Asad Correctional Power Series Method (ACPS)

• ACPS is a fractional power-series method that uses a correction condition to force the defect and its fractional derivatives to vanish at the expansion point.
• It handles both linear and nonlinear equations by substituting a fractional polynomial expansion into the governing Caputo equation and solving for coefficients.
• Validated on a fractional SIR epidemic model, ACPS shows rapid convergence and accuracy comparable to classical methods like RK4 in the integer-order limit.

The Asad Correctional Power Series Method (ACPS) is a fractional defect-corrected power-series method for solving initial-value problems in fractional differential equations, particularly equations with the Caputo fractional derivative. In its canonical form, ACPS assumes a fractional polynomial expansion, substitutes that expansion into the governing equation, constructs a defect or residual, and determines the unknown coefficients by enforcing vanishing conditions on the defect and its fractional derivatives at the expansion point. The method was introduced as a framework for linear and nonlinear fractional differential equations and was illustrated on a fractional SIR epidemic model, together with a side argument against the conformable fractional derivative as a valid definition of fractional differentiation (Freihet et al., 29 Sep 2025).

1. Definition and methodological identity

ACPS is formulated for nonlinear Caputo-type fractional differential equations of the form

$${}^{C}D_{t_0}^{\alpha} y(t)=f(t,y(t)), \qquad 0<\alpha\le 1,$$

subject to the initial condition

$y(t_0)=y_0.$

Its defining feature is not merely the use of a fractional power series, but the use of a correction condition imposed on the residual after substitution. In this sense, ACPS is a fractional power series method with a correction/defect enforcement procedure (Freihet et al., 29 Sep 2025).

The paper introducing ACPS presents the method as a response to limitations attributed to the Adomian Decomposition Method (ADM), Homotopy Analysis Method (HAM), and Variational Iteration Method (VIM), especially for nonlinear equations. Within that presentation, ACPS is described as combining algebraic manipulation, iterative correction/refinement, and fractional polynomial approximation. The same paper characterizes the resulting approximation as a rapidly convergent power series, but it does not provide a general convergence theorem or a general stability proof (Freihet et al., 29 Sep 2025).

A central terminological point is that ACPS is tied specifically to the Caputo derivative in the formulation actually developed. The paper states a broader connection to fractional calculus and functional analysis, but the technical development is dominated by fractional-calculus tools—Gamma functions, Beta functions, the Riemann–Liouville integral, the Caputo derivative, and fractional monomials—rather than by operator-theoretic results from functional analysis. This suggests that the phrase functional analysis is programmatic rather than structurally essential in the published derivation (Freihet et al., 29 Sep 2025).

2. Fractional-polynomial framework and derivative formulas

The basic ansatz in ACPS is a fractional polynomial centered at $t=t_0$: $P(\beta,n)(t)=\sum_{i=0}^{n} c_i (t-t_0)^{i\beta},$ with $0<\beta\le 1$. For ACPS itself, the specialization $\beta=\alpha$ is used, so the approximate solution is written as

$$y(t)=P(\alpha,n)(t)=\sum_{i=0}^{n} c_i (t-t_0)^{i\alpha}.$$

The initial condition enforces

$c_0=y_0.$

The paper places this ansatz in the standard Caputo setting. For $\alpha>0$, with $n\in\mathbb N$ such that $y(t_0)=y_0.$0, the Caputo derivative is

$y(t_0)=y_0.$1

The associated Riemann–Liouville fractional integral is

$y(t_0)=y_0.$2

A central formal ingredient is the paper’s “Asad-Type Caputo Derivative of Power Functions.” For

$y(t_0)=y_0.$3

with $y(t_0)=y_0.$4, $y(t_0)=y_0.$5, and $y(t_0)=y_0.$6 denoting the smallest integer $y(t_0)=y_0.$7, the paper derives

$y(t_0)=y_0.$8

The same derivation also yields the standard Caputo power rule

$y(t_0)=y_0.$9

The paper’s own exposition makes clear that this is a rewritten form of the usual Caputo power formula rather than a new derivative concept (Freihet et al., 29 Sep 2025).

For the fractional polynomial, the Caputo derivative becomes

$t=t_0$0

This explicit Gamma-ratio formula is the computational core of ACPS. It makes the basis $t=t_0$1 closed under one Caputo derivative of order $t=t_0$2, shifting the exponent from $t=t_0$3 to $t=t_0$4 (Freihet et al., 29 Sep 2025).

