Smart irrigation system for environmental sustainability in Africa: An Internet of Everything (IoE) approach

Favour Adenugba; Sanjay Misra; Rytis Maskeliūnas; Robertas Damaševičius; Egidijus Kazanavičius; Favour Adenugba; Sanjay Misra; Rytis Maskeliūnas; Robertas Damaševičius; Egidijus Kazanavičius

doi:10.3934/mbe.2019273

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 5490-5503. doi: 10.3934/mbe.2019273

Previous Article Next Article

Research article Special Issues

Smart irrigation system for environmental sustainability in Africa: An Internet of Everything (IoE) approach

1.
Department of Electrical and Information Engineering, Covenant University, Ota 0123, Nigeria
2.
Department of Computer Engineering, Atilim University, Incek 06836, Ankara, Turkey
3.
Department of Multimedia Engineering, Kaunas University of Technology, Kaunas, Lithuania
4.
Department of Software Engineering, Kaunas University of Technology, Kaunas, Lithuania
5.
Centre of Real Time Computer Systems, Kaunas University of Technology, Kaunas, Lithuania

Received: 21 December 2018 Accepted: 30 May 2019 Published: 13 June 2019

Water and food are two of the most important commodities in the world, which makes agriculture crucial to mankind as it utilizes water (irrigation) to provide us with food. Climate change and a rapid increase in population have put a lot of pressure on agriculture which has a snowball effect on the earth's water resource, which has been proven to be crucial for sustainable development. The need to do away with fossil fuel in powering irrigation systems cannot be over emphasized due to climate change. Smart Irrigation systems powered by renewable energy sources (RES) have been proven to substantially improve crop yield and the profitability of agriculture. Here we show how the control and monitoring of a solar powered smart irrigation system can be achieved using sensors and environmental data from an Internet of Everything (IoE). The collected data is used to predict environment conditions using the Radial Basis Function Network (RBFN). The predicted values of water level, weather forecast, humidity, temperature and irrigation data are used to control the irrigation system. A web platform was developed for monitoring and controlling the system remotely.

Keywords:

Citation: Favour Adenugba, Sanjay Misra, Rytis Maskeliūnas, Robertas Damaševičius, Egidijus Kazanavičius. Smart irrigation system for environmental sustainability in Africa: An Internet of Everything (IoE) approach[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 5490-5503. doi: 10.3934/mbe.2019273

Related Papers:

[1]	Shumoua F. Alrzqi, Fatimah A. Alrawajeh, Hany N. Hassan . An efficient numerical technique for investigating the generalized Rosenau–KDV–RLW equation by using the Fourier spectral method. AIMS Mathematics, 2024, 9(4): 8661-8688. doi: 10.3934/math.2024420
[2]	Xiaoli Wang, Lizhen Wang . Traveling wave solutions of conformable time fractional Burgers type equations. AIMS Mathematics, 2021, 6(7): 7266-7284. doi: 10.3934/math.2021426
[3]	Zhengang Zhao, Yunying Zheng, Xianglin Zeng . Finite element approximation of fractional hyperbolic integro-differential equation. AIMS Mathematics, 2022, 7(8): 15348-15369. doi: 10.3934/math.2022841
[4]	Ailing Zhu, Yixin Wang, Qiang Xu . A weak Galerkin finite element approximation of two-dimensional sub-diffusion equation with time-fractional derivative. AIMS Mathematics, 2020, 5(5): 4297-4310. doi: 10.3934/math.2020274
[5]	Jian-Gen Liu, Jian Zhang . A new approximate method to the time fractional damped Burger equation. AIMS Mathematics, 2023, 8(6): 13317-13324. doi: 10.3934/math.2023674
[6]	Weiwen Wan, Rong An . Convergence analysis of Euler and BDF2 grad-div stabilization methods for the time-dependent penetrative convection model. AIMS Mathematics, 2024, 9(1): 453-480. doi: 10.3934/math.2024025
[7]	Shanhao Yuan, Yanqin Liu, Yibin Xu, Qiuping Li, Chao Guo, Yanfeng Shen . Gradient-enhanced fractional physics-informed neural networks for solving forward and inverse problems of the multiterm time-fractional Burger-type equation. AIMS Mathematics, 2024, 9(10): 27418-27437. doi: 10.3934/math.20241332
[8]	Xin Zhao, Xin Liu, Jian Li . Convergence analysis and error estimate of finite element method of a nonlinear fluid-structure interaction problem. AIMS Mathematics, 2020, 5(5): 5240-5260. doi: 10.3934/math.2020337
[9]	Zhichao Fang, Ruixia Du, Hong Li, Yang Liu . A two-grid mixed finite volume element method for nonlinear time fractional reaction-diffusion equations. AIMS Mathematics, 2022, 7(2): 1941-1970. doi: 10.3934/math.2022112
[10]	Muhammad Asim Khan, Norma Alias, Umair Ali . A new fourth-order grouping iterative method for the time fractional sub-diffusion equation having a weak singularity at initial time. AIMS Mathematics, 2023, 8(6): 13725-13746. doi: 10.3934/math.2023697

Abstract

1. Introduction

In this article, we consider the following time-fractional generalized Rosenau-RLW-Burgers equation:

$\begin{equation} u_t-_{0}^{C}\!\!D_t^\alpha u_{xx}+_{0}^{C}\!\!D_t^\beta u_{xxxx}+u_x-u_{xx} +f(u)_x = g(x,t),\ (x,t)\in\Omega\times J, \end{equation}$

(1.1)

with boundary conditions

$\begin{equation} u(x,t) = u_{xx}(x,t) = 0,\ (x,t)\in\partial\Omega\times\bar{J}, \end{equation}$

(1.2)

and initial condition

$\begin{equation} u(x,0) = u_0(x),\ x\in\Omega, \end{equation}$

(1.3)

where $\Omega = (a, b)$ is the spatial domain, $J = (0, T]$ is the time interval with $T\in (0, \infty)$ , and $g(x, t)$ is a known source term function. The nonlinear term $f(u)$ satisfies the assumption condition $|f(u)|\le c_f(u)|u|$ , where $c_f(u)$ is a positive constant on $u$ . $_{0}^{C}\!D_t^\alpha u$ and $_{0}^{C}\!D_t^\beta u$ are both Caputo fractional derivatives with $0 < \alpha, \beta < 1$ . Since $_{0}^{C}\!D_t^\gamma u = \frac{\partial^{\gamma}(u-u_0)}{\partial t^{\gamma}}$ , all of the above Caputo fractional derivatives can be converted into the Riemann-Liouville fractional derivative, note that

$\begin{equation} \frac{\partial^{\gamma} u}{\partial t^{\gamma}} = \frac{1}{\Gamma(1-\gamma)}\frac{\partial}{\partial t}\displaystyle {\int}_0^t\frac{u(x,s)}{(t-s)^{\gamma}}\mathrm{d}s,\; 0 < \gamma < 1. \end{equation}$

(1.4)

Specifically, when $\alpha = 1$ , $\beta = 1$ , (1.1) degenerates into the generalized Rosenau-RLW-Burgers equation which can be seen as the combined system between the generalized Rosenau-RLW equation and the generalized Rosenau-Burgers equation.

The RLW equation, the Rosenau equation, and their combined systems with other equations are significant mathematical and physical equations that effectively describe nonlinear wave behaviors. These equations have become interesting topics in the study of nonlinear dispersion dynamics. Since obtaining analytical solutions for these equations is challenging, studying their numerical methods is paramount. Over the years, there has been extensive research on numerical methods for solving this type of equation. In ^[1], Atouani and Omrani discussed the numerical solution of the Rosenau-RLW (RRLW) equation based on the Galerkin finite element method. In ^[2], He and Pan developed a three-level, linearly implicit finite difference method for solving the generalized Rosenau-Kawahara-RLW equation. In ^[3], Wongsaijai and Poochinapan developed a pseudo-compact finite difference scheme for solving the generalized Rosenau-RLW-Burgers equation. In ^[4], Mouktonglang et al. analyzed a generalized Rosenau-RLW-Burgers equation with periodic initial-boundary value. For more papers on related equations, please refer to ^[5,6,7,8]. It is worth noting that the literature on the fractional generalized Rosenau-RLW-Burgers equation is relatively scarce, and its analytical solution is difficult to obtain. Therefore, we have to consider effective numerical methods such as finite element methods ^[9,10,11,12], finite difference methods ^[13,14,15], finite volume methods ^[16], spectral methods ^[17,18,19], and mixed finite element methods ^[20,21,22]. In addition, the existence of time-fractional derivatives increases the difficulty of studying numerical methods. Therefore, it is crucial to choose an appropriate high-order approximation formula for the fractional derivative to establish a stable numerical scheme for (1.1).

In 1986, Lubich ^[23] proposed the convolution quadrature (CQ) formula for Riemann Liouville fractional operators using the discrete convolution. In ^[24], Chen et al. developed an alternating direction implicit fractional trapezoidal rule type to solve a two-dimensional fractional evolution equation. In ^[25], Jin et al. proposed a corrected approximation formula for high-order BDFs through appropriate initial modifications to discretize fractional evolution equations. Based on the CQ formula, in ^[26], Liu et al. developed the shifted convolution quadrature (SCQ) theory, which extended the CQ formula at $x_{n-\theta}$ and discussed the constraints of parameter $\theta$ . In ^[27], Yin et al. studied the generalized BDF2- $\theta$ with the finite element method for solving the fractional mobile/immobile transport model, and also developed a correction scheme by adding the starting part to restore convergence order. For more related papers, please refer to ^{[28,29,30,31,32,33]}.

In this article, we develop the generalized BDF2- $\theta$ in time combined with the mixed finite element method in space to solve (1.1). The focuses of this article are as follows:

● It is noted that the time-fractional generalized Rosenau-RLW-Burgers equation containing two time-fractional operators is studied.

● The stability of the time-fractional generalized Rosenau-RLW-Burgers equation (1.1) based on the mixed finite element method is given.

● Based on a comprehensive analysis of some numerical examples, the numerical method's feasibility and effectiveness have been extensively validated. Specifically, the issue of decreasing the convergence rate of nonsmooth solutions is solved by adding correction terms.

