
The goal of this study is twofold: first, to understand the rationales of public policies and possible outcomes on energy systems design behind supporting national hydrogen strategies in three major economic blocs (the EU, UK and USA) and possible outcomes on energy systems design; second, to identify differences in policy approaches to decarbonization through H2 promotion. Large-scale expansion of low-carbon H2 demands careful analysis and understanding of how public policies can be fundamental drivers of change. Our methodological approach was essentially economic, using the International Energy Agency (IEA) policy database as a main information source. First, we identified all regional policies and measures that include actions related to H2, either directly or indirectly. Then, we reclassified policy types, sectors and technologies to conduct a comparative analysis which allowed us to reduce the high degree of economic ambiguity in the database. Finally, we composed a detailed discussion of our findings. While the EU pushed for renewable H2, the UK immediately targeted low-carbon H2 solutions, equally considering both blue and green alternatives. The USA pursues a clean H2 economy based on both nuclear and CCS fossil technology. Although there is a general focus on fiscal and financing policy actions, distinct intensities were identified, and the EU presents a much stricter regulatory framework than the UK and USA. Another major difference between blocs concerns target sectors: While the EU shows a broad policy strategy, the UK is currently prioritizing the transport sector. The USA is focusing on H2 production and supply as well as the power and heat sectors. In all cases, policy patterns and financing options seem to be in line with national hydrogen strategies, but policies' balances reflect diverse institutional frameworks and economic development models.
Citation: João Moura, Isabel Soares. Financing low-carbon hydrogen: The role of public policies and strategies in the EU, UK and USA[J]. Green Finance, 2023, 5(2): 265-297. doi: 10.3934/GF.2023011
[1] | Edward Y. Uechi . Determining a proportion of labor and equipment to achieve optimal production: A model supported by evidence of 19 U.S. industries from 2000 to 2020. National Accounting Review, 2024, 6(2): 266-290. doi: 10.3934/NAR.2024012 |
[2] | Tinghui Li, Zimei Huang, Benjamin M Drakeford . Statistical measurement of total factor productivity under resource and environmental constraints. National Accounting Review, 2019, 1(1): 16-27. doi: 10.3934/NAR.2019.1.16 |
[3] | Timo Tohmo . Estimating SFLQ-based regional input-output tables for South Korean regions. National Accounting Review, 2025, 7(1): 125-142. doi: 10.3934/NAR.2025006 |
[4] | Evgenii Lukin, Tamara Uskova . Development of production cooperation in Russia: Quantitative measurement. National Accounting Review, 2023, 5(4): 322-337. doi: 10.3934/NAR.2023019 |
[5] | Ming He, Barnabé Walheer . Technology intensity and ownership in the Chinese manufacturing industry: A labor productivity decomposition approach. National Accounting Review, 2020, 2(2): 110-137. doi: 10.3934/NAR.2020007 |
[6] | Guido Ferrari, José Mondéjar Jiménez, Yanyun Zhao . The statistical information for tourism economics. The National Accounts perspective. National Accounting Review, 2022, 4(2): 204-217. doi: 10.3934/NAR.2022012 |
[7] | Jean-Marie Le Page . Structural rate of unemployment, hysteresis, human capital, and macroeconomic data. National Accounting Review, 2022, 4(2): 135-146. doi: 10.3934/NAR.2022008 |
[8] | Jinhong Wang, Yanting Xu . Factors influencing the transition of China's economic growth momentum. National Accounting Review, 2024, 6(2): 220-244. doi: 10.3934/NAR.2024010 |
[9] | Tinghui Li, Xue Li . Does structural deceleration happen in China? Evidence from the effect of industrial structure on economic growth quality. National Accounting Review, 2020, 2(2): 155-173. doi: 10.3934/NAR.2020009 |
[10] | Tran Van Hoa, Jo Vu, Pham Quang Thao . Vietnam's sustainable tourism and growth: a new approach to strategic policy modelling. National Accounting Review, 2020, 2(4): 324-336. doi: 10.3934/NAR.2020019 |
The goal of this study is twofold: first, to understand the rationales of public policies and possible outcomes on energy systems design behind supporting national hydrogen strategies in three major economic blocs (the EU, UK and USA) and possible outcomes on energy systems design; second, to identify differences in policy approaches to decarbonization through H2 promotion. Large-scale expansion of low-carbon H2 demands careful analysis and understanding of how public policies can be fundamental drivers of change. Our methodological approach was essentially economic, using the International Energy Agency (IEA) policy database as a main information source. First, we identified all regional policies and measures that include actions related to H2, either directly or indirectly. Then, we reclassified policy types, sectors and technologies to conduct a comparative analysis which allowed us to reduce the high degree of economic ambiguity in the database. Finally, we composed a detailed discussion of our findings. While the EU pushed for renewable H2, the UK immediately targeted low-carbon H2 solutions, equally considering both blue and green alternatives. The USA pursues a clean H2 economy based on both nuclear and CCS fossil technology. Although there is a general focus on fiscal and financing policy actions, distinct intensities were identified, and the EU presents a much stricter regulatory framework than the UK and USA. Another major difference between blocs concerns target sectors: While the EU shows a broad policy strategy, the UK is currently prioritizing the transport sector. The USA is focusing on H2 production and supply as well as the power and heat sectors. In all cases, policy patterns and financing options seem to be in line with national hydrogen strategies, but policies' balances reflect diverse institutional frameworks and economic development models.
Studying the theory of fractional differential equations (FDEs) and fractional integral equations (FIEs) is crucial because they are used in many modeling applications. Fractional equations are essential for many areas of fundamental analysis and their applications in economics, physics, and other disciplines. Quadratic integral equations (QIE), in particular, tend to be helpful in describing a wide range of everyday issues, such as theory of radiative transfer, the theory of neutron transport, the kinetic theory of gases, the queuing theory, and the traffic theory (see, for example, [1,2,3,4]).
One of the studied QIEs is called the hybrid integral equation (HIE); see [5]. This issue has received great attention in the last few years; see [6,7,8,9]. As we see, the hybrid fixed point theory is used to develop the existing solution of the hybrid equations; see [10,11,12,13]. Other researchers focused on the analysis of QIEs in Orlicz spaces [14], equations of QIEs with fractional order arising in the queuing theory and biology [15], and the analysis of QIEs depending on both Schauder and Schauder–Tychonoff fixed point principles [16].
The Chandrasekhar quadratic integral equation (CQIE) occurs in the theory of radiative transfer in a plane-parallel atmosphere [2]. The radiative transfer process and the integral equation for the scattering function and transmitted functions were developed by Chandrasekhar's work in the 1950s; see [14]. This work quickly turned into a major scientific topic in both mathematics and astrophysics, see [15,16]. The radiative transfer process and the simultaneous integral equation for the transmitted and scattering functions were developed in Chandrasekhar's seminal work from 1960. In [17], the simultaneous integral equation of Chandrasekhar was presented, along with the iteration scheme for the transmitted and scattering functions.
CQIE takes the form
{Φ(ϱ)=φ(ϱ)+ϱΦ(ϱ)∫ϱ0ϱϱ+sˆg1(s,Φ(s))ds+f1(ϱ,Φ(ϱ))RLJμˆg2(ϱ,Φ(ϱ)),ϱ∈ˆI,ˆI=[0,b],μ∈(0,1), | (1.1) |
where RLJμ denotes the Riemann-Liouville fractional integral (RLFI) of order μ, f1(ϱ,Φ(ϱ))∈C(ˆI×R,R−{0}), and ˆgκ(ϱ,Φ(ϱ))∈C(ˆI×R,R), for κ=1,2.
As a result of the applications of these equations, researchers were interested in studying them, and as a result of the difficulty of finding exact solutions, CQIEs are solved using the ADM and the PM. The ADM provides many advantages, including the ability to solve a variety of linear and nonlinear equations in deterministic or stochastic fields effectively and present an analytical solution for all of these equation types without requiring linearization or discretization. Additionally, it is reliable and provides faster convergence than other classical methods. Moreover, we use measures of weak noncompactness to study existence and uniqueness results. These results demonstrate that the two solutions provide nearly equal accuracy; however, when comparing the time required in each case, the ADM is found to take less time than the PM.
It can be summarized as follows: first, the second section introduces the basic concepts of the measure of noncompactness and the hypothesis. In the third section, we show that if the solution exists, then it will be unique and convergent. After that, we solve some nonlinear Chandraseker QIEs with fractional orders with a comparison between ADM and PM techniques. Finally, graphs are also constructed to illustrate the effectiveness of these two approaches and to compare them.
In the complement of this work, the classical Banach space C(ˆI)=C[0,b] is used, which contains all real continuous functions defined on ˆI having the norm
‖Φ‖=max{|Φ(ϱ)|:ϱ∈ˆI}. | (2.1) |
Let us recall the MNC definition in C(ˆI) which is used in this investigation and fix a bounded nonempty subset Ω of C(ˆI). For η∈Ω and ϵ>0, the modulus of continuity of the function η on the interval ˆI is defined by
ω(η,ϵ)=sup{|η(ϱ)−η(s)|:ϱ,s∈ˆI,|ϱ−s|≤ϵ} | (2.2) |
and
ω(Ω,ϵ)=sup{ω(η,ϵ):η∈Ω},ω0(Ω)=limϵ→0ω(Ω,ϵ). | (2.3) |
It is well-known that ω0(Ω) is a measure of noncompactness inC(ˆI)such that the Hausdorff measure χ may be expressed by the formula
χ(Ω)=12ω0(Ω), | (2.4) |
see [18]. We use the following theorem to prove our investigation.
