Advanced neural network approaches for coupled equations with fractional derivatives

Abstract

We investigate numerical solutions and compare them with Fractional Physics-Informed Neural Network (FPINN) solutions for a coupled wave equation involving fractional partial derivatives. The problem explores the evolution of functions u and v over time t and space x. We employ two numerical approximation schemes based on the finite element method to discretize the system of equations. The effectiveness of these schemes is validated by comparing numerical results with exact solutions. Additionally, we introduce the FPINN method to tackle the coupled equation with fractional derivative orders and compare its performance against traditional numerical methods.

Key findings reveal that both numerical approaches provide accurate solutions, with the FPINN method demonstrating competitive performance in terms of accuracy and computational efficiency. Our study highlights the significance of employing FPINNs in solving fractional differential equations and underscores their potential as alternatives to conventional numerical methods. The novelty of this work lies in its comparative analysis of traditional numerical techniques and FPINNs for solving coupled wave equations with fractional derivatives, offering insights into advancing computational methods for complex physical systems.

1 Introduction

In this work, we numerically solve a coupled diffusion equation with a fractional derivative order of the form

$$ \left \{ \textstyle\begin{array}{c@{\quad}c} _{0}D^{\alpha}_{t} u(x, t) -\,_{0}D^{\beta}_{x} u(x, t) + v(x, t) &= f_{1}(x,t), \quad (x,t)\in (0, L)\times (0, +\infty ), \\ _{0}D^{\alpha}_{t} v(x, t) -\,_{0}D^{\beta}_{x} v(x, t) + u(x, t) &= f_{2}(x,t), \quad (x,t)\in (0, L)\times (0, +\infty ), \end{array}\displaystyle \right . $$

(1)

with initial and boundary conditions

$$ u(x, 0) = u_{0}(x), \quad v(x, 0) = v_{0}(x), \quad x \in \Omega := [0, L], $$

and boundary conditions

$$ \begin{gathered} u(0, t) = v(0, t) = 0, \quad t \in I := [0, T], \\ u(L, t) = v(L, t) = 0, \quad t \in I := [0, T], \end{gathered} $$

(2)

where $0 < \alpha < 1$, $1 < \beta < 2$, $_{0}D^{\alpha}_{t} u$, $_{0}D^{\alpha}_{t} v$, and $_{0}D^{\beta}_{x} u$, $_{0}D^{\beta}_{x} v$ denote the Caputo or Riemann–Liouville fractional derivatives of orders α and β, respectively. Here $f_{i} : \Omega \times I \longrightarrow \mathbb{R}$, $i = 1, 2$, and $u_{0}$ and $v_{0}$ are given functions.

Equations (1) describe the interaction between two dependent variables $u(x, t)$ and $v(x, t)$ under the influence of source terms $f_{1}(x,t)$ and $f_{2}(x,t)$. The coupling terms $v(x, t)$ and $u(x, t)$ reflect the interaction between the processes. The form of these equations is inspired by the literature on fractional diffusion equations, which demonstrate the effectiveness of fractional models in capturing complex dynamics. References [2–5, 7, 13–15, 20, 21] provide a collection of published works that offer support for the mathematical formulation of problems concerning fractional differential equations and numerical methods. See also the references of [9–12, 16–19] for further existing results related to the stability and Numerically analysis of (1).

Fractional partial differential equations have applications in diverse fields such as electromagnetic theory, viscoelastic mechanics, fractal media, mathematical biology, and chemistry. Analytical solutions for these equations have been investigated using the Fourier and Laplace transforms, as well as Green’s functions [13, 15]. Additionally, various numerical methods have been proposed to solve space- and time-fractional partial differential equations [5, 7, 14]. Zeng et al. [21] considered two finite difference methods to approximate the time-fractional subdiffusion equation with the Caputo fractional derivative:

$$ _{0}^{C}D^{\beta}_{t} u = _{0}^{C}D^{2}_{x} u + f(x,t), \quad (x,t) \in [a,b] \times [0, \infty ]. $$

Using the finite difference method, Yokus [20] presented a numerical solution for the space- and time-fractional-order Burgers-type equation

$$ \frac{\delta ^{\beta} u}{\delta t^{\beta}} - u(x,t) \frac{\delta ^{\alpha} u}{\delta x^{\alpha}} + \frac{\delta ^{2} u}{\delta x^{2}} = 0. $$

Boutiba et al. [4] considered the finite element method for solving space-time partial differential equations involving fractional derivatives:

$$ _{0}^{RL}D^{\alpha}_{t} u = _{0}^{RL}D^{\beta}_{x} u + f(x,t), \quad (x,t) \in [0,1] \times [0,1]. $$

This work expands the use of Physics-Informed Neural Networks (PINNs) to solve coupled partial differential equations (PDEs) involving fractional derivatives, focusing on the interaction between two dependent variables $u(x, t)$ and $v(x, t)$. This approach provides a more accurate representation of memory and hereditary properties in physical processes, which are crucial in fields such as viscoelasticity, anomalous diffusion, and complex fluid dynamics.

This paper is organized as follows. In Sect. 3, we theoretically study the existence of a unique solution for a class of partial differential equations involving fractional derivatives. In Sect. 4, we numerically analyze the problem using the finite element method with two different time discretization methods using the Riemann–Liouville and Caputo derivatives, along with the Grünwald–Letnikov approximation.

Finally, in Sect. 5, we provide a numerical example to verify the consistency of our theoretical and numerical results.

2 Preliminaries

We now introduce some notations and definitions of functional spaces equipped with their norms, seminorms, and inner products, which will be used hereafter.

Let Λ be a domain representing I, Ω, or $\mathcal{O}$. The space $L^{2}(\Lambda )$ consists of measurable functions with Lebesgue-integrable squares over Λ. Its inner product and norm are defined as

$$ (u, v)_{\Lambda }= \int _{\Lambda} uv \, d\Lambda , \quad \|u\|_{0, \Lambda} = \sqrt{(u, u)_{\Lambda}} $$

for $u, v \in L^{2}(\Lambda )$.

For a nonnegative real number s, by $H^{s}(\Omega )$ and $H^{s}_{0}(\Omega )$ we denote the usual Sobolev spaces with norms denoted by $\|\cdot \|_{s,\Omega}$ (see [1]). For a Sobolev space X with norm $\|\cdot \|_{X}$, we define

$$ H^{s}(I;X) := \{ v \mid \|v(\cdot , t)\|_{X} \in H^{s}(I) \} $$

equipped with the norm

$$ \|v\|_{H^{s}(I;X)} := \|v(\cdot , t)\|_{X}. $$

Particularly, when X is $H^{\sigma}(\Omega )$ or $H_{0}^{\sigma}(\Omega )$ for $\sigma > 0$, the norm of the space $H^{s}(I;X)$ will be denoted by $\|\cdot \|_{\sigma ,s,Q}$.

Now we recall some definitions of fractional integrals and fractional derivatives, as detailed in [8].

Definitions 1

• 1. For any positive integer n and $\alpha \in (n-1,n]$, the fractional integral is defined by

  $$ I^{\alpha}_{t} f(t) = \frac{1}{\Gamma (\alpha )} \int _{0}^{t} (t- \tau )^{\alpha -1} f(\tau ) \, d\tau \quad \text{for all } t \in [0,T], $$

  where Γ is Euler’s gamma function.

• 2. The Riemann fractional derivative of a function $f(t)$ is defined as follows:

  $$ ^{RL}_{0}D^{\alpha}_{t} f(t) = \frac{1}{\Gamma (n-\alpha )} \frac{d^{n}}{dt^{n}} \int _{0}^{t} (t-\tau )^{n-\alpha -1} f(\tau ) \, d\tau $$

  for any positive integer n and $\alpha \in (n-1,n]$.

• 3. The Caputo fractional derivative of a function f(t) is defined as follows:

  $$ ^{C}_{0}D^{\alpha}_{t} f(t) = \frac{1}{\Gamma (n-\alpha )} \int _{0}^{t} (t-\tau )^{n-\alpha -1} f^{(n)}(\tau ) \, d\tau $$

  for any positive integer n and $\alpha \in (n-1,n]$.

