# Numerically Efficient Methods for Variational Fractional Wave Equations: An Explicit Four-Step Scheme

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Theorem**

**1**(Dissipation of energy)

**.**

**Proof.**

**Corollary**

**1**

**Corollary**

**2**(Boundedness).

**Proof.**

**Example**

**1.**

- (a)
- Suppose that $G\left(v\right)=1-cos\left(v\right)$ and ${G}_{1}\left(v\right)=\frac{1}{2}{v}^{2}$, for each $v\in \mathbb{R}$. Then, the resulting partial differential equation in Equation (6) is the fractional sine-Gordon equation, which is a well-known physical model that appears in relativistic quantum mechanics.
- (b)
- The fractional form of the nonlinear Klein–Gordon equation is obtained from the partial differential equation of Equation (6) when $G\left(v\right)=\frac{1}{2!}{v}^{2}-\frac{1}{4!}{v}^{4}$ for each $v\in \mathbb{R}$, and ${G}_{1}$ is as in(a). The Klein–Gordon equation is also a useful model in relativistic quantum mechanics and particle physics.
- (c)
- If ${G}_{1}$ is as in(a)and $G\left(v\right)=1-\frac{1}{3}cosv-\frac{1}{6}cos\left(2v\right)$ for each $v\in \mathbb{R}$, then the partial differential equation resulting in Equation (6) is the fractional double sine-Gordon equation.
- (d)

## 3. Numerical Method

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Lemma**

**1**

- (a)
- The coefficients ${\left({g}_{k}^{\left(\alpha \right)}\right)}_{k=-\infty}^{\infty}$ satisfy$${g}_{0}^{\left(\alpha \right)}=\frac{\mathsf{\Gamma}(\alpha +1)}{\mathsf{\Gamma}{(\alpha /2+1)}^{2}},\phantom{\rule{2.em}{0ex}}{g}_{k+1}^{\left(\alpha \right)}=\left(1-\frac{\alpha +1}{\alpha /2+k+1}\right){g}_{k},\phantom{\rule{1.em}{0ex}}\forall k\in \mathbb{N}\cup \left\{0\right\}.$$
- (b)
- ${g}_{0}^{\left(\alpha \right)}\ge 0$.
- (c)
- ${g}_{k}^{\left(\alpha \right)}={g}_{-k}^{\left(\alpha \right)}<0$ for all $k\ge 1$, and
- (d)
- $\sum _{k=-\infty}^{\infty}{g}_{k}^{\left(\alpha \right)}=0$. It follows that $g}_{0}^{\left(\alpha \right)}=-\sum _{\begin{array}{c}k=-\infty \\ k\ne 0\end{array}}^{\infty}{g}_{k}^{\left(\alpha \right)$.

**Definition**

**8.**

**Lemma**

**2**

- (a)
- $\parallel {\delta}_{x}^{(\alpha /2)}{V\parallel}_{2}^{2}\le {g}_{h}^{\left(\alpha \right)}{\parallel V\parallel}_{2}^{2}$, for each $V\in {\mathcal{V}}_{h}$, and
- (b)
- $\parallel {\delta}_{x}^{\left(\alpha \right)}{V\parallel}_{2}^{2}\le ({g}_{h}^{\left(\alpha \right)}{\parallel V\parallel}_{2}{)}^{2}$, for each $V\in {\mathcal{V}}_{h}$.

