Citation: Aristotelis Koinis, Emmanouil Velonakis, Chara Tzavara, Foteini Tzavella, Styliani Tziaferi. Psychometric properties of the workplace psychologically violent behaviors-WPVB instrument. Translation and validation in Greek Health Professionals[J]. AIMS Public Health, 2019, 6(1): 79-95. doi: 10.3934/publichealth.2019.1.79
| [1] | Yuhua Zhu . A local sensitivity and regularity analysis for the Vlasov-Poisson-Fokker-Planck system with multi-dimensional uncertainty and the spectral convergence of the stochastic Galerkin method. Networks and Heterogeneous Media, 2019, 14(4): 677-707. doi: 10.3934/nhm.2019027 |
| [2] | Karoline Disser, Matthias Liero . On gradient structures for Markov chains and the passage to Wasserstein gradient flows. Networks and Heterogeneous Media, 2015, 10(2): 233-253. doi: 10.3934/nhm.2015.10.233 |
| [3] | L.L. Sun, M.L. Chang . Galerkin spectral method for a multi-term time-fractional diffusion equation and an application to inverse source problem. Networks and Heterogeneous Media, 2023, 18(1): 212-243. doi: 10.3934/nhm.2023008 |
| [4] | Kexin Li, Hu Chen, Shusen Xie . Error estimate of L1-ADI scheme for two-dimensional multi-term time fractional diffusion equation. Networks and Heterogeneous Media, 2023, 18(4): 1454-1470. doi: 10.3934/nhm.2023064 |
| [5] | Yves Achdou, Victor Perez . Iterative strategies for solving linearized discrete mean field games systems. Networks and Heterogeneous Media, 2012, 7(2): 197-217. doi: 10.3934/nhm.2012.7.197 |
| [6] | Hirotada Honda . On Kuramoto-Sakaguchi-type Fokker-Planck equation with delay. Networks and Heterogeneous Media, 2024, 19(1): 1-23. doi: 10.3934/nhm.2024001 |
| [7] | Leqiang Zou, Yanzi Zhang . Efficient numerical schemes for variable-order mobile-immobile advection-dispersion equation. Networks and Heterogeneous Media, 2025, 20(2): 387-405. doi: 10.3934/nhm.2025018 |
| [8] | Yin Yang, Aiguo Xiao . Dissipativity and contractivity of the second-order averaged L1 method for fractional Volterra functional differential equations. Networks and Heterogeneous Media, 2023, 18(2): 753-774. doi: 10.3934/nhm.2023032 |
| [9] | Shui-Nee Chow, Xiaojing Ye, Hongyuan Zha, Haomin Zhou . Influence prediction for continuous-time information propagation on networks. Networks and Heterogeneous Media, 2018, 13(4): 567-583. doi: 10.3934/nhm.2018026 |
| [10] | Ioannis Markou . Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Networks and Heterogeneous Media, 2017, 12(4): 683-705. doi: 10.3934/nhm.2017028 |
In the present paper, we consider numerical solution of the time fractional Fokker-Planck equations (TFFPEs):
| $ {∂αtu−Δu+p⋅(∇u)+q(x,t)u=f(x,t),(x,t)∈Ω×(0,T],u(x,0)=u0(x),x∈Ω,u(x,t)=0,x∈∂Ω,t∈(0,T], $ | (1.1) |
where $ \Omega\in \mathbb{R}^d\; (d = 1, 2, 3) $, $ \boldsymbol{x} = {(x_1, x_2, \ldots, x_d)} $, $ u_0(\boldsymbol{x}) $ is smooth on $ \Omega $, $ \boldsymbol{p}: = (p_1, p_2, \ldots, p_d) $ with $ p_i: = p_i(\boldsymbol{x}, t) \; (i = 1, 2, \ldots, d) $ and $ q: = q(\boldsymbol{x}, t) $ are continuous functions. $ {\partial}_t^{\alpha}u $ represents the Caputo derivative of order $ \alpha \in (0, 1) $. When $ \alpha = 1 $ in Eq (1.1), the corresponding equations are a class of very useful models of statistical physics to describe some practical phenomena. TFFPEs are widely used in statistical physics to describes the probability density function of position and the evolution of the velocity of a particle, see e.g., [1,2,3,4]. The TFFPEs also represent the continuous limit of a continuous time random walk with a Mittag-Leffler residence time density. For a deeper understanding of TFFPEs, we refer the readers to [5,6]. In addition, the regularity of the solutions of the TFFPE (1.1) can be found in [7].
For the past few years, many numerical methods were used to solve the TFFPEs. For example, Deng [8] proposed an efficient predictor-corrector scheme. Vong and Wang [9] constructed a compact finite difference scheme. Mahdy [10] used two different techniques to study the approximate solution of TFFPEs, namely the fractional power series method and the new iterative method. Yang et al. [11] proposed a nonlinear finite volume format to solve the two-dimensional TFFPEs. More details can refer to [12,13,14,15]. Besides, it is difficult that analysing the convergence and stability properties of the numerical schemes for TFFPEs, when convective and diffusion terms exist at the same time. In the study of TFFPEs, the conditions imposed on $ \boldsymbol{p} $ and $ q $ were somewhat restrictive. For example, for solving the one-dimensional TFFPEs, Deng [16] proved the stability and convergence under the conditions that $ p_1 $ was a monotonically decreasing function and $ q \leq 0 $. Chen et al. [17] obtained the stability and convergence properties of the method with the conditions that $ p_1 $ was monotone or a constant and $ q $ was a constant.