• 4. The generalized Caputo fractional derivative of order $\alpha \in (n-1,n]$ is defined as follows:

  $$ ^{C}_{0}D^{\alpha ,\eta}_{t} f(t) = \frac{1}{\Gamma (n-\alpha )} \int _{0}^{t} (t-\tau )^{n-\alpha -1} e^{-\eta (t-\tau )} f^{(n)}( \tau ) \, d\tau $$

  for any positive integer n.

• 5. For any positive integer n and $\alpha \in (n-1,n)$, the Grünwald–Letnikov fractional derivative of a function f is defined by

  $$ _{0}^{GL}D_{\alpha }^{t} \ f(t) = \lim _{\Delta t \to 0} \frac{1}{\Delta t^{\alpha}} \sum _{k=0}^{n} \delta _{k}^{\alpha } f(t-k \Delta t), $$

  where $\delta _{k}^{\alpha } = (-1)^{k} \frac{\Gamma (\alpha +1)}{\Gamma (\alpha +1-k) \Gamma (k+1)}$.

• 6. In the usual Sobolev space $H^{s}_{0}(\Lambda )$, we define the seminorm (see [7])

  $$ |v|^{\ast}_{H^{s}_{0}(\Lambda )} := \left ( \frac{\left ( ~^{R}D^{s}_{z}v,\,^{R}D^{s}_{z}v\right )_{ \Lambda}}{\cos (\pi s)} \right )^{\frac{1}{2}} $$

  (3)

  for all $v \in H^{s}_{0}(\Lambda )$.

We recall also the following lemmas (see [6, 7] for proofs).

Lemma 1

[7] For $s > 0$, $s \neq n - 1/2$, the spaces ${}^{l}H^{s}_{0}(\Lambda )$, ${}^{r}H^{s}_{0}(\Lambda )$, and $H^{s}_{0}(\Lambda )$ are equal, and their seminorms are all equivalent to $|\cdot |_{{\ast}H^{s}_{0}(\Lambda )}$.

Lemma 2

[6] For $0 < \alpha < 2$, $\alpha \neq 1$, and $w, v \in H^{\frac{\alpha}{2}}_{0}(\mathcal{O})$, we have

$$ \bigg( ~^{R}D^{\alpha}_{z} w, v \bigg)_{\mathcal{O}} = \bigg( ~^{R}D^{ \frac{\alpha}{2}}_{z} w, ~^{R}D^{\frac{\alpha}{2}}_{z} v \bigg)_{ \mathcal{O}}. $$

3 Existence and uniqueness of the solution

We define the space $B^{s, \sigma}\left ( \mathcal{O} \right ) = H^{s}\left (I, L^{2}( \Omega )\right ) \cap L^{2}\left (I, H^{s}(\Omega )\right )$ with the norm

$$ \Vert u \Vert ^{2}_{B^{s, \sigma}} = \Vert u \Vert ^{2}_{H^{s}\left (I, L^{2}(\Omega )\right )} + \Vert u \Vert ^{2}_{L^{2}\left (I, H^{s}( \Omega )\right )}, $$

where $\mathcal{O} = \Omega \times I$. The Riemann–Liouville weak formulation of our problem (1)–(2) is defined as follows: for $f_{1}, f_{2} \in B^{\frac{\alpha}{2}, \frac{\beta}{2}}(\mathcal{O})'$, find $u, v \in B^{\frac{\alpha}{2}, \frac{\beta}{2}}(\mathcal{O})$ such that

$$ \mathcal{A}\left ((u,v),(\chi ,\xi )\right ) = \mathcal{L}(\chi ,\xi ), \quad \forall \chi , \xi \in B^{\frac{\alpha}{2}, \frac{\beta}{2}}( \mathcal{O}), $$

(4)

where the bilinear form $\mathcal{A}(\cdot , \cdot )$ is defined as

$$ \textstyle\begin{array}{l@{\quad}l} \mathcal{A}\left ((u,v),(\chi ,\xi )\right ) := & \bigg( ~_{0}D^{ \frac{\alpha}{2}}_{t} u, ~_{0}D^{\frac{\alpha}{2}}_{t}\chi \bigg)_{ \mathcal{O}} - \bigg( ~_{0}D^{\frac{\beta}{2}}_{x} u, ~_{0}D^{ \frac{\beta}{2}}_{x}\chi \bigg)_{\mathcal{O}} + \bigg( u, \chi \bigg)_{ \mathcal{O}} \\ & + \bigg( ~_{0}D^{\frac{\alpha}{2}}_{t} v, ~_{0}D^{\frac{\alpha}{2}}_{t} \xi \bigg)_{\mathcal{O}} - \bigg( ~_{0}D^{\frac{\beta}{2}}_{x} v, ~_{0}D^{ \frac{\beta}{2}}_{x}\xi \bigg)_{\mathcal{O}} + \bigg( v, \xi \bigg)_{ \mathcal{O}}, \end{array} $$

(5)

and the functional $\mathcal{L}(\chi ,\xi )$ is given by

$$ \mathcal{L}(\chi ,\xi ) := \bigg( f_{1}, \chi \bigg)_{\mathcal{O}} + \bigg( f_{2}, \xi \bigg)_{\mathcal{O}}. $$

This formulation will be instrumental in proving the existence theorem for the solution of problem (1)–(2).

Theorem 1

For $0 < \alpha < 1$, $1 < \beta < 2$, and $f_{1}, f_{2} \in L^{2}(\mathcal{O})$, problem (1)–(2) has a unique solution. Furthermore, we have the following stability result:

$$ \|u\|_{B^{\frac{\alpha}{2}, \frac{\beta}{2}}(\mathcal{O})} \leq C \|f_{1} \|_{L^{2}(\mathcal{O})} \quad \textit{and} \quad \|v\|_{B^{ \frac{\alpha}{2}, \frac{\beta}{2}}(\mathcal{O})} \leq C \|f_{2}\|_{L^{2}( \mathcal{O})}. $$

Proof

The existence of the unique solution is established by the well-known Lax–Milgram lemma. It involves proving the coercivity and continuity of the bilinear form $\mathcal{A}$ and the continuity of the linear functional $\mathcal{F}$, which is straightforward. First, it follows from Lemma 1 that for $(u, v), (\chi , \xi ) \in B^{\frac{\alpha}{2}, \frac{\beta}{2}}( \mathcal{O}) \times B^{\frac{\alpha}{2}, \frac{\beta}{2}}(\mathcal{O})$, we have

$$ \textstyle\begin{array}{r@{\quad}l} \mathcal{A}\left ((u, v), (\chi , \xi )\right ) \leq & \displaystyle \Vert ~_{0}D^{\frac{\alpha}{2}}_{t} u \Vert \Vert ~_{0}D^{ \frac{\alpha}{2}}_{t} \chi \Vert _{\mathcal{O}} + \Vert ~_{0}D^{ \frac{\alpha}{2}}_{t} v \Vert \Vert ~_{0}D^{\frac{\alpha}{2}}_{t} \xi \Vert _{\mathcal{O}} + \Vert u \Vert _{\mathcal{O}} \Vert \chi \Vert _{\mathcal{O}} \\ +& \displaystyle \Vert ~_{0}D^{\frac{\beta}{2}}_{x} u \Vert \Vert ~_{0}D^{ \frac{\beta}{2}}_{x} \chi \Vert _{\mathcal{O}} + \Vert ~_{0}D^{ \frac{\beta}{2}}_{x} v \Vert \Vert ~_{0}D^{\frac{\beta}{2}}_{x} \xi \Vert _{\mathcal{O}} + \Vert v \Vert _{\mathcal{O}} \Vert \xi \Vert _{ \mathcal{O}} \\ \leq & \displaystyle \Vert u \Vert _{H^{\frac{\alpha}{2}} \left ( I, L^{2}( \Omega ) \right )} \Vert \chi \Vert _{H^{\frac{\alpha}{2}} \left ( I, L^{2}( \Omega ) \right )} + \Vert v \Vert _{H^{\frac{\alpha}{2}} \left ( I, L^{2}( \Omega ) \right )} \Vert \xi \Vert _{H^{\frac{\alpha}{2}} \left ( I, L^{2}( \Omega ) \right )} \\ +& \displaystyle \Vert u \Vert _{\mathcal{O}} \Vert \chi \Vert _{ \mathcal{O}} + \Vert u \Vert _{L^{2} \left ( I, H^{\frac{\beta}{2}} ( \Omega ) \right )} \Vert \chi \Vert _{L^{2} \left ( I, H^{ \frac{\beta}{2}} (\Omega ) \right )} \\ +& \displaystyle \Vert v \Vert _{L^{2} \left ( I, H^{\frac{\beta}{2}} ( \Omega ) \right )} \Vert \xi \Vert _{L^{2} \left ( I, H^{ \frac{\beta}{2}} (\Omega ) \right )} + \Vert v \Vert _{\mathcal{O}} \Vert \xi \Vert _{\mathcal{O}} \\ \leq & \displaystyle \| u \|_{B^{\frac{\alpha}{2}, \frac{\beta}{2}} ( \mathcal{O})} \| \chi \|_{B^{\frac{\alpha}{2}, \frac{\beta}{2}} ( \mathcal{O})} + \| v \|_{B^{\frac{\alpha}{2}, \frac{\beta}{2}} ( \mathcal{O})} \| \xi \|_{B^{\frac{\alpha}{2}, \frac{\beta}{2}} ( \mathcal{O})}. \end{array} $$

