A FPS centered at x=0 with a fractional exponent step 1/n (n∈N) is defined as: (5)y(x)=∑j=0∞cjxj/n,cj∈R.

This series is assumed to converge for 0≤x<R for some radius of convergence R>0. For operational convenience, particularly when applying the Caputo derivative, it is often useful to express this series using a specific basis [25]: (6)y(x)=∑j=0∞vjwj(n),wherewj(n)=xj/nΓ(1+j/n).

The coefficients are related by vj=cjΓ(1+j/n). The set of such convergent series is denoted by F(1/n)C.

Addition of two FPS as well as multiplication of FPS by a scalar is defined in a standard way, while multiplication of two FPS f(x)=∑bjwj(n)∈F(1/n)C and g(x)=∑cjwj(n)∈F(1/n)C is defined in the Cauchy sense: (7)f(x)⋅g(x)=∑j=0∞(∑k=0j(j/nk/n)bkcj−k)wj(n),using the property (8)wj(n)wk(n)=((j+k)/nk/n)wj+k(n).

The binomial coefficient of non-integer arguments α,β is defined as: (9)(αβ)=Γ(α+1)Γ(β+1)Γ(α−β+1).

A Caputo differentiation operator of order 1n acts on basis elements wj(n) as [25]: (10)CD(1/n)wj(n)={0,if j=0wj−1(n),if j≥1.

Thus, the iterated application of the Caputo derivative of order 1n follows naturally: (11)(CD(1/n))kwj(n)={0,if j<kwj−k(n),if j≥k.

This property simplifies the application of fractional operators to series solutions.

Previous work demonstrated a methodology for transforming FDEs involving the operator CD(1/n) (12)CD(1/n)y(x)=F(y(x)),subject to y(0)=v0, into an equivalent higher-order FDE involving the n-th iteration of the operator, (CD(1/n))n [26].

This transformation utilizes the properties of FPS and introduces auxiliary relations to manage the fractional differentiation of nonlinear functions of y(x). For example, when dealing with a quadratic term like y2 and n=2, the application of the operator requires a specific relation: (13)CD(1/2)(z2)=2z(CD(1/2)z)+Θz(x),where Θz(x) is an auxiliary FPS whose coefficients depend on the coefficients of z(x). This relation accounts for the deviation from the integer-order chain rule. Using such auxiliary relations derived within the FPS framework, the original FDE (12) can be systematically transformed into the (CD(1/n))n-type FDE: (14)(CD(1/n))ny(x)=G(y(x))+Ψ(x).

Here, G(y(x)) is a closed-form function, while Ψ(x) is the resulting infinite FPS that encapsulates the residual effects [26].

For a general analytic nonlinearity F(y), where y(x)=∑vjwj(n), the function F(y(x)) also has an FPS representation. The auxiliary relations are derived by applying the operator D(1/n)C to this series and relating it to the derivative of the argument, D(1/n)Cy(x). This process, which utilizes the Cauchy product rule for FPS multiplication, systematically separates the resulting expression into terms that depend on y(x) and its fractional derivatives, and a residual FPS, such as Θz(x) in (13). While the specific form of the auxiliary relation depends on the function F, the procedure is generalizable to any analytic function, thereby providing a deterministic method to perform the transformation for a given FDE.

Subsequently, the resulting (CD(1/n))n-type FDE is reducible to a first-order ODE via the substitution: (15)t=x1/n.

Letting y^(t)=y(tn), the equivalent ODE takes the form: (16)dy^dt=ntn−1(G(y^(t))+Ψ(tn))+∑j=1n−1jvjΓ(1+jn)tj−1,with y^(0)=v0. The summation term arises from the initial conditions associated with the lower-order fractional derivatives vj=(CD(1/n))jy(x)|x=0 for j=1,…,n−1 [26].

For computation, the infinite series Ψ(tn) is truncated to (17)ΨM(tn)=∑j=0Mψjtj/Γ(1+j/n),which introduces an approximation error, the magnitude of which depends on the truncation order M and the convergence properties of Ψ(x).

To further investigate the robustness of the optimal model architecture, a parameter space analysis is conducted. The optimization procedure is systematically applied across a grid of FDE coefficients, with a0 and a1 each varying over the range [−1,1]. For each pair of (a0,a1) coefficients, the optimal polynomial degree NG∈{1,2,3,4} is determined by identifying the model that yields the minimum normalized target value, Tnorm. The initial conditions are held constant at v0=1.5 and v1=2, and all other computational parameters are maintained as specified in Table 1. Note that the objective of this analysis is to demonstrate the stability of the optimal architecture across a representative, bounded domain of system parameters, rather than to conduct a computationally exhaustive global exploration. Naturally, a broader analysis of the parameter space would be a valuable direction for future work.