**Definition**

**9.**

**Theorem**

**2**(Existence and uniqueness of solutions).

**Definition**

**10.**

**Lemma**

**3.**

- (a)
- $\langle {\widehat{\delta}}_{t}^{\left(2\right)}{U}^{n+\frac{1}{2}},{\delta}_{t}{U}^{n+\frac{1}{2}}\rangle =\frac{1}{2}{\delta}_{t}\langle {\delta}_{t}{U}^{n+\frac{1}{2}},{\delta}_{t}{U}^{n-\frac{1}{2}}\rangle $;
- (b)
- $\langle {\delta}_{t}{U}^{n+\frac{1}{2}},{\delta}_{t}{U}^{n-\frac{1}{2}}\rangle ={\mu}_{t}\parallel {\delta}_{t}{U}^{n-\frac{1}{2}}{\parallel}_{2}^{2}-{\textstyle \frac{1}{2}}{\tau}^{2}{\parallel {\delta}_{t}^{\left(2\right)}{U}^{n}\parallel}_{2}^{2}$;
- (c)
- $\langle {\delta}_{U}G\left({U}^{n+\frac{1}{2}}\right),{\delta}_{t}{U}^{n+\frac{1}{2}}\rangle ={\delta}_{t}\langle G\left({U}^{n+\frac{1}{2}}\right),1\rangle $; and
- (d)
- $\langle -{\delta}_{x}^{({\alpha}_{j}/2)}\left[{\delta}_{{\delta}_{x}^{({\alpha}_{j}/2)}U}{G}_{j}\left({U}_{m}^{n+\frac{1}{2}}\right)\right],{\delta}_{t}{U}^{n+\frac{1}{2}}\rangle ={\delta}_{t}\langle {G}_{j}\left({\delta}_{x}^{({\alpha}_{j}/2)}{U}^{n+\frac{1}{2}}\right),1\rangle $.

**Proof.**

**Theorem**

**3**(Dissipation of discrete energy)

**.**

**Proof.**

## 4. Numerical Properties

**Definition**

**11.**

**Theorem**

**4**

**Proof.**

- (a)
- If $U,V\in {\mathcal{V}}_{h}$, then $2\left|\langle U,V\rangle \right|\le {\parallel U\parallel}_{2}^{2}+{\parallel V\parallel}_{2}^{2}$.
- (b)
- If $n\in \mathbb{N}$ and ${U}^{1},{U}^{2},\dots ,{U}^{n}\in {\mathcal{V}}_{h}$, then$$\parallel {U}^{1}+{U}^{2}+\dots +{U}^{n}{\parallel}_{2}^{2}\le n(\parallel {U}^{1}{\parallel}_{2}^{2}+\parallel {U}^{2}{\parallel}_{2}^{2}+\dots +\parallel {U}^{n}{\parallel}_{2}^{2}).$$
- (c)
- If ${\left({U}^{n}\right)}_{n=0}^{N}\subseteq {\mathcal{V}}_{h}$ and $n\in {I}_{N}$, then$$\parallel {U}^{n}{\parallel}_{2}^{2}\le 2\parallel {U}^{0}{\parallel}_{2}^{2}+2T\tau \sum _{k=0}^{n-1}{\parallel {\delta}_{t}{U}^{k}\parallel}_{2}^{2},\phantom{\rule{2.em}{0ex}}\forall n\in {I}_{N}.$$

**Lemma**

**4**

**.**Let $G\in {\mathcal{C}}^{2}\left(\mathbb{R}\right)$, and assume that ${\left({R}^{n+\frac{1}{2}}\right)}_{n=0}^{N-1}$ is a sequence in ${\mathcal{V}}_{h}$. Moreover, suppose that ${\left({V}^{n}\right)}_{n=0}^{N}$ and ${\left({W}^{n}\right)}_{n=0}^{N}$ are two solutions of Equation (30) corresponding to ${\mathsf{\Phi}}_{V}$ and ${\mathsf{\Phi}}_{W}$, respectively. Let ${\epsilon}^{n}={V}^{n}-{W}^{n}$ for each $n\in {\overline{I}}_{N}$, and define

**Lemma**

**5.**

**Proof.**

**Lemma**

**6**

**Theorem**

**5**(Stability)

**.**

**Proof.**

**Theorem**

**6**(Convergence)

**.**

**Proof.**

## 5. Computer Simulations

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

## 6. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Forward-difference stencil for the approximation to the exact solution of the partial differential equation of Equation (6) at the time ${t}_{n}$, using the finite-difference scheme in Equation (30). The black circles represent the known approximations at the times ${t}_{n-1}$, ${t}_{n}$ and ${t}_{n+1}$, while the cross denotes the unknown approximation at the time ${t}_{n+2}$.