(6)

This proves the continuity of $\mathcal{A}$. Then for the coercivity, from the same lemma, for all $(u, v) \in B^{\frac{\alpha}{2}, \frac{\beta}{2}}(\mathcal{O})$, we have

$$ \textstyle\begin{array}{l@{\quad}l@{\quad}l} \mathcal{A}\left ((u, v), (u, v)\right )& = & \bigg( ~_{0}D^{ \frac{\alpha}{2}}_{t} u, ~_{0}D^{\frac{\alpha}{2}}_{t} u \bigg)_{ \mathcal{O}} - \bigg( ~_{0}D^{\frac{\beta}{2}}_{x} u, ~_{0}D^{ \frac{\beta}{2}}_{x} u \bigg)_{\mathcal{O}} + \bigg( u, u \bigg)_{ \mathcal{O}} \\ & +& \bigg( ~_{0}D^{\frac{\alpha}{2}}_{t} v, ~_{0}D^{\frac{\alpha}{2}}_{t} v \bigg)_{\mathcal{O}} - \bigg( ~_{0}D^{\frac{\beta}{2}}_{x} v, ~_{0}D^{ \frac{\beta}{2}}_{x} v \bigg)_{\mathcal{O}} + \bigg( v, v \bigg)_{ \mathcal{O}} \\ & \cong & \cos \left ( \frac{\pi \alpha}{2} \right ) \bigg( ~_{0}D^{ \frac{\alpha}{2}}_{t} u, ~_{0}D^{\frac{\alpha}{2}}_{t} u \bigg)_{ \mathcal{O}} - \cos \left ( \frac{\pi \beta}{2} \right ) \bigg( ~_{0}D^{ \frac{\beta}{2}}_{x} u, ~_{0}D^{\frac{\beta}{2}}_{x} u \bigg)_{ \mathcal{O}} \\ &+& \Vert u \Vert _{\mathcal{O}}^{2} + \cos \left ( \frac{\pi \alpha}{2} \right ) \bigg( ~_{0}D^{\frac{\alpha}{2}}_{t} v, ~_{0}D^{ \frac{\alpha}{2}}_{t} v \bigg)_{\mathcal{O}} \\ &-& \cos \left ( \frac{\pi \beta}{2} \right ) \bigg( ~_{0}D^{ \frac{\beta}{2}}_{x} v, ~_{0}D^{\frac{\beta}{2}}_{x} v \bigg)_{ \mathcal{O}} + \Vert v \Vert _{\mathcal{O}}^{2}. \end{array} $$

(7)

As $1 < \beta < 2$, we can conclude that

$$ \mathcal{A}\left ((u, v), (u, v)\right ) \geq C \left ( \| u \|_{B^{ \frac{\alpha}{2}, \frac{\beta}{2}} (\mathcal{O})} + \| v \|_{B^{ \frac{\alpha}{2}, \frac{\beta}{2}} (\mathcal{O})} \right )^{2} \geq C \| (u, v) \|_{B^{\frac{\alpha}{2}, \frac{\beta}{2}} (\mathcal{O})}^{2}. $$

To prove stability, we take $(\chi , \xi ) = (u, v)$ in (4), and using the coercivity result we get

$$ \| (u, v) \|_{B^{\frac{\alpha}{2}, \frac{\beta}{2}} (\mathcal{O})} \leq C \left ( \| f_{1} \|_{L^{2} (\mathcal{O})} + \| f_{2} \|_{L^{2} ( \mathcal{O})} \right ). $$

(8)

 □

4 Numerical approach

4.1 Finite element method of Caputo’s derivative (FEMC1)

Step 1. Base functions and discretization domain. Let $\Omega = [0,L]$ be a finite domain. Let $\Omega _{e}$ be a uniform partition of Ω with uniform grid

$$ 0 = x_{0} < x_{1} < \cdots < x_{m-1} < x_{m} = L, $$

so that $\Omega = \bigcup _{i=0}^{m-1} \Omega _{i}$, where $\Omega _{i} = [x_{i}, x_{i+1}]$. The time discretization of the interval $I = [0, T]$ is given by

$$ 0 = t_{0} < t_{1} < \cdots < t_{n-1} < t_{n} = T, $$

where m and n are positive integers, $\Delta x = x_{i} - x_{i-1} = \frac{L}{m}$, so that $x_{i} = i\Delta x$ for $i = 1, \ldots , m$, and $\Delta t = t_{j} - t_{j-1} = \frac{T}{n}$, so that $t_{j} = j\Delta t$ for $j = 1, \ldots , n$.

Denote by $u(x_{i}, t_{j}) = u_{i}^{j}$, $v(x_{i}, t_{j}) = v_{i}^{j}$, $f_{1}(x_{i}, t_{j}) = f_{1}^{j}$, and $f_{2}(x_{i}, t_{j}) = f_{2}^{j}$ the values of the functions u, v, $f_{1}$, and $f_{2}$ at the point $x_{i}$ and instant $t_{j}$. We also define the space $S_{k}$ as the set of piecewise linear functions associated with this partition:

$$ S_{k} = \{ u; \, u|_{\Omega _{i}} \in P_{1}(\Omega _{i}), \, u \in \mathcal{C}(\Omega ) \}, $$

where $P_{1}(\Omega _{i})$ is the space of linear polynomials defined on $\Omega _{i}$.

The base functions $h_{i}$ of $S_{k}$ for each $\Omega _{i}$ changing from the real base to the reference base are given by

$$ \mathcal{B}=\left \{h_{1} = \dfrac{1}{x_{2} - x_{1}}(x_{2} - x),\; h_{2} = \dfrac{1}{x_{2} - x_{1}}(x - x_{1}) \right \}. $$

Denoting by $u^{j}$, $v^{j}$ the approximations of $u(t_{j})$, $v(t_{j})$, we have

$$ u^{j}:=\displaystyle \sum _{i=0}^{m} u_{i}^{j}h_{i}(x), \quad \quad v^{j}:= \displaystyle \sum _{i=0}^{m} v_{i}^{j}h_{i}(x), $$

(9)

where

$$\begin{aligned} &h_{i}(x)= \textstyle\begin{cases} \dfrac{1}{\Delta x}(x-x_{i-1}), & x \in [x_{i-1} , x_{i}], \\ \dfrac{1}{\Delta x}(x_{i+1}-x), & x \in [x_{i} , x_{i+1}], \\ 0 & \text{elsewhere,} \end{cases}\displaystyle \qquad \dfrac{\partial }{\partial x}h_{i}(x)= \textstyle\begin{cases} \dfrac{1}{\Delta x}, & x \in [x_{i-1} , x_{i}], \\ \dfrac{-1}{\Delta x}, & x \in [x_{i}, x_{i+1}], \\ 0 & \text{elsewhere,} \end{cases}\displaystyle \\ &\displaystyle h_{0}(x)= \textstyle\begin{cases} \dfrac{1}{\Delta x}(x_{1}-x), & x \in [x_{0} , x_{1}], \\ 0, & \text{elsewhere,} \end{cases}\displaystyle \\ &\text{and } \displaystyle h_{m}(x)= \textstyle\begin{cases} \dfrac{1}{\Delta x}(x-x_{m-1}), & x \in [x_{m-1} , x_{m}], \\ 0 & \text{elsewhere.} \end{cases}\displaystyle \end{aligned}$$

In Fig. 1, we show the distribution of test functions across elements.

