
The media plays a dual role of "supervision" and "collusion" in governance mechanisms. This study investigates the impact of media attention and economic policy uncertainty on green innovation by analyzing A-share industrial listed enterprises data between 2011 and 2020. The results show that media attention can effectively promote green innovation and that this impact is significantly heterogeneous. Media attention significantly affects green innovation in non-state-owned enterprises and manufacturing companies positively, but it is insignificant for state-owned enterprises and mining and energy supply industries. Moreover, the results indicate that external economic policy uncertainty can lead enterprises to take early measures to hedge risks, thereby positively regulating the promotion effect of media attention on green innovation during economic fluctuations. Finally, media attention can promote green innovation by increasing environmental regulation intensity, reducing corporate financing constraints, and enhancing corporate social responsibility. Therefore, paying full attention to the media as an institutional subject outside of laws and regulations, gradually forming a pressure-driven mechanism for corporate green innovation, and reducing information opacity, is a pivotal way to promote enterprises' green innovation.
Citation: Yang Xu, Conghao Zhu, Runze Yang, Qiying Ran, Xiaodong Yang. Applications of linear regression models in exploring the relationship between media attention, economic policy uncertainty and corporate green innovation[J]. AIMS Mathematics, 2023, 8(8): 18734-18761. doi: 10.3934/math.2023954
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The media plays a dual role of "supervision" and "collusion" in governance mechanisms. This study investigates the impact of media attention and economic policy uncertainty on green innovation by analyzing A-share industrial listed enterprises data between 2011 and 2020. The results show that media attention can effectively promote green innovation and that this impact is significantly heterogeneous. Media attention significantly affects green innovation in non-state-owned enterprises and manufacturing companies positively, but it is insignificant for state-owned enterprises and mining and energy supply industries. Moreover, the results indicate that external economic policy uncertainty can lead enterprises to take early measures to hedge risks, thereby positively regulating the promotion effect of media attention on green innovation during economic fluctuations. Finally, media attention can promote green innovation by increasing environmental regulation intensity, reducing corporate financing constraints, and enhancing corporate social responsibility. Therefore, paying full attention to the media as an institutional subject outside of laws and regulations, gradually forming a pressure-driven mechanism for corporate green innovation, and reducing information opacity, is a pivotal way to promote enterprises' green innovation.
In recent years, fractional differential equations have gained prominence due to their proven usefulness in several unrelated scientific and engineering fields. For example, the nonlinear oscillations of an earthquake can be characterized by a fractional derivative, and the fractional derivative of the traffic fluid dynamics model can solve the insufficiency resulting from the assumption of continuous traffic flows [1,3]. Numerous chemical processes, mathematical biology, engineering, and scientific problems [4,5,6,7] are also modeled with fractional differential equations. Nonlinear partial differential equations (NPDEs) characterize various physical, biological, and chemical phenomena. Current research is focused on developing precise traveling wave solutions for such equations. Exact and explicit solutions help scientists understand the complicated physical phenomena and dynamic processes portrayed by NPDEs [8,9,10]. In the past four decades, numerous essential methodologies for attaining accurate solutions to NPDEs have been proposed [11,12].
The Helmholtz equation (HE) derives from the elliptic and wave equations. In a multi-dimensional nonhomogeneous isotropic standard with velocity c, the wave result is υ(ξ,ψ), which corresponds to a source of harmonic (ξ,ψ) vibrating at a given frequency and satisfying the Helmholtz equation in the area R. The classical order HE is
D2ξυ(ξ,ψ)+D2ψυ(ξ,ψ)+ευ(ξ,ψ)=−υ(ξ,ψ). | (1.1) |
Here, υ is a suitable boundary differentiable term of R, is a known function, and the wave number with wavelength 2/ξ=0 renders Eq (1.1) homogeneous. If (1.1) is expressed as
D2ξυ(ξ,ψ)+D2ψυ(ξ,ψ)−ευ(ξ,ψ)=−υ(ξ,ψ). |
Then it explains mass transfer with density biochemical processes of the 1st order. Equation (1.1) is investigated using the decomposition method [13], the finite element approach [14], the differential transform method [15], the Trefftz method [16], and the spectral collocation method [17], among others [3,4,5].
The Helmholtz equation is a partial differential equation that describes wave phenomena in various fields of physics, such as electromagnetism, acoustics, and fluid mechanics. Traditionally, the Helmholtz equation has been formulated using integer-order derivatives. However, in recent years, there has been a growing interest in the use of fractional-order derivatives to describe complex phenomena more accurately. In particular, fractional-order space Helmholtz equations are derived directly from mathematical formulas that involve fractional derivatives, rather than being generalized from integer-order space derivative Helmholtz equations. These equations can provide a more accurate description of wave propagation in complex media, such as porous materials, biological tissues, and fractal structures. Fractional-order space Helmholtz equations have attracted significant attention due to their potential applications in a wide range of fields, including medical imaging, geophysics, and telecommunications. They offer a promising avenue for understanding the behavior of waves in complex media and developing new technologies for wave-based sensing and imaging [6,7,8].
It is advantageous to utilize fractional differential equations in physical problems due to their nonlocal features. Non-locality characterizes fractional-order derivatives, whereas locality characterizes integer-order derivatives [24,25,26,27]. It demonstrates that the future state of the physical system depends on all of its previous states in addition to its current state. Consequently, fractional models are more accurate. In fractional differential equations, the response expression has a parameter that specifies the fractional derivative of the variable order, which may vary to achieve many responses [9,10,11].
Standard HEs can be generalized to fractional-order Helmholtz equations by extending the Caputo fractional-order space derivative to the integer-order space derivative. The fractional Helmholtz equation in space is
Dϱξυ(ξ,ψ)+D2ψυ(ξ,ψ)+ευ(ξ,ψ)=−ψ(ξ,ψ), |
with υ(0,ψ)=g(ψ) as the initial condition (IC). Gupta et al. [31] solved the multi-dimensional fractional Helmholtz equation using the homotopy perturbation approach. In contrast, Abuasad et al. [14] recently solved a fractional model of the Helmholtz problem using the reduced differential transform method.