The structure of this article is as follows: In Section 2, the generalized BDF2- $\theta$ is introduced, and the fully discrete mixed finite element scheme is provided. In Section 3, the existence and uniqueness theorem for the fully discrete mixed finite element scheme is given. In Section 4, the stability of the scheme is proved. In Section 5, some numerical examples with smooth and nonsmooth solutions based on the discrete scheme are presented. In Section 6, some conclusions are given.

2. Numerical scheme

In this section, we present the fully discrete mixed finite element scheme for (1.1) in space, which combines the generalized BDF2- $\theta$ in time. The generalized BDF2- $\theta$ with the starting part is introduced in ^[27]. Further, we divide the time interval $[0, T]$ into $0 = t_0 < t_1 < \cdots < t_{N-1} < t_{N} = T$ , and let $t_{n} = n\tau\; (n = 1, 2, \cdots, N)$ , where $\tau$ is time step length size and $N$ is a positive integer.

For the convenience of research, set $\hat{u}: = u-u_0$ , and assume that $\hat{u}$ has the following form:

$\begin{equation} \hat{u}(x,t) = \hat{u}_1(x,t)+\hat{u}_2(x,t): = \sum\limits_{j = 1}^{\kappa}c_jt^{\sigma_j}+t^{\sigma_{\kappa+1}}\phi(x,t), \end{equation}$

(2.1)

where $c_j = c(x)$ , $1 < \sigma_1 < \sigma_2 < \cdots < \sigma_{\kappa} < \sigma_{\kappa+1}$ and $\phi(x, t)$ is sufficiently differentiable with respect to $t$ .

Using $\hat{u}: = u-u_0$ , we can write (1.1)–(1.3) as

$\begin{equation} \hat{u}_t-\frac{\partial^{\alpha} \hat{u}_{xx}}{\partial t^{\alpha}}+\frac{\partial^{\beta} \hat{u}_{xxxx}}{\partial t^{\beta}}+\hat{u}_x-\hat{u}_{xx} +f(\hat{u})_x = \hat{g}(x,t),\; (x,t)\in\Omega\times J, \end{equation}$

(2.2)

with boundary conditions

$\begin{equation} \hat{u}(x,t) = \hat{u}_{xx}(x,t) = 0,\; (x,t)\in\partial\Omega\times\bar{J}, \end{equation}$

(2.3)

and initial condition

$\begin{equation} \hat{u}(x,0) = 0,\; x\in\Omega, \end{equation}$

(2.4)

where $\hat{g}(x, t) = g(x, t)+(u_0)_x-(u_0)_{xx}$ .

Now, we introduce an auxiliary variable $q = \hat{u}_{xx}$ to obtain the following coupled system:

$\begin{equation} \hat{u}_t-\frac{\partial^{\alpha} \hat{u}_{xx}}{\partial t^{\alpha}}+\frac{\partial^{\beta} q_{xx}}{\partial t^{\beta}}+\hat{u}_x-\hat{u}_{xx} +f(\hat{u})_x = \hat{g}(x,t), \end{equation}$

(2.5)

and

$\begin{equation} q = \hat{u}_{xx}. \end{equation}$

(2.6)

Multiplying (2.5) and (2.6) by $v \in H_0^1$ and $w \in H_0^1$ , respectively, integrating the result equations, and using integration by parts, we obtain the following weak form:

$\begin{equation} (\hat{u}_t,v)+(\frac{\partial^{\alpha} \hat{u}_{x}}{\partial t^{\alpha}},v_x)-(\frac{\partial^{\beta} q_{x}}{\partial t^{\beta}},v_x)-(\hat{u},v_x)+(\hat{u}_{x},v_x)-(f(\hat{u}),v_x) = (\hat{g},v), \; \forall v\in H_0^1, \end{equation}$

(2.7)

and

$\begin{equation} (q,w)+(\hat{u}_{x},w_x) = 0,\; \forall w\in H_0^1. \end{equation}$

(2.8)

To provide the fully discrete numerical scheme, we first introduce the relevant formulas and lemmas for the generalized BDF2- $\theta$ .

For smooth functions $\hat{u}$ and $q$ in $[0, T]$ , we let $\hat{u}^n = \hat{u}(\cdot, t_n)$ , $q^n = (\cdot, t_n)$ . The approximation formula for the Riemann-Liouville fractional derivative at time $t_{n-\theta}$ with the generalized BDF2- $\theta$ is

$\begin{equation} \begin{split} \frac{\partial^{\gamma} \hat{u}^{n-\theta}}{\partial t^{\gamma}}& = \tau^{-\gamma}\sum\limits_{j = 0}^{n}\omega^{(\gamma)}_j \hat{u}^{n-j}+\tau^{-\gamma}\sum\limits_{j = 1}^{\kappa}\omega^{(\gamma)}_{n,j} \hat{u}^{j}+R^{n-\theta}_\gamma\\ &: = \Psi_\tau^{\gamma,n}\hat{u}+S_{\tau,\kappa}^{\gamma,n}\hat{u}+R^{n-\theta}_\gamma, \end{split} \end{equation}$

(2.9)

where $|R^{n-\theta}_\gamma|\leq C\tau^2$ .

The discrete convolution part is denoted as

$\begin{equation} \Psi_\tau^{\gamma,n}\hat{u}: = \tau^{-\gamma}\sum\limits_{j = 0}^{n}\omega^{(\gamma)}_j \hat{u}^{n-j}, \end{equation}$

(2.10)

and the starting part is

$\begin{equation} S_{\tau,\kappa}^{\gamma,n}\hat{u}: = \tau^{-\gamma}\sum\limits_{j = 1}^{\kappa}\omega^{(\gamma)}_{n,j} \hat{u}^{j}. \end{equation}$

(2.11)

The convolution weights $\{\omega^{(\gamma)}_{j}\}^{\infty}_{j = 0}$ in (2.10) are generated by the following generating function:

$\begin{equation} \omega^{(\gamma)}(\xi ) = \left(\frac{3\gamma-2\theta}{2\gamma} -\frac{2\gamma-2\theta}{\gamma}\xi+\frac{\gamma-2\theta}{2\gamma}\xi^2 \right)^\gamma. \end{equation}$

(2.12)

Lemma 2.1. ^[27] We give the convolution weights $\{\omega^{(\gamma)}_j\}_{j = 0}^{\infty}$ of the generalized BDF2- $\theta$ as follows:

$\begin{equation} \begin{split} &\omega_0^{(\gamma)} = \left(\frac{3\gamma-2\theta}{2\gamma}\right)^{\gamma},\quad \omega_1^{(\gamma)} = 2(\theta-\gamma)\left(\frac{2\gamma}{3\gamma-2\theta}\right)^{1-\gamma},\\ &\omega_j^{(\gamma)} = \frac{2\gamma}{j(3\gamma-2\theta)}\left[2(\gamma-\theta)\left(\frac{j-1}{\gamma}-1\right)\omega_{j-1}^{(\gamma)}+(\gamma-2\theta)\left(1-\frac{j-2}{2\gamma}\right)\omega_{j-2}^{(\gamma)}\right],\quad j\geq2. \end{split} \end{equation}$

(2.13)

Lemma 2.2. ^[27] The starting weights $\{\omega^{(\gamma)}_{n, j}\}_{j = 1}^{\kappa}$ of the generalized BDF2- $\theta$ are given as the following:

$\begin{equation} \begin{split} \sum\limits_{j = 1}^{\kappa}\omega^{(\gamma)}_{n,j}j^{\ell} = \frac{\Gamma(\ell+1)}{\Gamma(\ell-\gamma+1)}(n-\theta)^{\ell-\gamma}-\sum\limits_{j = 1}^{n}\omega^{(\gamma)}_{n-j}j^{\ell},\quad \ell = \sigma_1,\sigma_2,\cdots,\sigma_{\kappa}. \end{split} \end{equation}$

(2.14)

Lemma 2.3. ^[12,15] For $\hat{u}\in C^4[0, \pi]$ , the following two approximate formulas at $t_{n-\theta}$ hold:

$\begin{equation} \begin{split} g(t_{n-\theta})& = g^{n-\theta}+O(\tau^2),\\ f(\hat{u}(t_{n-\theta}))& = f(\hat{u}^{n-\theta})+O(\tau^2), \end{split} \end{equation}$

(2.15)

where $g^{n-\theta}: = (1-\theta)g^{n}+\theta g^{n-1}$ and $f(\hat{u}^{n-\theta}): = (2-\theta)f(\hat{u}^{n-1})-(1-\theta)f(\hat{u}^{n-2})$ .

Next, we have the following approximate formula:

$\begin{equation} \begin{split} \hat{u}(t_{n-\theta})& = \hat{u}^{n-\theta}+S_{\tau,\kappa}^{0,n}\hat{u}+O(\tau^2)\\ &: = (1-\theta)\hat{u}^{n}+\theta \hat{u}^{n-1}+S_{\tau,\kappa}^{0,n}\hat{u}+O(\tau^2). \end{split} \end{equation}$

(2.16)

Without considering the starting part, we can obtain the weak form of (2.5) and (2.6) at $t_{n-\theta}$ :

$\begin{equation} \begin{split} &(\Psi_{\tau}^{1,n} \hat{u},v)+(\Psi_{\tau}^{\alpha,n} \hat{u}_{x},v_x)-(\Psi_{\tau}^{\beta,n}q_{x},v_x)-(\hat{u}^{n-\theta},v_x)+(\hat{u}^{n-\theta}_{x},v_x) \\ = &(f(\hat{u}^{n-\theta}),v_x)+(\hat{g}^{n-\theta},v)-(R_1^{n-\theta},v), \end{split} \end{equation}$

(2.17)

and

$\begin{equation} (q^{n-\theta},w)+(\hat{u}^{n-\theta}_{x},w_x) = -(R_2^{n-\theta},w), \end{equation}$

(2.18)

where $R_1^{n-\theta} = O(\tau^2)$ and $R_2^{n-\theta} = O(\tau^2)$ .

