# A Discrete Grönwall Inequality and Energy Estimates in the Analysis of a Discrete Model for a Nonlinear Time-Fractional Heat Equation

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## Abstract

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**16**: 892–910 (2013)]. In that article, the authors established the stability and consistency of the discrete model using arguments from Fourier analysis. As opposed to that work, in the present work, we use the method of energy inequalities to show that the scheme is stable and converges to the exact solution with order $\mathcal{O}({\tau}^{2-\alpha}+{h}^{4})$, in the case that $0<\alpha <1$ satisfies ${3}^{\alpha}\ge \frac{3}{2}$, which means that $0.369\u2a85\alpha \le 1$. The novelty of the present work lies in the derivation of suitable energy estimates, and a discrete fractional Grönwall inequality, which is consistent with the discrete approximation of the Caputo fractional derivative of order $0<\alpha <1$ used for that scheme at ${t}_{k+1/2}$.

## 1. Introduction

## 2. Compact Difference Scheme

**Lemma**

**1.**

**Lemma**

**2.**

**Lemma**

**3.**

**Proof.**

## 3. Auxiliary Lemmas

**Definition**

**1.**

**Lemma**

**4.**

**Lemma**

**5.**

**Proof.**

**Definition**

**2.**

**Lemma**

**6.**

**Proof.**

**Lemma**

**7.**

**Lemma**

**8.**

**Corollary**

**1.**

**Lemma**

**9.**

- (a)
- If $1\le k\le j$ then $0<{p}_{j}<1$ and ${\sum}_{r=k}^{j}{p}_{j-r}{c}_{r-k}^{k+1}={\left(\frac{1}{2}\right)}^{1-\alpha}$.
- (b)
- If $m\ge 1$ then$$\begin{array}{cc}\hfill \frac{\mathsf{\Gamma}(2-\alpha )}{\mathsf{\Gamma}(1+(m-1\left)\alpha \right)}\sum _{r=1}^{j}{p}_{j-r}{r}^{(m-1)\alpha}& \le \frac{{\left(\frac{1}{2}\right)}^{1-\alpha}{j}^{m\alpha}}{\mathsf{\Gamma}(1+m\alpha )}.\hfill \end{array}$$

**Lemma**

**10.**

**Theorem**

**1**

**Proof.**

## 4. Numerical Properties

**Theorem**

**2**

**Proof.**

**Definition**

**3.**

**Theorem**

**3**

**Proof.**

**Definition**

**4.**

**Theorem**

**4**

**Proof.**

**Example**

**1.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Numerical solution of the problem (74) using $L=T=1$ and $\alpha =\frac{1}{2}$. We used the method presented in this work with computational parameters $h=0.01$ and $\tau =0.001$.

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**MDPI and ACS Style**

Hendy, A.S.; Macías-Díaz, J.E.
A Discrete Grönwall Inequality and Energy Estimates in the Analysis of a Discrete Model for a Nonlinear Time-Fractional Heat Equation. *Mathematics* **2020**, *8*, 1539.
https://doi.org/10.3390/math8091539

**AMA Style**

Hendy AS, Macías-Díaz JE.
A Discrete Grönwall Inequality and Energy Estimates in the Analysis of a Discrete Model for a Nonlinear Time-Fractional Heat Equation. *Mathematics*. 2020; 8(9):1539.
https://doi.org/10.3390/math8091539

**Chicago/Turabian Style**

Hendy, Ahmed S., and Jorge E. Macías-Díaz.
2020. "A Discrete Grönwall Inequality and Energy Estimates in the Analysis of a Discrete Model for a Nonlinear Time-Fractional Heat Equation" *Mathematics* 8, no. 9: 1539.
https://doi.org/10.3390/math8091539