This section describes the properties of the fractional derivatives and a few essential details concerning the Yang transform.
Definition 2.1. The fractional derivative in terms of Caputo is as follows
Dϱψυ(ξ,ψ)=1Γ(k−ϱ)∫ψ0(ψ−ϱ)k−ϱ−1υ(k)(ξ,ϱ)dϱ,k−1<ϱ≤k,k∈N. | (2.1) |
Definition 2.2. The YT is represented as follows
Y{υ(ψ)}=M(u)=∫∞0e−ψuυ(ψ)dψ, ψ>0, u∈(−ψ1,ψ2), | (2.2) |
having inverse YT as follows
Y−1{M(u)}=υ(ψ). | (2.3) |
Definition 2.3. The nth derivative YT is stated as follows
Y{υn(ψ)}=M(u)un−n−1∑k=0υk(0)un−k−1, ∀ n=1,2,3,⋯ | (2.4) |
Definition 2.4. The YT of derivative having fractional-order is stated as follows
Y{υϱ(ψ)}=M(u)uϱ−n−1∑k=0υk(0)uϱ−(k+1), 0<ϱ≤n. | (2.5) |
Consider the general fractional partial differential equations,
Dϱψυ(ξ,ψ)+Mυ(ξ,ψ)+Nυ(ξ,ψ)=h(ξ,ψ),ψ>0,0<ϱ≤1,υ(ξ,0)=g(ξ),ν∈ℜ. | (3.1) |
Using Yang transform of Eq (3.1), we get
Y[Dϱyυ(ξ,ψ)+Mυ(ξ,ψ)+Nυ(ξ,ψ)]=Y[h(ξ,ψ)],ψ>0,0<ϱ≤1,υ(ξ,ψ)=sg(ξ)+sϱY[h(ξ,ψ)]−sϱY[Mυ(ξ,ψ)+Nυ(ξ,ψ)]. | (3.2) |
Now, applying inverse Yang transform, we have
υ(ξ,ψ)=F(ξ,ψ)−Y−1[sϱY{Mυ(ξ,ψ)+Nυ(ξ,ψ)}], | (3.3) |
where
F(ξ,ψ)=Y−1[sg(ξ)+sϱY[h(ξ,ψ)]]=g(ν)+Y−1[sϱY[h(ξ,ψ)]]. | (3.4) |
The parameter p is perturbation technique and p∈[0,1] defined as
υ(ξ,ψ)=∞∑k=0pkυk(ξ,ψ), | (3.5) |
The nonlinear function is expressed as
Nυ(ξ,ψ)=∞∑k=0pkHk(υk), | (3.6) |
where Hn are He's polynomials in term of υ0,υ1,υ2,⋯,υn, and can be calculated as
Hn(υ0,υ1,⋯,υn)=1ϱ(n+1)Dıp[N(∞∑ı=0pıυı)]p=0, | (3.7) |
where Dıp=∂ı∂pı.
Putting Eqs (3.6) and (3.7) in Eq (3.3), we achieved as
∞∑ı=0pıυı(ξ,ψ)=F(ξ,ψ)−p×[Y−1{sϱY{M∞∑ı=0pıυı(ξ,ψ)+∞∑ı=0pıHı(υı)}}]. | (3.8) |
Comparison both sides of coefficient p, we get
p0:υ0(ξ,ψ)=F(ξ,ψ),p1:υ1(ξ,ψ)=Y−1[sϱY(Mυ0(ξ,ψ)+H0(υ))],p2:υ2(ξ,ψ)=Y−1[sϱY(Mυ1(ξ,ψ)+H1(υ))],⋮pı:υı(ξ,ψ)=Y−1[sϱY(Mυı−1(ξ,ψ)+Hı−1(υ))],ı>0,ı∈N. | (3.9) |
Finally, present the obtained solution and check it with any available analytical or numerical solutions for the given PDE. The υı(ξ,ψ) components can be calculated easily which quickily converges to series form. We can get p→1,
υ(ξ,ψ)=limM→∞M∑ı=1υı(ξ,ψ). | (3.10) |
Problem 4.1. Consider the space fractional Helmholtz equation
∂ϱυ(ξ,ψ)∂ξϱ+∂2υ(ξ,ψ)∂ψ2−υ(ξ,ψ)=0,1<ϱ≤2, | (4.1) |
with the ICs
υ(0,ψ)=ψ andυξ(0,ψ)=0. | (4.2) |
Using the Yang transform of Eq (4.1), we obtained as
1sϱY[υ(ξ,ψ)]=υ(0,ψ)s1−ϱ−Y{∂2υ(ξ,ψ)∂ψ2−υ(ξ,ψ)}, | (4.3) |
Y[υ(ξ,ψ)]=sυ(0,ψ)−sϱY{∂2υ(ξ,ψ)∂ψ2−υ(ξ,ψ)}, | (4.4) |
Taking inverse Yang Transformation, we have
Y[υ(ξ,ψ)]=ψ−Y−1[sϱY{∂2υ(ξ,ψ)∂ψ2−υ(ξ,ψ)}], | (4.5) |
Implemented HPM in Eq (4.5), we can achieve as
∞∑ı=0pıυı(ξ,ψ)=ψ−p[Y−1{sϱY{(∞∑ı=0pıυı(ξ,ψ))ψψ−∞∑ı=0pıυı(ξ,ψ)}}]. | (4.6) |
On both sides comparing coefficients of p, we get
p0:υ0(ξ,ψ)=ψ,p1:υ1(ξ,ψ)=−Y−1[sϱY{∂2υ0(ξ,ψ)∂ψ2−υ0(ξ,ψ)}]=ξϱΓ(ϱ+1)ψ,p2:υ2(ξ,ψ)=−Y−1[sϱY{∂2υ1(ξ,ψ)∂ψ2−υ1(ξ,ψ)}]=ξ2ϱΓ(2ϱ+1)ψ,p3:υ3(ξ,ψ)=−Y−1[sϱY{∂2υ2(ξ,ψ)∂ψ2−υ2(ξ,ψ)}]=ξ3ϱΓ(3ϱ+1)ψ,p4:υ4(ξ,ψ)=−Y−1[sϱY{∂2υ3(ξ,ψ)∂ψ2−υ3(ξ,ψ)}]=ξ4ϱΓ(4ϱ+1)ψ,⋮ | (4.7) |
The series type result of the first problem example is