3. Defect formulation and coefficient determination

The correctional mechanism in ACPS is implemented through a defect function

$t=t_0$5

The coefficients $t=t_0$6 are then determined by the conditions

$t=t_0$7

These are the defining correction conditions of ACPS. They force not only the residual itself, but also successive fractional derivatives of the residual, to vanish at the expansion point (Freihet et al., 29 Sep 2025).

Operationally, ACPS follows a fixed sequence. One first chooses the truncation degree $t=t_0$8, writes

$t=t_0$9

computes

$P(\beta,n)(t)=\sum_{i=0}^{n} c_i (t-t_0)^{i\beta},$0

forms the defect, and solves the resulting algebraic equations produced by the vanishing-defect conditions. The method is therefore semi-analytic: it yields an analytic truncated series, but the coefficients are found through symbolic or algebraic elimination rather than by closed universal recurrence formulas (Freihet et al., 29 Sep 2025).

In the integer-order limit $P(\beta,n)(t)=\sum_{i=0}^{n} c_i (t-t_0)^{i\beta},$1, the construction reduces formally to an ordinary defect-based power-series procedure. The paper explicitly notes that in this case the fractional derivative conditions reduce to ordinary differentiation. This identifies ACPS as a genuine extension of classical power-series logic to the Caputo setting, rather than a method unrelated to standard series methods (Freihet et al., 29 Sep 2025).

At the same time, the paper does not provide a general coefficient recurrence theorem for arbitrary $P(\beta,n)(t)=\sum_{i=0}^{n} c_i (t-t_0)^{i\beta},$2. The method is algorithmic, but the general theory stops at the defect conditions and derivative identities. A plausible implication is that ACPS is easiest to use when the algebraic structure of $P(\beta,n)(t)=\sum_{i=0}^{n} c_i (t-t_0)^{i\beta},$3 makes repeated substitution and evaluation at $P(\beta,n)(t)=\sum_{i=0}^{n} c_i (t-t_0)^{i\beta},$4 tractable.

4. Fractional SIR formulation and numerical validation

The principal application in the ACPS paper is the fractional SIR epidemic model

$P(\beta,n)(t)=\sum_{i=0}^{n} c_i (t-t_0)^{i\beta},$5

$P(\beta,n)(t)=\sum_{i=0}^{n} c_i (t-t_0)^{i\beta},$6

$P(\beta,n)(t)=\sum_{i=0}^{n} c_i (t-t_0)^{i\beta},$7

The initial data are

$P(\beta,n)(t)=\sum_{i=0}^{n} c_i (t-t_0)^{i\beta},$8

with parameters

$P(\beta,n)(t)=\sum_{i=0}^{n} c_i (t-t_0)^{i\beta},$9

For this system, the paper uses degree-4 fractional polynomials

$0<\beta\le 1$0

with

$0<\beta\le 1$1

The corresponding defect functions are

$0<\beta\le 1$2

$0<\beta\le 1$3

$0<\beta\le 1$4

and the coefficients are determined from

$0<\beta\le 1$5

The leading ACPS terms reported for the SIR variables are

$0<\beta\le 1$6

$0<\beta\le 1$7

$0<\beta\le 1$8

The higher-order printed expressions are reported with typographical corruption in the source summary, but the displayed leading coefficients are clear (Freihet et al., 29 Sep 2025).

For validation, the paper compares ACPS with the classical fourth-order Runge–Kutta (RK4) method in the special case $0<\beta\le 1$9. It also provides degree-9 classical series for $\beta=\alpha$0, $\beta=\alpha$1, and $\beta=\alpha$2. At $\beta=\alpha$3, the reported absolute errors are approximately

$\beta=\alpha$4

$\beta=\alpha$5

and

$\beta=\alpha$6

The paper states that the ACPS and RK4 curves practically overlap for $\beta=\alpha$7, and that as $\beta=\alpha$8, the fractional solution curves approach the classical ones (Freihet et al., 29 Sep 2025).

The epidemiological interpretation adopted in the paper is standard for Caputo models: the fractional derivative is taken to encode memory and hereditary effects through its nonlocal integral form. This supports the paper’s claim that fractional-order SIR dynamics can be more realistic than integer-order dynamics in settings where historical dependence matters.

5. Relation to adjacent power-series and series-correction methods

ACPS belongs to a broader family of methods that extend, continue, or restructure truncated series, but it is not interchangeable with earlier power-series frameworks. Several nearby methods are directly relevant.

