# On Λ-Fractional Differential Equations

## Abstract

**:**

## 1. Introduction

## 2. Fractional Calculus

## 3. Geometry in the Λ-Fractional Space

^{2}y

^{2}, 0 < x < 1, 0 < y < 1

^{1.714}Y

^{1.714}.

## 4. The Fractional Field Theorems

- a.
- Green’s theorem. Let Q
_{x}(x,y), Q_{y}(x,y), be smooth real functions in a domain Ω, with its boundary being a smooth closed curve $\partial \Omega $. Then,$${{\displaystyle \int}}_{\partial \Omega}^{}\left({Q}_{x}dx+{Q}_{y}dy\right)={{\displaystyle \iint}}_{\Omega}^{}dxdy\left(\frac{d{Q}_{x}}{dy}-\frac{d{Q}_{y}}{dx}\right)$$

_{x}(x,y), Q

_{y}(x,y), are derived by a potential function $\Phi $(x,y) with ${Q}_{x}=\frac{d\Phi}{dx},{Q}_{y}=\frac{d\Phi}{dy},$ the RHS of Equation (22) becomes zero. That means that the curvilinear integral along a closed smooth boundary is zero.

- b.
- Stoke’s theorem:

**F**defined on a simple surface Ω with the boundary $\partial \Omega $, Stoke’s theorem is expressed by:

- c.
- The Gauss’ (divergence) theorem:

**F**over Ω is equal to the surface integral of

**F**over the boundary $\partial \Omega $:

## 5. Fractional Multiple Integrals and Calculus of Variations

## 6. Existence and Uniqueness Theorems of Λ-Fractional Differential Equations

_{0}when X = X

_{0}. The solution may be transferred into the initial space through the relation:

_{0}is a given number and Z

_{0}= G(Y,0), Q

_{0}=${G}^{\prime}\left({Y}_{0}\right)$, and if F(X,Y,Z,G) and all its partial derivatives are continuous in the region S defined by:

- (a)
- Φ(Χ,Υ) and all its partial derivatives are continuous in the region R defined by:$$\left|X-{X}_{0}\right|<{\mathsf{\Delta}}_{1},\left|{\rm Y}-{{\rm Y}}_{0}\right|{\mathsf{\Delta}}_{2}.$$

- (b)
- For all (X,Y) in R, Z = Φ(X,Y) is a solution of the equation:$$\frac{\partial Z}{\partial X}=F(X,Y,Z,\frac{\partial Z}{\partial Y}).$$

- (c)
- For all values of Y in the interval $\left|{\rm Y}-{{\rm Y}}_{0}\right|<{\mathsf{\Delta}}_{1}$, Φ(X
_{0},Y) = G(Y).

## 7. Linear Oscillations with Fractional Dissipation

## 8. The Wave Equation

_{1}only. The second is the additional fractional order γ

_{2}of the space x co-ordinate to simulate the non-homogeneous media, such as porous or composite materials.

_{1}, the solution in the initial space is presented for fractional order γ

_{1}= 0.8 in Figure 10.

_{1}with additional fractional order γ

_{2}of the space x co-ordinate to simulate the non-homogeneous media, such as porous or composite materials.

_{1}= 0.5 and space fractional order γ

_{2}= 0.5 corresponding to various non-uniformities of the medium.

_{1}= 0.5 and space fractional order γ

_{2}= 0.8 is shown in Figure 12.

_{1}= 0.8, the solution y(x,t) for space-fractional order γ

_{2}= 0.5 is shown in Figure 13.

_{1}= 0.8, the solution y(x,t) for space-fractional order γ

_{2}= 0.8 is shown in Figure 14.

## 9. The Λ-Fractional Diffusion Equation

_{1}only. In that case,

_{1}= 0.8. in the initial space.

_{1}with additional fractional order γ

_{2}of the space x co-ordinate to simulate the non-homogeneous media, such as porous or composite materials. In that case,

_{1}= 0.5 and space fractional order γ

_{2}= 0.5.

_{1}= 0.5 and space fractional order γ

_{2}= 0.8.

_{1}= 0.8 and space fractional order γ

_{2}= 0.5.

_{1}= 0.8 and space fractional order γ

_{2}= 0.8.

## 10. Branching of the Λ-Fractional Differential Equations

## 11. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 16.**The solution of the diffusion equation in the initial space for fractional order γ

_{1}= 0.5.

**Figure 17.**The solution of the diffusion equation in the initial space for fractional order γ

_{1}= 0.8.

**Figure 19.**The solution of the diffusion equation in the initial space for time fractional order γ

_{1}= 0.5 and space fractional order γ

_{2}= 0.5.

**Figure 20.**The solution of the diffusion equation in the initial space for time fractional order γ

_{1}= 0.5 and space fractional order γ

_{2}= 0.8.

**Figure 21.**The solution of the diffusion equation in the initial space for time fractional order γ

_{1}= 0.8 and space fractional order γ

_{2}= 0.5.

**Figure 22.**The solution of the diffusion equation in the initial space for time fractional order γ

_{1}= 0.8 and space fractional order γ

_{2}= 0.8.

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Lazopoulos, K.A.
On Λ-Fractional Differential Equations. *Foundations* **2022**, *2*, 726-745.
https://doi.org/10.3390/foundations2030050

**AMA Style**

Lazopoulos KA.
On Λ-Fractional Differential Equations. *Foundations*. 2022; 2(3):726-745.
https://doi.org/10.3390/foundations2030050

**Chicago/Turabian Style**

Lazopoulos, Konstantinos A.
2022. "On Λ-Fractional Differential Equations" *Foundations* 2, no. 3: 726-745.
https://doi.org/10.3390/foundations2030050