υ(ξ,ψ)=υ0(ξ,ψ)+υ1(ξ,ψ)+υ2(ξ,ψ)+υ3(ξ,ψ)+υ4(ξ,ψ)+⋯υ(ξ,ψ)=ψ[1+ξϱΓ(ϱ+1)+ξ2ϱΓ(2ϱ+1)+x3ϱΓ(3ϱ+1)+ξ4ϱΓ(4ϱ+1)+⋯]. | (4.8) |
The exact solution is
υ(ξ,ψ)=ψcoshξ. |
Similarly y-space can be calculated as:
∂ϱυ(ξ,ψ)∂ψϱ+∂2υ(ξ,ψ)∂ξ2−υ(ξ,ψ)=0, | (4.9) |
with the IC
υ(ξ,0)=ξ. | (4.10) |
Thus, the solution of the above Eq (4.9) is obtain as
υ(ξ,ψ)=ξ(1+ψϱΓ(ϱ+1)+ψ2ϱΓ(2ϱ+1)+ψ3ϱΓ(3ϱ+1)+ψ4ϱΓ(4ϱ+1)+…), |
in the case when ϱ=2, then the solution through HPTM is
υ(ξ,ψ)=ξcoshψ. | (4.11) |
Figure 1 illustrates the exact and HPTM solutions in two-dimensional plots for various values of ϱ ranging from 2 to 1.5, with ξ values ranging from 0 to 1 and ψ set to 1. The solutions are presented in Figure 1a and b for the exact and HPTM methods, respectively. Figure 2 displays the 3-dimensional plots of the exact and HPTM solutions for ϱ=2 and analyzes the point of intersection between the two solutions. Figure 2c and d depict the HPTM solutions at ϱ=1.8 and 1.6 respectively for Problem 4.1. The fractional results were also evaluated for their convergence towards an integer-order result for each problem. Similarly, the figures for the ψ-space can also be generated using the same approach.
Exact result | ||||
(0.2, 0.1) | 0.2102371 | 0.2100072 | 0.2100000 | 0.2100000 |
(0.4, 0.1) | 0.4104630 | 0.4100141 | 0.4100000 | 0.4100000 |
(0.6, 0.1) | 0.6106889 | 0.6100209 | 0.6100000 | 0.6100000 |
(0.2, 0.2) | 0.2103355 | 0.2100121 | 0.2100000 | 0.2100000 |
(0.4, 0.2) | 0.4106550 | 0.4100237 | 0.4100000 | 0.4100000 |
(0.6, 0.2) | 0.6109746 | 0.6100353 | 0.6100000 | 0.6100000 |
(0.2, 0.3) | 0.2104110 | 0.2100164 | 0.2100000 | 0.2100000 |
(0.4, 0.3) | 0.4108025 | 0.4100321 | 0.4100000 | 0.4100000 |
(0.6, 0.3) | 0.6111940 | 0.6100478 | 0.6100000 | 0.6100000 |
(0.2, 0.4) | 0.2104747 | 0.2100204 | 0.2100000 | 0.2100000 |
(0.4, 0.4) | 0.4109269 | 0.4100399 | 0.4100000 | 0.4100000 |
(0.6, 0.4) | 0.6113790 | 0.6100593 | 0.6100000 | 0.6100000 |
(0.2, 0.5) | 0.2105309 | 0.2100241 | 0.2100000 | 0.2100000 |
(0.4, 0.5) | 0.4110365 | 0.4100471 | 0.4100000 | 0.4100000 |
(0.6, 0.5) | 0.6115421 | 0.6100701 | 0.6100000 | 0.6100000 |
Problem 4.2. Consider the space-fractional HE
∂ϱυ(ξ,ψ)∂ξϱ+∂2υ(ξ,ψ)∂ψ2+5υ(ξ,ψ)=0,1<ϱ≤2, | (4.12) |
with the ICs
υ(0,ψ)=ψ andυξ(0,ψ)=0. | (4.13) |
Using the Yang transform of Eq (4.12), we obtain as
1sϱY[υ(ξ,ψ)]=υ(0,ψ)s1−ϱ−Y{∂2υ(ξ,ψ)∂ψ2+5υ(ξ,ψ)}, | (4.14) |
Y[υ(ξ,ψ)]=sυ(0,ψ)−sϱY{∂2υ(ξ,ψ)∂ψ2+5υ(ξ,ψ)}. | (4.15) |
Applying the inverse Yang Transform, we get
Y[υ(ξ,ψ)]=ψ−Y−1[sϱY{∂2υ(ξ,ψ)∂ψ2+5υ(ξ,ψ)}], | (4.16) |
Using the HPM in Eq (4.16), we obtained as
∞∑ı=0pıυı(ξ,ψ)=ψ−p[Y−1{sϱY{(∞∑ı=0pıυı(ξ,ψ))ψψ+5∞∑ı=0pıυı(ξ,ψ)}}]. | (4.17) |
On both sides comparing coefficients of p, we get
p0:υ0(ξ,ψ)=ψ,p1:υ1(ξ,ψ)=−Y−1[sϱY{∂2υ0(ξ,ψ)∂ψ2+5υ0(ξ,ψ)}]=−5ψξϱΓ(ϱ+1),p2:υ2(ξ,ψ)=−Y−1[sϱY{∂2υ1(ξ,ψ)∂ψ2+5υ1(ξ,ψ)}]=25ψξ2ϱΓ(2ϱ+1),p3:υ3(ξ,ψ)=−Y−1[sϱY{∂2υ2(ξ,ψ)∂ψ2+5υ2(ξ,ψ)}]=−125ξ3ϱΓ(3ϱ+1),p4:υ4(ξ,ψ)=−Y−1[sϱY{∂2υ3(ξ,ψ)∂ψ2+5υ3(ξ,ψ)}]=625ψξ4ϱΓ(4ϱ+1),⋮ | (4.18) |
The series type result of second problem as