Theorem 1. [18] Let Q be a bounded, nonempty, and closed convex subset in the space E. Also, let ˆH:Q→Q be a continuous operator such that χ(ˆHΩ)≤l χ(Ω) to any nonempty subset Ω of Q, as l∈[0,1) is constant, so ˆH has a fixed point in the set Q.
Considering the hypothesis:
i. φ :ˆI→R is continuous.
ii. f1:ˆI×R→R is continuous and the function f1(ϱ,0)∈ C(R),R is the space of all bounded continuous functions, and there exists a positive constant M=supϱ∈ˆI|f1(ϱ,0)|.
iii. |f1(ϱ,Φ)−f1(ϱ,ζ)|≤L|Φ−ζ| for any ϱ∈ˆI for all Φ,ζ∈R.
iv. ˆgκ:ˆI×R→R,κ =1,2 satisfies the Carathéodory condition (CC) so, it is measurable in ϱ for all Φ∈R and continuous in Φ for all ϱ∈ˆI, and there exist functions mκ, κ =1,2∈L1 such that:
|ˆgκ(ϱ,Φ)|≤mκ(ϱ) for all (ϱ,Φ)∈ˆI×R. | (2.5) |
v. RLJγm2(ϱ)<M2, γ≤μ, C≥0, and ∫b01ϱ+sm1(s)ds≤M1.
vi. There exists a number R0>0, such that
R0=[‖φ‖+M2Lbμ−γΓ(μ−γ+1)+M2Mbμ−γΓ(μ−γ+1)][1−b2M1]−1. | (2.6) |
Define the ball BR0 as
BR0={Φ∈C(ˆI):‖Φ‖≤R0}. | (3.1) |
Theorem 2. Using the hypotheses (i)−(vi), if [M2Lbμ−γΓ(μ−γ+1)+M1b2]<1, then there exists at least a solution Φ∈C(ˆI) for HIE (1.1).
Proof. Let the operator ˆH defined on C(ˆI) be
(ˆHΦ)(ϱ)=φ(ϱ)+ϱΦ(ϱ)∫ϱ0ϱϱ+sˆg1(s,Φ(s))ds+f1(ϱ,Φ(ϱ))∫ϱ0(ϱ−s)μ−1Γ(μ)ˆg2(s,Φ(s))ds,ϱ∈ˆI. |
From the hypotheses (i)−(vi), the function ˆHΦ is continuous on ˆI for any Φ∈BR0. Further, applying the given hypothesis, we derive the following estimate:
|ˆHΦ(ϱ)|≤|φ(ϱ)|+|ϱΦ(ϱ)|∫ϱ0ϱϱ+sm1(s)ds+|f1(ϱ,Φ)|∫ϱ0(ϱ−s)μ−1Γ(μ)m2(s)ds≤‖φ‖+|ϱΦ(ϱ)|∫ϱ0ϱϱ+sm1(s)ds+[|f1(ϱ,Φ)−f1(ϱ,0)|+|f1(ϱ,0)|]∫ϱ0(ϱ−s)μ−1Γ(μ)m2(s)ds≤‖φ‖+|b2Φ(ϱ)|∫b01ϱ+sm1(s)ds +[|f1(ϱ,Φ)−f1(ϱ,0)|+|f1(ϱ,0)|]RLJμ−γRLJγm2(ϱ)≤‖φ‖+b2R0M1+M2LR0bμ−γΓ(μ−γ+1)+M2Mbμ−γΓ(μ−γ+1). | (3.2) |
So ˆHΦ is bounded on the interval ˆI. Also, we get
‖ˆHΦ‖≤‖φ‖+b2R0M1+M2LR0bμ−γΓ(μ−γ+1)+M2Mbμ−γΓ(μ−γ+1), |
for the operator ˆH, which transforms the ball BR0 into itself, and R0 is:
R0=[‖φ‖+M2Mbμ−γΓ(μ−γ+1)][1−b2M1+M2Lbμ−γΓ(μ−γ+1)]−1>0. |
Now, we are going to show that the operator ˆH is continuous on the ball BR0. So, we need to prove that the operator G2 defined by
G2Φ(ϱ)=∫ϱ0(ϱ−s)μ−1Γ(μ)g2(s,Φ(s))ds,ϱ∈ˆI, |
is continuous on BR0. To do this, fix ϵ>0, let Φ0∈BR0, and from hypothesis (ii), we find δ>0 such that ||Φ−Φ0||≤δ, and then we have |ˆg2(s,Φ)−ˆg2(s,Φ0)|≤ϵ for s ∈ˆI, where Φ is any arbitrary element in BR0. For arbitrary fixed ϱ∈ˆI, we get
|G2Φ(ϱ)−G2Φ0(ϱ)|=1Γ(μ)∫ϱ0(ϱ−s)μ−1|ˆg2(s,Φ(s))−ˆg2(s,Φ0(s))|ds≤ϵΓ(μ)∫ϱ0(ϱ−s)μ−1ds≤ϵΓ(μ+1). |
The operator f1Φ(ϱ)=f1(ϱ,Φ(ϱ)) is continuous, and then the operator f1.G2 is continuous on BR0, and similarly, we can prove that the operator
G1Φ(ϱ)=ϱΦ(ϱ)∫ϱ0ϱϱ+sˆg1(s,Φ(s))ds,ϱ∈ˆI, |
is continuous on BR0. This shows that ˆH is continuous on BR0. Let χ be a nonempty subset of BR0. Fix ϵ>0, and choose Φ∈χ and ϱ1,ϱ2∈ˆI such that |ϱ2−ϱ1|≤ϵ. Let ϱ1≤ϱ2, and then
|(ˆHΦ)(ϱ2)−(ˆHΦ)(ϱ1)|=|φ(ϱ2)−φ(ϱ1) |
+ϱ2Φ(ϱ2)∫ϱ20ϱ2ϱ2+sˆg1(s,Φ(s))ds−ϱ1Φ(ϱ1)∫ϱ10ϱ1ϱ1+sˆg1(s,Φ(s))ds+ϱ2Φ(ϱ2)∫ϱ10ϱ1ϱ1+sˆg1(s,Φ(s))ds−ϱ2Φ(ϱ2)∫ϱ10ϱ1ϱ1+sˆg1(s,Φ(s))ds+f1(ϱ2,Φ(ϱ2))RLJμˆg2(ϱ2,Φ(ϱ2))−f1(ϱ1,Φ(ϱ1))RLJμˆg2(ϱ1,Φ(ϱ1))+f1(ϱ1,Φ(ϱ1))RLJμˆg2(ϱ2,Φ(ϱ2))−f1(ϱ1,Φ(ϱ1))RLJμˆg2(ϱ2,Φ(ϱ2))|. |
Hence,
|(ˆHΦ)(ϱ2)−(ˆHΦ)(ϱ1)|≤|φ(ϱ2)−φ(ϱ1)|+|ϱ2Φ(ϱ2)|∫ϱ20|ϱ2ϱ2+s−ϱ1ϱ1+s|m1(s)ds+|ϱ2Φ(ϱ2)−ϱ1Φ(ϱ1)|∫ϱ10ϱ1ϱ1+sm1(s)ds+|ϱ2Φ(ϱ2)|∫ϱ2ϱ1|ϱ2ϱ2+s|m1(s)ds+[f1(ϱ2,Φ(ϱ2))−f1(ϱ2,Φ(ϱ1))]RLJμˆg2(ϱ2,Φ(ϱ2))+[f1(ϱ2,Φ(ϱ1))−f1(ϱ1,Φ(ϱ1))]RLJμˆg2(ϱ2,Φ(ϱ2))+f1(ϱ1,Φ(ϱ1))[RLJμˆg2(ϱ2,Φ(ϱ2))−RLJμˆg2(ϱ1,Φ(ϱ1))], |
and so
|RLJμˆg2(ϱ2,Φ(ϱ2))−RLJμˆg2(ϱ1,Φ(ϱ1))|=|∫ϱ10(ϱ2−s)μ−1Γ(μ)ˆg2(s,Φ(s))ds+∫ϱ2ϱ1(ϱ2−s)μ−1Γ(μ)ˆg2(s,Φ(s))ds−∫ϱ10(ϱ1−s)μ−1Γ(μ)ˆg2(s,Φ(s))ds|≤|∫ϱ10(ϱ2−s)μ−1Γ(μ)ˆg2(s,Φ(s))ds+∫ϱ2ϱ1(ϱ2−s)μ−1Γ(μ)ˆg2(s,Φ(s))ds−∫ϱ10(ϱ2−s)μ−1Γ(μ)ˆg2(s,Φ(s))ds|≤|∫ϱ2ϱ1(ϱ2−s)μ−1Γ(μ)ˆg2(s,Φ(s))ds|. |
Then
|RLJμˆg2(ϱ2,Φ(ϱ2))−RLJμˆg2(ϱ1,Φ(ϱ1))|≤RLJμϱ1|ˆg2(ϱ2,Φ(ϱ2))|≤RLJμϱ1m2(ϱ2)≤RLJμ−γϱ1 RLJγϱ1m2(ϱ2)≤M2(ϱ2−ϱ1)μ−γΓ(μ−γ+1). |
Then