**Figure 2.**Graphs of the approximate solution (

**a**,

**c**,

**e**) and respective energy densities (

**b**,

**d**,

**f**) of the system in Equation (6) versus t and x. The fractional partial differential equation considers $\gamma =0$ and $\mathsf{\Omega}=(0,125)\times (0,500)$. The initial data were defined by the functions $\varphi \left(x\right)=\psi \left(x\right)=\chi \left(x\right)=sin(2\pi x/125)$, for each $x\in (0,125)$. Computationally, we fixed $h=1$ and $\tau =0.025$. We employed $\alpha ={\alpha}_{1}$, with $\alpha =2$ (

**a**,

**b**), $\alpha =1.6$ (

**c**,

**d**) and $\alpha =1.2$ (

**e**,

**f**). The remaining parameters are $p=1$, $G\equiv 0$, and ${G}_{1}$ is given by Equation (73).

**Figure 3.**Graphs of the dynamics of the total energy of the system in Equation (6) for various values of $\alpha ={\alpha}_{1}$. The fractional partial differential equation considers $\gamma =0$ and $\mathsf{\Omega}=(0,125)\times (0,500)$. The initial data were defined by the functions $\varphi \left(x\right)=\psi \left(x\right)=\chi \left(x\right)=sin(2\pi x/125)$, for each $x\in (0,125)$. Computationally, we fixed $h=1$ and $\tau =0.025$. The remaining parameters are $p=1$, $G\equiv 0$, and ${G}_{1}$ is given by Equation (73).

**Figure 4.**Graphs of the dynamics of the total energy of the system in Equation (6) for various values of $\alpha ={\alpha}_{1}$. The fractional partial differential equation considers $\mathsf{\Omega}=(0,125)\times (0,500)$, and different values of $\gamma $. The initial data were defined by the functions $\varphi \left(x\right)=\psi \left(x\right)=\chi \left(x\right)=sin(2\pi x/125)$, for each $x\in (0,125)$. Computationally, we fixed $h=1$ and $\tau =0.025$. The remaining parameters are $p=1$, $G\equiv 0$, and ${G}_{1}$ is given by Equation (73).

**Figure 5.**Graphs of the approximate solution (

**a**,

**c**,

**e**) and respective energy densities (

**b**,

**d**,

**f**) of the system in Equation (6) versus t and x. The parameters employed are $p=1$, $G\equiv 0$, and ${G}_{1}$ is given by Equation (75). We used $\u03f5=0.75$, $\gamma =0$ and $\mathsf{\Omega}=(0,128)\times (0,500)$. The initial data were defined by the functions $\varphi \left(x\right)=v(x,0)$, $\psi \left(x\right)=v(x,\tau )$ and $\chi \left(x\right)=v(x,2\tau )$, for each $x\in (0,32)$. Here, v is the function defined by Equation (76). Computationally, we fixed $h=1$ and $\tau =0.025$. We employed $\alpha ={\alpha}_{1}$ with $\alpha =2$ (

**a**,

**b**), $\alpha =1.6$ (

**c**,

**d**) and $\alpha =1.2$ (

**e**,

**f**).

**Figure 6.**Graphs of the dynamics of the total energy of the system in Equation (6) for $\alpha ={\alpha}_{1}$ with (

**a**) $\alpha =1.8$, (

**b**) $\alpha =1.6$, (

**c**) $\alpha =1.4$ and (

**d**) $\alpha =1.2$. The parameters employed are $p=1$, $G\equiv 0$, and ${G}_{1}$ is given by Equation (75). We used $\u03f5=0.75$ and $\mathsf{\Omega}=(0,128)\times (0,500)$. The initial data were defined by the functions $\varphi \left(x\right)=v(x,0)$, $\psi \left(x\right)=v(x,\tau )$ and $\chi \left(x\right)=v(x,2\tau )$, for each $x\in (0,32)$. Here, v is the function defined by Equation (76). Computationally, we fixed $h=1$ and $\tau =0.025$. In each case, we used the values of $\gamma $ shown within the legend at the bottom of each graph.