To establish the fully discrete mixed finite element scheme, we introduce the following finite element space:

$V_h = \{v_h|v_h\in H_0^1,v_h|_{I_i}\in P_k(I_i), \; \forall I_i\in \mathcal{T} _h,k\geq 1 \},$

where $\mathcal{T}_h$ is a subdivision of $\bar{\Omega} = [a, b]$ into $M$ subintervals $I_i = [x_{i-1}, x_{i}]$ , with $h_i = x_{i}-x_{i-1}$ , $h = \mathop{\max}\limits_{1\leq i\leq M}h_i$ , and $P_k(I_i)$ represent the polynomials with a degree less than or equal to $k$ in $I_i$ .

Next, we provide linear basis functions $\{\varphi_i\}_{i = 1}^{M}$ of finite element space $V_h$ as follows:

$\begin{equation} \begin{aligned} \varphi_i(x) = \left\{\begin{matrix} 1+\frac{x-x_i}{h_i},& x\in I_i,\\ 1-\frac{x-x_i}{h_{i+1}}, & x\in I_{i+1},\\ 0, & \text{others}, \end{matrix}\right. \end{aligned} \end{equation}$

(2.19)

$\begin{equation} \begin{aligned} \varphi_M(x) = \left\{\begin{matrix} 1+\frac{x-x_M}{h_M},& x\in I_M,\\ 0, & \text{others}. \end{matrix}\right. \end{aligned} \end{equation}$

(2.20)

Based on the above finite element space, we find $\{U^{n-\theta}, Q^{n-\theta}\}\in V_h\times V_h$ satisfying

$\begin{equation} \begin{split} &(\Psi_{\tau}^{1,n} U,V)+(\Psi_{\tau}^{\alpha,n} U_{x},V_x)-(\Psi_{\tau}^{\beta,n}Q_{x},V_x)-(U^{n-\theta},V_x)+(U_{x}^{n-\theta },V_x)\\ = &(f(U^{n-\theta}),V_x) +(\hat{g}^{n-\theta},V),\; \forall V\in V_h, \end{split} \end{equation}$

(2.21)

and

$\begin{equation} (Q^{n-\theta},W)+(U_{x}^{n-\theta},W_x) = 0,\; \forall W\in V_h . \end{equation}$

(2.22)

3. Existence and uniqueness of numerical solution

Theorem 3.1. The solution of the fully discrete mixed finite element scheme (2.21) and (2.22) is uniquely solvable.

Proof. Taking basis functions $\{\varphi_i\}_{i = 1}^{M}$ of finite element space $V_h$ , we have

$\begin{equation} \begin{split} U^{n} = \sum\limits_{i = 1}^{M}u_i^{n}\varphi_i,\quad Q^{n} = \sum\limits_{i = 1}^{M}q_i^{n}\varphi_i. \end{split} \end{equation}$

(3.1)

Taking $V = \varphi_j$ and $W = \varphi_j$ from (2.21) and (2.22), we have

$\begin{equation} \begin{split} &\tau^{-1}\omega_0^{(1)}A \mathbf{U^n}+\tau^{-\alpha}\omega_0^{(\alpha)}B \mathbf{U^n}+(1-\theta)B\mathbf{U^n}-(1-\theta)C\mathbf{U^n}-\tau^{-\beta}\omega_0^{(\beta)}B\mathbf{Q^n}\\ = &\mathbf{F}^{n-\theta}+\mathbf{G}^{n-\theta}-\tau^{-1}\sum\limits_{k = 1}^{n} \omega_{k}^{(1)}A \mathbf{U^{n-k}}-\tau^{-\alpha}\sum\limits_{k = 1}^{n} \omega_{k}^{(\alpha)}B \mathbf{U^{n-k}}\\ &-\theta B \mathbf{U^{n-1}}+\theta C \mathbf{U^{n-1}}+\tau^{-\beta}\sum\limits_{k = 1}^{n} \omega_{k}^{(\beta)}B \mathbf{Q^{n-k}}, \end{split} \end{equation}$

(3.2)

and

$\begin{equation} \begin{split} &(1-\theta)B\mathbf{U^{n}}+(1-\theta)A\mathbf{Q^{n}} = -\theta B\mathbf{U^{n}}-\theta A\mathbf{Q^{n}}, \end{split} \end{equation}$

(3.3)

where

$\begin{align*} &A = \left[(\varphi_i,\varphi_j)\right]^T_{1\le i,j\le M},\; B = \left[(\varphi_{ix},\varphi_{jx})\right]^T_{1\le i,j\le M},\; C = \left[(\varphi_{i},\varphi_{jx})\right]^T_{1\le i,j\le M},\\ &\mathbf{F^{n-\theta}} = \left[\left(f(U^{n-\theta}),\varphi_{1x}\right),\cdots,\left(f(U^{n-\theta}),\varphi_{Mx}\right)\right]^T,\; \mathbf{G^{n-\theta}} = \left[\left(g^{n-\theta},\varphi_{1}\right),\cdots,\left(g^{n-\theta},\varphi_{M}\right)\right]^T. \end{align*}$

Obviously, $A$ and $B$ are symmetric and positive definite. Further, processing the boundary and simplifying the right-hand term, we have

$\begin{equation} \left(\tau^{-1}\omega_0^{(1)}\widetilde{A}+\tau^{-\alpha}\omega_{0}^{(\alpha)}\widetilde{B}+(1-\theta)\widetilde{B}-(1-\theta)\widetilde{C}\right)\mathbf{U}^{n}-\tau^{-\beta}\omega_0^{(\beta)}\widetilde{B}\mathbf{Q}^{n} = \mathbf{H}_1^n, \end{equation}$

(3.4)

and

$\begin{equation} (1-\theta)\widetilde{B}\mathbf{U}^n+(1-\theta)\widetilde{A}\mathbf{Q}^n = \mathbf{H}_2^n, \end{equation}$

(3.5)

where

$\begin{align*} \mathbf{H}_1^n = &\mathbf{F}^{n-\theta}+\mathbf{G}^{n-\theta}-\tau^{-1}\sum\limits_{k = 1}^{n} \omega_{k}^{(1)}A \mathbf{U^{n-k}}-\tau^{-\alpha}\sum\limits_{k = 1}^{n} \omega_{k}^{(\alpha)}B \mathbf{U^{n-k}}-\theta B \mathbf{U^{n-1}}\\ &+\theta C \mathbf{U^{n-1}}+\tau^{-\beta}\sum\limits_{k = 1}^{n} \omega_{k}^{(\beta)}B \mathbf{Q^{n-k}},\\ \mathbf{H}_2^n = &-\theta B\mathbf{U^{n}}-\theta A\mathbf{Q^{n}}. \end{align*}$

Multiplying (3.4) by $\tau \widetilde{A}^{-1}$ , we have

$\begin{equation} \begin{split} \left(\omega_0^{(1)}E+\tau^{1-\alpha}\omega_{0}^{(\alpha)}\widetilde{A}^{-1}\widetilde{B}+\tau(1-\theta)\widetilde{A}^{-1}\widetilde{B}-\tau(1-\theta)\widetilde{A}^{-1}\widetilde{C}\right)\mathbf{U}^{n} -\tau^{1-\beta}\omega_0^{(\beta)}\widetilde{A}^{-1}\widetilde{B}\mathbf{Q}^{n} = \tau\widetilde{A}^{-1}\mathbf{H}_1^n. \end{split} \end{equation}$

(3.6)

Further, rewrite (3.5) as

$\begin{equation} \begin{split} \mathbf{Q}^{n} = \mathbf{H}_3^n, \end{split} \end{equation}$

(3.7)

where $\mathbf{H}_3^n = (1-\theta)^{-1}\widetilde{A}^{-1}\mathbf{H}_2^n-\widetilde{A}^{-1}\widetilde{B}\mathbf{U}^{n}$ .

Substitute (3.7) into (3.6) to obtain

$\begin{equation} \begin{split} &\mathbf{K}\mathbf{U}^{n} = \mathbf{H}_4^n, \end{split} \end{equation}$

(3.8)

where

$\mathbf{K} = \omega_0^{(1)}E+\tau^{1-\alpha}\omega_{0}^{(\alpha)}\widetilde{A}^{-1}\widetilde{B}+\tau(1-\theta)\widetilde{A}^{-1}\widetilde{B} -\tau(1-\theta)\widetilde{A}^{-1}\widetilde{C}+\tau^{1-\beta}\omega_0^{(\beta)}\widetilde{A}^{-1}\widetilde{B}\widetilde{A}^{-1}\widetilde{B},$

$\mathbf{H}_4^n = \tau \widetilde{A}^{-1}\mathbf{H}_1^n+\tau^{1-\beta}(1-\theta)^{-1}\omega_0^{(\beta)}\widetilde{A}^{-1}\widetilde{B}\widetilde{A}^{-1}\mathbf{H}_2^n.$

It is easy to see that (3.7) and (3.8) are equivalent to (3.4) and (3.5). Due to $\tau$ being small enough and $E$ being an identity matrix, the matrix $\mathbf{K}$ is invertible. Additionally, since $U^k\; (k = 0, 1, \cdots, n-1)$ is known, after multiple iterations, (3.7) and (3.8) have a unique solution. □

Remark 3.1. Since we introduce the auxiliary variable $q = \hat{u}_{xx}$ to transform (2.2) into a first-order system (2.5) and (2.6), according to ^[34,35], the mixed finite element scheme (2.21) and (2.22) do not need to satisfy the LBB condition. In ^[36], the LBB condition is a condition for the problem to be well posed. From this perspective, typically satisfying the LBB condition is to obtain the existence and uniqueness of a solution. Although the mixed finite element scheme in this article does not need to satisfy the LBB condition, it still satisfies the existence and uniqueness of a solution.