|(ˆHΦ)(ϱ2)−(ˆHΦ)(ϱ1)|≤|φ(ϱ2)−φ(ϱ1)|+b2R0∫ϱ10|ϱ2−ϱ1|ϱ1+s(ϱ2+s)m1(s)ds+b2R0∫ϱ2ϱ11ϱ2+sm1(s)ds+b[|ϱ2Φ(ϱ2)−ϱ2Φ(ϱ1)|+|ϱ2Φ(ϱ1)−ϱ1Φ(ϱ1)|]∫b01ϱ1+sm1(s)ds+|f1(ϱ2,Φ(ϱ2))−f1(ϱ2,Φ(ϱ1))|RLJμ−γ RLJγm2(ϱ2)+|f1(ϱ2,Φ(ϱ1))−f1(ϱ1,Φ(ϱ1))|RLJμ−γRLJγm2(ϱ2)+|f1(ϱ1,Φ(ϱ1))−f1(ϱ1,0)|M2(ϱ2−ϱ1)μ−γΓ(μ−γ+1)+|f1(ϱ1,0)|M2(ϱ2−ϱ1)μ−γΓ(μ−γ+1). |
We get
|(ˆHΦ)(ϱ2)−(ˆHΦ)(ϱ1)|≤|φ(ϱ2)−φ(ϱ1)|+bR0|ϱ2−ϱ1|∫b01ϱ1+sm1(s)ds+b2R0∫ϱ2ϱ1m1(s)ds+M1b2|Φ(ϱ2)−Φ(ϱ1)|+bR0|ϱ2−ϱ1|+L|Φ(ϱ2)−Φ(ϱ1)|RLJμm2(ϱ2)+η(f1,ϵ)RLJμm2(ϱ2)+L|Φ(ϱ1)|M|ϱ2−ϱ1|μ−ηΓ(μ−γ+1)+M2M|ϱ2−ϱ1|μ−γΓ(μ−γ+1), |
so,
|(ˆHΦ)(ϱ2)−(ˆHΦ)(ϱ1)|≤|φ(ϱ2)−φ(ϱ1)|+bR0M1|ϱ2−ϱ1|+b2R0∫ϱ2ϱ1m1(s)ds+M1b2|Φ(ϱ2)−Φ(ϱ1)|+bR0M1|ϱ2−ϱ1|+|Φ(ϱ2)−Φ(ϱ1)|M2Lbμ−γΓ(μ−γ+1)+M2Ψ(f1,ϵ)bμ−γΓ(μ−γ+1)+L|Φ(ϱ1)|M|ϱ2−ϱ1|μ−γΓ(μ−γ+1)+M2M|ϱ2−ϱ1|μ−ηΓ(μ−γ+1), |
where Ψ(f1,ϵ)=sup{|f1(ϱ2,Φ(ϱ1)−f1(ϱ1,Φ(ϱ1))|:ϱ1,ϱ2∈ˆI,|ϱ2−ϱ1|≤ϵ,Φ∈BR0}.
Knowing that f1 is uniformly continuous on the set ˆI ×BR0, we derive the inequality
ω0(ˆHΦ)≤[M2Lbμ−γΓ(μ−γ+1)+M1b2]ω0(Φ). |
Then
ω0(ˆHΦ)≤[M2Lbμ−γΓ(μ−γ+1)+M1b2]ω0(Φ). |
From (2.4), this inequality leads to
χ(ˆHΦ)≤[M2Lbμ−γΓ(μ−γ+1)+M1b2]χ(Φ). |
If [M2Lbμ−γΓ(μ−γ+1)+M1b2]<1 and using Theorem 1, there exists at least a solution Φ∈C(ˆI) for HIE (1.1).
The following hypotheses are satisfied if there exist ˆgκ, κ =1,2, such that (iv)∗ ˆgκ: ˆI×R→R, κ =1,2, satisfies CC, so it can be measurable in ϱ for all Φ∈R and continuous in Φ for all ϱ∈ˆI. Then
|ˆgκ(ϱ,Φ)−ˆgκ(ϱ,ζ)|≤Lκ|Φ−ζ|, κ=1,2, | (3.3) |
for all ϱ∈ˆI and Φ,ζ∈ R. Let Φ1 and Φ2 be two solutions for the HIE (1.1), and hence
|Φ1(ϱ)−Φ2(ϱ)|=|ϱΦ1(ϱ)∫ϱ0ϱϱ+sˆg1(s,Φ1(s))ds−ϱΦ2(ϱ)∫ϱ0ϱϱ+sˆg1(s,Φ2(s))ds+ϱΦ2(ϱ)∫ϱ0ϱϱ+sˆg1(s,Φ1(s))ds−ϱΦ2(ϱ)∫ϱ0ϱϱ+sˆg1(s,Φ1(s))ds+f1(ϱ,Φ1(ϱ))∫ϱ0(ϱ−s)μ−1Γ(μ)ˆg2(s,Φ1(s))ds−f1(ϱ,Φ2(ϱ))∫ϱ0(ϱ−s)μ−1Γ(μ)ˆg2(s,Φ2(s))ds+f1(ϱ,Φ1(ϱ))∫ϱ0(ϱ−s)μ−1Γ(μ)ˆg2(s,Φ2(s))ds−f1(ϱ,Φ1(ϱ))∫ϱ0(ϱ−s)μ−1Γ(μ)ˆg2(s,Φ2(s))ds| | (3.4) |
≤ϱ|Φ1(ϱ)−Φ2(ϱ)|∫ϱ0ϱϱ+sm1(s)ds+ϱ|Φ2(ϱ)|∫ϱ0ϱϱ+sL1|Φ1(s)−Φ2|ds+|f1(ϱ,Φ1(ϱ))−f1(ϱ,Φ2(ϱ))|∫ϱ0(ϱ−s)μ−1Γ(μ)ˆg2(s,Φ2(s))ds+|f1(ϱ,Φ1(ϱ))|∫ϱ0(ϱ−s)μ−1Γ(μ)ˆg2(s,Φ2(s))−ˆg2(s,Φ1(s)))ds≤(b2M1+L1b2R0+LM2bμ−γΓ(μ−γ+1)+L2bμ(LR0+M)Γ(μ+1))|Φ1(ϱ)−Φ2(ϱ)|≤Υ|Φ1(ϱ)−Φ2(ϱ)|, | (3.5) |
where Υ=(b2M1+L1b2R0+LM2bμ−γΓ(μ−γ+1)+L2bμ(LR0+M)Γ(μ+1)). Then we get the theorem:
Theorem 3. Assume that the hypotheses (i)–(vi) are satisfied, and Υ<1. Then, the solution Φ∈C(ˆI) of (1.1) is unique.
In the 1980s, Adomian presented the ADM [19,20,21], which is an analytical method used to solve a lot of different equations such as DEs, IEs, integro-differential equations, and partial DEs [22,23,24,25]. The obtained solution is an infinite series that converges to the exact solution. An important benefit of the ADM is that there is no linearization or perturbation that can change the main problem that has been solved, which is serious. A lot of researchers are interested in using the ADM, as it is successfully applied to many applications that appear in applied sciences [26,27,28]. In this research, the ADM is used as the first method to solve the HIE (1.1).
Applying the ADM to (1.1), the ADM solution algorithm is
Φ0(ϱ)=φ(ϱ), | (4.1) |
Φκ(ϱ)=ϱˆAκ−1(ϱ)+ˇDκ−1(ϱ), | (4.2) |
where ˆAκ, and ˇDκ are Adomian polynomials of the nonlinear terms ˆg1(ϱ,Φ),f1(ϱ,Φ), and ˆg2(ϱ,Φ) which take the forms
ˆAκ=1κ![dκdλκ(∞∑κ=0λκΦκ∫ϱ0ϱϱ+sˆg1(s,∞∑κ=0λκΦκ(s))ds)]λ=0, | (4.3) |
ˇDκ=1κ![dκdλκ(f1(ϱ,∞∑κ=0λκΦκ)RLJμˆg2(ϱ,∞∑κ=0λκΦκ(ϱ)))]λ=0. | (4.4) |
Finally, the ADM solution is
Φ(ϱ)=∞∑κ=0Φκ(ϱ). | (4.5) |
Theorem 4. If hypotheses (i)−(vi) are satisfied, Υ2<1, and |Φ1(ϱ)|<k, where k is a positive constant, then the series solution (4.5) of (1.1) using the ADM is convergent.
Proof. Define the sequence {ˆSρ} such that ˆSρ=ρ∑κ=0Φκ(ϱ) is a sequence of partial sums taken from the series (4.5), and
Φ(ϱ)∫ϱ0ϱϱ+sˆg1(s,Φ(s))ds=∞∑κ=0ˆAκ,f1(ϱ,Φ(ϱ))RLJμˆg2(ϱ,Φ(ϱ))=∞∑κ=0ˇDκ. |
Let ˆSρ and ˆSθ be two partial sums of the ADM series solution such that ρ>θ. We want to prove that {ˆSρ} is a Cauchy sequence (CS) in this Banach space (Bs).