To solve the time Caputo fractional equations, one of the keys is the treatment of the Caputo derivative, which raised challenges in both theoretical and numerical aspects. Under the initial singularity of the solutions of the equations, many numerical schemes are only proved to be of $ \tau^\alpha $ in temporal direction, e.g., convolution quadrature (CQ) BDF method [18], CQ Euler method [19], uniform $ L1 $ method et al. [20,21]. Here $ \tau $ represents the temporal stepsize. Considering the singularity of solutions, different numerical formats were established to obtain high convergence orders, e.g., the Alikhanov scheme (originally proposed in [22]) and the $ L1 $ scheme (see e.g., [23]) by employing the graded mesh (i.e., $ t_n = T(n/K)^r, n = 1, 2, \ldots, K $, $ r $ is mesh parameter). It was proved that the optimal convergence of those methods can be $ 2 $ and $ 2-\alpha $ iff $ r\geq 2/\alpha $ and $ r\geq(2-\alpha)/\alpha $, respectively (see e.g., [24,25,26,27,28,29]). The $ \overline{{\rm L}1} $ scheme studied in [30,31,32] was another high-order scheme for Caputo fractional derivative. There were also some fast schemes for Caputo fractional derivative, see [33,34,35,36]. When $ \alpha $ was small, the grids at the beginning would become very dense. It may lead to the so-called round-off errors. Recently, taking the small $ \alpha $ and the initial singularity into account, Li et al. [37] introduced the transformation $ s = t^{\alpha} $ for the time variable, and derived and analyzed the equivalent fractional differential equation. They constructed the $ TL1 $ discrete scheme, and obtained that the convergence order of the $ TL1 $ scheme is of $ 2-\alpha $. Based on the previous research, Qin et al. [38] studied the nonlinear fractional order problem, and established the discrete fractional order Grönwall inequality. Besides, discontinuous Galerkin methods were also effective to solve the similar problems with weak singular solutions [39,40,41].
Much of the past study of TFFPEs (i.e., in [16,17,42,43]) has been based on many restrictions on $ q $ and $ p_i, \; i = 1, 2, \ldots, d $. This reduces the versatility of the equations. In the paper, we consider the more general TFFPE (1.1), i.e., $ q $ and $ p_i, \; i = 1, 2, \ldots, d, $ are variable coefficients, and $ q $ is independent of $ p_i $. We draw on the treatment of the Caputo derivative in [37], introduce variable substitution, and construct the $ TL1 $ Legendre-Galerkin spectral scheme to solve the equivalent $ s $-fractional equation. For time discreteness, we take into account the initial singularity, and obtain that the optimal convergence order is $ 2-\alpha $. In terms of spatial discreteness, unlike other schemes [16,17], which impose restrictions on coefficients, the Legendre-Galerkin spectral scheme does not require $ p_i $ and $ q $ to be constants or to be monotonic. Besides, we obtain the following theoretical results. The order of convergence in $ L^2 $-norm of the method is exponential order convergent in spatial direction and ($ 2-\alpha $)-th order convergent in the temporal direction. And the scheme is valid for equations with small parameter $ \alpha $.
The structure of the paper is as follows. In Section 2, we propose the $ TL1 $ Legendre-Galerkin spectral scheme for solving TFFPEs. In Section 3, the detailed proof of our main results is presented. In Section 4, two numerical examples are given to verify our obtained theoretical results. Some conclusion remarks are shown in Section 5.
We denote $ W^{m, p}(\Omega) $ and $ ||\cdot||_{W^{m, p}\;(\Omega)} $ as the Sobolev space of any functions defined on $ \Omega $ and the corresponding Sobolev norm, respectively, where $ m\geq0 $ and $ 1\leq p\leq \infty $. Especially, denote $ L^2(\Omega): = W^{0, 2}(\Omega) $ and $ H^m(\Omega): = W^{m, 2}(\Omega) $. Define $ C^{\infty}_0(\Omega) $ as the space of infinitely differentiable functions which are nonzero only on a compact subset of $ \Omega $ and $ H^1_0(\Omega) $ as the completion of $ C^{\infty}_0(\Omega) $. For convenience, denote $ ||\cdot||_0: = ||\cdot||_{L^2(\Omega)} $, $ ||\cdot||_m: = ||\cdot||_{H^m(\Omega)} $.
For simplicity, we suppose that $ \Omega = (-1, 1)^d $, and $ u(\boldsymbol{x}, t)\in H_0^1(\Omega)\cap H^m(\Omega) $ for $ 0\leq t\leq T $. First of all, we introduce $ TL1 $ scheme to discrete the Caputo fractional derivative. Introducing the change of variable as follows [21,37,44]:
| $ t=s1/α,w(x,s)=u(x,s1/α). $ | (2.1) |
By this, then the Caputo derivative of $ u(\boldsymbol{x}, t) $ becomes
| $ ∂αtu(x,t)=1Γ(1−α)∫t0∂u(x,r)∂r1(t−r)αdr=1Γ(1−α)∫s0∂w(x,r)∂r1(s1/α−r1/α)αdr=Dαsw(x,s). $ | (2.2) |
Hence, Eq (1.1) can be rewritten as
| $ Dαsw(x,s)−Δw+˜p⋅(∇w)+˜q(x,s)w=˜f(x,s),(x,s)∈Ω×(0,Tα], $ | (2.3) |
