Dimensional consistency in fractional differential equations with non singular kernels

This paper proposes a simple change of variables to resolve dimensional consistency issues when replacing ordinary time derivatives with fractional derivatives in equations featuring non-singular kernels, demonstrating the method's validity through a specific example.

The Gist

The Big Problem: Mixing Apples and Oranges

Imagine you are baking a cake. The recipe calls for "1 cup of flour" and "2 eggs." These are dimensionally consistent; they fit together to make a cake.

Now, imagine someone tries to rewrite the recipe using a new, fancy measuring tool called a "Fractional Cup." This tool doesn't measure in whole cups; it measures in "half-cups" or "three-quarters of a cup."

The problem is: If you just swap "1 cup" for "1 fractional cup" without changing anything else, your math breaks. You are trying to add "cups" to "eggs" in a way that doesn't make physical sense. In physics, this is called a dimensional inconsistency.

In the world of physics, scientists use Fractional Calculus to model things that have "memory" or "friction" (like how a spring slowly stops bouncing, or how electricity flows through a circuit with a bit of a delay). Traditional math uses whole numbers (1st derivative, 2nd derivative). Fractional math uses decimals (0.5 derivative, 0.8 derivative).

The author of this paper, Gabriel González, noticed that when scientists tried to use these "fractional" tools to describe real-world physics (like an electrical circuit), the units didn't match up. It was like trying to drive a car where the speedometer says "miles" but the gas pedal is calibrated in "liters." The car wouldn't work right.

The Old Solution: The "Magic Number" Fix

Previously, when scientists hit this problem, they would just invent a "magic number" (a scaling parameter) to force the units to match.

• Analogy: Imagine your recipe calls for "1 cup," but your new tool measures in "0.8 cups." To fix it, you just say, "Okay, let's pretend 1 cup is actually 1.25 of these new cups."
• The Flaw: This magic number often didn't have a clear physical meaning. It was just a mathematical trick to make the equation balance, but it didn't explain why the physics worked that way.

The New Solution: The "Time Travel" Map

González proposes a smarter way to fix the units, specifically for a modern type of fractional math called the Caputo-Fabrizio derivative (which uses a smooth, non-singular kernel). Think of this as a smoother, more realistic way to model memory than the old, jagged methods.

Instead of just throwing a magic number at the problem, he suggests rewriting the map of time itself.

1. The Old Time vs. The New Time: Imagine you are walking down a hallway. In "Normal Time" (integer calculus), you walk at a steady pace. In "Fractional Time," the hallway stretches and shrinks depending on how much "memory" the system has.
2. The Variable Change: González introduces a new function, let's call it $\phi$ (phi). This function acts like a translator. It takes the "Normal Time" we live in and converts it into "Fractional Time" in a way that respects the physical units (seconds, volts, etc.).
3. The Result: By using this translator, the equation automatically balances itself. You don't need a magic number; the math naturally knows that "1 second" in the fractional world is slightly different from "1 second" in the normal world, and it adjusts the units accordingly.

The Test Drive: The RC Circuit

To prove his idea works, the author tested it on a classic physics problem: The RC Circuit.

• What is it? A simple electrical circuit with a Resistor (R) and a Capacitor (C). Think of it like a water tank (capacitor) filling up through a narrow pipe (resistor).
• The Goal: Predict how fast the water level (voltage) rises.
• The Test: He replaced the standard "filling speed" equation with his new "Fractional filling speed" equation.
• The Outcome:
  • When he set the fractional number ($\alpha$) to 1, the math perfectly matched the real-world, standard physics (the water fills up normally).
  • When he set $\alpha$ to 0.8 or 0.9, the math showed a "sluggish" filling process. This represents a system with internal friction or "memory"—the water doesn't just flow; it hesitates and remembers its past flow.
  • Crucially, the units (Volts, Seconds) stayed consistent the whole time. No magic numbers were needed.

The Takeaway

This paper is essentially a guide on how to speak "Fractional" without losing your accent.

• Before: Scientists tried to speak Fractional by forcing the words to fit, which made the sentences sound weird and physically impossible.
• Now: González gives us a dictionary (the new variable change) that translates the concepts correctly.

The Metaphor:
If classical physics is a straight road, fractional physics is a winding mountain trail.

• The old way of doing math tried to measure the trail with a ruler meant for a straight road, which gave wrong distances.
• González's method is like giving you a GPS that recalculates the distance based on the winding path. It ensures that when you say "I walked 5 miles," it actually means 5 miles of that specific trail, not 5 miles of a straight highway.

This ensures that when we model complex real-world things (like how heat spreads, how diseases spread, or how electricity flows in new materials), our math remains physically honest and consistent.

Technical Summary

1. Problem Statement

The paper addresses a fundamental issue in fractional calculus: dimensional inconsistency when replacing ordinary integer-order derivatives with fractional-order derivatives in physical models.