ˆSρ−ˆSθ=ρ∑κ=0Φκ−θ∑κ=0Φκ=ϱρ∑κ=0ˆAκ−1(ϱ)+ρ∑κ=0ˇDκ−1(ϱ)−ϱθ∑κ=0ˆAκ−1(ϱ)−θ∑κ=0ˇDκ−1(ϱ), |
hence,
ˆSρ−ˆSθ=[ϱ(ρ∑κ=0ˆAκ−1(ϱ)−θ∑κ=0ˆAκ−1(ϱ))]+[ρ∑κ=0ˇDκ−1(ϱ)−θ∑κ=0ˇDκ−1(ϱ)]=[ϱ(ρ∑κ=θ+1ˆAκ−1(ϱ))]+[ρ∑κ=θ+1ˇDκ−1(ϱ)]. |
Thus, by applying |⋅| to both sides, we find
|ˆSρ−ˆSθ|=|[ϱ(ρ∑κ=θ+1ˆAκ−1(ϱ))]+[ρ∑κ=θ+1ˇDκ−1(ϱ)]| |
≤|ϱρ−1∑κ=θˆAκ(ϱ)|+|ρ−1∑κ=θˇDκ(ϱ)| |
≤|ϱ(ˆSρ−1−ˆSθ−1)|∫ϱ0|ϱϱ+s[ˆg1(s,(ρ−1∑κ=θΦκ))]|ds+|f1(ϱ,(ρ−1∑κ=θΦκ))RLJμˆg2(ϱ,(ρ−1∑κ=θΦκ))|≤b2|(ˆSρ−1−ˆSθ−1)|∫b01ϱ+s|ˆg1(s,Φ)|ds+|f1(ϱ,ˆSρ−1)−f1(ϱ,ˆSθ−1)|RLJμ|ˆg2(ϱ,Φ)|, |
|ˆSρ−ˆSθ|≤b2|(ˆSρ−1−ˆSθ−1)|∫b01ϱ+s|ˆg1(s,Φ)|ds+L|ˆSρ−1−ˆSθ−1|RLJμ−γ RLJγm2(ϱ)≤b2|(ˆSρ−1−ˆSθ−1)|M1+LM2|ˆSρ−1−ˆSθ−1|RLJμ−γ(1) |
‖ˆSρ−ˆSθ‖≤[b2M1+LM2(b)μ−γΓ(μ−γ+1)]‖ˆSρ−1−ˆSθ−1‖≤Υ2‖ˆSρ−1−ˆSθ−1‖, |
where Υ2=[b2M1+LM2(b)μ−γΓ(μ−γ+1)]. Let ρ=θ+1, and we get
‖ˆSθ+1−ˆSθ‖≤Υ2‖ˆSθ−ˆSθ−1‖≤Υ22‖ˆSθ−1−ˆSθ−2‖≤⋯≤Υθ2‖ˆS1−ˆS0‖. |
Using the triangle inequality, we arrive at
‖ˆSρ−ˆSθ‖≤‖ˆSθ+1−ˆSθ‖+‖ˆSθ+2−ˆSθ+1‖+⋯+‖ˆSρ−ˆSρ−1‖ ≤[Υθ2+Υθ+12+⋯+Υρ−12]‖ˆS1−ˆS0‖≤Υθ2[1+Υ2+⋯+Υρ−θ−12]‖ˆS1−ˆS0‖≤Υ2[1−Υρ−θ21−Υ2]‖Φ1‖. |
If 0<Υ2<1 and ρ>θ, this leads to (1−Υρ−θ2)≤1. Then
‖ˆSρ−ˆSθ‖≤Υθ21−Υ2‖Φ1‖≤Υθ21−Υ2maxϱ∈ˆI|Φ1(ϱ)|. |
If |Φ1(ϱ)|<k, θ→∞, then ‖ˆSρ−ˆSθ‖→0, which leads to {ˆSρ} being a CS in this BS and the series (4.5) is convergent.
Theorem 5. The maximum absolute error of the ADM series solution (4.5) is
maxϱ∈ˆI|Φ(ϱ)−θ∑κ=0Φκ(ϱ)|≤Υθ21−Υ2maxϱ∈ˆI|Φ1(ϱ)|. | (4.6) |
Proof. In Theorem 2, we see that
‖ˆSρ−ˆSθ‖≤Υθ21−Υ2maxϱ∈ˆI|Φ1(ϱ)|, |
and ˆSρ=ρ∑κ=0Φκ(ϱ), ρ→∞. Then, ˆSρ→Φ(ϱ), and hence
‖Φ(ϱ)−ˆSθ‖≤Υθ21−Υ2maxϱ∈ˆI|Φ1(ϱ)|, |
and the maximum absolute error is written as
\begin{equation*} {\max\limits_{{ \varrho }\in \;\hat{I}}}\left\vert \Phi { \left( { \varrho }\right) }-\sum\limits_{{ \kappa } = 0}^{\theta }\Phi _{{ \kappa }}{ \left( { \varrho }\right) }\right\vert \leq \frac{\Upsilon _{2}^{\theta }}{1-\Upsilon _{2}}\;{\max\limits_{ { \varrho }\in \;\hat{I}}}\;\left\vert \Phi _{1}{ \left( { \varrho }\right) }\right\vert . \end{equation*} |
The method of successive approximations (PM) was presented by Emile Picard in 1891. PM and ADM methods were first compared by Rach and Bellomo in 1987 [26,29]. In 1999, Golberg deduced that these two methods were equivalent for linear differential equations [30]. But this equivalence is not achieved in the nonlinear case. In 2010, El-Sayed et al. used them to solve QIE [31]. In 2012, El-Sayed et al. used them to solve a coupled system of fractional QIEs [32]. In 2014, El-Sayed et al. used them to solve FQIE [33]. In 2024, Ziada used them to solve a nonlinear FDE system containing the Atangana–Baleanu derivative [34]. In this research, we use them to get the solution for a nonlinear HDE and compare their results.
Applying the PM to the QIE (1.1), the solution is a sequence constructed by
\begin{eqnarray} \Phi _{0}{ \left( { \varrho }\right) } & = &\varphi { \left( { \varrho }\right) }, \\ \Phi _{{ \kappa }}{ \left( { \varrho }\right) } & = &\Phi _{0} { \left( { \varrho }\right) }+{ \varrho }\Phi _{{ \kappa } -1}{ \left( { \varrho }\right) }\int_{0}^{{ \varrho }}\frac{ { \varrho }}{{ \varrho }+s}\hat{g}_{1}\left( s, \Phi _{{ \kappa } -1}\left( s\right) \right) \mathrm{d}s \\ &&+f_{1}\left( { \varrho }, \Phi _{{ \kappa }-1}{ \left( { \varrho }\right) }\right) \;^{RL}J^{\mu }\hat{g}_{2}\left( { \varrho }, \Phi _{{ \kappa }-1}{ \left( { \varrho }\right) }\right) . \end{eqnarray} | (4.7) |
All the functions \Phi _{{ \kappa }}\left({ \varrho }\right) are continuous functions and \Phi _{{ \kappa }} are the sum of successive differences
\begin{equation} \Phi _{{ \kappa }}{ \left( { \varrho }\right) } = \Phi _{0} { \left( { \varrho }\right) }+\sum\limits_{{ \kappa } = 1}^{\infty }\left( \Phi _{{ \kappa }}-\Phi _{\kappa -1}\right) . \end{equation} | (4.8) |
Therefore, the sequence \Phi _{{ \kappa }} convergence is the same as the infinite series \sum \left(\Phi _{{ \kappa }}-\Phi _{{ \kappa }-1}\right) convergence. The final PM solution takes the form
\begin{equation} \Phi { \left( { \varrho }\right) } = \underset{{ \kappa } \rightarrow \infty }{\lim }\;\Phi _{_{{ \kappa }}}{ \left( { \varrho }\right) } . \end{equation} | (4.9) |
From the above relations, we can deduce that if the series \sum \left(\Phi _{{ \kappa }}-\Phi _{\kappa -1}\right) is convergent, then the sequence \Phi _{{ \kappa }}\left({ \varrho }\right) is convergent to \Phi \left({ \varrho }\right) . To prove that the sequence \left\{ \Phi _{{ \kappa }}{ \left({ \varrho } \right) }\right\} is informally convergent, consider the associated series
\begin{equation} \sum\limits_{{ \kappa } = 1}^{\infty }\left[ \Phi _{{ \kappa }} { \left( { \varrho }\right) }-\Phi _{{ \kappa }-1}{ \left( { \varrho }\right) }\right] . \end{equation} | (4.10) |
From (4.7) for \kappa = 1 , we get
\begin{equation} \Phi _{1}{ \left( { \varrho }\right) }-\Phi _{0}{ \left( { \varrho }\right) } = { \varrho }\Phi _{0}{ \left( { \varrho } \right) }\int_{0}^{{ \varrho }}\frac{{ \varrho }}{{ \varrho }+s} \hat{g}_{1}\left( s, \Phi _{0}\left( s\right) \right) \mathrm{d}s+f_{1}\left( { \varrho }, \Phi _{0}{ \left( { \varrho }\right) }\right) ^{RL}J^{\mu }\hat{g}_{2}\left( { \varrho }, \Phi _{0}{ \left( { \varrho }\right) }\right) . \end{equation} | (4.11) |
So, we have
\begin{eqnarray} \left\vert \Phi _{1}{ \left( { \varrho }\right) }-\Phi _{0}{ \left( { \varrho }\right) }\right\vert & = &\left\vert { \varrho }\Phi _{0}{ \left( { \varrho }\right) }\int_{0}^{{ \varrho }}\frac{ { \varrho }}{{ \varrho }+s}\hat{g}_{1}\left( s, \Phi _{0}\left( s\right) \right) \mathrm{d}s+f_{1}\left( { \varrho }, \Phi _{0}{ \left( { \varrho }\right) }\right) \;^{RL}J^{\mu }\;\hat{g} _{2}\left( { \varrho }, \Phi _{0}{ \left( { \varrho }\right) } \right) \right\vert \\ &\leq &\left\vert { \varrho }\right\vert \;\left\vert \Phi _{0} { \left( { \varrho }\right) }\right\vert \int_{0}^{{ \varrho } }\left\vert \frac{{ \varrho }}{{ \varrho }+s}\right\vert \left\vert \hat{g}_{1}\left( s, \Phi _{0}\left( s\right) \right) \right\vert \mathrm{d}s \\ &&+\left\vert f_{1}\left( { \varrho }, \Phi _{0}{ \left( { \varrho }\right) }\right) \right\vert \;^{RL}J^{\mu }\hat{g}_{2}\left( { \varrho }, \Phi _{0}{ \left( { \varrho }\right) }\right) . \end{eqnarray} | (4.12) |