Figure 1

Distribution of test functions across elements

To summarize the principle of the finite element method, we multiply equations (1)₁ and (1)₂ by h, and integrating by parts over Ω, we obtain

$$ \left \{ \textstyle\begin{array}{l@{\quad}l} \bigg( ~^{C}D^{\alpha}_{t} u^{j},h\bigg)_{\Omega} -\bigg( ~^{C}D^{ \beta}_{x} u^{j},h\bigg)_{\Omega} +\bigg(v^{j},h\bigg)_{\Omega} &= \bigg(f_{1}^{j},h\bigg)_{\Omega}, \\ \bigg( ~^{C}D^{\alpha}_{t} v^{j},h\bigg)_{\Omega} -\bigg( ~^{C}D^{ \beta}_{x} v^{j},h\bigg)_{\Omega} +\bigg(u^{j},h\bigg)_{\Omega} &= \bigg(f_{2}^{j},h\bigg)_{\Omega}, \end{array}\displaystyle \right . $$

(10)

for all $j=\overline{1,n}$.

The weak formulation of the problem can also be expressed by choosing each test function h as $h_{i}$ on $\Omega = \displaystyle \cup _{e=0}^{m-1} \Omega _{e}$ in (10), so we have

$$ \left \{ \textstyle\begin{array}{l@{\quad}l} \displaystyle \sum _{e=0}^{m-1}\big( ~^{C}D^{\alpha}_{t} u^{j},h_{i} \big)_{\Omega _{e}} +\displaystyle \sum _{e=0}^{m-1}\big( ~^{C}D^{ \lambda}_{x} u^{j},\dfrac{\partial h_{i}}{\partial x}\big)_{\Omega _{e}} +\displaystyle \sum _{e=0}^{m-1}\big(v^{j},h_{i}\big)_{\Omega _{e}} &= \displaystyle \sum _{e=0}^{m-1}\big(f_{1}^{j},h_{i}\big)_{\Omega _{e}}, \\ \displaystyle \sum _{e=0}^{m-1}\big( ~^{C}D^{\alpha}_{t} v^{j},h_{i} \big)_{\Omega _{e}} +\displaystyle \sum _{e=0}^{m-1}\big( ~^{C}D^{ \lambda}_{x} v^{j},\dfrac{\partial h_{i}}{\partial x}\big)_{\Omega _{e}} +\displaystyle \sum _{e=0}^{m-1}\big(u^{j},h_{i}\big)_{\Omega _{e}} &= \displaystyle \sum _{e=0}^{m-1}\big(f_{2}^{j},h_{i}\big)_{\Omega _{e}}, \end{array}\displaystyle \right . $$

(11)

where $\lambda =\beta -1$. The Caputo fractional derivative can be approximated by the formulas

$$ ~^{C}D^{\alpha}_{t} u^{j}=\frac{1}{\Gamma (1-\alpha )}\sum _{s=0}^{j-1} \int _{t_{s}}^{t_{s+1}}(t_{j}-r)^{-\alpha} \left ( \dfrac{u^{s+1}-u^{s}}{\Delta t}\right )\;dr $$

(12)

and

$$ ~^{C}D^{\lambda}_{x} u^{j}=\frac{1}{\Gamma (1-\lambda )}\Bigg[\sum _{s=0}^{i-1} \int _{x_{s}}^{x_{s+1}}(x-r)^{-\lambda} \dfrac{\partial u^{j}(r)}{\partial r}\;dr+\int _{x_{i}}^{x}(x-r)^{- \lambda} \dfrac{\partial u^{j}(r)}{\partial r}\;dr\Bigg]. $$

(13)

With element-by-element $\Omega _{e}$, $e=\overline{0,m-1}$, we get

$$ \left \{ \textstyle\begin{array}{l@{\quad}l} \bigg( ~^{C}D^{\alpha}_{t} u^{j},h_{i}\bigg)_{\Omega _{e}} +\bigg( ~^{C}D^{ \lambda}_{x} u^{j},\dfrac{\partial h_{i}}{\partial x}\bigg)_{\Omega _{e}} +\bigg(v^{j},h_{i}\bigg)_{\Omega _{e}} &= \bigg(f_{1}^{j},h_{i}\bigg)_{ \Omega _{e}}, \\ \bigg( ~^{C}D^{\alpha}_{t} v^{j},h_{i}\bigg)_{\Omega _{e}} +\bigg( ~^{C}D^{ \lambda}_{x} v^{j},\dfrac{\partial h_{i}}{\partial x}\bigg)_{\Omega _{e}} +\bigg(u^{j},h_{i}\bigg)_{\Omega _{e}} &= \bigg(f_{2}^{j},h_{i}\bigg)_{ \Omega _{e}}. \end{array}\displaystyle \right . $$

(14)

In Fig. 2 we show the pattern mesh of u and v using the discretization of the intervals $(0,L)$ and $0,T$).

Figure 2

The finite difference grid

In the following, we reformulate system (14) in abstract form as follows:

$$ K_{e}^{j}\times U = F_{e}^{j}, $$

(15)

where

$$\begin{aligned}& U = (u_{0}^{j}, \ldots , u_{i+1}^{j}, u_{i}^{0},\ldots, u_{i}^{j-1}, u_{i+1}^{0},\ldots, u_{i+1}^{j-1}, v_{0}^{j},\ldots, v_{i+1}^{j}, v_{i}^{0},\ldots, v_{i}^{j-1}, v_{i+1}^{0},\ldots, v_{i+1}^{j-1})^{T},\\& {F_{e}^{j} = \bigg((f_{1}^{j}, h_{i})_{\Omega _{e}}, (f_{1}^{j}, h_{i+1})_{\Omega _{e}}, (f_{2}^{j}, h_{i})_{\Omega _{e}}, (f_{2}^{j}, h_{i+1})_{ \Omega _{e}}\bigg)^{T}}, \end{aligned}$$

and $K_{e}^{j}$ represents the element e at $t_{j}$ defined by

$$ K_{e}^{j} = \left ( \textstyle\begin{array}{c@{\quad}c} A_{i}^{j} & B_{i}^{j} \\ B_{i}^{j} & A_{i}^{j} \end{array}\displaystyle \right ), $$

