# Analytical Solution of Linear Fractional Systems with Variable Coefficients Involving Riemann–Liouville and Caputo Derivatives

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Lemma**

**1.**

**Lemma**

**2.**

**Theorem**

**1**

**.**Suppose that $X,Y\subseteq \mathbb{R}$ are intervals. Suppose also that the function $f:X\times Y\to \mathbb{R}$ satisfies the following conditions:

- (a)
- For every fixed $y\in Y$, the function $f(\xb7,y)$ is measurable on X.
- (b)
- The partial derivative $\frac{\partial}{\partial y}f(x,y)$ exists for every interior point $(x,y)\in X\times Y$.
- (c)
- There exists a non-negative integrable function g such that $\left|\frac{\partial}{\partial y}f(x,y)\right|\le g\left(x\right)$ for every interior point $(x,y)\in X\times Y$.
- (d)
- There exists ${y}_{0}\in Y$ such that $f(x,{y}_{0})$ is integrable on X.

**Corollary**

**1.**

## 3. Homogeneous System of Linear FDEs with Variable Coefficients Involving Riemann–Liouville Derivatives

**Definition**

**1.**

**Assumption**

**1.**

**Lemma**

**3.**

**Theorem**

**2.**

**Remark**

**1.**

#### Example

## 4. Inhomogeneous System of Linear FDEs with Variable Coefficients Involving Riemann–Liouville Derivatives

**Lemma**

**4.**

- (a)
- For every fixed $t\in I$, the function $\tilde{\phi}(t,s)={}_{s}^{t}{J}_{t}^{1-\alpha}\phi (t,s)$ is measurable on I and integrable on I w.r.t. s for some ${t}^{*}\in I$.
- (b)
- The partial derivative ${}_{s}^{t}{D}_{t}^{\alpha}\phi (t,s)$ exists for every interior point $(t,s)\in \stackrel{\u02da}{I}\times \stackrel{\u02da}{I}$.
- (c)
- There exists a non-negative integrable function g such that $\left|{}_{s}^{t}{D}_{t}^{\alpha}\phi (t,s)\right|\le g\left(s\right)$ for every interior point $(t,s)\in \stackrel{\u02da}{I}\times \stackrel{\u02da}{I}$.

**Proof.**

**Theorem**

**3.**

**Proof.**

#### Example

## 5. Homogeneous System of Linear FDEs with Variable Coefficients Involving Caputo Derivatives

**Definition**

**2.**

**Assumption**

**2.**

**Lemma**

**5.**

**Theorem**

**4.**

**Remark**

**2.**

#### Example

## 6. Inhomogeneous System of Linear FDEs with Variable Coefficients Involving Caputo Derivatives

**Lemma**

**6.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Remark**

**3.**

#### Example

## 7. Conclusions

## Funding

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Matychyn, I.
Analytical Solution of Linear Fractional Systems with Variable Coefficients Involving Riemann–Liouville and Caputo Derivatives. *Symmetry* **2019**, *11*, 1366.
https://doi.org/10.3390/sym11111366

**AMA Style**

Matychyn I.
Analytical Solution of Linear Fractional Systems with Variable Coefficients Involving Riemann–Liouville and Caputo Derivatives. *Symmetry*. 2019; 11(11):1366.
https://doi.org/10.3390/sym11111366

**Chicago/Turabian Style**

Matychyn, Ivan.
2019. "Analytical Solution of Linear Fractional Systems with Variable Coefficients Involving Riemann–Liouville and Caputo Derivatives" *Symmetry* 11, no. 11: 1366.
https://doi.org/10.3390/sym11111366