Thus,
\begin{eqnarray} \left\vert \Phi _{1}{ \left( { \varrho }\right) }-\Phi _{0}{ \left( { \varrho }\right) }\right\vert &\leq &b^{2}\left\vert \Phi _{0} { \left( { \varrho }\right) }\right\vert \int_{0}^{b}\frac{1}{{ \varrho }+s}m_{1}\left( s\right) \mathrm{d}s \\ &&+\left[ \left\vert f_{1}\left( { \varrho }, \Phi _{0}{ \left( { \varrho }\right) }\right) )-f_{1}\left( { \varrho }, 0\right) +f_{1}\left( { \varrho }, 0\right) \right\vert \right] {\ }^{RL}J^{\mu }\; ^{RL}J^{\eta }m_{2}{ \left( { \varrho }\right) } \\ &\leq &\left[ b^{2}{ \mathbb{R} }_{0}M_{1}+\frac{M_{2}b^{\mu -\eta }\left( L{ \mathbb{R} }_{0}+M\right) }{\Gamma \left( \mu -\eta +1\right) }\right] : = \psi . \end{eqnarray} | (4.13) |
Now, we get an estimate for \Phi _{{ \kappa }}\left({ \varrho } \right) -\Phi _{\kappa -1}\left({ \varrho }\right), \kappa \geqslant 2 :
\begin{eqnarray} \Phi _{{ \kappa }}{ \left( { \varrho }\right) }-\Phi _{{ \kappa }-1}{ \left( { \varrho }\right) } & = &{ \varrho }\Phi _{ { \kappa }-1}{ \left( { \varrho }\right) }\int_{0}^{{ \varrho }}\frac{{ \varrho }}{{ \varrho }+s}\hat{g}_{1}\left( s, \Phi _{\kappa -1}\left( s\right) \right) \mathrm{d}s+f_{1}\left( { \varrho } , \Phi _{\kappa -1}{ \left( { \varrho }\right) }\right) \; ^{RL}J^{\mu }\;\hat{g}_{2}\left( { \varrho }, \Phi _{\kappa -1}{ \left( { \varrho }\right) }\right) \\ &&-{ \varrho }\Phi _{\kappa -2}{ \left( { \varrho }\right) } \int_{0}^{{ \varrho }}\frac{{ \varrho }}{{ \varrho }+s}\hat{g} _{1}\left( s, \Phi _{\kappa -2}\left( s\right) \right) \mathrm{d} s+f_{1}\left( { \varrho }, \Phi _{\kappa -2}{ \left( { \varrho } \right) }\right) \;^{RL}J^{\mu }\;\hat{g}_{2}\left( { \varrho }, \Phi _{\kappa -2}{ \left( { \varrho }\right) }\right) \\ & = &{ \varrho }\Phi _{\kappa -1}{ \left( { \varrho }\right) } \int_{0}^{{ \varrho }}\frac{{ \varrho }}{{ \varrho }+s}\hat{g} _{1}\left( s, \Phi _{\kappa -1}\left( s\right) \right) \mathrm{d} s+f_{1}\left( { \varrho }, \Phi _{\kappa -1}{ \left( { \varrho } \right) }\right) \;^{RL}J^{\mu }\;\hat{g}_{2}\left( { \varrho }, \Phi _{\kappa -1}{ \left( { \varrho }\right) }\right) \\ &&-{ \varrho }\Phi _{\kappa -2}{ \left( { \varrho }\right) } \int_{0}^{{ \varrho }}\frac{{ \varrho }}{{ \varrho }+s}\hat{g} _{1}\left( s, \Phi _{\kappa -2}\left( s\right) \right) \mathrm{d}s \\ &&-f_{1}\left( { \varrho }, \Phi _{\kappa -2}{ \left( { \varrho } \right) }\right) \;^{RL}J^{\mu }\;\hat{g}_{2}\left( { \varrho }, \Phi _{\kappa -2}{ \left( { \varrho }\right) }\right) , \end{eqnarray} | (4.14) |
and
\begin{eqnarray} \Phi _{{ \kappa }}{ \left( { \varrho }\right) }-\Phi _{{ \kappa }-1}{ \left( { \varrho }\right) } & = &{ \varrho }\Phi _{\kappa -1}{ \left( { \varrho }\right) }\int_{0}^{{ \varrho }} \frac{{ \varrho }}{{ \varrho }+s}\hat{g}_{1}\left( s, \Phi _{\kappa -1}\left( s\right) \right) \mathrm{d}s+{ \varrho }\Phi _{\kappa -1}{ \left( { \varrho }\right) }\int_{0}^{{ \varrho }}\frac{{ \varrho } }{{ \varrho }+s}\hat{g}_{1}\left( s, \Phi _{\kappa -2}\left( s\right) \right) \mathrm{d}s \\ &&-{ \varrho }\Phi _{\kappa -1}{ \left( { \varrho }\right) } \int_{0}^{{ \varrho }}\frac{{ \varrho }}{{ \varrho }+s}\hat{g} _{1}\left( s, \Phi _{\kappa -2}\left( s\right) \right) \mathrm{d}s-{ \varrho }\Phi _{\kappa -2}{ \left( { \varrho }\right) }\int_{0}^{ { \varrho }}\frac{{ \varrho }}{{ \varrho }+s}\hat{g}_{1}\left( s, \Phi _{\kappa -2}\left( s\right) \right) \mathrm{d}s \\ &&+f_{1}\left( { \varrho }, \Phi _{\kappa -1}{ \left( { \varrho } \right) }\right) \;^{RL}J^{\mu }\;\hat{g}_{2}\left( { \varrho }, \Phi _{\kappa -1}{ \left( { \varrho }\right) }\right) +f_{1}\left( { \varrho }, \Phi _{\kappa -1}{ \left( { \varrho } \right) }\right) \;^{RL}J^{\mu }\;\hat{g}_{2}\left( { \varrho }, \Phi _{\kappa -2}{ \left( { \varrho }\right) }\right) \\ &&-f_{1}\left( { \varrho }, \Phi _{\kappa -1}{ \left( { \varrho } \right) }\right) \;^{RL}J^{\mu }\;\hat{g}_{2}\left( { \varrho }, \Phi _{\kappa -2}{ \left( { \varrho }\right) }\right) \\ &&-f_{1}\left( { \varrho }, \Phi _{\kappa -2}{ \left( { \varrho } \right) }\right) \;^{RL}J^{\mu }\;\hat{g}_{2}\left( { \varrho }, \Phi _{\kappa -2}{ \left( { \varrho }\right) }\right) . \end{eqnarray} | (4.15) |
Thus,
\begin{eqnarray} \Phi _{{ \kappa }}{ \left( { \varrho }\right) }-\Phi _{{ \kappa }-1}{ \left( { \varrho }\right) } & = &{ \varrho }\Phi _{\kappa -1}{ \left( { \varrho }\right) }\int_{0}^{{ \varrho }} \frac{{ \varrho }}{{ \varrho }+s}\left[ \hat{g}_{1}\left( s, \Phi _{\kappa -1}\left( s\right) \right) -\hat{g}_{1}\left( s, \Phi _{\kappa -2}\left( s\right) \right) \right] \mathrm{d}s \\ &&+{ \varrho }\left[ \Phi _{\kappa -1}{ \left( { \varrho }\right) }-\Phi _{\kappa -2}{ \left( { \varrho }\right) }\right] \int_{0}^{ { \varrho }}\frac{{ \varrho }}{{ \varrho }+s}\hat{g}_{1}\left( s, \Phi _{\kappa -2}\left( s\right) \right) \mathrm{d}s \\ &&+f_{1}\left( { \varrho }, \Phi _{\kappa -1}{ \left( { \varrho } \right) }\right) \;^{RL}J^{\mu }\;\left[ \hat{g}_{2}\left( { \varrho }, \Phi _{\kappa -1}{ \left( { \varrho }\right) } \right) -\hat{g}_{2}\left( { \varrho }, \Phi _{\kappa -2}{ \left( { \varrho }\right) }\right) \right] \\ &&+\left[ f_{1}\left( { \varrho }, \Phi _{\kappa -1}{ \left( { \varrho }\right) }\right) -f_{1}\left( \varrho , \Phi _{\kappa -2}\left( { \varrho }\right) \right) \right] \;^{RL}J^{\mu }\;\hat{g} _{2}\left( { \varrho }, \Phi _{\kappa -2}{ \left( { \varrho } \right) }\right) . \end{eqnarray} | (4.16) |
From the hypotheses (\mathrm{ii}) and (\mathrm{iii}), we have