4. Stability

Lemma 4.1. ^[12,14] For $\forall U^m\in V_h$ , satisfying $U^m = 0\; (m < 0)$ , we have

$(\Psi_t^{1,m} U,U^{m-\theta})\ge \frac{1}{4\tau}(\mathbb{H}[U^m]-\mathbb{H}[U^{m-1}]),\ m\ge 1,$

where

$\mathbb{H}[U^m] = (3-2\theta)\|U^{m}\|^2-(1-2\theta)\|U^{m-1}\|^2+(2-\theta)(1-2\theta)\|U^m-U^{m-1}\|^2,$

and

$\mathbb{H}[U^m]\ge\frac{1}{1-\theta}\|U^m\|^2,\ m\ge1.$

Lemma 4.2. ^[27] } {For any vector $(v^0, v^1, \cdots, v^{n-1})\in \mathbb{R}^n$ , defining $\{\omega_k^{(\gamma)}\}_{k = 0}^{\infty}\; (0 < \gamma < 1)$ be a sequence of coefficients of the generating function $\omega^{(\gamma)}(\xi)$ in (2.12) and $0\le\theta\le\min\{\gamma, \frac{1}{2}\}$ , we have

$\sum\limits_{m = 1}^{n-1}v^m\sum\limits_{k = 1}^{m}\omega_{m-k}^{(\gamma)} v^k\ge0,\ n\ge1.$

Theorem 4.1. Let $u^n_h = U^n+\bar{u}_h^0$ , where $\bar{u}_h^0$ is an approximation of $u_0$ , the following stability of the fully discrete scheme (2.21) and (2.22) holds:

$\begin{equation} \|u_h^L\|^2\ \le\ C\left(\|\bar{u}_h^0 \|^2+\tau\sum\limits_{n = 1}^{L}\|g^{n-\theta} \|^2\right),\ 1\le L \le N, \end{equation}$

(4.1)

where $C$ is a positive constant independent of $h$ and $\tau$ .

Proof. Taking $V = U^{n-\theta}$ , $W = \Psi_{\tau}^{\beta, n}Q$ , (2.21) and (2.22) can be written as

$\begin{equation} \begin{split} &(\Psi_{\tau}^{1,n} U,U^{n-\theta})+(\Psi_{\tau}^{\alpha,n} U_{x},U^{n-\theta}_x)-(\Psi_{\tau}^{\beta,n}Q_{x},U^{n-\theta}_x)+\|U_{x}^{n-\theta }\|^2\\ = &(U^{n-\theta},U^{n-\theta}_x)+(f(U^{n-\theta}),U^{n-\theta}_x)+(\hat{g}^{n-\theta},U^{n-\theta}), \end{split} \end{equation}$

(4.2)

and

$\begin{equation} (Q^{n-\theta},\Psi_{\tau}^{\beta,n}Q)+(U^{n-\theta}_{x},\Psi_{\tau}^{\beta,n}Q_x) = 0. \end{equation}$

(4.3)

Adding (4.2) and (4.3), we have

$\begin{equation} \begin{split} &(\Psi_{\tau}^{1,n} U,U^{n-\theta})+(\Psi_{\tau}^{\alpha,n} U_{x},U^{n-\theta}_x)+(Q^{n-\theta},\Psi_{\tau}^{\beta,n}Q)+\|U_{x}^{n-\theta }\|^2\\ = &(U^{n-\theta},U^{n-\theta}_x)+(f(U^{n-\theta}),U^{n-\theta}_x)+(\hat{g}^{n-\theta},U^{n-\theta}). \end{split} \end{equation}$

(4.4)

Using Lemma 4.1, we obtain

$\begin{equation} \begin{split} &\frac{1}{4\tau}(\mathbb{H}[U^n]-\mathbb{H}[U^{n-1}])+(\Psi_{\tau}^{\alpha,n} U_{x},U^{n-\theta}_x)+(Q^{n-\theta},\Psi_{\tau}^{\beta,n}Q)+\|U_{x}^{n-\theta }\|^2\\ \le&(U^{n-\theta},U^{n-\theta}_x)+(f(U^{n-\theta}),U^{n-\theta}_x)+(\hat{g}^{n-\theta},U^{n-\theta}). \end{split} \end{equation}$

(4.5)

Multiply (4.5) by $4\tau$ and sum it with respect to $n$ from $1$ to $L$ to get

$\begin{equation} \begin{split} &\mathbb{H}[U^L]-\mathbb{H}[U^{0}]+ 4\tau\sum\limits_{n = 1}^{L}(\Psi_t^{\alpha,n} U_{x},U^{n-\theta}_x)+ 4\tau\sum\limits_{n = 1}^{L}(Q^{n-\theta},\Psi_{\tau}^{\beta,n}Q)+ 4\tau\sum\limits_{n = 1}^{L}\|U_{x}^{n-\theta }\|^2\\ \le&4\tau\left(\sum\limits_{n = 1}^{L}(U^{n-\theta},U^{n-\theta}_x)+\sum\limits_{n = 1}^{L}(f(U^{n-\theta}),U^{n-\theta}_x)+\sum\limits_{n = 1}^{L}(\hat{g}^{n-\theta},U^{n-\theta})\right). \end{split} \end{equation}$

(4.6)

By the Hölder inequality and Young inequality, the three terms on the right-hand side of (4.6) can be expanded to

$\begin{equation} \sum\limits_{n = 1}^{L}(U^{n-\theta},U^{n-\theta}_x)\le\frac{1}{2}\sum\limits_{n = 1}^{L}\|U^{n-\theta}\|^2+\frac{1}{2}\sum\limits_{n = 1}^{L}\|U_{x}^{n-\theta }\|^2, \end{equation}$

(4.7)

$\begin{equation} \begin{split} \sum\limits_{n = 1}^{L}(f(U^{n-\theta}),U^{n-\theta}_x) \le&\sum\limits_{n = 1}^{L}\|c_f(U^{n-\theta})\|_{\infty}\|U^{n-\theta}\|\|U^{n-\theta}_x\| \\ \le& C\sum\limits_{n = 1}^{L}\|U^{n-\theta}\|^2+\frac{1}{2}\sum\limits_{n = 1}^{L}\|U_{x}^{n-\theta }\|^2, \end{split} \end{equation}$

(4.8)

$\begin{equation} \sum\limits_{n = 1}^{L}(\hat{g}^{n-\theta},U^{n-\theta})\le\frac{1}{2}\sum\limits_{n = 1}^{L}\|\hat{g}^{n-\theta}\|^2 +\frac{1}{2}\sum\limits_{n = 1}^{L}\|U^{n-\theta}\|^2, \end{equation}$

(4.9)

where we use the bounded condition $\|c_f(U^{n-\theta})\|_{\infty}\leq C$ .

Substituting (4.7)–(4.9) into (4.6), we arrive at

$\begin{equation} \begin{split} & \mathbb{H}[U^L]-\mathbb{H}[U^{0}]+ 4\tau\sum\limits_{n = 1}^{L}(\Psi_t^{\alpha,n} U_{x},U^{n-\theta}_x)+ 4\tau\sum\limits_{n = 1}^{L}(Q^{n-\theta},\Psi_{\tau}^{\beta,n}Q)\\ \le& C\tau \left(\sum\limits_{n = 1}^{L}\|\hat{g}^{n-\theta}\|^2+\sum\limits_{n = 1}^{L}\|U^{n-\theta}\|^2\right). \end{split} \end{equation}$

(4.10)

In what follows, using Lemmas 4.1 and 4.2 and the Gronwall inequality, we have

$\begin{equation} \|U^L\|^2-\|U^0\|^2\le C\tau\sum\limits_{n = 1}^{L}\|\hat{g}^{n-\theta}\|^2. \end{equation}$

(4.11)

Since $U^0 = 0$ , we obtain

$\begin{equation} \|U^L\|^2\le C\tau\sum\limits_{n = 1}^{L}\|\hat{g}^{n-\theta}\|^2. \end{equation}$

(4.12)

Noting that $U^L = u^L_h-\bar{u}_h^0$ and using the triangle inequality, the conclusion of this theorem is derived. □

5. Numerical results

In this section, we present numerical simulation results for both smooth and nonsmooth solutions to verify the effectiveness of the numerical scheme. Next, we set the nonlinear term $f(u) = u^2$ , the spatial domain $\Omega = (0, 1)$ , and the time interval $J = (0, 1]$ .

Example 5.1 The exact solution is $u(x, t) = t^{2}\sin(2\pi x)$ satisfying $u(x, 0) = 0$ , and the known source function $g(x, t)$ is given by

$\begin{equation} g(x,t) = \sin(2\pi x)\left(2t+\frac{8\pi ^2t^{2-\alpha}}{\Gamma(3-\alpha)}+\frac{32\pi^4t^{2-\beta}}{\Gamma(3-\beta)}+4\pi^2t^2\right)+2\pi t^2\cos(2\pi x)+2\pi t^4\sin(4\pi x ). \end{equation}$

(5.1)

In , fixing $\tau = 1/1000$ and choosing $h = 1/10, 1/20, 1/40, 1/80$ , we provide the $L^2$ -errors and the spatial convergence rates for $u$ and $q$ with different parameters $\alpha$ , $\beta$ , and $\theta$ , where $\theta\le\min\{\alpha, \beta, \frac{1}{2}\}$ . Similarly, in , taking $h = 1/1000$ , we calculate the $L^2$ -errors and the time convergence rates with $\tau = 1/10, 1/20, 1/40, 1/80$ . From and , one can see that the convergence rates in both space and time are close to 2 when the exact solution is smooth. In , if $\theta > \min\{\alpha, \beta, \frac{1}{2}\}$ , the convergence accuracy will be unstable, which verifies the range of $\theta$ values from a numerical perspective. To observe the effect of numerical simulation more clearly, we provide the comparison images between numerical solutions and exact solutions. In , we show distinct comparison images of the numerical solutions of $u_h$ and $q_h$ and the exact solutions of $u$ and $q$ with $\tau = 1/1000$ , $h = 1/80$ , $\alpha = 0.2$ , $\beta = 0.8$ , and $\theta = 0.2$ .

Table 1. Spatial convergence results with

$\tau = 1/1000$ .