υ(ξ,ψ)=υ0(ξ,ψ)+υ1(ξ,ψ)+υ2(ξ,ψ)+υ3(ξ,ψ)+υ4(ξ,ψ)+⋯υ(ξ,ψ)=ψ[1−5ξϱΓ(ϱ+1)+25ξ2ϱΓ(2ϱ+1)−125x3ϱΓ(3ϱ+1)+625ξ4ϱΓ(4ϱ+1)+⋯]. | (4.19) |
The exact solution is
υ(ξ,ψ)=ψcos√5ξ. |
Now similarly, the result of y-space can be calculated with the help of homotopy perturbation
∂ϱυ(ξ,ψ)∂ψϱ+∂2υ(ξ,ψ)∂ξ2+5υ(ξ,ψ)=0, | (4.20) |
with the IC
υ(ξ,0)=ξ. | (4.21) |
The solution of the Eq (4.20) is expressed as
υ(ξ,ψ)=ξ(1−5ψϱΓ(ϱ+1)+25ψ2ϱΓ(2ϱ+1)−125ψ3ϱΓ(3ϱ+1)+625ψ4ϱΓ(4ϱ+1)+⋯). |
The exact solution is
υ(ξ,ψ)=ξcos√5ψ. | (4.22) |
Figure 3 illustrates the solutions of exact and HPTM in two-dimensional plots, as shown in Figure 3a and b for different values of ϱ, ranging from 2 to 1.5, respectively. The interval considered for ξ is [0, 1], while ψ is constant at 1. The results obtained from the fractional-order model converge to the integer-order solution of the problem. In Figure 4, the 3-dimensional plots of exact and HPTM solutions are presented in Figures (a) and (b), respectively, for ϱ=2. The closed contact of the two solutions is analyzed. Additionally, Figure 4c and d depict the HPTM solutions at ϱ=1.8 and 1.6, respectively, for Problem 4.2. Similarly, graphs for ψ-space can also be generated.
Problem 4.3. Consider the space-fractional HE
∂ϱυ(ξ,ψ)∂ξϱ+∂2υ(ξ,ψ)∂ψ2−2υ(ξ,ψ)=(12ξ2−3ξ4)sinψ,1<ϱ≤2,0≤ψ≤2π, | (4.23) |
with the ICs
υ(0,ψ)=0 andυξ(0,ψ)=0. | (4.24) |
Applying the Yang transform of Eq (4.23), we achieve
1sϱY[υ(ξ,ψ)]=υ(0,ψ)s1−ϱ−Y{∂2υ(ξ,ψ)∂ψ2−2υ(ξ,ψ)}, | (4.25) |
Y[υ(ξ,ψ)]=sυ(0,ψ)−sϱY{∂2υ(ξ,ψ)∂ψ2−2υ(ξ,ψ)}. | (4.26) |
Implementing inverse Yang transform, we get
Y[υ(ξ,ψ)]=(ξ4−ξ610)sinψ−Y−1[sϱY{∂2υ(ξ,ψ)∂ψ2−2υ(ξ,ψ)}]. | (4.27) |
Applying Homotopy perturbation method in Eq (4.27), we achieved as
∞∑ı=0pıυı(ξ,ψ)=ψ−p[Y−1{sϱY{(∞∑ı=0pıυı(ξ,ψ))ψψ−2∞∑ı=0pıυı(ξ,ψ)}}]. | (4.28) |
Both sides on comparison coefficients of p, we obtain
p0:υ0(ξ,ψ)=(ξ4−ξ610)sinψ,p1:υ1(ξ,ψ)=−Y−1[sϱY{∂2υ0(ξ,ψ)∂ψ2−2υ0(ξ,ψ)}]=3(ξϱ+4Γ(ϱ+5)−72ξϱ+6Γ(ϱ+7))sinψ,p2:υ2(ξ,ψ)=−Y−1[sϱY{∂2υ1(ξ,ψ)∂ψ2−2υ1(ξ,ψ)}]=3(ξ2ϱ+4Γ(2ϱ+5)−216ξ2ϱ+6Γ(2ϱ+7))sinψ,p3:υ3(ξ,ψ)=−Y−1[sϱY{∂2υ2(ξ,ψ)∂ψ2−2υ2(ξ,ψ)}]=3(ξ3ϱ+4Γ(3ϱ+5)−648ξ3ϱ+6Γ(3ϱ+7))sinψ,p4:υ4(ξ,ψ)=−Y−1[sϱY{∂2υ3(ξ,ψ)∂ψ2−2υ3(ξ,ψ)}]=3(ξ4ϱ+4Γ(4ϱ+5)−1944ξ2ϱ+6Γ(2ϱ+7))sinψ,⋮ | (4.29) |
The series type result of the third problem is
υ(ξ,ψ)=υ0(ξ,ψ)+υ1(ξ,ψ)+υ2(ξ,ψ)+υ3(ξ,ψ)+υ4(ξ,ψ)+⋯υ(ξ,ψ)=(ξ4−ξ610)sinψ+3(ξϱ+4Γ(ϱ+5)−72ξϱ+6Γ(ϱ+7))sinψ+3(ξ2ϱ+4Γ(2ϱ+5)−216ξ2ϱ+6Γ(2ϱ+7))sinψ+3(ξ3γ+4Γ(3ϱ+5)−648ξ3ϱ+6Γ(3ϱ+7))sinψ+3(ξ4ϱ+4Γ(4ϱ+5)−1944ξ2ϱ+6Γ(2ϱ+7))sinψ+⋯. | (4.30) |
The exact solution is
υ(ξ,ψ)=ξ4sinψ. |
Figure 5a and b display the exact and HPTM solutions, respectively, in a 3-dimensional plot at ϱ=2. The closed contact between the exact and HPTM solutions is examined. Figure 6 depicts the exact and HPTM solutions in two-dimensional plot for various values of ϱ=2,1.9,1.8,1.7,1.6,1.5 for ξ∈[0,1] and ψ=1. The fractional results are observed to approach an integer-order solution of the problem. Similarly, the graphs for ψ-space fractional-order derivative can also be plotted.