\begin{eqnarray} \left\vert \Phi _{\kappa }{ \left( { \varrho }\right) }-\Phi _{\kappa -1}{ \left( { \varrho }\right) }\right\vert &\leq &\left\vert { \varrho }\Phi _{\kappa -1}{ \left( { \varrho } \right) }\right\vert \int_{0}^{{ \varrho }}\frac{{ \varrho }}{{ \varrho }+s}\left\vert \hat{g}_{1}\left( s, \Phi _{\kappa -1}\left( s\right) \right) -\hat{g}_{1}\left( s, \Phi _{\kappa -2}\left( s\right) \right) \right\vert \mathrm{d}s \\ &&+{ \varrho }\left\vert \Phi _{\kappa -1}{ \left( { \varrho } \right) }-\Phi _{\kappa -2}{ \left( { \varrho }\right) } \right\vert \int_{0}^{{ \varrho }}\frac{{ \varrho }}{{ \varrho }+s} \left\vert \hat{g}_{1}\left( s, \Phi _{\kappa -2}\left( s\right) \right) \right\vert \mathrm{d}s \\ &&+\left\vert f_{1}\left( { \varrho }, \Phi _{\kappa -1}{ \left( { \varrho }\right) }\right) \right\vert \;^{RL}J^{\mu }\left\vert \hat{g}_{2}\left( { \varrho }, \Phi _{\kappa -1}{ \left( { \varrho }\right) }\right) -\hat{g}_{2}\left( { \varrho }, \Phi _{\kappa -2}{ \left( { \varrho }\right) }\right) \right\vert \\ &&+\left\vert f_{1}\left( \varrho , \Phi _{\kappa -1}\left( { \varrho } \right) \right) -f_{1}\left( { \varrho }, \Phi _{\kappa -2}{ \left( { \varrho }\right) }\right) \right\vert \;^{RL}J^{\mu }\; \left\vert \hat{g}_{2}\left( { \varrho }, \Phi _{\kappa -2}{ \left( { \varrho }\right) }\right) \right\vert , \end{eqnarray} | (4.17) |
so,
\begin{eqnarray} \left\vert \Phi _{\kappa }{ \left( { \varrho }\right) }-\Phi _{\kappa -1}{ \left( { \varrho }\right) }\right\vert &\leq &b^{2} { \mathbb{R} }_{0}L_{1}\left\vert \Phi _{\kappa -1}{ \left( { \varrho }\right) } -\Phi _{\kappa -2}{ \left( { \varrho }\right) }\right\vert \int_{0}^{b}\frac{1}{{ \varrho }+s}\mathrm{d}s \\ &&+b^{2}\left\vert \Phi _{\kappa -1}{ \left( { \varrho }\right) } -\Phi _{\kappa -2}{ \left( { \varrho }\right) }\right\vert \int_{0}^{b}\frac{1}{{ \varrho }+s}m_{1}\left( s\right) \mathrm{d}s \\ &&+\left\vert f_{1}\left( { \varrho }, \Phi _{\kappa -1}{ \left( { \varrho }\right) }\right) -f_{1}\left( { \varrho }, 0\right) +f_{1}\left( { \varrho }, 0\right) \right\vert \\ &&+L_{2}\left\vert \Phi _{\kappa -1}{ \left( { \varrho }\right) } -\Phi _{\kappa -2}{ \left( { \varrho }\right) }\right\vert \; ^{RL}J^{\mu }\;\left( 1\right) \\ &&+L\left\vert \Phi _{{ \kappa }-1}{ \left( { \varrho }\right) }-\Phi _{{ \kappa }-2}{ \left( { \varrho }\right) } \right\vert {\ }^{RL}J^{\mu }\;^{RL}J^{\eta }m_{2}({ \varrho }) \\ &\leq &\left[ b^{2}{ \mathbb{R} }_{0}L_{1}+b^{2}M_{1}+\frac{L_{2}b^{\mu }\left( L{ \mathbb{R} }_{0}+M\right) }{\Gamma \left( \mu +1\right) }+\frac{LM_{2}b^{\mu -\eta }}{ \Gamma \left( \mu -\eta +1\right) }\right] \;\left\vert \Phi _{\kappa -1}{ \left( { \varrho }\right) }-\Phi _{\kappa -2}{ \left( { \varrho }\right) }\right\vert \\ &\leq &\left[ b^{2}{ \mathbb{R} }_{0}M_{1}+\frac{M_{2}b^{\mu -\eta }\left( L{ \mathbb{R} }_{0}+M\right) }{\Gamma \left( \mu -\eta +1\right) }\right] \left\vert \Phi _{\kappa -1}{ \left( { \varrho }\right) }-\Phi _{\kappa -2}{ \left( { \varrho }\right) }\right\vert \\ &\leq &\Upsilon _{1}\left\vert \Phi _{\kappa -1}{ \left( { \varrho } \right) }-\Phi _{\kappa -2}{ \left( { \varrho }\right) } \right\vert , \end{eqnarray} | (4.18) |
where \Upsilon _{1} = \left[b^{2}{ \mathbb{R} }_{0}M_{1}+\frac{M_{2}b^{\mu -\eta }\left(L{ \mathbb{R} }_{0}+M\right) }{\Gamma \left(\mu -\eta +1\right) }\right].
In the above relation, if we put \kappa = 2 and use (4.13), we get
\begin{eqnarray} \left\vert \Phi _{2}{ \left( { \varrho }\right) }-\Phi _{1}{ \left( { \varrho }\right) }\right\vert &\leq &\Upsilon _{1}\left\vert \Phi _{1}{ \left( { \varrho }\right) }-\Phi _{0}{ \left( { \varrho }\right) }\right\vert \\ \left\vert \Phi _{2}-\Phi _{1}\right\vert &\leq &\Upsilon _{1}\;\psi . \end{eqnarray} | (4.19) |
Doing the same for \kappa = 3, 4, \cdots gives us
\begin{eqnarray*} \left\vert \Phi _{3}-\Phi _{2}\right\vert &\leq &\Upsilon _{1}\left\vert \Phi _{2}{ \left( { \varrho }\right) }-\Phi _{1}{ \left( { \varrho }\right) }\right\vert \\ &\leq &\Upsilon _{1}^{2}\;\psi \\ \left\vert \Phi _{4}-\Phi _{3}\right\vert &\leq &\Upsilon _{1}\left\vert \Phi _{3}{ \left( { \varrho }\right) }-\Phi _{2}{ \left( { \varrho }\right) }\right\vert \\ &\leq &\Upsilon _{1}^{3}\;\psi \\ &&\vdots \end{eqnarray*} |
Then the general form of this relation is
\begin{equation} \left\vert \Phi _{{ \kappa }}-\Phi _{{ \kappa }-1}\right\vert \leq \Upsilon _{1}^{{ \kappa }-1}\;\psi . \end{equation} | (4.20) |
Since \Upsilon _{1} < 1, then the series
\begin{equation} \sum\limits_{{ \kappa } = 1}^{\infty }\left[ \Phi _{{ \kappa }} { \left( { \varrho }\right) }-\Phi _{{ \kappa }-1}{ \left( { \varrho }\right) }\right] \end{equation} | (4.21) |
is uniformly convergent. Hence, the sequence \left\{ \Phi _{{ \kappa } }{ \left({ \varrho }\right) }\right\} is uniformly convergent. Since \hat{g}_{1}\left(\varrho, \Phi \left({ \varrho }\right) \right), \hat{g}_{2}\left(\varrho, \Phi \left({ \varrho }\right) \right) , and f_{1}\left(\varrho, \Phi \left({ \varrho }\right) \right) are continuous in \Phi , then
\begin{eqnarray} \Phi { \left( { \varrho }\right) } & = &\underset{{ \kappa } \rightarrow \infty }{\lim }{ \varrho }\Phi _{_{{ \kappa }}}{ \left( { \varrho }\right) }\int_{0}^{{ \varrho }}\frac{{ \varrho } }{{ \varrho }+s}\hat{g}_{1}\left( s, \Phi _{{ \kappa }}\left( s\right) \right) \mathrm{d}s+f_{1}\left( { \varrho }, \Phi _{{ \kappa }}{ \left( { \varrho }\right) }\right) \;^{RL}J^{\mu }\hat{g} _{2}\left( { \varrho }, \Phi _{{ \kappa }}{ \left( { \varrho }\right) }\right) \\ & = &{ \varrho }\Phi { \left( { \varrho }\right) }\int_{0}^{{ \varrho }}\frac{{ \varrho }}{{ \varrho }+s}\hat{g}_{1}\left( s, \Phi \left( s\right) \right) \mathrm{d}s+f_{1}\left( { \varrho }, \Phi { \left( { \varrho }\right) }\right) \;^{RL}J^{\mu }\hat{g} _{2}\left( { \varrho }, \Phi { \left( { \varrho }\right) }\right) . \end{eqnarray} | (4.22) |
Hence, the solution exists.
