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Fractal Sumudu Transform and Economic Models

This paper introduces a new fractal derivative with a nonsingular kernel, analyzes its fundamental properties, and demonstrates its effectiveness by applying it alongside the Caputo fractal derivative to study economic models.

Original authors: Krishna Mani Nath, Bipan Hazarika, Hemanta Kalita

Published 2026-02-23

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Krishna Mani Nath, Bipan Hazarika, Hemanta Kalita

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Why "Rough" Math Matters

Imagine you are looking at a coastline on a map. If you zoom in, you don't see a smooth line; you see bays, rocks, and tiny pebbles. The more you zoom in, the more jagged it gets. This is a fractal.

For a long time, mathematicians and economists only liked "smooth" things (like perfect circles or straight lines) because they were easy to calculate. They ignored the "rough" stuff. But the real world—nature, stock markets, and human behavior—is rarely smooth. It's jagged, unpredictable, and full of tiny details.

This paper argues that to understand the economy better, we need math that can handle these "rough" edges. The authors introduce a new mathematical tool to do exactly that.

The New Tool: The "Memory" Calculator

The core of this paper is a new type of math called a Fractal Derivative.

The Analogy: The Forgetful vs. The Remembering Driver

Old Math (Classical Calculus): Imagine a driver who only looks at the road directly in front of their bumper. If they hit a bump right now, they react. They have no memory of the potholes they hit five minutes ago. This is how traditional economics often works: it assumes the market reacts only to the current price.
New Math (Fractal Derivative): Now, imagine a driver who remembers every bump, every turn, and every speed bump they've encountered for the last hour. Their reaction to a current bump is influenced by the memory of the previous ones.

The authors propose a new "memory" calculator (specifically, one with a nonsingular kernel, which is just a fancy way of saying the memory fades out smoothly rather than exploding into infinity). They call this new tool the wsk-fractal derivative.

They also use a specific mathematical technique called the Sumudu Transform. Think of this as a "magic translator." It takes a complicated, messy equation (like a chaotic market) and translates it into a simpler language where the answer is easy to find. Once the answer is found, it translates it back into the real world.

The Real-World Test: The Price of Apples

To prove their new math works, the authors applied it to a classic economic problem: Price Adjustment.

The Scenario:
Imagine a market for apples.

Demand: If apples are cheap, people buy more.
Supply: If apples are expensive, farmers grow more.
Equilibrium: The "perfect price" is where the amount people want to buy equals the amount farmers want to sell.

Usually, prices don't jump instantly to this perfect price. They wiggle and adjust over time.

The Experiment:
The authors ran two simulations using their new math:

The "No-Expectation" Model: People just react to the price right now.
The "With-Expectation" Model: People look at the past and guess what the price will be tomorrow. They adjust their buying based on their "memory" of the market.

The Results:

Using the old math (Caputo fractal derivative), they got a solution that looked like a complex, wavy curve (involving something called the Mittag-Leffler function). It showed the price slowly settling down, but with a "fractal" texture—meaning the path wasn't a smooth slide; it was a jagged, realistic walk.
Using their new math (wsk-fractal derivative), they got a solution that looked like a smooth exponential curve (like a standard cooling cup of coffee).

Why does this matter?
The paper suggests that by using these new "fractal" tools, economists can describe market behavior more accurately.

If the market is jagged and unpredictable (like a stormy sea), the fractal model captures that chaos better.
If the market is smooth and predictable (like a calm lake), the new model simplifies the math to show that clearly.

The Takeaway

This paper is like giving economists a new pair of glasses.

Old Glasses: Showed the economy as a smooth, straight line. Good for simple problems, but blurry for real life.
New Glasses (Fractal Sumudu): Show the economy as it really is—rough, detailed, and full of history.

The authors conclude that by using these new "memory-aware" math tools, we can better understand how prices find their balance, how people form expectations, and how to predict market crashes or booms more accurately. It's a step toward making economic models feel less like abstract theory and more like the messy, beautiful reality of human behavior.

1. Problem Statement

The paper addresses two primary challenges:

Mathematical Modeling of Irregular Systems: Classical calculus often fails to accurately describe natural phenomena and economic systems that exhibit irregular, non-smooth, or fractal structures. While fractal geometry provides a framework for these shapes, there is a need for robust fractional calculus operators (specifically those with non-singular kernels) to model dynamic processes within these structures.
Economic Price Adjustment Dynamics: Traditional economic models often assume smooth, continuous price adjustments. However, real-world markets involve complex, memory-dependent behaviors and agent expectations. The authors aim to solve price adjustment equations in competitive markets using advanced fractal calculus tools to better capture market equilibrium dynamics, specifically comparing models with and without agent expectations.