$\alpha$	$\beta$	$\theta$	$h$	$\\|u_h-u\\|$	Rate	$\\|q_h-q\\|$	Rate
			1/10	2.2165E-02	-	2.6449E-02	-
		0.2	1/20	5.6375E-03	1.9752	6.1401E-03	2.1069
			1/40	1.4151E-03	1.9941	1.5121E-03	2.0217
			1/80	3.5399E-04	1.9992	3.8250E-04	1.9830
			1/10	2.2165E-02	-	2.6470E-02	-
0.2	0.8	-0.5	1/20	5.6369E-03	1.9753	6.1607E-03	2.1032
			1/40	1.4146E-03	1.9945	1.5327E-03	2.0070
			1/80	3.5346E-04	2.0008	4.0308E-04	1.9269
			1/10	2.2164E-02	-	2.6490E-02	-
		-1	1/20	5.6364E-03	1.9754	6.1810E-03	2.0996
			1/40	1.4141E-03	1.9949	1.5530E-03	1.9928
			1/80	3.5293E-04	2.0024	4.2346E-04	1.8747
			1/10	2.1828E-02	-	4.0463E-02	-
		0.5	1/20	5.5499E-03	1.9756	9.6844E-03	2.0629
			1/40	1.3932E-03	1.9941	2.3964E-03	2.0148
			1/80	3.4861E-04	1.9987	5.9910E-04	2.0000
			1/10	2.1828E-02	-	4.0465E-02	-
0.5	0.5	0.2	1/20	5.5499E-03	1.9757	9.6868E-03	2.0626
			1/40	1.3931E-03	1.9941	2.3988E-03	2.0137
			1/80	3.4854E-04	1.9989	6.0151E-04	1.9956
			1/10	2.1826E-02	-	4.0523E-02	-
		-1	1/20	5.5484E-03	1.9759	9.7445E-03	2.0561
			1/40	1.3916E-03	1.9953	2.4564E-03	1.9880
			1/80	3.4706E-04	2.0035	6.5921E-04	1.8977
			1/10	2.1399E-02	-	5.8275E-02	-
		0.2	1/20	5.4386E-03	1.9763	1.4195E-02	2.0375
			1/40	1.3651E-03	1.9943	3.5279E-03	2.0085
			1/80	3.4156E-04	1.9988	8.8248E-04	1.9992
			1/10	2.1399E-02	-	5.8276E-02	-
0.8	0.2	0	1/20	5.4386E-03	1.9763	1.4196E-02	2.0374
			1/40	1.3650E-03	1.9943	3.5290E-03	2.0082
			1/80	3.4153E-04	1.9989	8.8360E-04	1.9978
			1/10	2.1396E-02	-	5.8410E-02	-
		-1	1/20	5.4352E-03	1.9769	1.4329E-02	2.0272
			1/40	1.3616E-03	1.9970	3.6619E-03	1.9683
			1/80	3.3812E-04	2.0097	1.0167E-03	1.8487

| Show Table

DownLoad: CSV

Table 2. Time convergence results with

$h = 1/1000$ .

$\alpha$	$\beta$	$\theta$	$\tau$	$\\|u_h-u\\|$	Rate	$\\|q_h-q\\|$	Rate
			1/10	1.9614E-03	-	7.7510E-02	-
		0.2	1/20	5.0017E-04	1.9714	1.9814E-02	1.9679
			1/40	1.2703E-04	1.9773	5.0533E-03	1.9712
			1/80	3.2128E-05	1.9832	1.2867E-03	1.9736
			1/10	7.2565E-03	-	2.8650E-01	-
0.2	0.8	-0.5	1/20	1.8309E-03	1.9867	7.2335E-02	1.9858
			1/40	4.6032E-04	1.9919	1.8206E-02	1.9903
			1/80	1.1551E-04	1.9946	4.5778E-03	1.9917
			1/10	1.2197E-02	-	4.8151E-01	-
		-1	1/20	3.1215E-03	1.9662	1.2329E-01	1.9655
			1/40	7.8542E-04	1.9907	3.1091E-02	1.9875
			1/80	1.9579E-04	2.0042	7.8135E-03	1.9924
			1/10	5.3041E-04	-	2.1042E-02	-
		0.5	1/20	1.3386E-04	1.9863	5.3622E-03	1.9724
			1/40	3.3676E-05	1.9910	1.3637E-03	1.9753
			1/80	8.4582E-06	1.9933	3.4761E-04	1.9720
			1/10	1.1457E-03	-	4.5326E-02	-
0.5	0.5	0.2	1/20	2.8884E-04	1.9879	1.1461E-02	1.9836
			1/40	7.2679E-05	1.9907	2.8911E-03	1.9871
			1/80	1.8260E-05	1.9929	7.2972E-04	1.9862
			1/10	1.5622E-02	-	6.1668E-01	-
		-1	1/20	4.0111E-03	1.9615	1.5838E-01	1.9611
			1/40	1.0095E-03	1.9904	3.9868E-02	1.9901
			1/80	2.5362E-04	1.9929	1.0017E-02	1.9928
			1/10	5.2861E-04	-	2.0885E-02	-
		0.5	1/20	1.3465E-04	1.9730	5.3245E-03	1.9717
			1/40	3.4107E-05	1.9811	1.3530E-03	1.9765
			1/80	8.5857E-06	1.9901	3.4270E-04	1.9812
			1/10	2.2763E-03	-	8.9851E-02	-
0.8	0.5	0	1/20	5.7402E-04	1.9875	2.2660E-02	1.9874
			1/40	1.4460E-04	1.9890	5.7086E-03	1.9889
			1/80	3.6405E-05	1.9899	1.4372E-03	1.9898
			1/10	1.5534E-02	-	6.1313E-01	-
		-1	1/20	3.9902E-03	1.9609	1.5749E-01	1.9609
			1/40	1.0033E-03	1.9917	3.9605E-02	1.9916
			1/80	2.5177E-04	1.9946	9.9391E-03	1.9945

| Show Table

DownLoad: CSV

Table 3. Time convergence results with

$h = 1/1000$ .

$\alpha$	$\beta$	$\theta$	$\tau$	$\\|u_h-u\\|$	Rate	$\\|q_h-q\\|$	Rate
			1/10	2.4519E-03	-	9.6803E-02	-
0.1	0.9	0.11	1/20	6.2095E-04	1.9813	2.4519E-02	1.9811
			1/40	5.2080E-04	0.2538	2.0647E-02	0.2480
			1/80	4.0699E+02	-19.5758	1.6068E+04	-19.5699
			1/10	1.8093E-03	-	7.1425E-02	-
0.5	0.5	0.51	1/20	4.5109E-04	2.0039	1.7808E-02	2.0039
			1/40	2.0018E-04	1.1721	7.8154E-03	1.1881
			1/80	5.1099E-04	-1.3520	2.0090E-02	-1.3621
			1/10	4.8552E-04	-	1.9119E-02	-
0.8	0.2	0.21	1/20	1.2438E-04	1.9647	4.8398E-03	1.9820
			1/40	5.8298E-05	1.0933	2.2123E-03	1.1294
			1/80	2.3759E-02	-8.6708	9.3790E-01	-8.7277

| Show Table

DownLoad: CSV

Figure 1.

$u_h$ ,

$q_h$ and

$u$ ,

$q$ with

$\tau = 1/1000$ ,

$h = 1/80$ ,

$\alpha = 0.2$ ,

$\beta = 0.8$ ,

$\theta = 0.2$ .

DownLoad: Full-Size Img PowerPoint

Example 5.2 In this example, we consider the case where the nonsmooth solution is taken as $u = (t^{\alpha+\beta}+t^3) \sin(2\pi x)$ , and the known source term $g(x, t)$ is

$\begin{equation} \begin{split} g(x,t) = &\sin(2\pi x)\left[(\alpha+\beta)t^{\alpha+\beta-1}+3t^2+4\pi^2\left(\frac{t^{\beta}\Gamma(\alpha+\beta+1)}{\Gamma(\beta+1)}+\frac{6t^{3-\alpha}}{\Gamma(4-\alpha)}\right)\right]\\ &+\sin(2\pi x)\left[16\pi^4\left(\frac{t^{\alpha}\Gamma(\alpha+\beta+1)}{\Gamma(\alpha+1)}+\frac{6t^{3-\beta}}{\Gamma(4-\beta)}\right)+4\pi^2(t^{\alpha+\beta}+t^3)\right]\\ &+2\pi (t^{\alpha+\beta}+t^3)\cos(2\pi x)+2\pi(t^{\alpha+\beta} +t^3)^2\sin(4\pi x ). \end{split} \end{equation}$

(5.2)

presents the $L^2$ -errors and the spatial convergence rates of $u$ and $q$ before and after adding the starting parts with $h = 1/10, 1/20, 1/40, 1/80$ , $\tau = 1/2000$ , where Error_o denotes the error before adding the starting parts and Error_c denotes the error after adding the starting parts.

Table 4. Spatial convergence results with

$\alpha = 0.9$ ,

$\beta = 0.2$ ,

$\tau = 1/2000$ .

		$\\|u_h-u\\|$				$\\|q_h-q\\|$
$\theta$	$h$	Error_o	Rate	Error_c	Rate	Error_o	Rate	Error_c	Rate
	1/10	4.2694E-02	-	4.2693E-02	-	1.2162E-01	-	1.2163E-01	-
0.2	1/20	1.0850E-02	1.9763	1.0850E-02	1.9763	2.9675E-02	2.0351	2.9679E-02	2.0349
	1/40	2.7235E-03	1.9942	2.7234E-03	1.9942	7.3708E-03	2.0094	7.3746E-03	2.0088
	1/80	6.8165E-04	1.9984	6.8155E-04	1.9985	1.8361E-03	2.0051	1.8400E-03	2.0029
	1/10	4.2693E-02	-	4.2692E-02	-	1.2165E-01	-	1.2168E-01	-
-0.5	1/20	1.0849E-02	1.9764	1.0849E-02	1.9764	2.9706E-02	2.0340	2.9728E-02	2.0331
	1/40	2.7227E-03	1.9945	2.7221E-03	1.9947	7.4017E-03	2.0048	7.4237E-03	2.0016
	1/80	6.8085E-04	1.9996	6.8028E-04	2.0005	1.8671E-03	1.9871	1.8891E-03	1.9744
	1/10	4.2691E-02	-	4.2690E-02	-	1.2172E-01	-	1.2177E-01	-
-1	1/20	1.0848E-02	1.9766	1.0846E-02	1.9767	2.9776E-02	2.0314	2.9822E-02	2.0297
	1/40	2.7209E-03	1.9952	2.7197E-03	1.9957	7.4714E-03	1.9947	7.5178E-03	1.9880
	1/80	6.7905E-04	2.0025	6.7786E-04	2.0044	1.9368E-03	1.9477	1.9833E-03	1.9224

| Show Table

DownLoad: CSV

The spatial convergence rate is almost unaffected before and after correction, based on a comparison of the data in . In and , we present the $L^2$ -errors and the time convergence rates of $u$ and $q$ before and after adding the starting parts. Without the addition of the starting parts, the time convergence rates are unstable and cannot reach the second-order convergence results computed by the generalized BDF2- $\theta$ . After adding the starting parts, the time convergence rates keep around $2$ , indicating that the starting part plays a major role in correcting the time convergence rates.