Example 1. For the HIE of Chandraseker type:
\begin{equation} \Phi { \left( { \xi }\right) } = \frac{1}{50}{ \xi }\Phi { \left( { \xi }\right) }\int_{0}^{{ \xi }}\frac{{ \xi }}{{ \xi }+s}\Phi ^{2}\left( { \xi }\right) \mathrm{d}s+ \frac{1}{20}\Phi { \left( { \xi }\right) }^{RL}J^{\mu }\; \Phi ^{3}{ \left( { \xi }\right) , \ \ \xi \left( 0\right) = 0, \ } \end{equation} | (5.1) |
where
\begin{equation*} \varphi \left( { \xi }\right) = \left[ { \xi }^{2}-\left( \frac{- \frac{7}{12}+\ln (2)}{50}\right) { \varrho }^{7}-\frac{\Gamma (7)}{ 20\Gamma (7+\mu )}{ \xi }^{8+\mu }\right] , \end{equation*} |
and its exact solution is \Phi \left({ \xi }\right) = \xi ^{2}.
Applying the ADM to Eq (5.1), we get
\begin{eqnarray} \Phi _{0}{ \left( { \xi }\right) } & = &\varphi \left( { \xi }\right) , \end{eqnarray} | (5.2) |
\begin{eqnarray} \Phi _{\kappa }{ \left( { \xi }\right) } & = &\frac{1}{50}{ \xi }\hat{A}_{\kappa -1}\left( { \xi }\right) +\frac{1}{20}\; \check{D}_{ { \kappa -1 }}{ \left( { \xi }\right) }, \kappa \geq 1. \end{eqnarray} | (5.3) |
Using the PM in Eq (5.1), the solution algorithm is
\begin{eqnarray} \Phi _{0}{ \left( { \xi }\right) } & = &\varphi { \left( { \xi }\right) }, \end{eqnarray} | (5.4) |
\begin{eqnarray} \Phi _{\kappa }{ \left( { \xi }\right) } & = &\Phi _{0}{ \left( { \xi }\right) }+\frac{1}{50}{ \xi }\Phi _{\kappa -1} { \left( { \xi }\right) }\int_{0}^{{ \xi }}\frac{{ \xi }}{{ \xi }+s}\Phi _{\kappa -1}^{2}\left( s\right) \mathrm{d}s+ \frac{1}{20}\Phi { \left( { \xi }\right) }^{RL}J^{\mu }\; \Phi _{\kappa -1}^{3}{ \left( { \xi }\right) }, \kappa \geq 1. \end{eqnarray} | (5.5) |
Figure 1 shows ADM solutions at different values of \mu ( \mu = 0.5, 0.6, 0.7, 0.8, 0.9, 1 ), and Figure 2 shows PM solutions at the same values.
Remark 1. A comparison between the absolute relative error (ARE) of ADM and PM solutions with the exact solution (where \mu = 0.5 ) is given in Table 1. It is clear from these results that the two solutions nearly give the same accuracy, but when a comparison is made between the time used in these two cases, it is found that the ADM takes less time than the PM (ADM time = 22 sec., PM time = 319.188 sec.). Figure 3(a) shows the ADM and the exact solution, while Figure 3(b) shows the PM with the exact solution.
\varrho | \left\vert \frac{\Phi _{ES}-\Phi _{ADM}}{\Phi _{ES}}\right\vert | \left\vert \frac{\Phi _{ES}-\Phi _{PM}}{\Phi _{ES}}\right\vert |
0.1 | 2.48379 \times 10^{-9} | 2.19628 \times 10^{-8} |
0.2 | 2.47045 \times 10^{-7} | 7.0281 \times 10^{-7} |
0.3 | 2.04146 \times 10^{-6} | 5.33709 \times 10^{-6} |
0.4 | 6.32111 \times 10^{-6} | 0.0000224935 |
0.5 | 5.3704 \times 10^{-6} | 0.0000686815 |
0.6 | 0.0000360447 | 0.000171175 |
0.7 | 0.000215013 | 0.000371487 |
0.8 | 0.000750286 | 0.000731218 |
0.9 | 0.0020904 | 0.00134688 |
1 | 0.00511876 | 0.00240123 |
1.1 | 0.0115464 | 0.00436134 |
1.2 | 0.024631 | 0.00870185 |
1.3 | 0.0503602 | 0.0201514 |
1.4 | 0.0990462 | 0.0519358 |
1.5 | 0.186618 | 0.132278 |
Example 2. For the HIE of Chandraseker type:
\begin{equation} \Phi \left( { \xi }\right) = \varphi \left( { \xi }\right) +\frac{ 1}{10}{ \xi }\Phi \left( { \xi }\right) \int_{0}^{{ \xi }} \frac{{ \xi }}{{ \xi }+s}\sqrt{\Phi \left( s\right) }\mathrm{d}s+ \frac{\Phi ^{2}({ \xi })}{50}\;^{RL}J^{\mu }\frac{{ \xi ^{3} }}{20}\left( 5+\Phi ^{4}\left( { \xi }\right) \right) , \ \ \xi \left( 0\right) = 0, \end{equation} | (5.6) |
where
\begin{equation*} \varphi \left( { \xi }\right) = \frac{2\xi ^{3}}{15}, \end{equation*} |
and its exact solution is \Phi \left({ \xi }\right) = \xi.
Applying the ADM to Eq (5.6), we get
\begin{eqnarray} \Phi _{0}\left( { \xi }\right) & = &\varphi \left( { \xi }\right) , \end{eqnarray} | (5.7) |
\begin{eqnarray} \Phi _{\kappa }\left( { \xi }\right) & = &\frac{1}{10}{ \xi }\hat{A }_{\kappa -1}\left( { \xi }\right) +\frac{1}{50}\check{D}_{\kappa -1}\left( { \xi }\right) , \;\kappa \geq 1. \end{eqnarray} | (5.8) |
Using the PM in Eq (5.6), we have
\begin{eqnarray} \Phi _{0}\left( { \xi }\right) & = &\varphi \left( { \xi }\right) , \\ \Phi _{\kappa }\left( { \xi }\right) & = &\Phi _{0}\left( { \xi } \right) +\frac{1}{10}{ \xi }\Phi _{\kappa -1}\left( { \xi } \right) \int_{0}^{{ \xi }}\frac{{ \xi }}{{ \xi }+s}\sqrt{ \Phi _{\kappa -1}\left( s\right) }\mathrm{d}s \end{eqnarray} | (5.9) |
\begin{equation} +\frac{1}{50}\Phi _{\kappa -1}^{2}\left( { \xi }\right) \; ^{RL}J^{\mu }\frac{{ \xi ^{3} }}{20}\left( 5+\Phi _{\kappa -1}^{4}\left( { \xi }\right) \right) , \kappa \geq 1. \end{equation} | (5.10) |
Figure 4(a) shows the ADM and exact solution, while Figure 4(b) shows the PM bwith the exact solution.
Remark 2. The absolute difference (AD) between ADM and PM solutions (where \mu = 0.9 ) is \left\vert \Phi _{PM}-\Phi _{ADM}\right\vert = 0 for \xi = 0.2, 0.4, ..., 2 . It is clear from these results that the two solutions are nearly the same, but when a comparison is made between the time used in these two cases, it is found that the ADM takes less time than the PM (ADM time = 42 sec., PM time = 253.2 sec.). Figure 5 shows ADM and PM solutions at ( \mu = 0.9) .
Example 3. For the HIE of Chandraseker type:
\begin{equation} \Phi \left( { \xi }\right) = \varphi \left( { \xi }\right) +\frac{ 1}{10}{ \xi }\Phi \left( { \xi }\right) \int_{0}^{{ \xi }} \frac{{ \xi }}{{ \xi }+s}\Phi \left( s\right) \;e^{s} \mathrm{d}s+\frac{1}{10}\frac{\Phi \left( { \xi }\right) }{1+\Phi \left( { \xi }\right) }\;^{RL}J^{0.5}{ \xi }^{2}\left( 1+\Phi \left( { \xi }\right) \right) , \ \ \xi \left( 0\right) = 0, \end{equation} | (5.11) |
where
\begin{equation*} \varphi \left( { \xi }\right) = \left[ \xi +\frac{1}{10}\frac{-\xi }{ 1+\xi }(0.601802\xi ^{2.5}+0.51583\xi ^{3.5})-\frac{1}{10}\frac{\xi ^{3}(1+ \;e^{\xi }(\xi -1))}{\xi +2}\right] , \end{equation*} |
and its exact solution is \Phi \left({ \xi }\right) = \xi.