2. Methodology

The authors employ a multi-step mathematical approach combining fractal calculus, integral transforms, and economic modeling:

Fractal Calculus Framework: The study utilizes the Fractal Calculus framework, defined on fractal sets $F$ with a specific order $\alpha$ . Key concepts include:
- Mass Function ( $\gamma_\alpha$ ): To measure the "mass" of a set on a fractal.
- Integral Staircase Function ( $S^\alpha_F$ ): A function that maps the fractal set to a continuous domain, acting as the variable for integration and differentiation.
- Fractal Derivatives: The paper reviews Riemann-Liouville and Caputo fractal derivatives.
Fractal Sumudu Transform: The authors utilize the Fractal Sumudu Transform ( $S^\alpha_F$ ), an integral transform adapted for functions defined on fractal supports. This transform is chosen for its ability to handle differential equations without changing the units of the variables, making it suitable for physical and economic modeling. The paper establishes relationships between the Fractal Sumudu Transform and the Fractal Laplace Transform.
Introduction of a New Operator: The core mathematical innovation is the definition of a new fractal derivative with a non-singular kernel, termed the wsk-fractal time derivative ( $wskD^\gamma_F$ ).
- Unlike the Caputo derivative which uses a power-law kernel (singular at the origin), the wsk derivative uses an exponential kernel: $e^{-\frac{\gamma}{1-\gamma}(S^\alpha_F(t) - S^\alpha_F(\tau))}$ .
- This operator is designed to model systems with non-local memory effects but without the singularity issues often found in fractional derivatives.
Economic Modeling: The authors formulate price adjustment equations for competitive markets. They define demand ( $q_d$ ) and supply ( $q_s$ ) functions based on price $p(t)$ and constants ( $d_0, d_1, s_0, s_1$ ). They solve these differential equations under two scenarios:
1. Without Expectations: Agents react only to current price deviations.
2. With Expectations: Agents react to anticipated future price changes (introducing a second derivative term or modified coefficients).

3. Key Contributions

Definition of the wsk-Fractal Derivative: The paper introduces a novel fractal derivative operator ( $wskD^\gamma_F$ ) featuring a non-singular exponential kernel. This fills a gap in the literature regarding non-local operators that avoid singularities while retaining memory properties.
Derivation of Transform Properties: The authors rigorously derive the Fractal Sumudu Transform for the new wsk derivative (Theorem 3.6), providing a closed-form expression in the transform domain. This enables the analytical solution of differential equations involving this new operator.
Analytical Solutions to Economic Models: The paper provides exact analytical solutions for price adjustment equations using:
- The Caputo fractal derivative (yielding solutions involving the Fractal Mittag-Leffler function).
- The new wsk-fractal derivative (yielding solutions involving Fractal Exponential functions).
Comparative Analysis: The study explicitly contrasts the behavior of market equilibrium under the two different derivative operators, showing how the choice of kernel (power-law vs. exponential) alters the dynamic path of price convergence.

4. Results

The authors successfully solved the price adjustment equations for four distinct cases:

Case 1 (Caputo, No Expectations):
The price $p(t)$ converges to equilibrium $p^*$ following a Mittag-Leffler function decay:
$p(t) = p(0)E^\alpha_F(-\lambda(d_1+s_1)t^\beta) + \frac{\lambda(d_0+s_0)}{d_1+s_1}(1 - E^\alpha_F(-\lambda(d_1+s_1)t^\beta))$
This indicates a power-law memory effect in price adjustment.
Case 2 (Caputo, With Expectations):
The solution remains in terms of the Mittag-Leffler function but with modified coefficients reflecting agent expectations, altering the rate of convergence.
Case 3 (wsk-Derivative, No Expectations):
The solution involves the standard exponential function (due to the non-singular kernel):
$p(t) = C_1 e^{-kt} + C_2(1 - e^{-kt})$
where the rate constant depends on the normalization function $N(\gamma)$ and the fractal order. This suggests a different memory decay profile compared to the Caputo model.
Case 4 (wsk-Derivative, With Expectations):
Similar to Case 3, the solution is exponential but with coefficients adjusted for expectations.

Key Finding: The wsk-fractal derivative yields solutions in terms of exponential functions, whereas the Caputo fractal derivative yields solutions in terms of Mittag-Leffler functions. This implies that the choice of kernel fundamentally changes the mathematical nature of the economic dynamics (e.g., the "memory" of the market).

5. Significance

Theoretical Advancement: The paper expands the toolkit of fractional calculus by introducing a non-singular kernel operator specifically adapted for fractal sets. This is crucial for modeling systems where singularities are physically unrealistic.
Economic Insight: By applying these advanced operators to economic models, the study demonstrates that market equilibrium is not a single fixed dynamic. Instead, the "memory" of the market (how past prices influence current adjustments) can be modeled with varying degrees of complexity (power-law vs. exponential decay).
Methodological Utility: The successful application of the Fractal Sumudu Transform to these new operators provides a robust method for solving complex differential equations in physics and economics that were previously difficult to solve analytically.
Future Applications: The framework proposed can be extended to other non-linear economic systems, biological models, and physical processes occurring on fractal domains, offering a more accurate representation of irregular real-world phenomena.