Table 5. Spatial convergence results with

$\alpha = 0.5$ ,

$\beta = 0.7$ ,

$\tau = 1/2000$ .

		$\\|u_h-u\\|$				$\\|q_h-q\\|$
$\theta$	$h$	Error_o	Rate	Error_c	Rate	Error_o	Rate	Error_c	Rate
	1/10	4.3949E-02	-	4.3948E-02	-	6.8838E-02	-	6.8891E-02	-
0.5	1/20	1.1177E-02	1.9753	1.1176E-02	1.9754	1.6266E-02	2.0814	1.6318E-02	2.0778
	1/40	2.8070E-03	1.9934	2.8057E-03	1.9940	3.9696E-03	2.0348	4.0219E-03	2.0205
	1/80	7.0358E-04	1.9963	7.0222E-04	1.9984	9.4708E-04	2.0674	9.9935E-04	2.0088
	1/10	4.3949E-02	-	4.3948E-02	-	6.8842E-02	-	6.8897E-02	-
0.2	1/20	1.1177E-02	1.9753	1.1176E-02	1.9754	1.6270E-02	2.0811	1.6324E-02	2.0774
	1/40	2.8069E-03	1.9935	2.8055E-03	1.9940	3.9739E-03	2.0336	4.0279E-03	2.0189
	1/80	7.0346E-04	1.9964	7.0206E-04	1.9986	9.5140E-04	2.0624	1.0054E-03	2.0023
	1/10	4.3948E-02	-	4.3946E-02	-	6.8877E-02	-	6.8974E-02	-
-1	1/20	1.1176E-02	1.9754	1.1174E-02	1.9756	1.6304E-02	2.0788	1.6401E-02	2.0722
	1/40	2.8061E-03	1.9938	2.8035E-03	1.9948	4.0077E-03	2.0244	4.1049E-03	1.9984
	1/80	7.0259E-04	1.9978	7.0007E-04	2.0017	9.8520E-04	2.0243	1.0824E-03	1.9231

| Show Table

DownLoad: CSV

Table 6. Time convergence results with

$\alpha = 0.9$ ,

$\beta = 0.2$ ,

$h = 1/2000$ .

		$\\|u_h-u\\|$				$\\|q_h-q\\|$
$\theta$	$\tau$	Error_o	Rate	Error_c	Rate	Error_o	Rate	Error_c	Rate
	1/10	2.1916E-03	-	4.2694E-02	-	8.6515E-02	-	1.2162E-01	-
0.2	1/20	1.6194E-03	0.4365	1.0850E-02	1.9763	6.3929E-02	0.4365	2.9675E-02	2.0351
	1/40	1.0890E-03	0.5725	2.7235E-03	1.9942	4.2990E-02	0.5725	7.3710E-03	2.0093
	1/80	6.8790E-04	0.6627	6.8165E-04	1.9984	2.7157E-02	0.6627	1.8364E-03	2.0050
	1/10	2.5540E-03	-	4.2693E-02	-	1.0123E-01	-	1.2164E-01	-
0	1/20	1.1111E-03	1.2007	1.0850E-02	1.9763	4.3864E-02	1.2065	2.9692E-02	2.0345
	1/40	7.7616E-04	0.5176	2.7231E-03	1.9944	3.0641E-02	0.5176	7.3878E-03	2.0069
	1/80	5.0234E-04	0.6277	6.8122E-04	1.9990	1.9831E-02	0.6277	1.8531E-03	1.9952
	1/10	2.4652E-02	-	4.2689E-02	-	9.7312E-01	-	1.2182E-01	-
-0.5	1/20	7.1878E-03	1.7781	1.0845E-02	1.9768	2.8376E-01	1.7779	2.9872E-02	2.0279
	1/40	1.9420E-03	1.8880	2.7184E-03	1.9962	7.6702E-02	1.8874	7.5673E-03	1.9809
	1/80	5.5906E-04	1.7965	6.7658E-04	2.0064	2.2071E-02	1.7971	2.0329E-03	1.8962

| Show Table

DownLoad: CSV

Table 7. Time convergence results with

$\alpha = 0.5$ ,

$\beta = 0.7$ ,

$h = 1/2000$ .

		$\\|u_h-u\\|$				$\\|q_h-q\\|$
$\theta$	$\tau$	Error_o	Rate	Error_c	Rate	Error_o	Rate	Error_c	Rate
	1/10	1.1456E-02	-	4.3948E-02	-	4.5228E-01	-	6.8880E-02	-
0.5	1/20	5.0680E-03	1.1767	1.1176E-02	1.9754	2.0008E-01	1.1767	1.6307E-02	2.0786
	1/40	2.2184E-03	1.1919	2.8060E-03	1.9939	8.7577E-02	1.1919	4.0110E-03	2.0235
	1/80	9.6796E-04	1.1965	7.0250E-04	1.9979	3.8213E-02	1.1965	9.8850E-04	2.0207
	1/10	5.4375E-03	-	4.3948E-02	-	2.1466E-01	-	6.8904E-02	-
0.2	1/20	2.5220E-03	1.1084	1.1176E-02	1.9754	9.9562E-02	1.1084	1.6332E-02	2.0769
	1/40	1.1387E-03	1.1472	2.8053E-03	1.9941	4.4952E-02	1.1472	4.0351E-03	2.0170
	1/80	5.0160E-04	1.1828	7.0188E-04	1.9989	1.9802E-02	1.1827	1.0126E-03	1.9946
	1/10	2.8636E-02	-	4.3948E-02	-	1.1304E+00	-	6.8904E-02	-
-1	1/20	7.7097E-03	1.8931	1.1176E-02	1.9754	3.0439E-01	1.8929	1.6332E-02	2.0769
	1/40	1.9098E-03	2.0133	2.8053E-03	1.9941	7.5437E-02	2.0126	4.0351E-03	2.0170
	1/80	6.3574E-04	1.5869	7.0188E-04	1.9989	2.5098E-02	1.5877	1.0126E-03	1.9946

| Show Table

DownLoad: CSV

In , we obtain the comparison images between the numerical solution and the exact solution with $\tau = 1/1000$ , $h = 1/80$ , $\alpha = 0.9$ , $\beta = 0.2$ , and $\theta = 0.2$ . In and , we present the space and time convergence rate images of $u_h$ and $q_h$ under different parameters $\alpha$ , $\beta$ , and $\theta$ . From Figure 4, one can see that the corrected scheme with the starting parts can effectively restore the second-order convergence rate for the nonsmooth problem.

Figure 2.

$u_h$ ,

$q_h$ and

$u$ ,

$q$ with

$\tau = 1/2000$ ,

$h = 1/80$ ,

$\alpha = 0.9$ ,

$\beta = 0.2$ ,

$\theta = 0.2$ .

DownLoad: Full-Size Img PowerPoint

Figure 3. The spatial convergence rates in

$L^2$ -errors with different parameters

$\alpha$ ,

$\beta$ , and

$\theta$ .

DownLoad: Full-Size Img PowerPoint

Figure 4. The time convergence rates in

$L^2$ -errors with different parameters

$\alpha$ ,

$\beta$ , and

$\theta$ .

DownLoad: Full-Size Img PowerPoint

Example 5.3. To better investigate the effect of changes of two fractional parameters $\alpha$ and $\beta$ on the convergence rates, we introduce the numerical example with two nonsmooth terms. Here, we take the nonsmooth solution $u$ with

$u = (t^{1+\alpha}+t^{1+\beta}+t^3) \sin(2\pi x),$

and the known source term

$\begin{equation} \begin{split} g(x,t) = &\sin(2\pi x)\left[(1+\alpha)t^\alpha+(1+\beta)t^\beta+3t^2+4\pi^2\left(t\Gamma(2+\alpha)+\frac{t^{1+\beta-\alpha}\Gamma(2+\beta)}{\Gamma(2+\beta-\alpha)}+\frac{6t^{3-\alpha}}{\Gamma(4-\alpha)}\right)\right]\\ &+\sin(2\pi x)\left[16\pi^4\left(\frac{t^{1+\alpha-\beta}\Gamma(2+\alpha)}{\Gamma(2+\alpha-\beta)}+t\Gamma(2+\beta)+\frac{6t^{3-\beta}}{\Gamma(4-\beta)}\right)+4\pi^2(t^{1+\alpha}+t^{1+\beta}+t^3)\right]\\ &+2\pi (t^{1+\alpha}+t^{1+\beta}+t^3)\cos(2\pi x)+2\pi(t^{1+\alpha}+t^{1+\beta}+t^3)^2\sin(4\pi x ). \end{split} \end{equation}$

(5.3)

In , we provide the errors of $\|u_h-u\|$ and $\|q_h-q\|$ and the spatial convergence rates under different parameters, which indicate that the corrected term hardly affects the spatial convergence rate.

Table 8. Spatial convergence results with

$\alpha = 0.5$ ,

$\beta = 0.6$ ,

$\tau = 1/2000$ .