| $ w(x,s)=0,(x,s)∈∂Ω×(0,Tα], $ | (2.4) |
| $ w(x,0)=u0(x),x∈Ω, $ | (2.5) |
where $ \tilde{\boldsymbol{p}} = (\tilde{p}_1, \tilde{p}_2, \ldots, \tilde{p}_d) $, $ \tilde{p}_d: = p_d(\boldsymbol{x}, s^{1/\alpha}), \; \tilde{q}: = q(\boldsymbol{x}, s^{1/\alpha}) $, and $ \tilde{f}(\boldsymbol{x}, s) = f(\boldsymbol{x}, s^{1/\alpha}) $. Let $ s_n = T^\alpha n/K, \; n = 0, 1, \ldots, K $, and the uniform mesh on $ [0, T^\alpha] $ with $ \tau_s = s_n-s_{n-1} $. For convenience, $ K_i $, $ i\geq 1 $ represent the positive constants independent of $ \tau_s $ and $ N $, where $ N $ represents polynomial degree. In addition, we define the following notations
| $ \tilde{p}^n_d: = \tilde{p}_d(\boldsymbol{x},s_n),\; \tilde{q}^n: = \tilde{q}(\boldsymbol{x},s_n),\; \tilde{f}^n: = \tilde{f}(\boldsymbol{x},s_n), $ |
| $ w^n: = w(\boldsymbol{x},s_n),\; \tilde{\boldsymbol{p}}^n: = (\tilde{p}_1^n,\tilde{p}_2^n,\ldots,\tilde{p}_d^n). $ |
Applying the $ TL1 $ approximation, we have
| $ Dαswn=1Γ(1−α)∫sn0∂w(x,r)∂r1(s1/αn−r1/α)αdr=1Γ(1−α)n∑l=1wl−wl−1τs∫slsl−1dr(s1/αn−r1/α)α+Qn=n∑l=1an,n−l(wl−wl−1)+Qn:=Dατwn+Qn. $ | (2.6) |
Here the coefficients $ a_{n, n-l} = \frac{1}{\tau_s\Gamma(1-\alpha)}\int_{s_{l-1}}^{s_l}\frac{dr}{(s_n^{1/\alpha}-r^{1/\alpha})^\alpha} $, and $ Q^n $ represents the truncation error. For more details, we refer to [37,38]. By Eq (2.6), then Eq (2.3) arrives at
| $ D_\tau^{\alpha}w^n- \Delta w^n +\tilde {\boldsymbol{p}}^n \cdot (\nabla w^n) +\tilde{q}^nw^n = \tilde{f}^n-Q^n. $ |
For spatial discretization, we introduce the following basis functions:
| $ \{\psi_{\boldsymbol{k}}(\boldsymbol{x})\} = \{\psi_{k_1}({x_1})\psi_{k_2}({x_2})\ldots \psi_{k_d}({x_d}),\; k_1,k_2,\ldots,k_d\in I_N\}, $ |
where $ \boldsymbol{k} = {(k_1, k_2, \ldots, k_d)} $, $ I_N = {\{0, 1, 2, \ldots, N-2\}} $. For $ \psi_{k_i}({x_i}), \; i = 1, 2, \ldots, d $, one has
| $ ψki(xi)=Lki(xi)−Lki+2(xi)for ∀ki∈IN, $ | (2.7) |
where $ \{L_j(x)\}_{j = 0}^N $ are the Legendre orthogonal polynomials, given by the following recurrence relationship [45]:
| $ {(j+1)Lj+1(x)=(2j+1)xLj(x)−jLj−1(x)for j≥1,L0(x)=1,L1(x)=x. $ | (2.8) |
Define the finite-dimensional approximation space
| $ X_{\boldsymbol{N}} = span{\{\psi_{\boldsymbol{k}}(\boldsymbol{x}),\; k_1,k_2,\ldots,k_d\in I_N\}}, $ |
where $ \boldsymbol{N} = (\underbrace {N, N, \ldots, N}_{d}) $. For any function $ w_{\boldsymbol{N}}(x) $, write
| $ w_{\boldsymbol{N}}(x) = \sum\limits_{k_1,k_2,\ldots,k_d\in I_N} \hat w_{\boldsymbol{k}} \psi_{\boldsymbol{k}}(\boldsymbol{x}). $ |
By Eqs (2.7) and (2.8), we have
| $ w_{\boldsymbol{N}}(x)|_{\partial\Omega} = 0 \quad \text{for } \forall w_{\boldsymbol{N}}(x)\in X_{\boldsymbol{N}}. $ |
Then, the $ TL1 $ Legendre-Galerkin spectral scheme is to seek $ W^{n}\in X_{\boldsymbol{N}} $, such that
| $ (DατWn,v)+(∇Wn,∇v)+(∇Wn,˜pnv)+(˜qnWn,v)=(˜fn,v)for ∀v∈XN. $ | (2.9) |
Here $ W^0 = \pi_{\boldsymbol{N}} w^0 $, and $ \pi_{\boldsymbol{N}} $ is the Ritz projection operator given in Lemma 2. For instance, if $ d = 1 $, we solve Eqs (2.3) and (2.4) by
| $ A1Dατˆwn+(A2+A3n+A4n)ˆwn=Fn, $ | (2.10) |
where $ \hat{\boldsymbol{w}}^n = (\hat{w}^n_0, \hat{w}^n_1, \hat{w}^n_2, \dots, \hat{w}^n_{N-2})^T $, $ \boldsymbol{A1}_{j, h} = (\psi_{h}(x), \psi_{j}(x)) $, $ j, h \in I_N $, $ \boldsymbol{A2}_{j, h} = (\psi'_{h}(x), \psi'_{j}(x)) $, $ \boldsymbol{A3}^n_{j, h} = (\tilde{p}^n\psi'_{h}(x), \psi_{j}(x)) $, $ \boldsymbol{A4}^n_{j, h} = (\tilde{q}^n\psi_{h}(x), \psi_{j}(x)) $, and $ \boldsymbol{F}^n_{j, 1} = (\tilde{f}^n, \psi_{j}(x)) $.
The typical solution of Eq (1.1) meets [18,46,47]
| $ \bigg|\bigg|\frac{{\partial}u}{\partial t}(\boldsymbol{x},t)\bigg|\bigg|_{0}\leq Ct^{\alpha-1}, $ |
then, with the help of the changes of variable (2.1), one has (see e.g., [38])
| $ ||∂lw∂sl(x,s)||0≤C(1+s1/α−l)<∞,l=1,2, $ | (2.11) |
where $ C > 0 $ is a constant independent of $ s $ and $ \boldsymbol{x} $. From [37, Lemma 2.2] and [38, Lemma 2.1], the solution becomes smoother at the beginning.
Now, the convergence results of $ TL1 $ Legendre-Galerkin spectral scheme (2.9) is given as follows.
Theorem 1. Assume that $ \tilde{q} $ and $ \tilde{p_i}, \; i = 1, 2, \ldots, d, $ in (2.3) are bounded, and that the unique solution $ w $ of Eqs (2.3) and (2.4) satisfying Eq (2.11) and $ w(\boldsymbol{x}, s)\in H_0^1(\Omega)\cap H^m(\Omega) $. Then, there exist $ N_0 > 0 $ and $ \tau_0 > 0 $ such that when $ N\geq N_0 $ and $ \tau_s\leq\tau_0 $, Eq (2.9) has a unique solution $ W^n\; (n = 0, 1, \ldots, K) $, which satisfies
| $ ||wn−Wn||0≤K∗(τ2−αs+N1−m), $ | (2.12) |
where $ K^* > 0 $ is a constant independent of $ \tau_s $ and $ N $.
We will present the detailed proof of Theorem 1 in this section. For this, we first introduce the following several lemmas.
