Fractal Sumudu Transform and Economic Models

In this paper, we present a new fractal derivative with a nonsingular kernel and analyze its fundamental properties. The effectiveness of the proposed operator is illustrated through the study of economic models using both the Caputo fractal derivative and the new fractal derivative.

💡 Research Summary

The manuscript introduces a novel fractal derivative operator that avoids the singular kernel typical of traditional fractional calculus, and investigates its properties through the framework of the fractal Sumudu transform. After reviewing the necessary preliminaries—mass functions, the staircase function SαF, and the definitions of fractal Laplace and Sumudu transforms—the authors establish a key relationship (Lemma 2.13) linking the two transforms, which allows results derived in one domain to be transferred to the other.

The core contribution is the definition of the “new fractal derivative” N a DγF f(t) (Equation 3.1). Unlike the Caputo or Riemann–Liouville derivatives, whose kernels contain a term (t−τ)^{−(1−β)} that becomes singular as τ→t, the proposed kernel uses (SαF(t)−SαF(τ))^γ, which remains bounded. The operator is normalized by the fractal Gamma function ΓαF(n−γ) and reduces to a simpler form for first‑order derivatives (Equation 3.2). The authors compute the operator on constant and power functions, showing that it annihilates constants and yields a closed‑form expression involving fractal Gamma functions for powers x^p when p>n−1. They also prove linearity (Theorem 3.4) and relate the operator to a fractal Beta function, establishing a connection with classical special functions.

A secondary operator, the “wsk‑fractal derivative” (Definition 3.5), modifies the kernel further by inserting an exponential factor e^{−γ(1−γ)SαF(t)} and a normalization function N(γ). Theorem 3.6 provides its Laplace and Sumudu transforms, revealing additional correction terms that disappear when γ→0 or γ→1, thereby ensuring consistency with integer‑order calculus.

The practical relevance is demonstrated by applying both the Caputo fractal derivative and the new wsk‑derivative to a price‑adjustment model in a competitive market. The Caputo‑based equation (3.16) incorporates a fractional order β∈(0,1) and a linear combination of demand and supply coefficients. By taking the Sumudu transform, the authors obtain an algebraic expression for the transformed price p̂(v) and invert it to retrieve p(t). A parallel analysis using the wsk‑derivative yields a modified algebraic equation, allowing a direct comparison of the two fractional dynamics.

While the theoretical development is mathematically elegant, several shortcomings limit the impact of the work. First, many proofs are sketched rather than fully detailed; for instance, the linearity proof does not explicitly state the function spaces required, and the derivation of the Beta–Gamma relationship assumes properties that are not verified for the fractal analogues. Second, the convergence domains of the fractal Sumudu transform are only briefly mentioned; rigorous conditions on the admissible functions are missing, which raises concerns about the applicability to real‑world data. Third, the economic application remains at a purely analytical level: no numerical simulations, parameter estimation, or empirical validation are provided. Consequently, the economic interpretation of the fractional orders β and γ, as well as the practical advantage of the non‑singular kernel over the traditional Caputo derivative, are not convincingly demonstrated.

In conclusion, the paper contributes a new non‑singular fractal derivative and integrates it with the fractal Sumudu transform, offering fresh analytical tools for fractional dynamic models. To move beyond a theoretical exposition, future research should (i) furnish complete proofs and clarify functional spaces, (ii) delineate precise convergence criteria for the transforms, (iii) explore numerical schemes for solving the resulting fractional differential equations, and (iv) apply the framework to empirical economic data to assess its predictive power and economic relevance.