(16)

where

$$\begin{aligned}& A_{i}^{j} = \left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 0 \ldots 0 & 0 & 0 & 0 \ldots 0 & 0 \ldots 0 \\ -P^{j}_{0} \,-P^{j}_{1} \ldots -P^{j}_{i-1} & -P^{j}_{i} + C^{j}_{i+1} & -P^{j}_{i+1} + C^{j}_{i} & C^{0}_{i+1} \ldots C^{j-1}_{i+1} & C^{0}_{i} \ldots C^{j-1}_{i} \end{array}\displaystyle \right ),\\& B_{i}^{j} = \frac{\Delta x}{6} \left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 0 \ldots 0 & 0 & 0 & 0 \ldots 0 \\ \overbrace{0 \ldots 0}^{(i-1)} & 1 & 2 & \overbrace{0 \ldots 0}^{2(j-1)} \end{array}\displaystyle \right ) \end{aligned}$$

for $j = \overline{1,n}$, and the element $\Omega _{0}$. We perform the same analysis using the test functions $h_{i}$, $i = \overline{1,m-1}$, and find

$$\begin{aligned}& A_{i}^{j} = \left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} P^{j}_{0} P^{j}_{1} \ldots P^{j}_{i-1} & P^{j}_{i} + C^{j}_{i} & P^{j}_{i+1} + C^{j}_{i+1} & C^{0}_{i} \ldots C^{j-1}_{i} \quad C^{0}_{i+1} \ldots C^{j-1}_{i+1} \\ -P^{j}_{0} -P^{j}_{1} \ldots -P^{j}_{i-1} & -P^{j}_{i} + C^{j}_{i+1} & -P^{j}_{i+1} + C^{j}_{i} & C^{0}_{i+1} \ldots C^{j-1}_{i+1} \quad C^{0}_{i} \ldots C^{j-1}_{i} \end{array}\displaystyle \right ),\\& B_{i}^{j} = \frac{\Delta x}{6} \left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \overbrace{0 \ldots 0}^{(i-1)} & 2 & 1 & \overbrace{0 \ldots 0}^{2(j-1)} \\ \overbrace{0 \ldots 0}^{(i-1)} & 1 & 2 & \overbrace{0 \ldots 0}^{2(j-1)} \end{array}\displaystyle \right ), \end{aligned}$$

and

$$\begin{aligned}& A_{i}^{j} = \left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} P^{j}_{0}, P^{j}_{1} \ldots P^{j}_{i-1} & P^{j}_{i} + C^{j}_{i} & P^{j}_{i+1} + C^{j}_{i+1} & C^{0}_{i} \ldots C^{j-1}_{i} & C^{0}_{i+1} \ldots C^{j-1}_{i+1} \\ 0 \ldots 0 & 0 & 0 & 0 \ldots 0 & 0 \ldots 0 \end{array}\displaystyle \right ),\\& B_{i}^{j} = \frac{\Delta x}{6} \left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \overbrace{0 \ldots 0}^{(i-1)} & 2 & 1 & \overbrace{0 \ldots 0}^{2(j-1)} \\ 0 \ldots 0 & 0 & 0 & 0 \ldots 0 \end{array}\displaystyle \right ) \end{aligned}$$

for $j = \overline{1,n}$, and the element $\Omega _{m}$. The following lemmas will be useful.

Lemma 3

For $0 < \alpha < 1$ and $x_{i} \leq x \leq x_{i+1}$, we have

$$ ~^{C}D^{\alpha}_{x} h_{i}(x) = -\mu \bigg((x - x_{i})^{1 - \alpha} - (x - x_{i+1})^{1 - \alpha}\bigg) $$

and

$$ ~^{C}D^{\alpha}_{x} h_{i+1}(x) = \mu \bigg((x - x_{i})^{1 - \alpha} - (x - x_{i+1})^{1 - \alpha}\bigg), $$

where $\mu = \dfrac{1}{\Delta x \Gamma (2-\alpha )}$.

Using equation (12) and Lemma 3, we obtain the following result.

Lemma 4

For $j=1, 2, \ldots , n$ and $0 < \alpha < 1$, we have

$$ \textstyle\begin{array}{l@{\quad}l@{\quad}l} ~^{C}D^{\alpha}_{x} u^{j} &=& \mu (u^{j}_{k} - u^{j}_{k-1}) \cdot (x - x_{k-1})^{(1 - \alpha )} \\ & +& \displaystyle \mu \sum _{i=0}^{k-1}(u^{j}_{i+1} - u^{j}_{i}) \cdot \bigg((x - x_{i})^{1 - \alpha} - (x - x_{i+1})^{1 - \alpha} \bigg), \end{array} $$

where $0 \leq x \leq x_{k}$ and $\mu = \dfrac{1}{\Delta x \Gamma (2-\alpha )}$.

For all $j = \overline{1,n}$ and element $\Omega _{i}$, $i = \overline{1,m-2}$, we get

$$ \bigg(~^{C}D^{\lambda}_{x} u^{j}, \dfrac{\partial h_{i}}{\partial x} \bigg)_{\Omega _{i}} = \sum _{k=0}^{i+1} P^{j}_{k} u^{j}_{k} $$

and

$$ \bigg(~^{C}D^{\lambda}_{x} u^{j}, \dfrac{\partial h_{i+1}}{\partial x}\bigg)_{\Omega _{i}} = \sum _{k=0}^{i+1} -P^{j}_{k} u^{j}_{k}. $$

The Algorithm 1 summarizes all the steps for the calculation of the coefficients of $P^{j}$.

Algorithm 1

Compute the coefficients of $P^{j}$

Using equation (13), we obtain the following results.

Lemma 5

For $j = 1, 2, \ldots , n$ and $0 < \alpha < 1$, we have

$$ ~^{C}D^{\alpha}_{t} u^{j} = B \sum _{k=0}^{j-1}(u^{k+1} - u^{k}) \bigg((t_{j} - t_{k})^{1 - \alpha} - (t_{j} - t_{k+1})^{1 - \alpha} \bigg), $$

where $B = \dfrac{1}{\Delta t \Gamma (2 - \alpha )}$ and $t_{j} - t_{k} = \Delta t (j - k)$.

Lemma 6

For $i = \overline{1, m-1}$, we have

$$ \big(h_{i}, h_{i}\big)_{\Omega _{i}} = \dfrac{\Delta x}{3}, \quad \quad \big(h_{i}, h_{i+1}\big)_{\Omega _{i}} = \dfrac{\Delta x}{6}, \quad \textit{and} \quad \big(h_{i+1}, h_{i+1}\big)_{\Omega _{i}} = \dfrac{\Delta x}{3}. $$

For all $j = \overline{1, n}$ and the element $\Omega _{i}$, $i = \overline{1, m-2}$, we get

$$ \bigg( ~^{C}D^{\alpha}_{t} u^{j}, h_{i}\bigg)_{\Omega _{i}} = \sum _{k=0}^{j} \bigg(C_{i}^{k} u^{k}_{i} + C_{i+1}^{k} u^{k}_{i+1}\bigg) $$

and

$$ \bigg( ~^{C}D^{\alpha}_{t} u^{j}, h_{i+1}\bigg)_{\Omega _{i}} = \sum _{k=0}^{j} \bigg(C_{i+1}^{k} u^{k}_{i} + C_{i}^{k} u^{k}_{i+1}\bigg). $$

The Algorithm 2 summarizes all the steps for the calculation of the coefficients of $C_{i}$.

Algorithm 2

Compute the coefficients of $C_{i}$

Recall that for all $j = \overline{1, n}$ and elements $\Omega _{i}$, $i = \overline{1, m-2}$, we get

$$ \bigg(v^{j}, h_{i}\bigg)_{\Omega _{i}} = \frac{\Delta x}{6} \big(2 \cdot v^{j}_{i} + v^{j}_{i+1}\big) $$

and

$$ \bigg(v^{j}, h_{i+1}\bigg)_{\Omega _{i}} = \frac{\Delta x}{6} \big(v^{j}_{i} + 2 \cdot v^{j}_{i+1}\big). $$

Step 2. Assembly of the global system. We construct the global system matrix K. The contribution to the global matrix from an element consisting of nodes i and j is placed in a submatrix of the global matrix formed by rows i and j and columns i and j.

For this purpose, the procedure requires the solution to be continuous and the unknown source terms $K_{i}^{j}$ be balanced at nodes common to several elements where they are connected to each other. We recall that

$$ K_{i}^{j} = \left ( \textstyle\begin{array}{c@{\quad}c} A_{i}^{j} & B_{i}^{j} \\ B_{i}^{j} & A_{i}^{j} \end{array}\displaystyle \right ), $$

(17)

where

$$\begin{aligned}& A_{i}^{j} = \left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} P^{j}_{0} P^{j}_{1} \ldots P^{j}_{i-1} & P^{j}_{i} + C^{j}_{i} & P^{j}_{i+1} + C^{j}_{i+1} & C^{0}_{i} \ldots C^{j-1}_{i} \quad C^{0}_{i+1} \ldots C^{j-1}_{i+1} \\ -P^{j}_{0} - P^{j}_{1} \ldots -P^{j}_{i-1} & -P^{j}_{i} + C^{j}_{i+1} & -P^{j}_{i+1} + C^{j}_{i} & C^{0}_{i+1} \ldots C^{j-1}_{i+1} \quad C^{0}_{i} \ldots C^{j-1}_{i} \end{array}\displaystyle \right ),\\& B_{i}^{j} = \frac{\Delta x}{6}\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \overbrace{0 \ldots 0}^{(i-1)} & 2 & 1 & \overbrace{0 \ldots 0 }^{2(j-1)} \\ \overbrace{0 \ldots 0 }^{(i-1)} & 1 & 2 & \overbrace{0 \ldots 0 }^{2(j-1)} \end{array}\displaystyle \right ). \end{aligned}$$

The system in the Fig. 3, contains the global matrix.