Lemma 1. [37,38] For $ n\geq 1 $, we get
| $ 0<an,n−1≤an,n−2≤⋯≤an,0. $ | (3.1) |
Lemma 2. If we given the Ritz projection operator $ \pi_{\boldsymbol{N}}:H_0^1(\Omega) \rightarrow X_{\boldsymbol{N}} $ by
| $ (\nabla (\pi_{\boldsymbol{N}}w-w),\nabla v) = 0 \quad \mathit{\text{for}}\;\forall v\in X_{\boldsymbol{N}}, $ |
then, one can get that [48]
| $ ||\pi_{\boldsymbol{N}} w-w||_l\leq C_{\Omega}N^{l-m}||w||_m \quad \mathit{\text{for}}\; \forall w\in H_0^1(\Omega)\cap H^m(\Omega) $ |
with $ d \leq m\leq N+1 $, where $ C_{\Omega} > 0 $ is a constant independent of $ N $.
Lemma 3. [49] For any $ s_K = T^{\alpha} > 0 $ and given nonnegative sequence $ \left\{\lambda_i\right\}^{K-1}_{i = 0} $, assume that there exists a constant $ \lambda^* > 0 $ independent of $ \tau_s $ such that $ \lambda^*\geq\sum^{K-1}_{i = 0}\lambda_i $. Assume also that the grid function $ \{{w^n|n\geq0}\} $ satisfies
| $ D_\tau^{\alpha}(w^n)^2\leq\sum\limits^{n}_{i = 1}\lambda_{n-i}(w^i)^2+w^n(Q^n+\xi) \quad \mathit{\text{for}}\; n\geq1, $ |
where $ \{Q^n|n\geq1\} $ is well defined in Eq (2.6). Then, there exists a constant $ \tau_s^* > 0 $ such that, when $ \tau_s\leq\tau_s^* $,
| $ w^j\leq 2E_\alpha(2\lambda^*s_j)\left[w^0+C_1^*(\tau_s^{2-\alpha}+\xi)\right] \quad \mathit{\text{for}}\; 1\leq j\leq K, $ |
where $ C_1^* $ is a constant and $ E_\alpha(x) = \sum_{k = 0}^\infty\frac{x^k}{\Gamma(1+k\alpha)} $.
We will offer the proof of Theorem 1 in this section. The projection $ \pi_{\boldsymbol{N}} w^n $ of the exact solution $ w^n $ satisfies
| $ (DατπNwn,v)=−(∇πNwn,∇v)−(∇πNwn,˜pnv)−(˜qnπNwn,v)+(˜fn,v)−(Qn,v)−(Rn,v)for ∀v∈XN. $ | (3.2) |
Here $ R^n = D_\tau^\alpha(w^n-\pi_{\boldsymbol{N}} w^n)-\Delta(w^n-\pi_{\boldsymbol{N}} w^n)+\tilde {\boldsymbol{p}}^n \cdot \nabla(w^n-\pi_{\boldsymbol{N}} w^n)+\tilde{q}^n(w^n-\pi_{\boldsymbol{N}} w^n) $, and $ Q^n $ is the truncation error for approximating the fractional derivative defined in Eq (2.6).
The error between numerical solution $ W^n $ and exact solution $ w^n $ can be divided into
| $ ||wn−Wn||0≤||wn−πNwn||0+||πNwn−Wn||0. $ | (3.3) |
Let
| $ e^n: = \pi_{\boldsymbol{N}}w^n-W^n \quad \text{for }n = 0,1,\ldots,K. $ |
Subtracting Eq (2.9) from Eq (3.2), we get that
| $ (Dατen,v)=−(∇en,∇v)−(∇en,˜pnv)−(˜qnen,v)−(Qn,v)−(Rn,v)for ∀v∈XN. $ | (3.4) |
Setting $ v = e^n $ in Eq (3.4), we obtain
| $ (Dατen,en)=−(∇en,∇en)−(∇en,˜pnen)−(˜qnen,en)−(Qn,en)−(Rn,en). $ | (3.5) |
By Lemma 1, we have
| $ (Dατen,en)=(n∑l=1an,n−l(el−el−1),en)=(an,0en−n−1∑l=1(an,n−l−1−an,n−l)el−an,n−1e0,en)≥12(an,0||en||20−n−1∑l=1(an,n−l−1−an,n−l)||el||20−an,n−1||e0||20)=12Dατ||en||20. $ | (3.6) |
By Cauchy-Schwartz inequality, one can obtain that
| $ −(∇en,∇en)−(∇en,˜pnen)−(˜qnen,en)≤−||∇en||20+K1|(∇en,en)|+K2||en||20≤−||∇en||20+||∇en||20+K214||en||20+K2||en||20≤(K214+K2)||en||20. $ | (3.7) |
Here $ K_1 = \mathop{\max}\limits_{0\leq n\leq K}\left\lbrace ||\tilde {\boldsymbol{p}}(\boldsymbol{x}, s_n)||_0 \right\rbrace $, and $ K_2 = \mathop{\max}\limits_{0\leq n\leq K}\left\lbrace \mathop{\max}\limits_{\boldsymbol{x}\in \Omega}|\tilde{q}(\boldsymbol{x}, s_n)| \right\rbrace $. Similarly, we see that
| $ −(Qn,en)≤||Qn||0||en||0. $ | (3.8) |
Noting that $ e^n\in X_{\boldsymbol{N}} $ and by Lemma 2, one has
| $ \left(\nabla (w^n-\pi_{\boldsymbol{N}} w^n),\nabla e^n\right) = 0. $ |
Then
| $ −(Rn,en)=−(Dατ(wn−πNwn),en)−(∇(wn−πNwn),∇en)−(∇(wn−πNwn),pnen)−(˜qn(wn−πNwn),en)≤||Dατ(wn−πNwn)||0||en||0+K1||∇(wn−πNwn)||0||en||0+K2||wn−πNwn||0||en||0≤CΩN−m||Dατwn||m||en||0+K1CΩN1−m||wn||m||en||0+K2CΩN−m||wn||m||en||0≤K3N1−m||en||0. $ | (3.9) |
Here $ K_3 = \mathop{\max}\limits_{0\leq n\leq K}\left\lbrace C_{\Omega}||D_\tau^\alpha w^n||_m, K_1C_{\Omega}||w^n||_m, K_2C_{\Omega}||w^n||_m\right\rbrace $, and Lemma 2 is applied. Substituting Eqs (3.6)–(3.9) into Eq (3.5), one gets
| $ Dατ||en||20≤2(K214+K2)||en||20+2(||Qn||0+K3N1−m)||en||0. $ |
Noting that $ e^0 = 0 $ and by Lemma 3, it follows that
| $ ||en||0≤4K3C∗1(τ2−αs+N1−m)Eα(4(K21/4+K2)sn). $ |
By Eq (3.3), we observe
| $ ||wn−Wn||0≤||wn−πNwn||0+||en||0≤CΩN−m||wn||m+4K3C∗1(τ2−αs+N1−m)Eα(4(K21/4+K2)sn)≤K∗(τ2−αs+N1−m), $ |
where $ K^* = \mathop{\max}\limits_{0\leq n\leq K}{\big\{C_{\Omega}||w^n||_m, \; 4K_3C_1^*E_\alpha\big(4(K_1^2/4+K_2)s_n\big)\big\}} $. This completes the proof.