In this study, fractional-order Helmholtz equations were solved using the Homotopy Perturbation Yang transform method. Due to the great agreement between the generated approximative solution and the precise solution, the homotopy perturbation Yang transform method was demonstrated to be a successful method for solving partial differential equations with Caputo operators. The computation size of the approach was compared to those required by other numerical methods to demonstrate how tiny it is. Additionally, the procedure's quick convergence demonstrates its dependability and marks a notable advancement in the way linear and non-linear fractional-order partial differential equations are solved.
This research has been funded by Deputy for Research & Innovation, Ministry of Education through Initiative of Institutional Funding at University of Ha'il–Saudi Arabia through project number IFP-22 064.
The authors declare that they have no competing interests.
[1] |
X. D. Yang, J. N. Zhang, S. Y. Ren, Q. Y. Ran, Can the new energy demonstration city policy reduce environmental pollution? Evidence from a quasi-natural experiment in China, J. Clean. Prod., 287 (2021), 125015. http://doi.org/10.1016/j.jclepro.2020.125015 doi: 10.1016/j.jclepro.2020.125015
![]() |
[2] |
M. Ellman, F. Germano, What do the papers sell? A model of advertising and media bias, The Economic Journal, 119 (2009), 680–704. https://doi.org/10.1111/j.1468-0297.2009.02218.x doi: 10.1111/j.1468-0297.2009.02218.x
![]() |
[3] |
H. T. Wu, Y. W. Li, Y. Hao, S. Y. Ren, P. F. Zhang, Environmental decentralization, local government competition, and regional green development: Evidence from China, Sci. Total Environ., 708 (2020), 135085. http://doi.org/10.1016/j.scitotenv.2019.135085 doi: 10.1016/j.scitotenv.2019.135085
![]() |
[4] |
Y. Liu, C. Ma, Z. Huang, Can the digital economy improve green total factor productivity? An empirical study based on Chinese urban data, Math. Biosci. Eng., 20 (2023), 6866–6893. http://doi.org/10.3934/mbe.2023296 doi: 10.3934/mbe.2023296
![]() |
[5] |
L. Katusiime, International monetary spillovers and macroeconomic stability in developing countries, National Accounting Review, 3 (2021), 310–329. http://doi.org/10.3934/NAR.2021016 doi: 10.3934/NAR.2021016
![]() |
[6] |
Y. Liu, L. Chen, L. Lv, P. Failler, The impact of population aging on economic growth: a case study on China, AIMS Mathematics, 8 (2023), 10468–10485. http://doi.org/10.3934/math.2023531 doi: 10.3934/math.2023531
![]() |
[7] |
K. C. Ho, X. X. Shen, C. Yan, X. Hu, Influence of green innovation on disclosure quality: mediating role of media attention, Technol. Forecast. Soc., 188 (2023), 122314. http://doi.org/10.1016/j.techfore.2022.122314 doi: 10.1016/j.techfore.2022.122314
![]() |
[8] |
S. Boulianne, J. Ohme, Pathways to environmental activism in four countries: social media, environmental concern, and political efficacy, J. Youth Stud., 25 (2022), 771–792. http://doi.org/10.1080/13676261.2021.2011845 doi: 10.1080/13676261.2021.2011845
![]() |
[9] |
C. C. Lee, M. L. Zeng, C. S. Wang, Environmental regulation, innovation capability, and green total factor productivity: new evidence from China, Environ. Sci. Pollut. Res., 29 (2022), 39384–39399. http://doi.org/10.1007/s11356-021-18388-0 doi: 10.1007/s11356-021-18388-0
![]() |
[10] |
S. Huang, K. T. Huat, Z. Zhou, The studies on Chinese traditional culture and corporate environmental responsibility: literature review and its implications, National Accounting Review, 4 (2022), 1–15. http://doi.org/10.3934/NAR.2022001 doi: 10.3934/NAR.2022001
![]() |
[11] |
J. M. Mazzarino, L. Turatti, S. T. Petter, Environmental governance: media approach on the united nations programme for the environment, Environ. Dev., 33 (2020), 100502. http://doi.org/10.1016/j.envdev.2020.100502 doi: 10.1016/j.envdev.2020.100502
![]() |
[12] |
Z. Li, L. Chen, H. Dong, What are bitcoin market reactions to its-related events?, Int. Rev. Econ. Financ., 73 (2021), 1–10. http://doi.org/10.1016/j.iref.2020.12.020 doi: 10.1016/j.iref.2020.12.020
![]() |
[13] |
P. Liu, Y. Zhao, J. Zhu, C. Yang, Technological industry agglomeration, green innovation efficiency, and development quality of city cluster, Green Finance, 4 (2022), 411–435. http://doi.org/10.3934/GF.2022020 doi: 10.3934/GF.2022020
![]() |
[14] |