		$\\|u_h-u\\|$				$\\|q_h-q\\|$
$\theta$	$h$	Error_o	Rate	Error_c	Rate	Error_o	Rate	Error_c	Rate
	1/10	6.5710E-02	-	6.5710E-02	-	1.1335E-01	-	1.1335E-01	-
0.5	1/20	1.6709E-02	1.9755	1.6709E-02	1.9755	2.7037E-02	2.0678	2.7037E-02	2.0678
	1/40	4.1946E-03	1.9940	4.1946E-03	1.9940	6.6773E-03	2.0176	6.6775E-03	2.0175
	1/80	1.0498E-03	1.9984	1.0498E-03	1.9984	1.6615E-03	2.0068	1.6617E-03	2.0066
	1/10	6.5710E-02	-	6.5710E-02	-	1.1336E-01	-	1.1336E-01	-
0.2	1/20	1.6708E-02	1.9755	1.6708E-02	1.9755	2.7042E-02	2.0676	2.7042E-02	2.0676
	1/40	4.1944E-03	1.9940	4.1944E-03	1.9940	6.6826E-03	2.0167	6.6825E-03	2.0167
	1/80	1.0497E-03	1.9986	1.0497E-03	1.9985	1.6668E-03	2.0033	1.6667E-03	2.0034
	1/10	6.5709E-02	-	6.5709E-02	-	1.1339E-01	-	1.1339E-01	-
-0.5	1/20	1.6708E-02	1.9756	1.6708E-02	1.9756	2.7078E-02	2.0661	2.7077E-02	2.0662
	1/40	4.1935E-03	1.9943	4.1935E-03	1.9943	6.7188E-03	2.0109	6.7178E-03	2.0110
	1/80	1.0487E-03	1.9995	1.0487E-03	1.9995	1.7031E-03	1.9801	1.7020E-03	1.9808

| Show Table

DownLoad: CSV

In –, fixing $\tau = 1/4000$ , choosing $h = 1/10, 1/20, 1/40, 1/80$ , and changing parameters $\alpha$ , $\beta$ , and $\theta$ , we provide the $L^2$ -errors and the time convergence rates for $u$ and $q$ based on the corrected scheme and uncorrected scheme. The impact of different fractional parameters on the time convergence rates of nonsmooth problems is evident from Tables 9–11. Furthermore, one can see that the corrected scheme with the starting part can effectively restore the second-order convergence rate.

Table 9. Time convergence results with

$\alpha = 0.1$ ,

$\beta = 0.9$ ,

$h = 1/4000$ .

		$\\|u_h-u\\|$				$\\|q_h-q\\|$
$\theta$	$\tau$	Error_o	Rate	Error_c	Rate	Error_o	Rate	Error_c	Rate
	1/10	8.9204E-03	-	6.2820E-03	-	3.5216E-01	-	2.4809E-01	-
0.1	1/20	5.2261E-03	0.7714	1.8535E-03	1.7609	2.0632E-01	0.7714	7.3209E-02	1.7608
	1/40	2.6562E-03	0.9764	4.9663E-04	1.9001	1.0486E-01	0.9764	1.9628E-02	1.8992
	1/80	1.2855E-03	1.0470	1.2782E-04	1.9580	5.0749E-02	1.0470	5.0645E-03	1.9544
	1/10	1.1291E-02	-	9.1163E-03	-	4.4575E-01	-	3.5998E-01
0	1/20	6.8182E-03	0.7277	2.7085E-03	1.7510	2.6917E-01	0.7277	1.0696E-01	1.7509
	1/40	3.4949E-03	0.9641	7.2773E-04	1.8960	1.3797E-01	0.9641	2.8751E-02	1.8954
	1/80	1.6987E-03	1.0408	1.8766E-04	1.9553	6.7062E-02	1.0408	7.4268E-03	1.9528
	1/10	2.1457E-02	-	2.6030E-02	-	1.1105E+00	-	1.7696E+00	-
-0.5	1/20	1.4552E-02	0.5603	8.3150E-03	1.6464	8.5207E-01	0.3822	6.2053E-01	1.5118
	1/40	7.7887E-03	0.9017	2.2980E-03	1.8554	4.7736E-01	0.8359	1.7762E-01	1.8047
	1/80	3.8350E-03	1.0222	6.0067E-04	1.9357	2.3888E-01	0.9988	4.7205E-02	1.9118

| Show Table

DownLoad: CSV

Table 10. Time convergence results with

$\alpha = 0.5$ ,

$\beta = 0.6$ ,

$h = 1/4000$ .

		$\\|u_h-u\\|$				$\\|q_h-q\\|$
$\theta$	$\tau$	Error_o	Rate	Error_c	Rate	Error_o	Rate	Error_c	Rate
	1/10	4.3621E-03	-	3.6704E-03	-	1.7226E-01	-	1.4498E-01	-
0.5	1/20	1.0876E-03	2.0039	9.5152E-04	1.9476	4.2941E-02	2.0042	3.7576E-02	1.9480
	1/40	3.3133E-04	1.7148	2.4178E-04	1.9765	1.3080E-02	1.7150	9.5356E-03	1.9784
	1/80	1.1132E-04	1.5735	6.1138E-05	1.9835	4.3948E-03	1.5735	2.3985E-03	1.9912
	1/10	1.4643E-03	-	1.0614E-03	-	5.8318E-02	-	4.2803E-02	-
0.2	1/20	4.8956E-04	1.5806	3.0032E-04	1.8214	1.9326E-02	1.5934	1.2126E-02	1.8197
	1/40	1.6670E-04	1.5542	7.8625E-05	1.9335	6.5811E-03	1.5541	3.1895E-03	1.9267
	1/80	5.7051E-05	1.5470	1.9768E-05	1.9918	2.2523E-03	1.5470	8.1546E-04	1.9676
	1/10	8.7609E-03	-	6.8360E-03	-	3.4601E-01	-	2.7011E-01	-
0	1/20	2.2404E-03	1.9673	1.9129E-03	1.8374	8.8510E-02	1.9669	7.5599E-02	1.8371
	1/40	5.6418E-04	1.9895	5.0061E-04	1.9340	2.2304E-02	1.9886	1.9798E-02	1.9330
	1/80	1.5985E-04	1.8194	1.2738E-04	1.9746	6.3107E-03	1.8214	5.0505E-03	1.9709

| Show Table

DownLoad: CSV

Table 11. Time convergence results with

$\alpha = 0.9$ ,

$\beta = 0.1$ ,

$h = 1/4000$ .

		$\\|u_h-u\\|$				$\\|q_h-q\\|$
$\theta$	$\tau$	Error_o	Rate	Error_c	Rate	Error_o	Rate	Error_c	Rate
	1/10	1.2779E-03	-	4.4043E-04	-	5.0452E-02	-	2.5629E-02	-
0.1	1/20	1.0267E-03	0.3157	1.1804E-04	1.8996	4.0534E-02	0.3158	7.0942E-03	1.8531
	1/40	7.5884E-04	0.4362	3.0793E-05	1.9386	2.9957E-02	0.4362	1.8662E-03	1.9265
	1/80	5.0089E-04	0.5993	8.1141E-06	1.9241	1.9774E-02	0.5993	4.7828E-04	1.9642
	1/10	4.3621E-03	-	1.4545E-03	-	7.7482E-02	-	6.1261E-02	-
0	1/20	1.8873E-03	0.6822	3.9496E-04	1.8807	4.6434E-02	0.7387	1.6763E-02	1.8697
	1/40	9.2034E-04	0.3539	1.0228E-04	1.9492	3.6333E-02	0.3539	4.3730E-03	1.9386
	1/80	6.1337E-04	0.5854	2.5740E-05	1.9904	2.4215E-02	0.5854	1.1169E-03	1.9692
	1/10	9.0746E-03	-	6.1035E-03	-	3.5896E-01	-	2.4209E-01	-
-0.1	1/20	2.3761E-03	1.9332	1.7550E-03	1.7982	9.4044E-02	1.9324	6.9621E-02	1.7980
	1/40	7.1647E-04	1.7296	4.6664E-04	1.9111	2.8284E-02	1.7333	1.8526E-02	1.9100
	1/80	5.9682E-04	0.2636	1.1984E-04	1.9612	2.3561E-02	0.2636	4.7711E-03	1.9572

| Show Table

DownLoad: CSV

To further validate the performance of the parameter $\theta$ in numerical simulations with nonsmooth solutions, we provide the computing data in , from which one can see that the parameter $\theta$ still needs to satisfy $\theta\le \min\{\alpha, \beta, \frac{1}{2}\}$ , whether before or after correction. Notably, when $\theta$ is negative, as long as it is not much less than 0, we can still obtain second-order convergence accuracy.

Table 12. Time convergence results with

$\alpha = 0.7$ ,

$\beta = 0.3$ ,

$h = 1/4000$ .

		$\\|u_h-u\\|$				$\\|q_h-q\\|$
$\theta$	$\tau$	Error_o	Rate	Error_c	Rate	Error_o	Rate	Error_c	Rate
	1/10	3.3675E-03	-	2.0604E-03	-	1.3333E-01	-	8.2027E-02	-
0.31	1/20	1.2977E-03	1.3757	5.3018E-04	1.9584	5.1287E-02	1.3783	2.1127E-02	1.9570
	1/40	1.9442E-03	-0.5833	1.4145E-04	1.9062	7.6738E-02	-0.5813	5.6232E-03	1.9096
	1/80	6.1733E-02	-4.9888	1.3510E-04	0.0663	2.4372E+00	-4.9892	5.3212E-03	0.0797
	1/10	4.5472E-02	-	3.0346E-02	-	1.7952E+00	-	1.1983E+00	-
-0.5	1/20	1.2127E-02	1.9067	9.1861E-03	1.7240	4.7880E-01	1.9066	3.6273E-01	1.7240
	1/40	3.1344E-03	1.9520	2.4851E-03	1.8861	1.2376E-01	1.9518	9.8141E-02	1.8859
	1/80	7.9608E-04	1.9772	6.4304E-04	1.9504	3.1447E-02	1.9766	2.5407E-02	1.9496
	1/10	1.1762E+00	-	3.4223E-01	-	4.6430E+01	-	1.3512E+01	-
-5	1/20	3.9521E-01	1.5734	1.9704E-01	0.7964	1.5601E+01	1.5734	7.7788E+00	0.7966
	1/40	1.2069E-01	1.7113	7.7857E-02	1.3396	4.7643E+00	1.7113	3.0736E+00	1.3396
	1/80	3.3458E-02	1.8509	2.4673E-02	1.6579	1.3208E+00	1.8509	9.7402E-01	1.6579

| Show Table

DownLoad: CSV

The time convergence rates of $u$ and $q$ are compared before and after correction with different parameters $\alpha$ , $\beta$ , and $\theta$ in Figure 5, where the slope of the line segment indicates the convergence rate. The slope of each line segment in the corrected images is the same regardless of the parameters chosen, indicating that the introduction of the starting part has a significant effect on the time convergence rates for the case with nonsmooth solutions.

Figure 5. The time convergence rates in

$L^2$ -errors with different parameters

$\alpha$ ,

$\beta$ , and

$\theta$ .

