Hybrid Sumudu Variational (HSV) Method

• Hybrid Sumudu Variational (HSV) Method is a hybrid analytical approach that combines Sumudu transforms with the Variational Iteration Method to solve delay, pantograph, and fractional growth equations.
• The method reformulates population dynamics and pantograph-type delay differential equations using transform-domain algebra and Adomian decomposition, ensuring effective iterative correction and convergence.
• HSV extends to fractional logistic models with ABC derivatives, offering practical insights for nonlocal memory effects and applications in biological, industrial, and market dynamics.

Hybrid Sumudu Variational (HSV) method is a hybrid analytical and semi-analytical methodology that combines the Sumudu transform with the Variational Iteration Method (VIM) to construct solutions of delay, pantograph, and fractional growth equations when direct closed-form integration is difficult or unavailable. In the literature considered here, HSV is formulated for population dynamics models written as pantograph-type delay differential equations and later extended to a fractional logistic growth model with Atangana–Baleanu in Caputo sense (ABC) derivatives and proportional delay. In both settings, the method uses transform-domain algebra to simplify the linear part of the governing equation and then recovers time-domain iterates through variational correction and inverse transformation (Aibinu et al., 2021, Aibinu et al., 22 Sep 2025).

1. Scope and model classes

HSV is presented as a method for equations of the canonical form

$L[u](t)+N[u](t)=g(t),$

with appropriate initial data and, when required, initial history. In the population-dynamics formulation, the target class includes pantograph-type models in which the state at time $t$ depends on the state at a proportional lag $\alpha t$ and possibly on a discrete delay $t-\tau$:

$$x'(t)=F\bigl(t,x(t),x(\alpha t),x(t-\tau)\bigr), \qquad t\ge 0.$$

This setting is used to treat modified Hutchinson/logistic and modified Nicholson blowflies equations, both of which are recast in forms suitable for HSV iteration (Aibinu et al., 2021).

The modified Hutchinson/logistic pantograph model is written as

$$x'(t)=r\,x(t)\left(1-\frac{x(\alpha t)}{K}\right), \qquad x(0)=x_0.$$

The special case $\alpha=0$ yields a modified exponential model, while $\alpha=1$ recovers the classical logistic equation. The modified Nicholson blowflies pantograph model is written as

$$x'(t)=P\,x(\alpha t)e^{-x(\alpha t)/x_0}-\delta x(t), \qquad x(0)=x_0.$$

These models establish HSV as a method for proportional-delay population dynamics rather than for ordinary autonomous systems alone (Aibinu et al., 2021).

A later extension applies HSV to a fractional logistic equation with proportional delay and memory effects:

$${}^{ABC}_0D^\mu z(t)=r\,z(t)\left(1-\frac{z(\lambda t)}{K}\right), \qquad z(0)=z_0,\qquad 0<\mu<1,\ 0\le \lambda\le 1.$$

Here the use of the ABC fractional derivative introduces a nonsingular Mittag-Leffler memory kernel, and the proportional delay $t$0 modifies the saturating feedback term. This places HSV in a broader class of nonlocal and delay-dependent growth models (Aibinu et al., 22 Sep 2025).

2. Sumudu-transform and variational foundations

The Sumudu transform used in HSV is defined for suitable functions by

$t$1

It is related to the Laplace transform through

$t$2

This relation underlies the inverse-transform formula

$t$3

which is used when explicit inverse Sumudu transforms are not directly available (Aibinu et al., 2021).

The operational rules emphasized in HSV are those that convert differentiation and delay into simple algebraic manipulations. For derivatives,

$t$4

and more generally,

$t$5

For proportional delays, the crucial identity is

$t$6

while for shifted delays with Heaviside handling the transform rule is

$t$7

These properties explain why HSV is particularly natural for pantograph equations: the delay term becomes argument scaling in the transform variable rather than an additional differential operator (Aibinu et al., 2021).

