# On Λ-Fractional Analysis and Mechanics

## Abstract

**:**

## 1. Introduction

## 2. The Fractional Calculus

## 3. The Homogenization of the Fractals Procedure

**R**, and the gamma function Γ. Further, the homogenization procedure is used to apply Green–Gauss field’s theorem, valid for local derivatives. For the one-dimensional fractals, the coefficient ${c}_{3}$ becomes:

## 4. The Λ-Fractional Analysis

- If f(z) is an analytic function of the complex variable z (or z = x a real variable), the derivative D
^{γ}(f(z)) is the analytic function of γ and z. - The operation D
^{γ}f must produce the same result as the ordinary differentiation when γ is a positive integer: D^{γ}f(x) = f^{γ}(x). If γ = −n, a negative integer, D^{γ}f(x) must produce the same result as ordinary n-fold integration, and g(x) = D^{(−n)}f(x) must vanish together with all its n-l derivatives at x = the lower terminal of integration. - The fractional operators must be linear.
- The operation of order zero leaves the function unchanged: D
^{o}f = f. - The law of exponents (indices) holds for integration of arbitrary order:

^{−μ}D

^{−v}f = D

^{−μ−ν}f, Re(μ) and R(v) > O.

^{3}.

^{3}and the green one is the conventional derivative. Furthermore, the yellow one is the well-known Riemann–Liouville fractional derivative. However, the red one is the function of the Λ-fractional derivative transferred into the initial space. The action of the Λ-fractional derivative is more intense than the other ones.

## 5. The Λ-Fractional Fractal Bar Extension

_{o}and the deformed ψ(x

_{ο}) placements of the bar in the initial space.

^{Λ}along the x-axis. Indeed:

#### Non-Local Action

## 6. The Extension of a Bar with Fractal Cross-Section Area Distribution

#### 6.1. The Left Λ-Fractional Analysis

_{H}= 1.5, and it is defined by:

^{Λ}defined by Equation (25).

#### 6.2. The Right Λ-Fractional Analysis

^{Λ}defined by Equation (25).

#### 6.3. The Two Side Λ-Fractional Analysis

## 7. Λ-Fractional Deformation with a Horizon

^{Λ}in the Λ-fractional space is defined by, see Equation (59):

^{Λ}in the Λ-fractional space is defined by:

^{ΛH}with horizon H is defined by:

## 8. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 8.**The non-dimensional displacement of the symmetric fractional fractal bar of order γ = 0.63.

**Figure 9.**The non-dimensional displacement of the symmetric fractional fractal bar of order γ = 0.80.

**Figure 27.**The average left and right Λ-fractional displacement of the Cantor rod in the initial space.

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Lazopoulos, K.A.
On Λ-Fractional Analysis and Mechanics. *Axioms* **2022**, *11*, 85.
https://doi.org/10.3390/axioms11030085

**AMA Style**

Lazopoulos KA.
On Λ-Fractional Analysis and Mechanics. *Axioms*. 2022; 11(3):85.
https://doi.org/10.3390/axioms11030085

**Chicago/Turabian Style**

Lazopoulos, Konstantinos A.
2022. "On Λ-Fractional Analysis and Mechanics" *Axioms* 11, no. 3: 85.
https://doi.org/10.3390/axioms11030085