Before ACPS was named, algebraic approximants based on Hermite–Padé polynomials were used to predict unknown coefficients of a power series by constructing an algebraic relation and continuing the series recursively. That approach is explicitly a series-correction / continuation mechanism, but it is built around algebraic approximants rather than Caputo residual conditions (Homeier, 2011). A different line of work, asymptotic approximants, corrected divergent, truncated, and underspecified series by embedding known far-field asymptotics into a closed-form ansatz; that framework is strongly related in spirit to ACPS as a power-series correction method, but it relies on asymptotic matching rather than defect cancellation at the expansion point (Barlow et al., 2017). Another closely related development introduced a generalized fractional power series with systematic exponent selection based on the derivative orders and monomial coefficient factors present in the equation; this is particularly relevant to ACPS-like reasoning about how the series basis should be chosen, yet it is not named ACPS and is centered on exponent lattices rather than residual enforcement (Assebbane et al., 2024). There are also recursive stream-decomposition methods for ODEs and PDEs in which the solution is written as a sum of “streams” driven by lower streams; these methods are power-series-like, but they do not define the correctional step used by ACPS (Ross, 2019).

Paper	Method	Relation to ACPS
(Homeier, 2011)	Algebraic approximants via Hermite–Padé polynomials	Coefficient continuation, not ACPS
(Barlow et al., 2017)	Asymptotic approximants	Series correction by asymptotic matching
(Assebbane et al., 2024)	Generalized fractional power series	Strong overlap in exponent selection
(Ross, 2019)	Stream-decomposition power series method	Recursive series refinement, not ACPS

This comparison clarifies a common misconception. ACPS is not a generic label for any corrected or extended power-series method. It denotes a specific Caputo-based procedure whose characteristic step is the imposition of vanishing defect derivatives at the expansion point. Earlier methods may be methodologically adjacent, but the sources summarized here explicitly distinguish them from ACPS.

6. Limitations, criticism, and disputed claims

The published ACPS formulation is constructive, but its theoretical support remains limited. The paper claims rapid convergence, greater accuracy, and computational efficiency, yet it does not provide a general convergence theorem, a general error bound, a radius-of-convergence analysis beyond mentioning $\beta=\alpha$9 in the fractional-polynomial definition, or a formal stability proof. It also does not derive a universal recurrence relation for the coefficients in the nonlinear scalar case, nor for the full SIR system. These absences are central to any technical assessment of the method (Freihet et al., 29 Sep 2025).

The empirical validation is also narrow. The only explicit quantitative benchmark is against RK4 for $$y(t)=P(\alpha,n)(t)=\sum_{i=0}^{n} c_i (t-t_0)^{i\alpha}.$$0, which is the classical, not genuinely fractional, regime. The paper does not provide head-to-head comparisons against trusted solvers for $$y(t)=P(\alpha,n)(t)=\sum_{i=0}^{n} c_i (t-t_0)^{i\alpha}.$$1, nor direct computational benchmarks against ADM, HAM, or VIM, despite presenting ACPS as advantageous relative to those methods. This suggests that the claims of superiority are at present better read as methodological proposals than as settled comparative conclusions.

A second debated aspect is the paper’s critique of the conformable fractional derivative. The paper argues that the conformable derivative is not a valid fractional derivative because it lacks the Gamma-factor scaling appearing in the Caputo power rule, does not reproduce Caputo results even for basic power functions, is local rather than nonlocal, and therefore lacks the memory effect associated with fractional calculus. In formulaic terms, the contrast is between

$$y(t)=P(\alpha,n)(t)=\sum_{i=0}^{n} c_i (t-t_0)^{i\alpha}.$$2

for $$y(t)=P(\alpha,n)(t)=\sum_{i=0}^{n} c_i (t-t_0)^{i\alpha}.$$3, and the conformable expression

$$y(t)=P(\alpha,n)(t)=\sum_{i=0}^{n} c_i (t-t_0)^{i\alpha}.$$4

The paper treats the missing Gamma ratio as decisive. This is a substantive claim about the foundations of fractional calculus, but within the ACPS article it functions mainly as a justification for using the Caputo derivative rather than as a theorem necessary for ACPS itself (Freihet et al., 29 Sep 2025).

Finally, the method rests on implicit regularity assumptions: that the local solution admits a fractional-power expansion in powers of $$y(t)=P(\alpha,n)(t)=\sum_{i=0}^{n} c_i (t-t_0)^{i\alpha}.$$5, that the defect conditions determine the coefficients uniquely, and that the truncated local series remains accurate over the interval of interest. The paper does not analyze singular cases, stiff systems, or problems whose local structure is not naturally captured by a basis of the form $$y(t)=P(\alpha,n)(t)=\sum_{i=0}^{n} c_i (t-t_0)^{i\alpha}.$$6. This suggests that ACPS should presently be understood as a promising local semi-analytic method whose scope is plausible but not yet exhaustively established.