The VIM component supplies the correction-functional structure. For

$t$8

VIM constructs iterates through

$t$9

where

$\alpha t$0

In HSV, the determination of the Lagrange multiplier is moved to the transform domain. For an $\alpha t$1th-order linear operator $\alpha t$2, the paper finds $\alpha t$3; for first-order models, $\alpha t$4. The combination of Sumudu algebra and transform-domain variational stationarity is the defining structural feature of HSV (Aibinu et al., 2021).

3. Canonical HSV construction

HSV proceeds by a fixed workflow. First, the governing equation is transformed by $\alpha t$5 so that the linear differential operator becomes algebraic in $\alpha t$6, while pantograph terms are rewritten through the scaling property. Second, the transformed equation is rearranged into a VIM-compatible correction formula in the $\alpha t$7-domain, and the transform-domain Lagrange multiplier is determined by imposing stationarity under restricted variation of the nonlinear terms. Third, the inverse Sumudu transform is applied to obtain an explicit iteration in time. Fourth, the iteration is continued until either a stopping criterion is met or a recognizable closed-form pattern emerges (Aibinu et al., 2021).

For ordinary first-order pantograph models, the initialization is

$\alpha t$8

while for an $\alpha t$9th-order problem the initialization is the polynomial built from the initial derivatives,

$t-\tau$0

The stopping criteria described in the source are based on either the successive-difference norm or the residual norm. The residual is

$t-\tau$1

This makes HSV operationally close to Picard-type iteration, but its correction step is determined by transform-domain variational calculus rather than direct time-domain quadrature (Aibinu et al., 2021).

Nonlinear terms are handled through Adomian decomposition. If

$t-\tau$2

then the nonlinear operator is expanded as a series of Adomian polynomials,

$t-\tau$3

In this way, the transformed correction formula remains explicit even when products, exponentials, or composed delay terms appear. The method therefore hybridizes three ingredients: Sumudu-transform algebra, VIM correction functionals, and Adomian-polynomial decomposition (Aibinu et al., 2021).

4. Fractional ABC-delay formulation

In the fractional extension, HSV is adapted to equations involving the Atangana–Baleanu derivative in Caputo sense. The ABC derivative of order $t-\tau$4 is defined by

$t-\tau$5

with $t-\tau$6, where the one-parameter Mittag-Leffler function is

$t-\tau$7

The generalized logistic equation studied in this framework is

$t-\tau$8

Its associated Volterra integral form is

$t-\tau$9

with

$$x'(t)=F\bigl(t,x(t),x(\alpha t),x(t-\tau)\bigr), \qquad t\ge 0.$$0

This reformulation makes the nonlocal memory kernel explicit and connects the HSV iteration to standard fixed-point reasoning for Volterra equations (Aibinu et al., 22 Sep 2025).

The transform-domain variational step changes in the ABC setting because the fractional derivative introduces the factor $$x'(t)=F\bigl(t,x(t),x(\alpha t),x(t-\tau)\bigr), \qquad t\ge 0.$$1. The transform-domain Lagrange multiplier becomes

$$x'(t)=F\bigl(t,x(t),x(\alpha t),x(t-\tau)\bigr), \qquad t\ge 0.$$2

With the initialization $$x'(t)=F\bigl(t,x(t),x(\alpha t),x(t-\tau)\bigr), \qquad t\ge 0.$$3, the concrete HSV iteration for the fractional logistic model is

$$x'(t)=F\bigl(t,x(t),x(\alpha t),x(t-\tau)\bigr), \qquad t\ge 0.$$4

where

$$x'(t)=F\bigl(t,x(t),x(\alpha t),x(t-\tau)\bigr), \qquad t\ge 0.$$5

The nonlinear term is again expanded by writing

$$x'(t)=F\bigl(t,x(t),x(\alpha t),x(t-\tau)\bigr), \qquad t\ge 0.$$6

and using Adomian polynomials $$x'(t)=F\bigl(t,x(t),x(\alpha t),x(t-\tau)\bigr), \qquad t\ge 0.$$7 tailored to the product structure. The first polynomials are