In this section, two numerical examples are given to verify our theoretical results. We define the maximal $ L^2 $ error and the convergence order in time, respectively, as
| $ e(K)=max0≤n≤K||wn−Wn||L2,order=log(e(K1)/e(K2))log(K2/K1). $ | (4.1) |
Example 1. Consider the one-dimensional TFFPEs:
| $ ∂αtu=uxx−2ux+t2u+f(x,t),u∈(−1,1)×(0,1], $ | (4.2) |
where the initial-boundary conditions and the forcing term function $ f $ are choosen by the analytical solution
| $ u(x,t) = (t^2+t^\alpha)(x^3+x^5)\sin(\pi x). $ |
In this case, $q$ is independent of $p_1$, furthermore, $p_1$ and $q$ are not monotone functions.
We solve this problem with the $ TL1 $ Legendre-Galerkin spectral method. Table 1 gives the maximal $ L^2 $ errors, the convergence orders in time and the CPU times with $ N = 14 $. The temporal convergence orders are close to $ 2-\alpha $ in Table 1. For the spatial convergence test, we set $ K = 8192 $. In Figure 1, we give the errors as a function of $ N $ with $ \alpha = 0.3, \; 0.5, \; 0.7 $ in logarithmic scale. We can observe that the errors indicate an exponential decay.
| $ \alpha=0.1 $ | $ \alpha=0.3 $ | $ \alpha=0.5 $ | |||||||||
| $ K $ | $ e(K) $ | order | CPU(s) | $ e(K) $ | order | CPU(s) | $ e(K) $ | order | CPU(s) | ||
| 4 | 5.3660e-03 | * | 1.12e-02 | 7.5697e-03 | * | 9.76e-03 | 8.5571e-03 | * | 9.92e-03 | ||
| 16 | 1.3833e-03 | 0.98 | 2.45e-02 | 1.1574e-03 | 1.35 | 2.19e-02 | 1.3367e-03 | 1.34 | 2.25e-02 | ||
| 64 | 1.7352e-04 | 1.50 | 8.06e-02 | 1.3606e-04 | 1.54 | 7.53e-02 | 1.8311e-04 | 1.43 | 7.40e-02 | ||
| 256 | 1.6850e-05 | 1.68 | 2.90e-01 | 1.4476e-05 | 1.62 | 3.00e-01 | 2.3859e-05 | 1.47 | 3.01e-01 | ||
DownLoad:
CSV
Example 2. Consider the two-dimensional TFFPEs:
| $ \partial_t^\alpha u=\Delta u+t^2 x^2 y^2\left(u_x+u_y\right)+\left(2 t^2 x y^2+2 t^2 x^2 y\right) u, u \in(-1,1)^2 \times(0,1], $ | (4.3) |
where the initial-boundary conditions and the forcing term function $ f $ are choosen by the analytical solution
| $ u(x,y,t) = E_\alpha(-t^\alpha)\sin(\pi x)\sin(\pi y). $ |
Table 2 gives the maximal $ L^2 $ errors, the convergence orders in time and the CPU times with $ N = 14 $. The temporal convergence orders are close to $ 2-\alpha $ in Table 2. For the spatial convergence test, we give the errors as a function of $ N $ for $ \alpha = 0.3, \; 0.5, \; 0.7 $ and $ K = 8192 $ in Figure 2. We use the logarithmic scale for the error-axis. Again, we observe that the errors indicate an exponential decay.
| $ \alpha=0.3 $ | $ \alpha=0.5 $ | $ \alpha=0.7 $ | |||||||||
| $ K $ | $ e(K) $ | order | CPU(s) | $ e(K) $ | order | CPU(s) | $ e(K) $ | order | CPU(s) | ||
| 32 | 7.0619e-05 | * | 2.08e-01 | 1.7386e-04 | * | 1.73e-01 | 3.1316e-04 | * | 1.69e-01 | ||
| 256 | 3.3124e-06 | 1.47 | 1.34e+00 | 9.7836e-06 | 1.38 | 1.28e+00 | 2.3617e-05 | 1.24 | 1.31e+00 | ||
| 2048 | 1.1965e-07 | 1.60 | 1.29e+01 | 4.6734e-07 | 1.46 | 1.28e+01 | 1.6199e-06 | 1.29 | 1.30e+01 | ||
| 8192 | 1.2339e-08 | 1.64 | 8.59e+01 | 5.9649e-08 | 1.48 | 8.54e+01 | 2.6824e-07 | 1.30 | 8.83e+01 | ||
DownLoad:
CSV
We present a $ TL1 $ Legendre-Galerkin spectral method to solve TFFPEs in this paper. The new scheme is convergent with $ O(\tau_s^{2-\alpha}+ N^{1-m}) $, where $ \tau_s $, $ N $ and $ m $ are the time step size, the polynomial degree and the regularity of the analytical solution, respectively. In addition, this $ TL1 $ Legendre-Galerkin spectral method still holds for problems with small $ \alpha $ and gives better numerical solutions near the initial time. The new scheme can achieve a better convergence result on a relatively sparse grid point.