**Figure 7.**Graphs of the total energy of the system in Equation (6) versus t. The parameters employed are $p=1$, $G\equiv 0$, and ${G}_{1}$ is given by Equation (75). We used $\u03f5=0.75$, $\alpha =1.5$, $\gamma =0$, $B=(0,128)$ and $T=1\times {10}^{6}$. The initial data were defined by the functions $\varphi \left(x\right)=v(x,0)$, $\psi \left(x\right)=v(x,\tau )$ and $\chi \left(x\right)=v(x,2\tau )$, for each $x\in (0,32)$. Here, v is the function defined by Equation (76). Computationally, we fixed $h=1$, and we employed: (

**a**) $\tau =4$; (

**b**) $\tau =2$; (

**c**) $\tau =1$; (

**d**) $\tau =0.5$; (

**e**) $\tau =0.25$; and (

**f**) $\tau =0.125$. The range of the scale of the y-axis is the interval $[0.17103,0.17108]$.

**Table 1.**Table of absolute errors in the maximum norm and temporal rates of convergence for various values of the parameters $\tau $ and h. The experiment corresponds to that described in Example 4.

$\mathit{h}=1$ | $\mathit{h}=0.5$ | $\mathit{h}=0.25$ | ||||
---|---|---|---|---|---|---|

$\mathit{\tau}$ | ${\mathit{\u03f5}}_{\mathit{t},\mathit{h}}$ | ${\mathit{\rho}}_{\mathit{\tau}}$ | ${\mathit{\u03f5}}_{\mathit{t},\mathit{h}}$ | ${\mathit{\rho}}_{\mathit{\tau}}$ | ${\mathit{\u03f5}}_{\mathit{t},\mathit{h}}$ | ${\mathit{\rho}}_{\mathit{\tau}}$ |

$0.1/{2}^{0}$ | $2.6805\times {10}^{-2}$ | − | $6.7596\times {10}^{-3}$ | − | $1.6854\times {10}^{-3}$ | − |

$0.1/{2}^{1}$ | $6.4470\times {10}^{-3}$ | $2.0558$ | $1.6881\times {10}^{-3}$ | $2.0015$ | $4.2704\times {10}^{-4}$ | $1.9807$ |

$0.1/{2}^{2}$ | $1.5595\times {10}^{-3}$ | $2.0475$ | $4.1406\times {10}^{-4}$ | $2.0275$ | $1.0711\times {10}^{-4}$ | $1.9952$ |

$0.1/{2}^{3}$ | $3.8014\times {10}^{-4}$ | $2.0365$ | $9.8730\times {10}^{-5}$ | $2.0683$ | $2.6528\times {10}^{-5}$ | $2.0136$ |

$0.1/{2}^{4}$ | $9.1697\times {10}^{-5}$ | $2.0516$ | $2.3789\times {10}^{-5}$ | $2.0532$ | $6.4825\times {10}^{-6}$ | $2.0329$ |

**Table 2.**Table of absolute errors in the maximum norm and spatial rates of convergence for various values of the parameters $\tau $ and h. The experiment corresponds to that described in Example 4.