$$x'(t)=F\bigl(t,x(t),x(\alpha t),x(t-\tau)\bigr), \qquad t\ge 0.$$8

This produces the component-wise iteration

$$x'(t)=F\bigl(t,x(t),x(\alpha t),x(t-\tau)\bigr), \qquad t\ge 0.$$9

The fractional version therefore preserves the basic HSV structure while replacing the first-order multiplier $$x'(t)=r\,x(t)\left(1-\frac{x(\alpha t)}{K}\right), \qquad x(0)=x_0.$$0 with an ABC-specific multiplier determined by the nonsingular kernel (Aibinu et al., 22 Sep 2025).

5. Representative solution structures

For the modified Hutchinson/logistic pantograph equation, the HSV construction in the 2021 study yields the sequence

$$x'(t)=r\,x(t)\left(1-\frac{x(\alpha t)}{K}\right), \qquad x(0)=x_0.$$1

$$x'(t)=r\,x(t)\left(1-\frac{x(\alpha t)}{K}\right), \qquad x(0)=x_0.$$2

$$x'(t)=r\,x(t)\left(1-\frac{x(\alpha t)}{K}\right), \qquad x(0)=x_0.$$3

$$x'(t)=r\,x(t)\left(1-\frac{x(\alpha t)}{K}\right), \qquad x(0)=x_0.$$4

The paper writes the resulting series as

$$x'(t)=r\,x(t)\left(1-\frac{x(\alpha t)}{K}\right), \qquad x(0)=x_0.$$5

For $$x'(t)=r\,x(t)\left(1-\frac{x(\alpha t)}{K}\right), \qquad x(0)=x_0.$$6, the model reduces to

$$x'(t)=r\,x(t)\left(1-\frac{x(\alpha t)}{K}\right), \qquad x(0)=x_0.$$7

with closed-form solution

$$x'(t)=r\,x(t)\left(1-\frac{x(\alpha t)}{K}\right), \qquad x(0)=x_0.$$8

For $$x'(t)=r\,x(t)\left(1-\frac{x(\alpha t)}{K}\right), \qquad x(0)=x_0.$$9, the paper identifies the classical logistic solution

$\alpha=0$0

These limits are used as verification cases for the HSV construction (Aibinu et al., 2021).

For the modified Nicholson blowflies pantograph model, the same paper derives

$\alpha=0$1

which sum to

$\alpha=0$2

The source explicitly remarks that this exponential form should be interpreted as the HSV approximation, or exact for the linearized case around $\alpha=0$3, consistent with the particular Adomian decomposition adopted there (Aibinu et al., 2021).

In the fractional logistic setting, the first HSV terms are

$\alpha=0$4

$\alpha=0$5

$\alpha=0$6

$\alpha=0$7

The corresponding approximation is the partial sum

$\alpha=0$8

and the authors show that the series has the form

$\alpha=0$9

For the special case $\alpha=1$0, the fractional-delay logistic equation becomes linear and admits the exact closed form

$\alpha=1$1

The source states that this solution reduces to classical exponential-type growth as $\alpha=1$2 and slows with stronger fractional memory, that is, smaller $\alpha=1$3 (Aibinu et al., 22 Sep 2025).

6. Existence, stability, and convergence properties

The theoretical treatment differs between the pantograph and fractional formulations. In the 2021 population-dynamics paper, no formal convergence theorem is given. Instead, the analysis follows the standard VIM rationale: if the nonlinear operator satisfies a Lipschitz condition on a suitable function space and the linear operator is invertible under the transform-based iteration, the HSV fixed-point mapping can be contractive on a sufficiently small time interval or under suitable parameter bounds. The residual norm $\alpha=1$4 or the successive-difference norm $\alpha=1$5 is then used as an empirical stopping criterion. The paper reports graphical agreement with benchmark models but does not provide numerical error norms (Aibinu et al., 2021).