The work of Yongtao Zhou is partially supported by the NSFC (12101037) and the China Postdoctoral Science Foundation (2021M690322).
The authors declare that they have no conflicts of interest.
| [1] |
Leymann H (1990) Mobbing and psychological terror at workplaces. Violence Vict 5: 119–126. doi: 10.1891/0886-6708.5.2.119
|
| [2] | Spiridakis M (2009) Power and mobbing in work. Athens, Dionikos (Book in Greek). |
| [3] | Bakella P, Yiagou E, Brahantini K (2013) Mobbing syndrome effect on nurses' professional life. Hellenic J Nurs Sci 6: 15–21 (article in Greek). |
| [4] |
Yildirim D, Yildirim A (2007) Mobbing in the workplace by peers and managers: mobbing experienced by nurses working in healthcare facilities in Turkey and its effect on nurses. J Clin Nurs 16: 1444–1453. doi: 10.1111/j.1365-2702.2006.01814.x
|
| [5] |
Zachariadou T, Zannetos S, Chira SE, et al. (2018) Prevalence and Forms of Workplace Bullying Among Health-care Professionals in Cyprus: Greek Version of "Leymann Inventory of Psychological Terror" Instrument. Saf Health Work 9: 339–346. doi: 10.1016/j.shaw.2017.11.003
|
| [6] |
Sahin B, Cetin M, Cimen M, et al. (2012) Assessment of Turkish junior male physicians' exposure to mobbing behavior. Croat Med J 53: 357–366. doi: 10.3325/cmj.2012.53.357
|
| [7] |
Pai HC, Lee S (2011) Risk factors for workplace violence in clinical registered nurses in Taiwan. J Clin Nurs 20: 1405–1412. doi: 10.1111/j.1365-2702.2010.03650.x
|
| [8] |
Erdogan V, Yildirim A (2017) Healthcare professionals' exposure to mobbing behaviors and the relation of mobbing with job satisfaction and organizational commitment. Procedia Comput Sci 120: 931–938. doi: 10.1016/j.procs.2017.11.328
|
| [9] | Durmus SC, Topcu I, Yildirim A (2018) Mobbing Behaviors Encountered by Nurses and their Effects on Nurses. Int J Caring Sci 11: 905–913. |
| [10] | Katsilaki EX (2017) Investigation of Mobbing Syndrome and the role of communication in Health Professionals (Doctors & Nurses) at a public hospital. Diploma thesis, EAP. |
| [11] |
Karatza C, Zyga S, Tziaferi S, et al. (2016) Workplace bullying and general health status among the nursing staff of Greek public hospitals. Ann Gen Psychiatry 15: 7. doi: 10.1186/s12991-016-0097-z
|
| [12] | Gavrielatos G (2015). The |
| [13] | Patsia A, Polychronopoulou A, Sakaretsanou AK (2017). The attitude of nurses in public hospitals for labor intimidation. Qualitative research. Thesis. Technological Educational Institution of Western Greece, School of Health and Welfare Professionals, Department of Nursing. |
| [14] | Mastora TM (2013). Exploring Nursing Students' Views on Intimidation in 2013: A Phenomenological Study. Thesis. Department of Nursing School of Health and Welfare Sciences. Technological Educational Institute of Epirus. Available from: http://apothetirio.teiep.gr/xmlui/bitstream/handle/123456789/547/nsl_2013002. pdf?sequence=1. |
| [15] |
Yildirim D (2009) Bullying among nurses and its effects. Int Nurs Rev 56: 504–511. doi: 10.1111/j.1466-7657.2009.00745.x
|
| [16] | Cinar HG, Korkmaz AC, Yılmaz D (2016) Mobbing in Nursing, J hum rhythm 2: 89–93. |
| [17] | da Silva Joao AL, Saldanha Portelada AF (2016) Mobbing and Its Impact on Interpersonal Relationships at the Workplace. J Interpers Violence 2016 Aug 1: 886260516662850. |
| [18] | Ekici D, Beder A (2014) The effects of workplace bullying on physicians and nurses. Aust J Adv Nurs 31: 24–33. |
| [19] |
Malinauskiene V, Einarsen S (2014) Workplace bullying and post-traumatic stress symptoms among family physicians in Lithuania: an occupation and region-specific approach. Int J Occup Med Environ Health 27: 919–932. doi: 10.2478/s13382-014-0328-y
|
| [20] |
Najafi F, Fallahi‑Khoshknab M, Ahmadi F, et al. (2018) Antecedents and consequences of workplace violence against nurses: A qualitative study. J Clin Nurs 27: e116–e128. doi: 10.1111/jocn.13884
|
| [21] |
Hassankhani H, Parizad N, Gacki-Smith J, et al. (2018) The consequences of violence against nurses working in the emergency department: A qualitative study. Int Emerg Nurs 39: 20–25. doi: 10.1016/j.ienj.2017.07.007
|
| [22] | Dikmetaş E, Top M, Ergin G (2011) An examination of mobbing and burnout of residents. Turk Psikiyatri Derg 22: 137–149. |
| [23] | Serafeimidou A, Dimou M (2016) Workplace bullying with special emphasis in the Greek public sector-A review article. 11th MIBES Conference-Heraklion, Crete, Greece, 22–24 June, 2016, 454−473. Available from: http://mibes.teilar.gr/proceedings/2016/Serafeimidou-Dimou.pdf. |
| [24] | Pantazis A, George I (2016) Workplace Bullying within Nursing Stuff [Greek]. Hellenic J Nurs 55: 320–328. |
| [25] | Fountouki A, Ourania P, Theofanidis D (2011) Nursing staff under heavy stress: focus on Greece. A critical review. Int J Caring Sci 4: 11–19. |
| [26] |