C. Luo, Z. Li, L. Liu, Does investor sentiment affect stock pricing? Evidence from seasoned equity offerings in China, National Accounting Review, 3 (2021), 115–136. http://doi.org/10.3934/NAR.2021006 doi: 10.3934/NAR.2021006
![]() |
[15] |
M. Akhtaruzzaman, S. Boubaker, Z. Umar, COVID-19 media coverage and ESG leader indices, Financ. Res. Lett., 45 (2022), 102170. http://doi.org/10.1016/j.frl.2021.102170 doi: 10.1016/j.frl.2021.102170
![]() |
[16] |
G. S. Miller, D. J. Skinner, The evolving disclosure landscape: How changes in technology, the media, and capital markets are affecting disclosure, J. Account. Res., 53 (2015), 221–239. http://doi.org/10.1111/1475-679x.12075 doi: 10.1111/1475-679x.12075
![]() |
[17] |
E. Assifuah-Nunoo, P. O. Junior, A. M. Adam, A. Bossman, Assessing the safe haven properties of oil in African stock markets amid the COVID-19 pandemic: a quantile regression analysis, Quant. Financ. Econ., 6 (2022), 244–269. http://doi.org/10.3934/QFE.2022011 doi: 10.3934/QFE.2022011
![]() |
[18] |
Y. X. Chen, J. Zhang, P. R. Tadikamalla, X. T. Gao, The relationship among government, enterprise, and public in environmental governance from the perspective of multi-player evolutionary game, Int. J. Environ. Res. Public Health, 16 (2019), 3351. http://doi.org/10.3390/ijerph16183351 doi: 10.3390/ijerph16183351
![]() |
[19] |
P. C. Tetlock, Giving content to investor sentiment: the role of media in the stock market, J. Financ., 62 (2007), 1139–1168. http://doi.org/10.1111/j.1540-6261.2007.01232.x doi: 10.1111/j.1540-6261.2007.01232.x
![]() |
[20] |
G. X. Zhang, Y. Q. Jia, B. Su, J. Xiu, Environmental regulation, economic development and air pollution in the cities of China: spatial econometric analysis based on policy scoring and satellite data, J. Clean. Prod., 328 (2021), 129496. http://doi.org/10.1016/j.jclepro.2021.129496 doi: 10.1016/j.jclepro.2021.129496
![]() |
[21] |
M. Irfan, A. Razzaq, A. Sharif, X. Yang, Influence mechanism between green finance and green innovation: exploring regional policy intervention effects in China, Technol. Forecast. Soc., 182 (2022), 121882. http://doi.org/10.1016/j.techfore.2022.121882 doi: 10.1016/j.techfore.2022.121882
![]() |
[22] |
M. Pichlak, A. R. Szromek, Eco-innovation, sustainability and business model innovation by open innovation dynamics, Journal of Open Innovation: Technology, Market, and Complexity, 7 (2021), 149. http://doi.org/10.3390/joitmc7020149 doi: 10.3390/joitmc7020149
![]() |
[23] |
P. A. Nylund, A. Brem, N. Agarwal, Enabling technologies mitigating climate change: the role of dominant designs in environmental innovation ecosystems, Technovation, 117 (2021), 102271. http://doi.org/10.1016/j.technovation.2021.102271 doi: 10.1016/j.technovation.2021.102271
![]() |
[24] |
Y. Xu, W. F. Ge, G. L. Liu, X. F. Su, J. N. Zhu, C. Y. Yang, et al., The impact of local government competition and green technology innovation on economic low-carbon transition: new insights from China, Environ. Sci. Pollut. Res., 30 (2022), 23714–23735. http://doi.org/10.1007/s11356-022-23857-1 doi: 10.1007/s11356-022-23857-1
![]() |
[25] | C. Fussler, P. James, Driving eco-innovation: a breakthrough discipline for innovation and sustainability, Financial Times/Prentice Hall, 1996. |
[26] |
J. Hartmann, Toward a more complete theory of sustainable supply chain management: the role of media attention, Supply Chain Management, 26 (2021), 532–547. http://doi.org/10.1108/scm-01-2020-0043 doi: 10.1108/scm-01-2020-0043
![]() |
[27] |
Z. H. Li, Z. M. Huang, Y. Y. Su, New media environment, environmental regulation and corporate green technology innovation: Evidence from China, Energ. Econ., 119 (2023), 106545. http://doi.org/10.1016/j.eneco.2023.106545 doi: 10.1016/j.eneco.2023.106545
![]() |
[28] |
E. R. Gray, J. M. T. Balmer, Managing corporate image and corporate reputation, Long Range Plann., 31 (1998), 695–702. https://doi.org/10.1016/S0024-6301(98)00074-0 doi: 10.1016/S0024-6301(98)00074-0
![]() |
[29] |
E. Blankespoor, E. deHaan, C. Zhu, Capital market effects of media synthesis and dissemination: evidence from robo-journalism, Rev. Account. Stud., 23 (2018), 1–36. http://doi.org/10.1007/s11142-017-9422-2 doi: 10.1007/s11142-017-9422-2
![]() |
[30] |
X. F. Jiang, C. X. Zhao, J. J. Ma, J. Q. Liu, S. H. Li, Is enterprise environmental protection investment responsibility or rent-seeking? Chinese evidence, Environ. Dev. Econ., 26 (2021), 169–187. http://doi.org/10.1017/s1355770x20000327 doi: 10.1017/s1355770x20000327