The fractional ABC-delay study states stronger analytical results for the underlying initial-value problem. Under standard assumptions that $\alpha=1$6 is continuous in $\alpha=1$7 and Lipschitz in $\alpha=1$8, the problem

$\alpha=1$9

admits a unique local solution on $$x'(t)=P\,x(\alpha t)e^{-x(\alpha t)/x_0}-\delta x(t), \qquad x(0)=x_0.$$0 for some $$x'(t)=P\,x(\alpha t)e^{-x(\alpha t)/x_0}-\delta x(t), \qquad x(0)=x_0.$$1. If the solution remains bounded on $$x'(t)=P\,x(\alpha t)e^{-x(\alpha t)/x_0}-\delta x(t), \qquad x(0)=x_0.$$2, it extends globally. The source attributes this to the equivalent Volterra integral equation with the ABC kernel and standard fixed-point arguments in appropriate Banach spaces (Aibinu et al., 22 Sep 2025).

The same paper also records Hyers–Ulam stability. If an approximate solution $$x'(t)=P\,x(\alpha t)e^{-x(\alpha t)/x_0}-\delta x(t), \qquad x(0)=x_0.$$3 satisfies

$$x'(t)=P\,x(\alpha t)e^{-x(\alpha t)/x_0}-\delta x(t), \qquad x(0)=x_0.$$4

then there exists an exact solution $$x'(t)=P\,x(\alpha t)e^{-x(\alpha t)/x_0}-\delta x(t), \qquad x(0)=x_0.$$5 such that

$$x'(t)=P\,x(\alpha t)e^{-x(\alpha t)/x_0}-\delta x(t), \qquad x(0)=x_0.$$6

for a constant $$x'(t)=P\,x(\alpha t)e^{-x(\alpha t)/x_0}-\delta x(t), \qquad x(0)=x_0.$$7 independent of $$x'(t)=P\,x(\alpha t)e^{-x(\alpha t)/x_0}-\delta x(t), \qquad x(0)=x_0.$$8 and $$x'(t)=P\,x(\alpha t)e^{-x(\alpha t)/x_0}-\delta x(t), \qquad x(0)=x_0.$$9. In that setting, the HSV series is reported to converge rapidly in the figures, and the geometric-like factor

$${}^{ABC}_0D^\mu z(t)=r\,z(t)\left(1-\frac{z(\lambda t)}{K}\right), \qquad z(0)=z_0,\qquad 0<\mu<1,\ 0\le \lambda\le 1.$$0

is said to suggest a sufficient smallness condition, in time windows or parameter regimes, for convergence of the series. The paper explicitly notes, however, that a formal contraction proof for HSV itself is not given (Aibinu et al., 22 Sep 2025).

The classical logistic equilibria are also used as a qualitative reference point. For

$${}^{ABC}_0D^\mu z(t)=r\,z(t)\left(1-\frac{z(\lambda t)}{K}\right), \qquad z(0)=z_0,\qquad 0<\mu<1,\ 0\le \lambda\le 1.$$1