• The Core Issue: In classical physics, the time derivative operator $d/dt$ has dimensions of $[Time]^{-1}$ (e.g., $s^{-1}$). However, standard fractional derivative operators (such as the Caputo or Caputo-Fabrizio derivatives) often result in dimensions of $[Time]^{-\alpha}$ (where $0 < \alpha < 1$).
• Consequence: Directly substituting $d/dt$ with a fractional operator in physical equations (like those describing electrical circuits or mechanical oscillators) creates a dimensional imbalance. This renders the equations physically meaningless unless corrective measures are taken.
• Limitations of Existing Solutions: Previous strategies often introduce an auxiliary scaling parameter $\sigma$ (with units of time) to restore homogeneity. While effective for singular kernels (like Riemann-Liouville or Caputo), these methods can be arbitrary or physically difficult to interpret. Furthermore, specific challenges arise with non-singular kernels (like the Caputo-Fabrizio derivative), which are dimensionless by definition, requiring a different approach to dimensional regularization.

2. Methodology

The author proposes a systematic method to construct fractional differential equations (FDEs) with non-singular kernels while maintaining strict dimensional consistency without arbitrary scaling parameters.

• Variable Transformation: The core of the methodology involves introducing a new time-like variable $\tau$ defined by an integral transformation:
  $$\tau(t, \alpha) = \int_0^t \frac{dt}{\phi(t, \alpha)}$$
  Here, $\phi(t, \alpha)$ is a function with dimensions of time that depends on the fractional order $\alpha$.
• Operator Replacement: The ordinary derivative is replaced via the chain rule:
  $$\frac{d}{dt} \rightarrow \frac{d\tau}{dt} \frac{d}{d\tau} = \frac{1}{\phi(t, \alpha)} \frac{d}{d\tau}$$
  This allows the fractional derivative operator ($CFD^\alpha_t$) to be applied to the transformed variable $\tau$, ensuring the resulting equation retains the correct physical dimensions.
• Boundary Condition: The function $\phi(t, \alpha)$ is constrained such that when $\alpha \to 1$, the transformation recovers the standard integer-order derivative ($\frac{1}{\phi(t, 1)} \frac{d}{d\tau} = \frac{d}{dt}$).
• Analytical Solution: The paper derives the general solution for a first-order linear FDE with constant coefficients using this framework, employing an integrating factor method adapted for the transformed variable.

3. Key Contributions

• Dimensional Regularization for Non-Singular Kernels: The paper provides a specific mathematical framework to handle the Caputo-Fabrizio (CF) derivative, which is inherently dimensionless. Unlike singular kernel methods that rely on a constant scaling factor $\sigma$, this method uses a time-dependent function $\phi(t, \alpha)$ to balance dimensions dynamically.
• General Solution Derivation: The author derives an analytical solution for the general first-order FDE:
  $$CFD^\alpha_\tau x(\tau) + P(\tau)x(\tau) = Q(\tau)$$
  The solution is expressed in terms of the original time $t$ and the fractional order $\alpha$, ensuring the result is physically interpretable.
• Recovery of Classical Limits: The methodology is proven to satisfy the condition that as $\alpha \to 1$, the fractional solution converges exactly to the classical integer-order solution.

4. Results: RC Circuit Analysis

To validate the methodology, the author applies the proposed framework to an RC circuit (Resistor-Capacitor) connected to a DC battery.

• Model Formulation: The standard Kirchhoff voltage law equation is transformed using the proposed $\phi(t, \alpha) = e^{-(1-\alpha)\Gamma t}/\Gamma$ (where $\Gamma = 1/RC$).
• Derived Solution: The charge $q(t, \alpha)$ and capacitor voltage $V_C(t, \alpha)$ are derived as:
  $$q(t, \alpha) = q_0 \left( 1 - e^{(1-\alpha)\Gamma t} \left( \frac{2-\alpha}{(1-\alpha) + e^{(1-\alpha)\Gamma t}} \right)^{1/(1-\alpha)} \right)$$
• Verification:
  • Limit Check: Taking the limit $\alpha \to 1$ in the derived equation yields $q(t, 1) = q_0(1 - e^{-\Gamma t})$, which is the exact classical solution for an RC circuit.
  • Physical Interpretation: The graphical analysis (Figure 2) shows that for $0 < \alpha < 1$, the system exhibits a dissipative behavior distinct from the classical case. The fractional order $\alpha$ represents a non-local effect of energy dissipation (internal friction).
• Observation: The results demonstrate that the proposed method yields a physically consistent model where the voltage evolution depends on the fractional order, offering a more nuanced description of memory effects in the circuit compared to classical models.

5. Significance

• Physical Interpretability: By resolving the dimensional inconsistency, the paper ensures that fractional differential equations with non-singular kernels can be used to model real-world physical systems without introducing arbitrary, non-measurable constants.
• Theoretical Rigor: It bridges the gap between the mathematical convenience of non-singular kernels (which avoid singularities at the origin and simplify initial condition handling) and the physical requirement of dimensional homogeneity.
• Applicability: The proposed systematic approach is not limited to RC circuits; the author notes it can be extended to higher-order derivatives and other physical systems (e.g., mechanical oscillators, diffusion processes), providing a robust tool for researchers in fractional modeling.

In conclusion, González presents a rigorous mathematical correction to the application of Caputo-Fabrizio derivatives in physics, ensuring that fractional models remain dimensionally sound and physically interpretable.