The results of this meta-analysis are summarized in Table 3, which presents a map of the optimal polynomial degree, NG∗, for each (a0,a1) pair. The analysis reveals a consistent pattern: the quadratic model (NG=2) is the optimal choice in 28 out of the 36 cases tested, representing over 77% of the parameter space.

A closer examination of the cases where NG=2 is not the optimal choice provides further insight. In the instances where a linear model (NG=1) is selected, a trade-off between solution accuracy and the magnitude of the residual series is typically observed. For example, for the configuration (a0,a1)=(−0.2,−1.0), the optimal NG=1 model produced a smaller residual series norm (‖ΨM‖2=6.55×10−1) compared to the NG=2 model (‖ΨM‖2=6.92×10−1). Conversely, the NG=2 model achieved a slightly lower RMSE (4.15×10−6) than the NG=1 model (4.86×10−6). This indicates that even in regions where the simpler linear model is marginally preferred by the normalized target function, the quadratic model remains a highly competitive alternative.

The single instance where NG=3 is optimal, at (a0,a1)=(−1.0,−0.2), also warrants comment. For this configuration, the final metrics for the NG=3 model (RMSE = 6.353×10−9, ‖ΨM‖2=5.229×10−2) are nearly identical to those of the NG=2 model (RMSE = 6.356×10−9, ‖ΨM‖2=5.229×10−2), confirming that the performance difference is negligible.

Collectively, these findings strongly support the conclusion that a quadratic ODE architecture provides a robust and broadly applicable representation for the linear fractional differential equation under study. The stability of this result across a wide range of system parameters suggests that the method is not merely finding isolated numerical artifacts but is successfully identifying a fundamental, underlying integer-order structure inherent to the fractional system.

To better understand the structure of the quadratic architecture (NG=2), we now examine how its optimal coefficients b∗=(b0,b1,b2) depend on the original FDE parameters. Fig. 4 visualizes this relationship across the same intervals for a0 and a1 as given in Table 3. Even though the discretization is coarse, it can be seen that the values of b0,b1,b2 do not oscillate wildly and depend smoothly on a0 and a1.

To assess whether the optimal ODE architecture is an intrinsic property of the FDE or is dependent on the specific solution trajectory, a second meta-analysis has been performed. In this set of experiments, the FDE coefficients are held constant at a0=−1.5 and a1=0.5, while the initial conditions v0=y(0) and v1=(CD(1/3)y)(0) are varied over the range [−2,2]. All other computational parameters are maintained as specified in Table 1. The same optimization and dynamic normalization procedure is applied to find the optimal polynomial degree NG∗ for each pair of initial conditions.

The results, presented in Table 4, demonstrate a high degree of stability in the optimal model architecture. Across the 36 different initial conditions tested, the quadratic model (NG=2) is identified as the optimal choice in 33 cases, representing approximately 92% of the tested space. This consistency suggests that the optimal ODE representation is largely independent of the specific solution trajectory and is instead determined by the inherent structure of the FDE itself.

Notably, the few instances where NG=1 has been selected as optimal occurred at or close to the boundaries of the tested parameter space. In these cases, the linear model is found to be genuinely superior across both performance metrics. For example, for the initial conditions (v0,v1)=(−2.0,2.0), the winning NG=1 model produces both a lower RMSE and a smaller residual series norm than the NG=2 model. However, the overwhelming dominance of the quadratic model across the vast majority of the parameter space remains the principal finding. Furthermore, the higher-complexity models (NG=3 and NG=4) consistently underperform, often yielding results that are orders of magnitude worse than the simpler models.

The combined results from the parameter space analyses of both the FDE coefficients and the initial conditions provide strong evidence for the central hypothesis. The methodology consistently identifies a quadratic ODE as the most suitable integer-order representation for the given linear FDE. This stability across different system parameters and solution trajectories indicates that the optimization framework is successfully uncovering a fundamental, intrinsic property of the fractional system’s dynamics, rather than artifacts of a specific configuration.

A sensitivity analysis has been conducted to evaluate the influence of the residual fractional series truncation order, M, on the performance of the optimization framework. For this study, the FDE coefficients and initial conditions are held constant at the values used in the first example (a0=−1.5,a1=0.5,v0=1.5,v1=2), while the parameter M is varied from 5 to 20. All other computational parameters are maintained as specified in Table 1. The primary objective is to determine if the choice of the optimal model complexity, NG∗, is stable with respect to the accuracy of the residual fractional series approximation.