DownLoad: Full-Size Img PowerPoint

6. Conclusions

In this article, the spatial mixed finite element method with the generalized BDF2- $\theta$ for solving the time-fractional generalized Rosenau-RLW-Burgers equation was presented. Detailed proofs of stability were shown. The numerical scheme's effectiveness and feasibility were verified by conducting numerical examples that included both smooth and nonsmooth solutions. The numerical examples with good regularity indicated that our algorithm with changed parameters $\alpha$ , $\beta$ , and $\theta$ can maintain second-order convergence in time. Especially, the nonsmooth examples demonstrated that adding the correction term could effectively solve the problem of reduced order caused by weak singularity.

Author contributions

N. Yang: Writing–original draft, Formal analysis, Software; Y. Liu: Methodology, Validation, Formal analysis, Funding acquisition, Supervision, Writing–review & editing. All authors have read and agreed to the published version of the manuscript.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (12061053), Young Innovative Talents Project of Grassland Talents Project and Program for Innovative Research Team in Universities of Inner Mongolia Autonomous Region (NMGIRT2413).

Conflict of interest

The authors declare that they have no conflicts of interest.

References

[1]	B. N. Fantu, B. Guush, M. Bart, et al., Agricultural Transformation in Africa? Assessing the Evidence in Ethopia, World Dev., 105 (2018), 286–298.
[2]	S. Trilles, J. Torres-Sospedra, Ó. Belmonte, et al., Development of an open sensorized platform in a smart agriculture context: A vineyard support system for monitoring mildew disease, Sustain. Comput. Infor., (2019), in press.
[3]	Intergovernmental Panel on Climate Change, Climate Change 2014: Synthesis Report. Contribution of Working Groups I, II and III to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change [Core Writing Team, R.K. Pachauri and L.A. Meyer (eds.)]. IPCC (2014), Geneva, Switzerland, 151 pp., [online] Available from: https://www.ipcc.ch/site/assets/uploads/2018/02/SYR_AR5_FINAL_full.pdf.
[4]	A. M. García, I. F. García, E. C. Poyato, et al., Coupling irrigation scheduling with solar energy production in a smart irrigation management system, J. Clean. Prod., 175 (2018), 670–682.
[5]	M. Kala, U. Sadrul and B. Steven, Solar photovoltaic water pumping-opportunities and challenges, Renew. Sust. Energ. Rev., 4 (2008), 1162–1175.
[6]	European Commission, Overview of CAP Reform 2014–2020, December 2013, [online] Available from: http://ec.europa.eu/agriculture/policy-perspectives/policy-briefs/05_en.pdf.
[7]	S. Biswajit, Green Computing, Int. J. Comput. Trends Technol., 14 (2014), 46–50.
[8]	S. Murugesan, Harnessing green IT: Principles and practices, IT Prof., 10 (2008), 24–33.
[9]	E. Okewu, S. Misra, R. Maskeliunas, et al., Optimizing green computing awareness for environmental sustainability and economic security as a stochastic optimization problem, Sustainability, 9 (2017), 1857.
[10]	E. Okewu, S. Misra, L. Fernandez-Sanz, et al., An e-environment system for socio-economic sustainability and national security, Probl. Ekorozw., 13 (2018), 121–132.
[11]	A. C. Orgerie, Green Computing and Sustainability, Journées scientifiques, 15, (2016), 23–27.
[12]	A. Al-Zamil and A. K. J. Saudagar, Drivers and challenges of applying green computing for sustainable agriculture: A case study, Sustain. Comput. Infor., (2018), in press.
[13]	A. Mansur, H. Ghassan, S. A. Syed, et al., A review of solar-powered water pumping systems, Renew. Sust. Energ. Rev., 87 (2018), 61–76.
[14]	N. Mehdi, M. Peyman, N. Mohammad, et al., Techno-economic feasibility of off-grid solar irrigation for a rice paddy in Guilan province in Iran: A case study, Sol. Energy, 150 (2017), 546–557.
[15]	P. E. Campana, H. L. Li and J. Y. Yan, Techno-economic feasibility of the irrigation system for the grassland and farmland conservation in China: Photovoltaic vs. wind power water pumping, Energ. Convers. Manage., 103 (2015), 311–320.
[16]	P. E. Campana, H. L. Li and J. Y. Yan, Dynamic modelling of a PV pumping system with special consideration on water demand, Appl. Energy, 112 (2013), 635–645.
[17]	Z. Gu, Z. Qi, L. Ma, et al., Development of an irrigation scheduling software based on model predicted crop water stress, Comput. Electron. Agric., 143 (2017), 208–221.
[18]	R. López-Luque, J. Reca and J. Martínez, Optimal design of a standalone direct pumping photovoltaic system for deficit irrigation of olive orchards, Appl. Energy, 149 (2015), 13–23.
[19]	G. Vellidis, M. Tucker, C. Perry, et al., A real-time wireless smart sensor array for scheduling irrigation, Comput. Electron. Agric., 61 (2008), 44–50.
[20]	T. Ojha, S. Misra and N.S. Raghuwanshi, Wireless sensor networks for agriculture: the state-of-the-art in practice and future challenges, Comput. Electron. Agric., 118 (2015), 66–84.
[21]	B. Keswani, A. G. Mohapatra, A. Mohanty, et al., Adapting weather conditions based IoT enabled smart irrigation technique in precision agriculture mechanisms, Neural Comput. Appl., 31(S1) (2018), 277–292.
[22]	S. T. Oliver, A. González-Pérez and J. H. Guijarro, An IoT proposal for monitoring vineyards called SEnviro for agriculture, Proceedings of 8th International Conference on the Internet of Things, (2018), 20. ACM.
[23]	S. A. M. Varman, A. R. Baskaran, S. Aravindh, et al., Deep learning and IoT for smart agriculture using WSN, IEEE International Conference on Computational Intelligence and Computing Research, ICCIC 2017, (2018).
[24]	A. Goap, D. Sharma, A. K. Shukla, et al., An IoT based smart irrigation management system using machine learning and open source technologies, Comput. Electron. Agric., 155 (2018), 41–49.
[25]	T. Kashiwao, K. Nakayama, S. Ando, et al., A neural network-based local rainfall prediction system using meteorological data on the Internet: A case study using data from the Japan Meteorological Agency, Appl. Soft Comput., 56 (2017), 317–330.
[26]	O. Adeyemi, I. Grove, S. Peets, et al., Dynamic neural network modelling of soil moisture content for predictive irrigation scheduling, Sensors, 18 (2018), 3408.
[27]	L. Huang, L. Chen, Q. Wang, et al., Regional short-term micro-climate air temperature prediction with CBPNN, E3S Web of Conferences, 53 (2018).
[28]	L. T. Yang, B. Di Martino and Q. Zhang, Internet of Everything, Mob. Inf. Syst., (2017).
[29]	S. R. Barkunan, V. Bhanumathi and J. Sethuram, Smart sensor for automatic drip irrigation system for paddy cultivation, Comput. Electr. Eng., 73 (2019), 180–193.
[30]	C. Chang and K. Lin, Smart agricultural machine with a computer vision-based weeding and variable-rate irrigation scheme, Robotics, 7 (2018), 38.
[31]	C. Corbari, R. Salerno, A. Ceppi, et al., Smart irrigation forecast using satellite LANDSAT data and meteo-hydrological modeling, Agric. Water Manag., 212 (2019), 283–294.
[32]	W. Difallah, K. Benahmed, B. Draoui, et al., Implementing wireless sensor networks for smart irrigation, Taiwan Water Conservancy, 65 (2017), 44–54.
[33]	S. Geetha and R. Sathya Priya, Smart agriculture irrigation control using wireless sensor networks, GSM and android phone, Asian J. Inf. Technol., 15 (2016), 3780–3786.
[34]	A. Goap, D. Sharma, A. K. Shukla, et al., An IoT based smart irrigation management system using machine learning and open source technologies, Comput. Electron. Agr., 155 (2018), 41–49.
[35]	N. Hema and K. Kant, Cost-effective smart irrigation controller using automatic weather stations, Int. J. Hydrol. Sci. Technol., 9 (2019), 1–27.
[36]	C. Kamienski, J. Soininen, M. Taumberger, et al., Smart water management platform: IoT-based precision irrigation for agriculture, Sensors, 19 (2019), 276.
[37]	S. Katyara, M. A. Shah, S. Zardari, et al., WSN based smart control and remote field monitoring of Pakistan's irrigation system using SCADA applications, Wireless Pers. Commun., 95 (2017), 491–504.
[38]	O. Abayomi-Alli, M. Odusami, D. Ojinaka, et al., Smart-Solar Irrigation System (SMIS) for Sustainable Agriculture, International Conference on Applied Informatics, ICAI 2018, (2018), 198–212.
[39]	A. G. Mohapatra, S. K. Lenka and B. Keswani, Neural network and fuzzy logic based smart DSS model for irrigation notification and control in precision agriculture, P. Natl. A. Sci. India A, 89 (2019), 67–76.
[40]	M. S. Munir, I. S. Bajwa, M. A. Naeem, et al., Design and implementation of an IoT system for smart energy consumption and smart irrigation in tunnel farming, Energies, 11 (2018), 3427.
[41]	X. Fan, W. Wei, M. Wozniak, et al., Low energy consumption and data redundancy approach of wireless sensor networks with bigdata, Inf. Technol. Control, 47 (2018), 406–418.
[42]	A. Venčkauskas, N. Jusas, E. Kazanavičius, et al., An energy efficient protocol for the internet of things, J. Electr. Eng., 66 (2015), 47–52.
[43]	W. Wei, Z. Sun, H. Song, et al., Energy balance-based steerable arguments coverage method in WSNs, IEEE Access, 6 (2018), 33766–33773.
[44]	C.M Bishop, Neural networks for pattern recognition, Oxford University Press, 1995.
[45]	Z. Boger and H. Guterman, Knowledge extraction from artificial neural network models, IEEE Systems, Man, and Cybernetics Conference, 4 (1997), 3030–3035.
[46]	N. Srivastava, G. Hinton, A. Krizhevsky, et al., Dropout: a simple way to prevent neural networks from overfitting, J. Mach. Learn. Res., 15 (2014), 1929–1958.

Reader Comments

Your name:*

Email:*
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)