$\mathit{\tau}=5\times {10}^{-5}$ | $\mathit{\tau}=2.5\times {10}^{-5}$ | $\mathit{\tau}=1.25\times {10}^{-5}$ | ||||
---|---|---|---|---|---|---|

$\mathit{h}$ | ${\mathit{\u03f5}}_{\mathit{t},\mathit{h}}$ | ${\mathit{\rho}}_{\mathit{h}}$ | ${\mathit{\u03f5}}_{\mathit{t},\mathit{h}}$ | ${\mathit{\rho}}_{\mathit{h}}$ | ${\mathit{\u03f5}}_{\mathit{t},\mathit{h}}$ | ${\mathit{\rho}}_{\mathit{h}}$ |

$4/{2}^{0}$ | $7.1887\times {10}^{-3}$ | − | $1.8178\times {10}^{-3}$ | − | $4.5327\times {10}^{-4}$ | − |

$4/{2}^{1}$ | $1.8250\times {10}^{-3}$ | $1.9778$ | $5.0045\times {10}^{-4}$ | $1.8609$ | $1.2482\times {10}^{-4}$ | $1.8605$ |

$4/{2}^{2}$ | $4.5941\times {10}^{-4}$ | $1.9900$ | $1.2724\times {10}^{-4}$ | $1.9756$ | $3.1762\times {10}^{-5}$ | $1.9745$ |

$4/{2}^{3}$ | $1.0807\times {10}^{-4}$ | $2.0877$ | $3.2302\times {10}^{-5}$ | $1.9779$ | $8.0961\times {10}^{-6}$ | $1.9720$ |

$4/{2}^{4}$ | $2.0730\times {10}^{-5}$ | $2.3822$ | $7.8347\times {10}^{-6}$ | $2.0437$ | $1.9029\times {10}^{-6}$ | $2.0890$ |

**Table 3.**Table of absolute errors in the maximum norm and temporal rates of convergence for various values of the parameters $\tau $ and h. The experiment corresponds to that described in Example 5.

$\mathit{h}=1/{2}^{4}$ | $\mathit{h}=1/{2}^{5}$ | $\mathit{h}=1/{2}^{6}$ | ||||
---|---|---|---|---|---|---|

$\mathit{\tau}$ | ${\mathit{\u03f5}}_{\mathit{t},\mathit{h}}$ | ${\mathit{\rho}}_{\mathit{\tau}}$ | ${\mathit{\u03f5}}_{\mathit{t},\mathit{h}}$ | ${\mathit{\rho}}_{\mathit{\tau}}$ | ${\mathit{\u03f5}}_{\mathit{t},\mathit{h}}$ | ${\mathit{\rho}}_{\mathit{\tau}}$ |

$1/{2}^{3}$ | $2.7368\times {10}^{-4}$ | − | $6.6597\times {10}^{-5}$ | − | $1.6593\times {10}^{-5}$ | − |

$1/{2}^{4}$ | $6.9872\times {10}^{-5}$ | $1.9697$ | $1.6889\times {10}^{-5}$ | $1.9793$ | $4.2431\times {10}^{-6}$ | $1.9674$ |

$1/{2}^{5}$ | $1.7664\times {10}^{-5}$ | $1.9839$ | $4.1418\times {10}^{-6}$ | $2.0278$ | $1.0723\times {10}^{-6}$ | $1.9844$ |

$1/{2}^{6}$ | $4.4770\times {10}^{-6}$ | $1.9802$ | $1.0341\times {10}^{-6}$ | $2.0018$ | $2.6944\times {10}^{-7}$ | $1.9927$ |

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Macías-Díaz, J.E.
Numerically Efficient Methods for Variational Fractional Wave Equations: An Explicit Four-Step Scheme. *Mathematics* **2019**, *7*, 1095.
https://doi.org/10.3390/math7111095

**AMA Style**

Macías-Díaz JE.
Numerically Efficient Methods for Variational Fractional Wave Equations: An Explicit Four-Step Scheme. *Mathematics*. 2019; 7(11):1095.
https://doi.org/10.3390/math7111095

**Chicago/Turabian Style**

Macías-Díaz, Jorge E.
2019. "Numerically Efficient Methods for Variational Fractional Wave Equations: An Explicit Four-Step Scheme" *Mathematics* 7, no. 11: 1095.
https://doi.org/10.3390/math7111095