The results, summarized in Table 5, reveal a distinct trend. For lower values of M (specifically, M≤7), the optimization procedure has selected a cubic polynomial (NG=3) as the optimal architecture. However, as M increases to 8 and beyond, the optimal model complexity consistently stabilizes at the quadratic model (NG=2).

This transition in the optimal model complexity is a significant finding. It suggests that when the residual fractional series ΨM(tn) is poorly approximated (i.e., when M is small), the optimization framework compensates by selecting a more complex polynomial G(y) to capture the dynamics that are missing from the truncated series. As the series approximation becomes more accurate with increasing M, this compensation is no longer necessary, and the framework is able to identify the underlying quadratic structure of the system.

Furthermore, an examination of the final un-normalized metrics reveals the practical limits of increasing the residual series order. While the final RMSE of the optimal solution improves substantially as M increases from 8 to 15, it converges to approximately 1.27×10−8 for M≥15. Similarly, the final residual series norm, ‖ΨM‖2, stabilizes around a value of 4.3×10−2. This indicates that beyond a certain point (M≈15 for this problem), further increasing the number of terms in the residual fractional series approximation yields no discernible improvement in the quality of the final solution.

In conclusion, this sensitivity analysis demonstrates the robustness of the proposed methodology. It shows that provided the residual fractional series is approximated with a sufficient number of terms, the optimization consistently converges to the same optimal ODE architecture. This reinforces the conclusion that the framework is successfully identifying an intrinsic and stable integer-order representation of the fractional system.

To verify the effectiveness and generality of the proposed framework over different integration intervals, the experiment on the nonlinear Riccati FDE was repeated for two longer domains: x∈[0,2] and x∈[0,3]. The objective of this analysis is to determine if the optimal model architecture identified in the previous section is a robust structural property of the FDE or an artifact of the specific integration horizon. All FDE and computational parameters are kept identical to those in Table 6, with only the domain of the problem being extended. The reference solution for the RMSE calculation on the extended domains was again obtained using Garrappa’s method.

The optimization results for the extended domains are presented in Tables 8 and 9. The analysis reveals a consistency in the optimal architecture. For both the x∈[0,2] and x∈[0,3] intervals, the cubic model (NG=3) is unambiguously identified as the superior choice, achieving the lowest normalized target value in each case. While the absolute RMSE values naturally increase as the integration interval becomes longer, the optimizer’s structural preference remains stable. This finding strongly supports the conclusion that the identified cubic architecture is not a feature of a specific problem configuration but rather an intrinsic property of the underlying dynamics of the fractional Riccati equation.

To ensure that the findings of this study are not specific to the choice of the optimization algorithm, the experiment on the nonlinear Riccati FDE presented at the beginning of this section (with the integration interval x∈[0,1]) was repeated using an alternative global optimizer. The Genetic Algorithm (GA), which is another well-established, population-based metaheuristic, was chosen for this comparative analysis. The GA was configured with parameters comparable to those used for the PSO; specifically, the population size was set to 100 and the maximum number of generations to 300, mirroring the swarm size and maximum iterations of the PSO, respectively. All other FDE and method parameters were kept identical to those listed in Table 6.

The optimization results obtained using the GA are presented in Table 10 and are consistent with the PSO results from Table 7. GA also unambiguously identifies the cubic model (NG=3) as the optimal architecture, achieving the lowest normalized target value. While the final numerical metrics differ slightly between the two algorithms, which is a common occurrence given their stochastic nature, the central conclusion remains the same. This successful replication of the result with a different optimizer provides strong evidence that the identified cubic architecture is a genuine feature of the FDE’s underlying dynamics and not an artifact of the optimization method used to find it.

To assess the robustness of the optimization outcomes, a sensitivity analysis was performed on the optimal solution found by the PSO algorithm for the nonlinear Riccati FDE (see Table 7). The objective of this analysis is to determine if the identified optimal architecture is stable with respect to small variations in the FDE’s parameters. For this study, the optimal cubic architecture (NG=3) and its corresponding coefficient vector b∗ were held fixed. The performance of this fixed model was then evaluated on a set of perturbed FDEs, where the coefficient a1 was varied in increments of 1% over the range of ±5% from its original value. For each perturbed FDE, a new high-fidelity reference solution was computed to ensure an accurate RMSE calculation.

The results of this analysis are presented in Table 11. The results show that the optimal solution exhibits structural stability, with performance metrics degrading continuously with respect to parameter perturbations. Both the final RMSE and the norm of the residual series are minimized for the unperturbed, original problem and increase monotonously and symmetrically as the perturbation magnitude grows. This demonstrates that the optimal solution represents a stable optimum within a local neighborhood of the parameter space, confirming the robustness of the optimization outcome.