the equilibria are $${}^{ABC}_0D^\mu z(t)=r\,z(t)\left(1-\frac{z(\lambda t)}{K}\right), \qquad z(0)=z_0,\qquad 0<\mu<1,\ 0\le \lambda\le 1.$$2 and $${}^{ABC}_0D^\mu z(t)=r\,z(t)\left(1-\frac{z(\lambda t)}{K}\right), \qquad z(0)=z_0,\qquad 0<\mu<1,\ 0\le \lambda\le 1.$$3, with $${}^{ABC}_0D^\mu z(t)=r\,z(t)\left(1-\frac{z(\lambda t)}{K}\right), \qquad z(0)=z_0,\qquad 0<\mu<1,\ 0\le \lambda\le 1.$$4 unstable and $${}^{ABC}_0D^\mu z(t)=r\,z(t)\left(1-\frac{z(\lambda t)}{K}\right), \qquad z(0)=z_0,\qquad 0<\mu<1,\ 0\le \lambda\le 1.$$5 asymptotically stable. The fractional and delayed model is reported to preserve the carrying-capacity-controlled saturation qualitatively, while the fractional order $${}^{ABC}_0D^\mu z(t)=r\,z(t)\left(1-\frac{z(\lambda t)}{K}\right), \qquad z(0)=z_0,\qquad 0<\mu<1,\ 0\le \lambda\le 1.$$6 and delay parameter $${}^{ABC}_0D^\mu z(t)=r\,z(t)\left(1-\frac{z(\lambda t)}{K}\right), \qquad z(0)=z_0,\qquad 0<\mu<1,\ 0\le \lambda\le 1.$$7 modulate the rate and smoothness of convergence (Aibinu et al., 22 Sep 2025).

7. Relation to adjacent methods and modeling significance

HSV is compared qualitatively with several other analytic and semi-analytic techniques. Relative to Adomian Decomposition Method, it is described as simplifying bookkeeping because the Sumudu transform converts derivatives into algebraic terms before Adomian polynomials are applied. Relative to Homotopy Perturbation Method, it avoids the introduction and tuning of an embedding parameter. Relative to pure VIM, it makes the Lagrange multiplier easier to determine because the variational calculation is performed in the transform domain. Relative to Laplace-based approaches, it retains the relation to Laplace inversion while using a transform that preserves units and handles pantograph scaling directly through $${}^{ABC}_0D^\mu z(t)=r\,z(t)\left(1-\frac{z(\lambda t)}{K}\right), \qquad z(0)=z_0,\qquad 0<\mu<1,\ 0\le \lambda\le 1.$$8 (Aibinu et al., 2021).

The fractional paper adds a kernel-level comparison. In its qualitative summary, an ABC formulation with Mittag-Leffler kernel is associated with smooth, gradually fading memory curves; a CFC formulation with exponential kernel is associated with faster initial rise but more rigid dynamics; and a classical Caputo formulation with singular kernel is associated with sharper transitions and heightened sensitivity to delay. No direct numerical comparison table is provided, but the paper uses this contrast to position HSV as a semi-analytical method that preserves the memory structure induced by the ABC operator (Aibinu et al., 22 Sep 2025).

The method also has stated limitations. When fully nonlinear delay terms and nontrivial initial histories are present, the history contribution must be incorporated carefully in the transformed equation. Inverse Sumudu transforms may require Laplace inversion machinery, and non-elementary transforms can introduce computational overhead. For strongly nonlinear terms, the time interval may need to be limited and stepwise continuation may be advisable. These are implementation constraints rather than failures of the formal framework (Aibinu et al., 2021).

In the applied interpretation offered for the fractional logistic model, smaller $${}^{ABC}_0D^\mu z(t)=r\,z(t)\left(1-\frac{z(\lambda t)}{K}\right), \qquad z(0)=z_0,\qquad 0<\mu<1,\ 0\le \lambda\le 1.$$9 strengthens memory effects and slows the approach to saturation, whereas larger $t$00 weakens memory and moves the dynamics toward classical logistic behavior. Smaller $t$01 enhances delay effects by increasing the influence of earlier states, while larger $t$02 reduces the delay effect. Joint variation of $t$03 and $t$04 is reported to tune growth rate, saturation smoothness, and transient behavior. The paper identifies this as relevant for industrial processes with lagged control, biological systems with delayed feedback, and social or market dynamics with adoption lags, and describes the integration of ABC derivatives, proportional delay, and HSV solutions as novel in this logistic setting (Aibinu et al., 22 Sep 2025).