Figure 3

Assembly of the global system for two elements

For each element, we have a starting point at $(i, j)$. Therefore we perform the following transformation:

$$ \left \{ \textstyle\begin{array}{l@{\quad}l@{\quad}l} u_{k}^{j} & \longrightarrow s_{p}:=j(m+1)+k & \text{for $k = \overline{0, i+1}$}, \\ u_{i}^{k} & \longrightarrow s_{p}:=k(m+1)+i & \text{for $k = \overline{0, j-1}$}, \\ u_{i+1}^{k} & \longrightarrow s_{p}:=k(m+1)+i+1 & \text{for $k = \overline{0, j-1}$}, \\ v_{k}^{j} & \longrightarrow s_{p}:=j(m+1)+k+c & \text{for $k = \overline{0, i+1}$}, \\ v_{i}^{k} & \longrightarrow s_{p}:=k(m+1)+i+c & \text{for $k = \overline{0, j-1}$}, \\ v_{i+1}^{k} & \longrightarrow s_{p}:=k(m+1)+i+1+c & \text{for $k = \overline{0, j-1}$}, \end{array}\displaystyle \right . $$

where $c=(m+1) \times (n+1)$, and p represents a counter or an index that increments automatically at each iteration of the loop.

After assembly of the global system matrix, system (11) is equivalent to

$$ K \times W = F, $$

(18)

where

$$ \left \{ \textstyle\begin{array}{l@{\quad}l} K(p_{k}, s) = K(p_{k}, s) + K_{k, i}^{j} & \text{for $k = \overline{0, 3}$}, \\ F(p_{k}) = F(p_{k}) + F_{k, i}^{j} & \text{for $k = \overline{0, 3}$}. \end{array}\displaystyle \right . $$

The Algorithm 3 assembles the global system.

Algorithm 3

Assembly of the Equation System:

4.2 Finite element method of Riemann’s derivative (FEMR2)

We follow the same steps as in Model (FEMC1). Multiplying equations (1)₁ and (1)₂ by h and integrating by parts element-by-element over Ω, we have

$$ \left \{ \textstyle\begin{array}{l@{\quad}l} \bigg( ~^{RL}D^{\alpha}_{t} u^{j},h\bigg)_{\Omega _{e}} -\bigg( ~^{RL}D^{ \beta}_{x} u^{j},h\bigg)_{\Omega _{e}} +\bigg(v^{j},h\bigg)_{\Omega _{e}}&= \bigg(f_{1}^{j},h\bigg)_{\Omega _{e}}, \\ \bigg( ~^{RL}D^{\alpha}_{t} v^{j},h\bigg)_{\Omega _{e}} -\bigg( ~^{RL}D^{ \beta}_{x} v^{j},h\bigg)_{\Omega _{e}} +\bigg(u^{j},h\bigg)_{\Omega _{e}}&= \bigg(f_{2}^{j},h\bigg)_{\Omega _{e}}, \end{array}\displaystyle \right . $$

(19)

for $e=\overline{0,m-1}$ and $j=\overline{1,n}$.

We need the following lemma to give the second scheme.

Lemma 7

For $0 < \alpha < 1$ and $0 < \lambda < 1$, we have

$$ {}^{RL}{D^{\alpha}_{t}} u = ^{C}D^{\alpha}_{t} u+ \frac{u_{0}(x)t^{-\alpha}}{\Gamma (1-\alpha )} $$

and

$$ {}^{RL}{\!D^{\lambda}_{x}} u=^{C}D^{\lambda}_{x} u. $$

This lemma establishes the connection between the Riemann–Liouville fractional derivative and the Caputo fractional derivative. Therefore, applying Lemma 7, system (19) can be written as

$$ \left \{ \textstyle\begin{array}{l@{\quad}l} \bigg( ~^{C}D^{\alpha}_{t} u^{j},h\bigg)_{\Omega _{e}} +\bigg( ~^{C}D^{ \lambda}_{x} u^{j},\dfrac{\partial h}{\partial x}\bigg)_{\Omega _{e}} + \bigg(v^{j},h\bigg)_{\Omega _{e}}&= \bigg(g_{1}^{j},h\bigg)_{\Omega _{e}}, \\ \bigg( ~^{C}D^{\alpha}_{t} v^{j},h\bigg)_{\Omega _{e}} +\bigg( ~^{C}D^{ \lambda}_{x} v^{j},\dfrac{\partial h}{\partial x}\bigg)_{\Omega _{e}} + \bigg(u^{j},h\bigg)_{\Omega _{e}}&= \bigg(g_{2}^{j},h\bigg)_{\Omega _{e}}, \end{array}\displaystyle \right . $$

(20)

where $g_{1}^{j}=f_{1}^{j}- \frac{u_{0}(x)t_{j}^{-\alpha}}{\Gamma (1-\alpha )}$ and $g_{2}^{j}=f_{2}^{j}- \frac{u_{0}(x)t_{j}^{-\alpha}}{\Gamma (1-\alpha )}$.

In the following, we reformulate system (20) in abstract form as follows:

$$ K_{e}^{j} \times U = F_{e}^{j}, $$

(21)

where

$$\begin{aligned}& U = (u_{0}^{j},\ldots, u_{i+1}^{j}, u_{i}^{0},\ldots, u_{i}^{j-1}, u_{i+1}^{0},\ldots, u_{i+1}^{j-1}, v_{0}^{j},\ldots, v_{i+1}^{j}, v_{i}^{0},\ldots, v_{i}^{j-1}, v_{i+1}^{0},\ldots, v_{i+1}^{j-1})^{T},\\& {F_{e}^{j} = \bigg((g_{1}^{j}, h_{i})_{\Omega _{e}}, (g_{1}^{j}, h_{i+1})_{\Omega _{e}}, (g_{2}^{j}, h_{i})_{\Omega _{e}}, (g_{2}^{j}, h_{i+1})_{ \Omega _{e}}\bigg)^{T}}. \end{aligned}$$

Finally, we apply the same previous results.

4.3 PINN approach

The Physics-Informed Neural Networks (PINNs) represent a novel category of neural networks that integrate machine learning and physical laws. This new algorithmic technology emerged relatively recently, in 2019, stemming from research laboratories.

To solve a system containing the Caputo fractional derivative, we employ both the PINN model and the Finite Difference Method (FDM) (see Fig. 4). The PINN captures the complex behaviors of the studied system, whereas the FDM discretizes the differential or integral equations, enabling a numerical approach to problem resolution. By combining these two approaches we obtain a scheme termed Fractional Physics-Informed Neural Networks (fPINNs), capable of efficiently solving a variety of mathematical and physical problems.

