A Numerically Efficient Dissipation-Preserving Implicit Method for a Nonlinear Multidimensional Fractional Wave Equation

Abstract

This work is motivated by the investigation of a fractional extension of a general nonlinear multidimensional wave equation with damping. The model under study considers partial derivatives of orders in $(0, 1) \cup (1, 2]$ with respect to the spatial variables. The undamped one-dimensional version of the model was previously used in the literature to investigate the presence of the nonlinear phenomenon of supratransmission in fractional relativistic wave equations. In agreement with the continuous counterpart, the capability of the method to preserve the total energy of the system was demonstrated then. In the present work, we show that the modified methodology is capable of preserving the dissipation of energy of the extended continuous model. We note that the discrete energy functional is positive, a fact which is in agreement with the positivity of the continuous energy. Moreover, the unique solubility of the method is established, and we show that our technique is second-order consistent, stable and quadratically convergent. Some bounds for the numerical solution are computed in the way, and various simulations are performed in order to assess the capability of the method to preserve or dissipate the energy of the system.

Model Derivation

The purpose of this appendix is to outline the derivation of the fractional partial differential equation (1.5) from the discrete equations of motion (1.1). To that end, we will follow closely the discussion in [31].

Let us consider a system of interacting particles with dynamics governed by the Eq. (1.1). The distance between consecutive particles is equal to $h > 0$, and each of the functions $I_n$ is given by (1.2). Moreover, let us assume that

• (a) $J (n , m) = J (n - m) = J (m - n)$ for all $m , n \in \mathbb {Z}$,

• (b) $\displaystyle {\sum _{n = 1}^\infty | J (n) |^2 < \infty }$.

It is easy to see that these conditions imply that $J (- n) = J (n)$ for each $n \in \mathbb {Z}$.

Definition 2

Suppose that conditions (a) and (b) above are satisfied, and let $\alpha > 0$. Then J is called an $\alpha$-interaction if the function

$$\begin{aligned} J_\alpha (k) = \sum _{\begin{array}{c} n = - \infty \\ n \ne 0 \end{array}}^\infty e^{- i k n} J (n) = 2 \sum _{n = 1}^\infty J (n) \cos (k n), \quad \forall k \in {\mathbb {R}}, \end{aligned}$$

(A.1)

satisfies

$$\begin{aligned} A_\alpha = \lim _{k \rightarrow 0} \frac{J_\alpha (k) - J_\alpha (0)}{|k|^\alpha } \in {\mathbb {R}} \setminus \{ 0 \}. \end{aligned}$$

(A.2)

Example 4

The following functions defined on $\mathbb {Z}$ are examples of $\alpha$-interactions (see [31]). For convenience, these functions may be defined as 0 for $n = 0$.

• 1. $J_1 (n) = (- 1)^n n^{- 2}$.

• 2. $J_2 (n) = | n |^{- (\beta + 1)}$, where $\beta \in {\mathbb {R}}^+ \setminus {\mathbb {N}}$.

• 3. $\displaystyle {J_3 (n) = \frac{(- 1)^n}{\varGamma (1 + \frac{\alpha }{2} + n) \varGamma (1 + \frac{\alpha }{2} - n)}}$, where $\alpha \in (0 , 1) \cup (1 , 2]$.

• 4. $\displaystyle {J_4 (n) = \frac{(- 1)^n}{a^2 - n^2}}$, where $a^2 \notin \mathbb {Z}$.

• 5. $\displaystyle {J_5 (n) = \frac{1}{|n| !}}$. $\square$

In the following, we will let ${\mathscr {F}}_h : u_n (t) \rightarrow \hat{u} (k , t)$ denote the Fourier series transform, let ${\mathscr {L}} : \hat{u} (k , t) \rightarrow \tilde{u} (k , t)$ be the passage to the limit when the distance between consecutive oscillators tend to zero, and let ${\mathscr {F}}^{- 1} : \tilde{u} (k , t) \rightarrow u (x , t)$ be the inverse Fourier transform. Finally, let $\circ$ represent the operation of composition of functions and ${\mathscr {T}} = {\mathscr {F}}^{- 1} \circ {\mathscr {L}} \circ {\mathscr {F}}_h$. Using these conventions, the next theorem establishes conditions under which (1.5) can be obtained from systems of oscillators with long-range interactions.

Theorem 8

(Tarasov [31]) Let $\alpha > 0$ and let J be an $\alpha$-interaction. Then ${\mathscr {T}}$ transforms the discrete equations of motion (1.1) into the fractional continuous equation (1.5), where the fractional derivative in space is the Riesz fractional derivative of order $\alpha$.

Proof

The theorem is a direct consequence of Proposition 1 [31]. Just let the constant g of that work be equal to 1.$\square$

Example 5

Consider the $\alpha$-interactions described in Example 4. We provide next the result of applying the transformation ${\mathscr {T}}$ to each of them (see [31]).

• 1. $\displaystyle {{\mathscr {T}} (J_1) = - \frac{1}{2} \frac{\partial ^2}{\partial x^2}}$.

• 2. $\displaystyle {{\mathscr {T}} (J_2) = - i \pi \frac{\partial }{\partial x}}$, where $\beta \in {\mathbb {R}}^+ \setminus {\mathbb {N}}$.

• 3. $\displaystyle {{\mathscr {T}} (J_3) = - 2 \varGamma (- \alpha ) \cos (\pi \alpha / 2) \frac{\partial ^\alpha }{\partial |x|^\alpha }}$, where $\alpha \in (0 , 1) \cup (1 , 2]$.

• 4. $\displaystyle {{\mathscr {T}} (J_4) = - \frac{a \pi }{2 \sin (\pi a)} \frac{\partial ^2}{\partial x^2}}$, where $a \in {\mathbb {R}}^+$ satisfies $a^2 \notin \mathbb {Z}$.

• 5. $\displaystyle {{\mathscr {T}} (J_5) = 4 i e \frac{\partial }{\partial x}}$. $\square$