Rodríguez-Carballeira A, Escartín Solanelles J, Visauta Vinacua B, et al. (2010) Categorization and hierarchy of workplace bullying strategies: A Delphi survey. Span J Psychol 13: 297–308. doi: 10.1017/S1138741600003875
|
| [27] | Koinis A, Velonakis E, Tzavella F, et al. (2017) The Phenomenon of Mobbing in the Greek Health Sector: A pilot study [Greek]. Hellenic J Nurs 56: 233–244. |
| [28] | Koinis A, Tziaferi SG (2018) The phenomenon of Mobbing among health professionals in a Greek public hospital: A qualitative study. Announcement at 32nd International Congress of Occupational Health (ICOH), Dublin, Ireland, 29 April–04 May, 2018. |
| [29] | Koinis A, Tzavella F, Velonakis E, et al. (2016) The mobbing in the health sector and its impact on quality of life of health professionals. A pilot study. Announcement at the 6th International FOHNEU Congress "New Occupational Health Horizons", Rotterdam, The Netherlands, 16–18 March, 2016. |
| [30] | Koinis A, Tzavella F, Velonakis E, et al. (2015) Effects of the phenomenon of mobbing on the quality of life of health professionals, in public and private hospitals of the first and sixth Health Region of Greece. A study protocol. Announcement at the 3rd International Congress on Occupational Health (ICOH)-Global Harmony for Occupational Health: Bridge the World. Seoul, Korea, 31 May–5 June, 2015. |
| [31] | Koinis A, Tziaferi SG (2014) Managing Ethical Harassment in the Workplace at Health Services, organized by the Medical and Nursing Service of the MN. Mesinia Rapporteur at the "Modern Nursing" Conference, Hematology News 9th Cycle, 20–21 September, 2014. |
| [32] | Drivas S (2002) The syndrome of mobbing at work. Petrol Refineries Chem Ind 9: 8–9. |
| [33] | Karakioulafi X (2003) Psychological and moral harassment in the workplace: definitions of the problem, institutional treatment and involvement of the employee representation bodies in managing and dealing with them. Labor Rel Rev 32: 21–37. |
| [34] | Konstantinidis M (2011) Labor harassment: A proposal for research and intervention. Occup Health Saf v.47. |
| [35] | Mueller R (2000) Basic principles of structural equation modeling. New York, USA: Springer. |
| [36] |
Hu L, Bentler PM (1999) Cut-off criteria for fit indices in covariance structure analysis: Conventional criteria versus new alternatives. Struct Equat Model 6: 1–55. doi: 10.1080/10705519909540118
|
| [37] |
Al-Omari H (2015) Physical and verbal workplace violence against nurses in Jordan. Int Nurs Rev 62: 111–118. doi: 10.1111/inr.12170
|
| [38] |
Jaradat Y, Nielsen MB, Kristensen P, et al. (2016) Workplace aggression, psychological distress, and job satisfaction among Palestinian nurses: A cross-sectional study. Appl Nurs Res 32: 190–198. doi: 10.1016/j.apnr.2016.07.014
|
| [39] |
Cheung T, Yip PS (2017) Workplace violence towards nurses in Hong Kong: prevalence and correlates. BMC Public Health 17: 196. doi: 10.1186/s12889-017-4112-3
|
| [40] | Al-Shamlan NA, Jayaseeli N, Al-Shawi MM, et al. (2017) Are nurses verbally abused? A cross-sectional study of nurses at a university hospital, Eastern Province, Saudi Arabia. J Fam Community Med 24: 173–180. |
| [41] |
Li Z, Yan CM, Shi L, et al. (2017) Workplace violence against the medical staff of Chinese children's hospitals: A cross-sectional study. PloS One 12: e0179373. doi: 10.1371/journal.pone.0179373
|
| [42] |
Mantzouranis G, Fafliora E, Bampalis VG, et al. (2015) Assessment and Analysis of Workplace Violence in a Greek Tertiary Hospital. Arch Environ Occup Health 70: 256–264. doi: 10.1080/19338244.2013.879564
|
| [43] |
Cortina LM, Magley VJ, Williams JH, et al. (2001) Incivility in the workplace: Incidence and impact. J Occup Health Psychol 6: 64–80. doi: 10.1037/1076-8998.6.1.64
|
| [44] |
Niedhammer I, David S, Degioanni S (2007) Economic activities and occupations at high risk for workplace bullying: Results from a large-scale cross-sectional survey in the general working population in France. Int Arch Occup Environ Health 80: 346–353. doi: 10.1007/s00420-006-0139-y
|
| [45] |
Jóhannsdóttir HL, Ólafsson RF (2004) Coping with bullying in the workplace: The effect of gender, age, and type of bullying. Brit J Guid Couns 32: 319–333. doi: 10.1080/03069880410001723549
|
| [46] |
Ortega A, Høgh A, Pejtersen JH, et al. (2009) Prevalence of workplace bullying and risk groups: A representative population study. Int Arch Occup Environ Health 82: 417–426. doi: 10.1007/s00420-008-0339-8
|
| [47] | Namie G (2003) Workplace bullying: Escalated incivility. Ivey Bus J 68: 1–6. |
| [48] | Iftikhar M, Qureshi MI (2014) Modeling the workplace bullying the mediator of "workplace climate-employee health" relationship. J Manage Info 4: 96–124. |
| [49] |