Figure 4

fPINNs to solve problem (1)

The predicted function values, denoted as $f1_{pred}$ and $f2_{pred}$, are defined as follows:

$$ \left \{ \textstyle\begin{array}{c} f1_{pred}(x,t) = \, ^{C}_{0}D^{\alpha}_{t} u(x, t) - \, ^{C}_{0}D^{ \beta}_{x} u(x, t) + v(x, t), \quad \text{in} \quad (x,t) \in (0, L) \times (0, +\infty ), \\ f2_{pred}(x,t) = \, ^{C}_{0}D^{\alpha}_{t} v(x, t) - \, ^{C}_{0}D^{ \beta}_{x} v(x, t) + u(x, t), \quad \text{in} \quad (x,t) \in (0, L) \times (0, +\infty ). \end{array}\displaystyle \right . $$

(22)

In this case, $u(x, t)$ and $v(x, t)$ will be approximated by a neural network. The latter has as an objective to minimize the following loss:

$$ MSE = MSE_{u} + MSE_{v}, $$

(23)

where

$$ \left \{ \textstyle\begin{array}{l@{\quad}l} MSE_{u} &= MSE_{u}^{0} + MSE_{u}^{0L} + MSE_{f_{1}}, \\ MSE_{v} &= MSE_{v}^{0} + MSE_{v}^{0L} + MSE_{f_{2}}. \end{array}\displaystyle \right . $$

(24)

Now we calculate the right-hand side of (24)₁ and (24)₂:

$$\begin{aligned}& \text{$MSE_{u}^{0}$} = \dfrac{1}{N_{0}} \sum _{i=0}^{N_{0}} \big(u(x^{0}_{i},0) - {u}^{0}_{i}\big)^{2}, \quad \text{$MSE_{v}^{0}$} = \dfrac{1}{N_{0}} \sum _{i=0}^{N_{0}} \big(v(x^{0}_{i},0) - {v}^{0}_{i}\big)^{2},\\& \text{$MSE_{u}^{0L}$} = \dfrac{1}{N_{b}} \sum _{j=0}^{N_{b}} \bigg( \big(u(0,t^{j}_{0}) - u^{j}_{0}\big)^{2} + \big(u(L,t^{j}_{0}) - u^{j}_{L} \big)^{2}\bigg),\\& \text{$MSE_{v}^{0L}$} = \dfrac{1}{N_{b}} \sum _{j=0}^{N_{b}} \bigg( \big(v(0,t^{j}_{0}) - v^{j}_{0}\big)^{2} + \big(v(L,t^{j}_{0}) - v^{j}_{L} \big)^{2}\bigg), \end{aligned}$$

and

$$ \text{$MSE_{f_{1}}$} = \dfrac{1}{N_{f}} \sum _{i=0}^{N_{f}} \big(f1_{pred}(x_{i}, t_{i}) - f_{1}(x_{i}, t_{i})\big)^{2}, \quad \text{$MSE_{f_{2}}$} = \dfrac{1}{N_{f}} \sum _{i=0}^{N_{f}} \big(f2_{pred}(x_{i}, t_{i}) - f_{2}(x_{i}, t_{i})\big)^{2}. $$

Here $\{x^{0}_{i}, u^{0}_{i}\}$ denotes the initial data at $t = 0$, $\{t^{j}_{0}, t^{j}_{L}\}$ the boundary data, and $\{x_{i}, t_{i}\}$ corresponds to collocation points on $f1_{pred}(x, t)$ and $f2_{pred}(x, t)$, where $N_{0}$, $N_{b}$, and $N_{f}$ are the numbers of available observations. Figure 5 shows the point cloud used for training the PINN and calculating the fractional derivative for each point $(x_{i},t_{j})$.

Figure 5

Point cloud used for training the PINN and calculating the fractional derivative for each point $(x_{i},t_{j})$

For $0 < \alpha < 1$ and the interval $[t_{0}, t_{j}]$ discretized into $j+1$ points, $0 = t_{0} < t_{1} < \cdots < t_{j}$, the Caputo fractional derivative of order α using the method of finite differences (MDF) is approximated as

$$\begin{aligned} D_{t}^{\alpha }u(x_{i},t_{j}) &= \sum _{k=0}^{j-1} \frac{u(x_{i},t_{k+1}) - u(x_{i},t_{k})}{\Delta t} \bigg((t_{j} - t_{k})^{1 - \alpha} - (t_{j} - t_{k+1})^{1 - \alpha}\bigg) \\ &= \sum _{k=0}^{i-1} \frac{\partial}{\partial t} u(x_{i},t_{k+1}) \bigg((t_{j} - t_{k})^{1 - \alpha} - (t_{j} - t_{k+1})^{1 - \alpha} \bigg). \end{aligned}$$

Lemma 8

Let q be the fractional order, where $q = n + \alpha$, and n is the integer part of q. The fractional derivative of order q can be expressed as

$$ D^{q}_{x} u(x,t) = D^{n+\alpha}_{x} u(x,t) = D^{n}_{x} \left ( D^{ \alpha}_{x} u(x,t) \right ), $$

where $D^{\alpha}_{x} u(x,t)$ represents the fractional derivative of order α of the function $u(x,t)$ and $D^{n}$ represents the integer-order derivative of order n. This expression shows that the fractional derivative of order q is obtained by first applying the fractional derivative of order $D^{\alpha}$, followed by the integer-order derivative of order n.

Then

$$ D^{\beta}_{x} u(x,t) = D^{1+\lambda}_{x} u(x,t) = D_{x} \left ( D^{ \lambda}_{x} u(x,t) \right ), $$

for $0 < \lambda < 1$ and the interval $[x_{0}, x_{i}]$ discretized into $i+1$ points $0 = x_{0} < x_{1} < \cdots < x_{i}$. Using the method of finite differences (MDF), the Caputo fractional derivative of order λ is approximated as

$$\begin{aligned} D_{x}^{\lambda }u(x_{i},t_{j}) &= \sum _{k=0}^{i-1} \frac{u(x_{k+1},t_{j}) - u(x_{k},t_{j})}{\Delta x} \bigg((x_{i} - x_{k})^{1 - \lambda} - (x_{i} - x_{k+1})^{1 - \lambda}\bigg) \\ &= \sum _{k=0}^{i-1} \frac{\partial}{\partial x} u(x_{k+1},t_{j}) \bigg((x_{i} - x_{k})^{1 - \lambda} - (x_{i} - x_{k+1})^{1 - \lambda} \bigg). \end{aligned}$$

The PINN calculates the integer-order partial derivative $\frac{\partial ^{n} u}{\partial ^{n} x}$ using automatic differentiation to obtain the gradients of the model predictions with respect to the inputs. For the FPINN with Riemann fractional derivative, we use Lemma 7:

$$ {}^{RL}{D^{\alpha}_{t}} u(x_{i},t_{j}) = ^{C}D^{\alpha}_{t} u(x_{i},t_{j}) + \frac{u(0,t_{j})t_{j}^{-\alpha}}{\Gamma (1-\alpha )}. $$

4.4 Numerical examples

4.4.1 Caputo derivative example

$$ \left \{ \textstyle\begin{array}{c@{\quad}c} ~^{C}D^{\alpha}_{t} u(x, t) -\,~^{C}D^{\beta}_{x} u(x, t) + v(x, t) &= f_{1}(x,t) \quad \text{in} \quad (x,t) \in (0, L) \times (0, +\infty ), \\ ~^{C}D^{\alpha}_{t} v(x, t) -\,~^{C}D^{\beta}_{x} v(x, t) + u(x, t) &= f_{2}(x,t) \quad \text{in} \quad (x,t) \in (0, L) \times (0, +\infty ). \end{array}\displaystyle \right . $$

The initial and boundary conditions are

$$ u(x, 0) = 0, \quad v(x, 0) = -x^{3}, \quad x \in \Omega := [0,1], $$

and the boundary conditions are

$$ \begin{gathered} u(0, t) = 0, \quad v(0, t) = t^{4}, \quad t \in I := [0, T], \\ u(1, t) = t^{2}, \quad v(1, t) = t^{4} - 1, \quad t \in I, \end{gathered} $$

(25)

where $0 < \alpha < 1$, $1 < \beta < 2$, and

$$\begin{aligned} f_{1}(x,t) &= t^{4} - x^{3} + \frac{2t^{2-\alpha}x^{4}}{\Gamma (3-\alpha )} - \frac{24t^{2}x^{4-\beta}}{\Gamma (5-\beta )}, \\ f_{2}(x,t) &= t^{2} x^{4} + \frac{6x^{3-\beta}}{\Gamma (4-\beta )} + \frac{24t^{4-\alpha}}{\Gamma (5-\alpha )}. \end{aligned}$$

The exact solution is $u(x,t) = t^{2} x^{4}$, $v(x,t) = t^{4} - x^{3}$. Figure 6 shows the value of $K_{0}^{1}$ and $K_{1}^{1}$ for $\alpha = 0.5$, $\beta = 1.75$, $\Delta x = 0.1$, and $\Delta t = 0.1$.