![]() |
[31] |
I. S. Farouq, N. U. Sambo, A. U. Ahmad, A. H. Jakada, I. A. Danmaraya, Does financial globalization uncertainty affect CO2 emissions? Empirical evidence from some selected SSA countries, Quant. Financ. Econ., 5 (2021), 247–263. http://doi.org/10.3934/QFE.2021011 doi: 10.3934/QFE.2021011
![]() |
[32] |
T. C. Chiang, Geopolitical risk, economic policy uncertainty and asset returns in Chinese financial markets, China Financ. Rev. Int., 11 (2021), 474–501. http://doi.org/10.1108/CFRI-08-2020-0115 doi: 10.1108/CFRI-08-2020-0115
![]() |
[33] |
J. L. Guan, H. J. Xu, D. Huo, Y. C. Hua, Y. F. Wang, Economic policy uncertainty and corporate innovation: Evidence from China, Pac.-Basin Financ. J., 67 (2021), 101542. http://doi.org/10.1016/j.pacfin.2021.101542 doi: 10.1016/j.pacfin.2021.101542
![]() |
[34] |
Y. Liu, J. Liu, L. Zhang, Enterprise financialization and R & D innovation: a case study of listed companies in China, Electron. Res. Arch., 31 (2023), 2447–2471. http://doi.org/10.3934/era.2023124 doi: 10.3934/era.2023124
![]() |
[35] |
T. C. Chiang, Can gold or silver be used as a hedge against policy uncertainty and COVID-19 in the Chinese market?, China Financ. Rev. Int., 12 (2022), 571–600. http://doi.org/10.1108/CFRI-12-2021-0232 doi: 10.1108/CFRI-12-2021-0232
![]() |
[36] |
L. H. Yin, C. Q. Wu, Promotion incentives and air pollution: from the political promotion tournament to the environment tournament, J. Environ. Manage., 317 (2022), 115491. http://doi.org/10.1016/j.jenvman.2022.115491 doi: 10.1016/j.jenvman.2022.115491
![]() |
[37] |
X. Yang, H. Wu, S. Ren, Q. Ran, J. Zhang, Does the development of the internet contribute to air pollution control in China? Mechanism discussion and empirical test, Struct. Change Econ. Dyn., 56 (2021), 207–224. http://doi.org/10.1016/j.strueco.2020.12.001 doi: 10.1016/j.strueco.2020.12.001
![]() |
[38] |
S. Y. Ren, Y. Hao, H. T. Wu, How does green investment affect environmental pollution? Evidence from China, Environ. Resource Econ., 81 (2022), 25–51. http://doi.org/10.1007/s10640-021-00615-4 doi: 10.1007/s10640-021-00615-4
![]() |
[39] |
Y. Li, X. D. Yang, Q. Y. Ran, H. T. Wu, M. Irfan, M. Ahmad, Energy structure, digital economy, and carbon emissions: evidence from China, Environ. Sci. Pollut. Res., 28 (2021), 64606–64629. http://doi.org/10.1007/s11356-021-15304-4 doi: 10.1007/s11356-021-15304-4
![]() |
[40] |
A. Biscione, R. Caruso, A. de Felice, Environmental innovation in European transition countries, Appl. Econ., 53 (2021), 521–535. http://doi.org/10.1080/00036846.2020.1808185 doi: 10.1080/00036846.2020.1808185
![]() |
[41] |
S. C. Zyglidopoulos, A. P. Georgiadis, C. E. Carroll, D. S. Siegel, Does media attention drive corporate social responsibility?, J. Bus. Res., 65 (2012), 1622–1627. http://doi.org/10.1016/j.jbusres.2011.10.021 doi: 10.1016/j.jbusres.2011.10.021
![]() |
[42] |
C. H. Yu, X. Wu, D. Zhang, S. Chen, J. Zhao, Demand for green finance: resolving financing constraints on green innovation in China, Energ. Policy, 153 (2021), 112255. https://doi.org/10.1016/j.enpol.2021.112255 doi: 10.1016/j.enpol.2021.112255
![]() |
[43] |
J. von Bloh, T. Broekel, B. Özgun, R. Sternberg, New(s) data for entrepreneurship research? An innovative approach to use Big Data on media coverage, Small Bus. Econ., 55 (2020), 673–694. http://doi.org/10.1007/s11187-019-00209-x doi: 10.1007/s11187-019-00209-x
![]() |
[44] |
G. B. Xiong, Y. D. Luo, Smog, media attention, and corporate social responsibility-empirical evidence from Chinese polluting listed companies, Environ. Sci. Pollut. Res., 28 (2021), 46116–46129. http://doi.org/10.1007/s11356-020-11978-4 doi: 10.1007/s11356-020-11978-4
![]() |
[45] |
S. R. Baker, N. Bloom, S. J. Davis, Measuring economic policy uncertainty, Quarterly Journal of Economics, 131 (2016), 1593–1636. http://doi.org/10.1093/qje/qjw024 doi: 10.1093/qje/qjw024
![]() |
[46] |
H. T. Wu, L. N. Xu, S. Y. Ren, Y. Hao, G. Y. Yan, How do energy consumption and environmental regulation affect carbon emissions in China? New evidence from a dynamic threshold panel model, Resour. Policy, 67 (2020), 101678. http://doi.org/10.1016/j.resourpol.2020.101678 doi: 10.1016/j.resourpol.2020.101678
![]() |
[47] |
T. Li, X. Li, G. Liao, Business cycles and energy intensity. Evidence from emerging economies, Borsa Istanb. Rev., 22 (2022), 560–570. http://doi.org/10.1016/j.bir.2021.07.005 doi: 10.1016/j.bir.2021.07.005
