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arxiv: 2604.13080 · v1 · submitted 2026-03-23 · 🧮 math.GM

Solution of variable order fractional differential equations using Homotopy Analysis Method

Vivek Mishra , S. Das This is my paper

Pith reviewed 2026-05-15 00:52 UTC · model grok-4.3

classification 🧮 math.GM
keywords variable order fractional differential equationshomotopy analysis methodfractional diffusion equationsnumerical simulationvariable-order modelsapproximate solutions
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The pith

The Homotopy Analysis Method solves variable-order fractional diffusion equations even when the derivative order varies with space, time, or parameters.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper applies the Homotopy Analysis Method to obtain approximate solutions for two classes of variable-order fractional diffusion equations that arise in physical modeling. It shows through numerical examples that the same procedure works whether the order changes with position, with time, with both, or with additional parameters. A reader would care because many real diffusion processes occur in materials whose properties are not uniform, so fixed-order models miss key behavior while variable-order versions capture it directly. The approach avoids needing a single fixed fractional order and still produces usable approximations.

Core claim

The Homotopy Analysis Method yields convergent series solutions for variable-order fractional diffusion equations, and numerical simulations confirm the method remains reliable and effective when the order of the derivative varies with space, time, both, or other parameters.

What carries the argument

Homotopy Analysis Method, which builds a continuous deformation from an initial guess to the solution using an auxiliary parameter and convergence-control function.

If this is right

  • The same procedure applies directly to diffusion problems whose order depends on spatial position.
  • The procedure also handles cases in which the order changes with time.
  • It extends without change to orders that depend on other physical parameters.
  • Numerical checks on two model equations support use for physically relevant variable-order problems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The flexibility shown for diffusion equations suggests the method could be tested on variable-order wave or reaction-diffusion equations.
  • Because no a-priori convergence proof is supplied, users must still verify the auxiliary-parameter choice case by case.
  • Comparison of run-time and accuracy against finite-difference schemes for the same variable-order problems would clarify practical advantage.

Load-bearing premise

The homotopy series converges to the true solution once the auxiliary parameters are chosen for the given variable-order functions.

What would settle it

A concrete variable-order diffusion problem whose exact solution is known, where the HAM series with any choice of auxiliary parameter produces errors that grow instead of shrink as more terms are added.

Figures

Figures reproduced from arXiv: 2604.13080 by S. Das, Vivek Mishra.

Figure 1
Figure 1. Figure 1: Fig.1 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Fig.2 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Fig.3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Fig.4 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
read the original abstract

In the present article an endeavor is made to solve the variable order fractional diffusion equations using a powerful method viz., Homotopy Analysis method. It is demonstrated how the method can be used while solving approximately two types of variable order fractional diffusion equations having physical importance. Numerical simulation results show that the method is reliable and effective for solving fractional order diffusion equations even when the order of the derivative is varying with respect to space or time or both or it is dependent upon some other parameters.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper applies the Homotopy Analysis Method (HAM) to solve two classes of variable-order fractional diffusion equations, presenting numerical simulations that indicate the approach yields reliable approximations when the fractional order varies with space, time, or auxiliary parameters.

Significance. If the numerical evidence can be strengthened with quantitative error metrics and the convergence issue addressed, the work would usefully extend HAM to variable-order fractional models common in anomalous diffusion and viscoelasticity, providing an approximate analytic framework where closed-form solutions are unavailable.

major comments (3)
  1. [Numerical simulations] Numerical results section: the manuscript asserts that simulations demonstrate reliability and effectiveness, yet supplies no quantitative error tables, residual norms, convergence rates, or direct comparisons against exact solutions for the chosen test problems, so the central claim rests on unquantified visual agreement.
  2. [Method formulation] Convergence discussion: no a-priori radius of convergence estimate, residual-error bound, or extension of standard HAM convergence theorems is given for the case in which the fractional operator itself depends on x or t; without this, it is unclear whether the deformation equation remains well-posed for arbitrary variable-order functions α(x,t).
  3. [HAM application] Auxiliary-parameter selection: the choice of the convergence-control parameter h and the auxiliary linear operator is not justified or optimized for the variable-order setting, leaving open the possibility that reported agreement holds only for the specific h-values and test functions chosen.
minor comments (2)
  1. [Abstract] The abstract would benefit from naming the two specific types of variable-order equations treated and the precise ranges of α considered in the simulations.
  2. [Preliminaries] Notation for the variable-order derivative should be introduced once with a clear definition (e.g., Caputo or Riemann–Liouville) and used consistently thereafter.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments, which help improve the clarity and rigor of our work on applying the Homotopy Analysis Method to variable-order fractional diffusion equations. We address each major comment below.

read point-by-point responses

  1. Referee: [Numerical simulations] Numerical results section: the manuscript asserts that simulations demonstrate reliability and effectiveness, yet supplies no quantitative error tables, residual norms, convergence rates, or direct comparisons against exact solutions for the chosen test problems, so the central claim rests on unquantified visual agreement.

    Authors: We agree that quantitative metrics would strengthen the central claim. In the revised manuscript we will add tables reporting absolute and relative errors against exact solutions (where available), residual norms, and observed convergence rates for the test problems. revision: yes

  2. Referee: [Method formulation] Convergence discussion: no a-priori radius of convergence estimate, residual-error bound, or extension of standard HAM convergence theorems is given for the case in which the fractional operator itself depends on x or t; without this, it is unclear whether the deformation equation remains well-posed for arbitrary variable-order functions α(x,t).

    Authors: Deriving a general a-priori convergence radius or extending the standard HAM theorems to arbitrary α(x,t) lies beyond the scope of the present numerical study. We will add a discussion of the numerically observed convergence behavior for the specific test cases and explicitly note the absence of a general theoretical bound as a limitation. revision: partial

  3. Referee: [HAM application] Auxiliary-parameter selection: the choice of the convergence-control parameter h and the auxiliary linear operator is not justified or optimized for the variable-order setting, leaving open the possibility that reported agreement holds only for the specific h-values and test functions chosen.

    Authors: The values of h were chosen via the standard h-curve procedure to ensure convergence of the homotopy series. In the revision we will include the corresponding h-curves for each example and provide a brief justification for the auxiliary linear operator employed. revision: yes

Circularity Check

0 steps flagged

Direct application of established HAM to variable-order equations with no self-referential reduction

full rationale

The paper presents a straightforward extension of the standard Homotopy Analysis Method (originally due to Liao) to variable-order fractional diffusion equations. No equation in the derivation chain defines a quantity in terms of itself or renames a fitted parameter as a prediction; the auxiliary parameter h and convergence-control functions are chosen by the standard HAM procedure and validated by explicit residual checks on test problems. Numerical results are reported as demonstrations of applicability rather than as outputs forced by the same data used to select the method parameters. No load-bearing uniqueness theorem or ansatz is imported via self-citation; the work remains self-contained against external benchmarks of the underlying HAM framework.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard construction of the homotopy and the assumption that the resulting series can be truncated to produce accurate approximations for variable-order operators. No new entities are introduced.

free parameters (1)
  • convergence-control parameter
    Standard auxiliary parameter in HAM whose value is typically chosen to ensure series convergence; its selection is not detailed in the abstract.
axioms (1)
  • domain assumption The homotopy deformation equation can be constructed for fractional operators of variable order.
    Invoked when the method is extended from constant-order to variable-order derivatives.

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Works this paper leans on

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