Figure 6

Calculating $K_{0}^{1}$ and $K_{1}^{1}$ for $\alpha = 0.5$, $\beta = 1.75$, $\Delta x = 0.1$, and $\Delta t = 0.1$

The data used in the graphic representations were obtained from the values obtained using appropriate algorithms. Figure 7 and Fig. 8 give the comparison between the exact solutions and the numerical solutions for u and v with Caputo derivative.

Figure 7

Exact solution

Figure 8

Numerical solution

We define Root Mean Square Error (RMSE) by

$$ \text{RMSE} = \sqrt{\frac{1}{N} \sum _{i=0}^{m}\sum _{j=0}^{n} ({u}^{j}_{i} - \hat{u}^{j}_{i})^{2}}. $$

(26)

The data presented in Table 1 and Table 2, were obtained by comparing the exact solutions u and v with their respective numerical approximations û and v̂ using different discretization values Δx and Δt. Specifically, we calculated the RMSE defined by Equation (26), which measures the discrepancy between u (or v) and û (or v̂). This comparison enabled us to assess the accuracy of the numerical method employed to approximate u and v.

Table 1 RMSE between the exact solution u and its numerical approximation û with $\Delta x = \Delta t = 0.1$

Table 2 RMSE between the exact solution v and its numerical approximation v̂ with $\Delta x = \Delta t = 0.1$

4.4.2 Riemann derivative example

$$ \left \{ \textstyle\begin{array}{c@{\quad}c} ~^{RL}D^{\alpha}_{t} u(x, t) -\, ~^{RL}D^{\beta}_{x} u(x, t) + v(x, t) &= f_{1}(x,t) \quad \text{in} \quad (x,t) \in (0, L) \times (0, + \infty ), \\ ~^{RL}D^{\alpha}_{t} v(x, t) -\, ~^{RL}D^{\beta}_{x} v(x, t) + u(x, t) &= f_{2}(x,t) \quad \text{in} \quad (x,t) \in (0, L) \times (0, + \infty ). \end{array}\displaystyle \right . $$

The initial and boundary conditions are

$$ u(x, 0) = 0, \quad v(x, 0) = x, \quad x \in \Omega := [0,1], $$

and the boundary conditions are

$$ \begin{gathered} u(0, t) = 0, \quad v(0, t) = t^{2}, \quad t \in I := [0, T], \\ u(1, t) = t, \quad v(1, t) = t^{2} + 1, \quad t \in I, \end{gathered} $$

(27)

where $0 < \alpha < 1$, $1 < \beta < 2$, and

$$\begin{aligned} f_{1}(x,t) &= \frac{t^{1 - \alpha} x^{2}}{\Gamma (2 - \alpha )} - \frac{2 t x^{2 - \beta}}{\Gamma (3 - \beta )} + t^{2} + x, \\ f_{2}(x,t) &= \frac{x \alpha ^{2} - 3 x \alpha + 2 t^{2} + 2 x}{t^{\alpha }\Gamma (3 - \alpha )} - \frac{x - \beta t^{2} + t^{2}}{x^{\beta }\Gamma (2 - \beta )} + t x^{2}. \end{aligned}$$

The exact solution is $u(x,t) = t x^{2}$, $v(x,t) = t^{2} + x$. Figure 9 and Fig. 10 give the comparison between the exact solutions and the numerical solutions for u and v with Reimann derivative.

Figure 9

Exact solution

Figure 10

Numerical solution

The data presented in Table 3 and Table 4, were obtained by comparing the exact solutions u and v with their respective numerical approximations û and v̂ using different discretization values Δx and Δt.

Table 3 RMSE between the exact solution u and its numerical approximation û with $\Delta x = \Delta t = 0.1$

Table 4 RMSE between the exact solution v and its numerical approximation v̂ with $\Delta x = \Delta t = 0.1$

4.4.3 FPINN example

$$ \left \{ \textstyle\begin{array}{c@{\quad}c} ~^{C}D^{\alpha}_{t} u(x, t) -\, ~^{C}D^{\beta}_{x} u(x, t) + v(x, t) &= f_{1}(x,t) \quad \text{in} \quad (x,t) \in (0, L) \times (0, +\infty ), \\ ~^{C}D^{\alpha}_{t} v(x, t) -\, ~^{C}D^{\beta}_{x} v(x, t) + u(x, t) &= f_{2}(x,t) \quad \text{in} \quad (x,t) \in (0, L) \times (0, +\infty ). \end{array}\displaystyle \right . $$

The initial and boundary conditions are

$$ u(x, 0) = 0, \quad v(x, 0) = x, \quad x \in \Omega := [0,1], $$

and the boundary conditions are

$$ \begin{gathered} u(0, t) = 0, \quad v(0, t) = t^{2}, \quad t \in I := [0, T], \\ u(1, t) = t, \quad v(1, t) = t^{2} + 1, \quad t \in I, \end{gathered} $$

(28)

where $0 < \alpha < 1$, $1 < \beta < 2$, and

$$\begin{aligned} f_{1}(x,t) &= x + t^{2} + \frac{t^{1-\alpha} x^{2}}{\Gamma (2-\alpha )} - \frac{2 t x^{2-\beta}}{\Gamma (3-\beta )}, \\ f_{2}(x,t) &= t x^{2} + \frac{2 t^{2-\alpha}}{\Gamma (3-\alpha )}. \end{aligned}$$

The exact solution is $u(x,t) = t x^{2}$, $v(x,t) = t^{2} + x$.

For $\alpha = 0.5$, $\beta = 1.75$, and $(x,t) \in [0,1] \times [0,1]$, we obtain the following results:

$$ \textstyle\begin{array}{@{}|l@{\quad}|@{\quad}l@{\quad}|@{\quad}l@{\quad}|@{\quad}l@{\quad}|} \hline \Delta x & \Delta t & \text{RMSE}_{u} & \text{RMSE}_{v} \\ \hline 0.1 & 0.1 & 0.0023 & 0.0031 \\ 0.05 & 0.05 & 0.0011 & 0.0016 \\ 0.01 & 0.01 & 0.0004 & 0.0006 \\ \hline \end{array} $$

The data presented in the tables above were obtained by comparing the exact solutions u and v with their respective numerical approximations û and v̂ using different discretization values Δx and Δt. Specifically, we calculated the RMSE defined by

$$ \text{RMSE} = \sqrt{\frac{1}{N} \sum _{i=0}^{m} \sum _{j=0}^{n} ({u}^{j}_{i} - \hat{u}^{j}_{i})^{2}}. $$

(29)

This comparison enabled us to assess the accuracy of the numerical method employed to approximate u and v.

Figures 11-14, were obtained by comparing the exact solutions u and v with their respective numerical FPINN approximations û and v̂.

Figure 11

Solution for 100 iterations

Figure 12

Solution for 1000 iterations

Figure 13

Solution for 5000 iterations

Figure 14

Solution for MSE < 0.01

The data presented in Table 5 and Table 6, were obtained by comparing the exact solutions u and v with their respective numerical FPINN approximations û and v̂ using different values of the order of the fractional derivatives α and β.

Table 5 RMSE between the exact solution u and its numerical FPINN approximation û

Table 6 RMSE between the exact solution v and its numerical FPINN approximation v̂

5 Conclusions

In this paper, we studied the well-posedness and the numerical solution of a coupled wave equation for solving a class of Caputo or Riemann–Liouville space-time fractional partial differential equations, subsequently comparing this solution with those obtained through various methods: the finite element method and Physics-Informed Neural Network (PINN).

This paper makes a significant contribution to the field of numerical analysis and applied mathematics by advancing the capabilities of PINNs in solving complex fractional and coupled PDEs, thus providing a powerful tool for researchers and practitioners dealing with sophisticated modeling challenges.

We conclude that PINNs emerge as a robust and promising tool for tackling complex PDEs, signaling a potentially transformative future alternative in this domain.