Zhang L, Wang A, Xie X, et al. (2017) Workplace violence against nurses: A cross-sectional study. Int J Nurs Stud 72: 8–14. doi: 10.1016/j.ijnurstu.2017.04.002
|
| [50] |
Davidoff F, Batalden P, Stevens D, et al. (2008) SQUIRE Development Group Publication Guidelines for Improvement Studies in Health Care: Evolution of the SQUIRE Project. Ann Intern Med 149: 670–676. doi: 10.7326/0003-4819-149-9-200811040-00009
|
| [51] |
Dilek Y, Aytolan Y (2008) Development and psychometric evaluation of workplace psychologically violent behaviors instrument. J Clin Nurs 17: 1361–1370. doi: 10.1111/j.1365-2702.2007.02262.x
|
| [52] | Economou C (2010) Greece: Health system review. Health Syst Transit 12: 1–180. |
| [53] | Pinar T, Acikel C, Pina, G, et al. (2015) Workplace violence in the health sector in Turkey: A national study. J Interpers Violence 32: 2345–2365. |
| [54] |
AbuAlRub RF, Al Khawaldeh AT (2014) Workplace physical violence among hospital nurses and physicians in underserved areas in Jordan. J Clin Nurs 23: 1937–1947. doi: 10.1111/jocn.12473
|
| [55] |
Hamdan M, Hamra AA (2015) Workplace violence towards workers in the emergency departments of Palestinian hospitals: a cross-sectional study. Hum Resour Health 13: 28. doi: 10.1186/s12960-015-0018-2
|
| 1. | Yanping Chen, Jixiao Guo, Unconditional error analysis of the linearized transformed L1 virtual element method for nonlinear coupled time-fractional Schrödinger equations, 2025, 457, 03770427, 116283, 10.1016/j.cam.2024.116283 | |
| 2. | Yongtao Zhou, Mingzhu Li, Error estimate of a transformed L1 scheme for a multi-term time-fractional diffusion equation by using discrete comparison principle, 2024, 217, 03784754, 395, 10.1016/j.matcom.2023.11.010 | |
| 3. | Asghar Ali, Jamshad Ahmad, Sara Javed, Rashida Hussain, Mohammed Kbiri Alaoui, Muhammad Aqeel, Numerical simulation and investigation of soliton solutions and chaotic behavior to a stochastic nonlinear Schrödinger model with a random potential, 2024, 19, 1932-6203, e0296678, 10.1371/journal.pone.0296678 | |
| 4. | Zemian Zhang, Yanping Chen, Yunqing Huang, Jian Huang, Yanping Zhou, A continuous Petrov–Galerkin method for time-fractional Fokker–Planck equation, 2025, 03770427, 116689, 10.1016/j.cam.2025.116689 |
Aristotelis Koinis, Emmanouil Velonakis, Chara Tzavara, Foteini Tzavella, Styliani Tziaferi. Psychometric properties of the workplace psychologically violent behaviors-WPVB instrument. Translation and validation in Greek Health Professionals[J]. AIMS Public Health, 2019, 6(1): 79-95. doi: 10.3934/publichealth.2019.1.79
| $ \alpha=0.1 $ | $ \alpha=0.3 $ | $ \alpha=0.5 $ | |||||||||
| $ K $ | $ e(K) $ | order | CPU(s) | $ e(K) $ | order | CPU(s) | $ e(K) $ | order | CPU(s) | ||
| 4 | 5.3660e-03 | * | 1.12e-02 | 7.5697e-03 | * | 9.76e-03 | 8.5571e-03 | * | 9.92e-03 | ||
| 16 | 1.3833e-03 | 0.98 | 2.45e-02 | 1.1574e-03 | 1.35 | 2.19e-02 | 1.3367e-03 | 1.34 | 2.25e-02 | ||
| 64 | 1.7352e-04 | 1.50 | 8.06e-02 | 1.3606e-04 | 1.54 | 7.53e-02 | 1.8311e-04 | 1.43 | 7.40e-02 | ||
| 256 | 1.6850e-05 | 1.68 | 2.90e-01 | 1.4476e-05 | 1.62 | 3.00e-01 | 2.3859e-05 | 1.47 | 3.01e-01 | ||
DownLoad:
CSV
| $ \alpha=0.3 $ | $ \alpha=0.5 $ | $ \alpha=0.7 $ | |||||||||
| $ K $ | $ e(K) $ | order | CPU(s) | $ e(K) $ | order | CPU(s) | $ e(K) $ | order | CPU(s) | ||
| 32 | 7.0619e-05 | * | 2.08e-01 | 1.7386e-04 | * | 1.73e-01 | 3.1316e-04 | * | 1.69e-01 | ||
| 256 | 3.3124e-06 | 1.47 | 1.34e+00 | 9.7836e-06 | 1.38 | 1.28e+00 | 2.3617e-05 | 1.24 | 1.31e+00 | ||
| 2048 | 1.1965e-07 | 1.60 | 1.29e+01 | 4.6734e-07 | 1.46 | 1.28e+01 | 1.6199e-06 | 1.29 | 1.30e+01 | ||
| 8192 | 1.2339e-08 | 1.64 | 8.59e+01 | 5.9649e-08 | 1.48 | 8.54e+01 | 2.6824e-07 | 1.30 | 8.83e+01 | ||
DownLoad:
CSV
| $ \alpha=0.1 $ | $ \alpha=0.3 $ | $ \alpha=0.5 $ | |||||||||
| $ K $ | $ e(K) $ | order | CPU(s) | $ e(K) $ | order | CPU(s) | $ e(K) $ | order | CPU(s) | ||
| 4 | 5.3660e-03 | * | 1.12e-02 | 7.5697e-03 | * | 9.76e-03 | 8.5571e-03 | * | 9.92e-03 | ||
| 16 | 1.3833e-03 | 0.98 | 2.45e-02 | 1.1574e-03 | 1.35 | 2.19e-02 | 1.3367e-03 | 1.34 | 2.25e-02 | ||
| 64 | 1.7352e-04 | 1.50 | 8.06e-02 | 1.3606e-04 | 1.54 | 7.53e-02 | 1.8311e-04 | 1.43 | 7.40e-02 | ||
| 256 | 1.6850e-05 | 1.68 | 2.90e-01 | 1.4476e-05 | 1.62 | 3.00e-01 | 2.3859e-05 | 1.47 | 3.01e-01 | ||
| $ \alpha=0.3 $ | $ \alpha=0.5 $ | $ \alpha=0.7 $ | |||||||||
| $ K $ | $ e(K) $ | order | CPU(s) | $ e(K) $ | order | CPU(s) | $ e(K) $ | order | CPU(s) | ||
| 32 | 7.0619e-05 | * | 2.08e-01 | 1.7386e-04 | * | 1.73e-01 | 3.1316e-04 | * | 1.69e-01 | ||
| 256 | 3.3124e-06 | 1.47 | 1.34e+00 | 9.7836e-06 | 1.38 | 1.28e+00 | 2.3617e-05 | 1.24 | 1.31e+00 | ||
| 2048 | 1.1965e-07 | 1.60 | 1.29e+01 | 4.6734e-07 | 1.46 | 1.28e+01 | 1.6199e-06 | 1.29 | 1.30e+01 | ||
| 8192 | 1.2339e-08 | 1.64 | 8.59e+01 | 5.9649e-08 | 1.48 | 8.54e+01 | 2.6824e-07 | 1.30 | 8.83e+01 | ||