![]() |
[48] |
C. J. Hadlock, J. R. Pierce, New evidence on measuring financial constraints: moving beyond the KZ index, Rev. Financ. Stud., 23 (2010), 1909–1940. http://doi.org/10.1093/rfs/hhq009 doi: 10.1093/rfs/hhq009
![]() |
[49] |
Z. X. He, C. S. Cao, C. Feng, Media attention, environmental information disclosure and corporate green technology innovations in China's heavily polluting industries, Emerg. Mark. Financ. Tr., 58 (2022), 3939–3952. http://doi.org/10.1080/1540496x.2022.2075259 doi: 10.1080/1540496x.2022.2075259
![]() |
[50] |
M. A. Khan, X. Z. Qin, K. Jebran, A. Rashid, The sensitivity of firms' investment to uncertainty and cash flow: evidence from listed state-owned enterprises and non-state-owned enterprises in China, Sage Open, 10 (2020), 17. http://doi.org/10.1177/2158244020903433 doi: 10.1177/2158244020903433
![]() |
[51] |
X. Chang, Impact of risks on forced CEO turnover, Quant. Financ. Econ., 6 (2022), 177–205. http://doi.org/10.3934/QFE.2022008 doi: 10.3934/QFE.2022008
![]() |
[52] |
Y. Yao, D. Hu, C. Yang, Y. Tan, The impact and mechanism of fintech on green total factor productivity, Green Finance, 3 (2021), 198–221. http://doi.org/10.3934/gf.2021011 doi: 10.3934/gf.2021011
![]() |
[53] |
Z. Y. Li, M. Tuerxun, J. H. Cao, M. Fan, C. Y. Yang, Does inclusive finance improve income: a study in rural areas, AIMS Mathematics, 7 (2022), 20909–20929. http://doi.org/10.3934/math.20221146 doi: 10.3934/math.20221146
![]() |
[54] |
Z. Li, C. Yang, Z. Huang, How does the fintech sector react to signals from central bank digital currencies?, Financ. Res. Lett., 50 (2022), 103308. http://doi.org/10.1016/j.frl.2022.103308 doi: 10.1016/j.frl.2022.103308
![]() |
[55] |
S. El Ghoul, O. Guedhami, R. Nash, A. Patel, New evidence on the role of the media in corporate social responsibility, J. Bus. Ethics, 154 (2019), 1051–1079. http://doi.org/10.1007/s10551-016-3354-9 doi: 10.1007/s10551-016-3354-9
![]() |
[56] |
T. Vanacker, D. P. Forbes, M. Knockaert, S. Manigart, Signal strength, media attention, and resource mobilization: evidence from new private equity firms, Acad. Manage. J., 63 (2020), 1082–1105. http://doi.org/10.5465/amj.2018.0356 doi: 10.5465/amj.2018.0356
![]() |
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Stability Analysis of Split‐Step θθ ‐Milstein Scheme for Stochastic Delay Integro‐Differential Equations,
2024,
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Exact result | ||||
(0.2, 0.1) | 0.2102371 | 0.2100072 | 0.2100000 | 0.2100000 |
(0.4, 0.1) | 0.4104630 | 0.4100141 | 0.4100000 | 0.4100000 |
(0.6, 0.1) | 0.6106889 | 0.6100209 | 0.6100000 | 0.6100000 |
(0.2, 0.2) | 0.2103355 | 0.2100121 | 0.2100000 | 0.2100000 |
(0.4, 0.2) | 0.4106550 | 0.4100237 | 0.4100000 | 0.4100000 |
(0.6, 0.2) | 0.6109746 | 0.6100353 | 0.6100000 | 0.6100000 |
(0.2, 0.3) | 0.2104110 | 0.2100164 | 0.2100000 | 0.2100000 |
(0.4, 0.3) | 0.4108025 | 0.4100321 | 0.4100000 | 0.4100000 |
(0.6, 0.3) | 0.6111940 | 0.6100478 | 0.6100000 | 0.6100000 |
(0.2, 0.4) | 0.2104747 | 0.2100204 | 0.2100000 | 0.2100000 |
(0.4, 0.4) | 0.4109269 | 0.4100399 | 0.4100000 | 0.4100000 |
(0.6, 0.4) | 0.6113790 | 0.6100593 | 0.6100000 | 0.6100000 |
(0.2, 0.5) | 0.2105309 | 0.2100241 | 0.2100000 | 0.2100000 |
(0.4, 0.5) | 0.4110365 | 0.4100471 | 0.4100000 | 0.4100000 |
(0.6, 0.5) | 0.6115421 | 0.6100701 | 0.6100000 | 0.6100000 |
Exact result | ||||
(0.2, 0.1) | 0.2102371 | 0.2100072 | 0.2100000 | 0.2100000 |
(0.4, 0.1) | 0.4104630 | 0.4100141 | 0.4100000 | 0.4100000 |
(0.6, 0.1) | 0.6106889 | 0.6100209 | 0.6100000 | 0.6100000 |
(0.2, 0.2) | 0.2103355 | 0.2100121 | 0.2100000 | 0.2100000 |
(0.4, 0.2) | 0.4106550 | 0.4100237 | 0.4100000 | 0.4100000 |
(0.6, 0.2) | 0.6109746 | 0.6100353 | 0.6100000 | 0.6100000 |
(0.2, 0.3) | 0.2104110 | 0.2100164 | 0.2100000 | 0.2100000 |
(0.4, 0.3) | 0.4108025 | 0.4100321 | 0.4100000 | 0.4100000 |
(0.6, 0.3) | 0.6111940 | 0.6100478 | 0.6100000 | 0.6100000 |
(0.2, 0.4) | 0.2104747 | 0.2100204 | 0.2100000 | 0.2100000 |
(0.4, 0.4) | 0.4109269 | 0.4100399 | 0.4100000 | 0.4100000 |
(0.6, 0.4) | 0.6113790 | 0.6100593 | 0.6100000 | 0.6100000 |
(0.2, 0.5) | 0.2105309 | 0.2100241 | 0.2100000 | 0.2100000 |
(0.4, 0.5) | 0.4110365 | 0.4100471 | 0.4100000 | 0.4100000 |
(0.6, 0.5) | 0.6115421 | 0.6100701 | 0.6100000 | 0.6100000 |