods and compares favorably to the algebraic convergence rates typical of finite-difference or low-order finite-element schemes, which would require significantly finer discretizations to achieve similar precision.

In comparison to other polynomial-based spectral approaches, the use of Hermite polynomials provides a natural advantage for problems whose solutions exhibit rapid decay or oscillatory behavior, which is common in fractional dynamical systems. Moreover, unlike most existing least-squares methods that are restricted to either Volterra or Fredholm operators (e.g., [25] for Volterra equations), our framework handles both integral types simultaneously within a single unified formulation. This eliminates the need for hybrid or iterative coupling of separate solvers, thereby simplifying implementation and reducing computational overhead.

When contrasted with recent hybrid techniques that combine integral transforms with polynomial expansions (e.g., [19]), the present method avoids the need for transform inversion and operates directly in the physical domain. Although the condition number of the resulting linear system grows with N—a common trait of global spectral methods—the least-squares formulation ensures stability, as evidenced by the controlled growth of the stability factor in all examples.

In summary, the proposed Hermite polynomial–based least-squares method offers a balanced combination of high accuracy, spectral convergence, and a unified treatment of fractional Volterra–Fredholm integro-differential equations. Its performance aligns with the best features of spectral methods while extending applicability to a broader and more challenging class of problems than previously addressed in the literature.

6.1 Future research directions

While the current work has established a solid foundation for solving linear Volterra–Fredholm fractional integro-differential equations using Hermite polynomial-based least-squares methods, several promising directions for future research emerge:

• Nonlinear Extensions: Developing computational frameworks for nonlinear VFIDEs, which would require iterative solution strategies and possibly Newton-type linearization techniques within the least-squares formulation.
• Multi-Term and Multi-Order Equations: Extending the methodology to handle equations involving multiple fractional derivatives of different orders, which are common in modeling complex physical processes with multiple memory mechanisms.
• Variable-Order Fractional Operators: Adapting the approach for problems with variable-order fractional derivatives, where the fractional order itself may be a function of time or space variables.
• Systems of Equations: Generalizing the method to solve coupled systems of fractional integro-differential equations, which would involve vector-valued Hermite polynomial expansions.
• Alternative Basis Functions: Investigating the performance of other orthogonal polynomial families (such as Laguerre, Jacobi, or generalized Hermite polynomials) for specific problem types or domain characteristics.
• High-Dimensional Problems: Developing efficient computational implementations and tensor-product formulations for problems in higher spatial dimensions, addressing the curse of dimensionality through sparse approximation techniques.

These research directions would significantly expand the applicability of spectral least-squares methods to broader classes of fractional differential equations, further establishing their value as computational tools in scientific computing and applied mathematics.

In summary, the Hermite polynomial-based least-squares method represents a valuable addition to the computational toolbox for fractional differential equations, offering an accurate and efficient alternative for solving challenging problems in mathematical physics and engineering applications.