Applying the ADM to Eq (5.11), we have
\begin{eqnarray} \Phi _{0}\left( { \xi }\right) & = &\varphi \left( { \xi }\right) , \end{eqnarray} | (5.12) |
\begin{eqnarray} \Phi _{\kappa }\left( { \xi }\right) & = &\frac{1}{10}{ \xi }\hat{A }_{\kappa -1}\left( { \xi }\right) +\frac{1}{10}\check{D}_{\kappa -1}\left( { \xi }\right) , \;\kappa \geq 1. \end{eqnarray} | (5.13) |
Using the PM in Eq (5.11), we get
\begin{eqnarray} \Phi _{0}\left( { \xi }\right) & = &\varphi \left( { \xi }\right) , \\ \Phi _{\kappa }\left( { \xi }\right) & = &\Phi _{0}\left( { \xi } \right) +\frac{1}{10}{ \xi }\Phi _{\kappa -1}\left( { \xi } \right) \int_{0}^{{ \xi }}\frac{{ \xi }}{{ \xi }+s}\Phi _{\kappa -1}\left( s\right) e^{s}\mathrm{d}s \end{eqnarray} | (5.14) |
\begin{equation} +\frac{1}{10}\frac{\Phi _{\kappa -1}\left( { \xi }\right) }{1+\Phi _{\kappa -1}\left( { \xi }\right) }\;^{RL}J^{0.5}{ \xi } ^{2}\left( 1+\Phi _{\kappa -1}\left( { \xi }\right) \right) , \kappa \geq 1. \end{equation} | (5.15) |
Remark 3. A comparison between the ARE of ADM and PM solutions with the exact solution is given in Table 2. It is clear from these results that the two solutions nearly give the same accuracy, but when a comparison is made between the time used in these two cases, it is found that the ADM takes less time than the PM (ADM time = 69.124 sec., PM time = 70.875 sec.). Figure 6 shows ADM solutions at different values of \mu ( \mu = 0.5, 0.6, 0.7, 0.8, 0.9, 1 ), and Figure 7 shows PM solutions at the same values.
\varrho | \left\vert \frac{\chi _{ES}-\chi _{ADM}}{\chi _{ES}}\right\vert | \left\vert \frac{\chi _{ES}-\chi _{PM}}{\chi _{ES}}\right\vert |
0.1 | 1.0964*10 ^{{-6}} | 0.0000724208 |
0.2 | 0.0000268761 | 0.00060454 |
0.3 | 0.000179457 | 0.00212539 |
0.4 | 0.000701695 | 0.0052377 |
0.5 | 0.00204302 | 0.0106117 |
0.6 | 0.00493079 | 0.0189737 |
0.7 | 0.0104472 | 0.0310892 |
0.8 | 0.0201116 | 0.0477376 |
0.9 | 0.0359674 | 0.0696801 |
1 | 0.0606745 | 0.0976178 |
In this research, two analytical methods (ADM and PM) are used to solve the fractional CQIE that was found in the nonlinear analysis and its applications. The existence of a unique solution and its convergence to the two methods are proved (see Theorems 2, 4, and 5). This article focused on making a comparison between them with the exact solution (see the results in Tables 1 and 2). It is observed from the obtained results that the difference between their accuracy is too small to consider, but when we compare their used time, it was clear that the ADM takes less time than the PM (it is more clear in Example 1). These results showed that the two methods satisfied certain criteria that were provided by the solutions.
IVP | Initial value problem |
ADM | Adomian decomposition method |
PM | Picard method |
FDEs | Fractional differential equations |
HDE | Hybrid differential equation |
CQIE | Chandrasekhar quadratic integral equation |
RLFI | Riemann--Liouville fractional integral |
MNC | Measure of noncompactness |
BS | Banach space |
CS | Cauchy sequence |
ARE | Absolute relative error |
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The authors would like to thank Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R406), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
Conceptualization, Eman A. A. Ziada and Hind Hashem; Formal analysis, Hind Hashem; Funding acquisition, Asma Al-Jaser; Investigation, Asma Al-Jaser and Osama Moaaz; Methodology, Eman A. A. Ziada and Monica Botros; Software, Asma Al-Jaser and Monica Botros; Writing—original draft, Eman A. A. Ziada and Monica Botros; Writing—review and editing, Osama Moaaz. All authors have read and agreed to the published version of the manuscript.
The authors declare there is no conflict of interest.
[1] |
Abbas AJ, Hassani H, Burby M, et al. (2021) An investigation into the volumetric flow rate requirement of hydrogen transportation in existing natural gas pipelines and its safety implications. Gases 1: 156–179. https://doi.org/10.3390/gases1040013 doi: 10.3390/gases1040013
![]() |
[2] |
Abujarad SY, Mustafa MW, Jamian JJ (2017) Recent approaches of unit commitment in the presence of intermittent renewable energy resources: A review. Renew Sust Energ Rev 70: 215–223. https://doi.org/10.1016/j.rser.2016.11.246 doi: 10.1016/j.rser.2016.11.246
![]() |
[3] |
Acemoglu D, Aghion P, Bursztyn L, et al. (2012) The Environment and Directed Technical Change. Am Econ Rev 102: 131–166. https://doi.org/10.1257/aer.102.1.131 doi: 10.1257/aer.102.1.131
![]() |
[4] |
Al-Refaie A, Lepkova N (2022) Impacts of Renewable Energy Policies on CO2 Emissions Reduction and Energy Security Using System Dynamics: The Case of Small-Scale Sector in Jordan. Sustainability 14: 5058. https://doi.org/10.3390/su14095058 doi: 10.3390/su14095058
![]() |
[5] | Anderson B, Cammeraat E, Dechezleprêtre A, et al. (2021) "Policies for a climate-neutral industry: Lessons from the Netherlands", OECD Science, Technology and Industry Policy Papers, 108, OECD Publishing, Paris, Available from: https://doi.org/10.1787/a3a1f953-en. |
[6] |
Antenucci A, Sansavini G (2019) Extensive CO2 recycling in power systems via Power-to-Gas and network storage. Renew Sust Energ Rev 100: 33–43. https://doi.org/10.1016/j.rser.2018.10.020 doi: 10.1016/j.rser.2018.10.020
![]() |
[7] |
Baker F (2022) Is the United Kingdom's Hydrogen Strategy an Effective Low Carbon Strategy? Int J Energ Prod Manag 7: 164–175. https://doi.org/10.2495/EQ-V7-N2-164-175 doi: 10.2495/EQ-V7-N2-164-175
![]() |
[8] |
Bale CS, Varga L, Foxon TJ (2015) Energy and complexity: New ways forward. Appl Energ 138: 150–159. https://doi.org/10.1016/j.apenergy.2014.10.057 doi: 10.1016/j.apenergy.2014.10.057
![]() |
[9] |
Ballo A, Valentin KK, Korgo B, et al. (2022) Law and Policy Review on Green Hydrogen Potential in ECOWAS Countries. Energies 15: 2304. https://doi.org/10.3390/en15072304 doi: 10.3390/en15072304
![]() |
[10] |
Bersalli G, Menanteau P, El-Methni J (2020) Renewable energy policy effectiveness: A panel data analysis across Europe and Latin America. Renew Sust Energ Rev 133: 110351. https://doi.org/10.1016/j.rser.2020.110351 doi: 10.1016/j.rser.2020.110351
![]() |
[11] | Bianco E, Blanco H (2020) Green hydrogen: a guide to policy making. IRENA. Available from: https://www.h2knowledgecentre.com/content/researchpaper1616. |
[12] |
Bölük G, Kaplan R (2022) Effectiveness of renewable energy incentives on sustainability: evidence from dynamic panel data analysis for the EU countries and Turkey. Environ Sci Pollut Res 2022: 1–18. https://doi.org/10.1007/s11356-021-17801-y doi: 10.1007/s11356-021-17801-y
![]() |
[13] | Cammeraat EA, Dechezleprêtre et G Lalanne (2022) « Innovation and industrial policies for green hydrogen », OECD Science, Technology and Industry Policy Papers, n 125, Éditions OCDE, Paris. https://doi.org/10.1787/f0bb5d8c-en |
[14] | Chakrabarty A (2022) UK: Applications open for first round of hydrogen funding. Sustainable Futures. Available from: https://sustainablefutures.linklaters.com/post/102htf2/uk-applications-open-for-first-round-of-hydrogen-funding. |
[15] |
Chen B, Xiong R, Li H, et al. (2019) Pathways for sustainable energy transition. J Clean Prod 228: 1564–1571. https://doi.org/10.1016/j.jclepro.2019.04.372 doi: 10.1016/j.jclepro.2019.04.372
![]() |
[16] |
Cheng W, Lee S (2022) How Green Are the National Hydrogen Strategies? Sustainability 14: 1930. https://doi.org/10.3390/su14031930 doi: 10.3390/su14031930
![]() |
[17] |
Chu KH, Lim J, Mang JS, et al. (2022) Evaluation of strategic directions for supply and demand of green hydrogen in South Korea. Int J Hydrogen Energ 47: 1409–1424. https://doi.org/10.1016/j.ijhydene.2021.10.107 doi: 10.1016/j.ijhydene.2021.10.107
![]() |
[18] | Clean Hydrogen Partnership (2022) European Partnership for Hydrogen Technologies. Available from: https://www.clean-hydrogen.europa.eu/index_en. |
[19] | Clifford C (2022) Why the EU didn't include nuclear energy in its plan to get off Russian gas. CNBC. Available from: https://www.cnbc.com/2022/03/09/why-eu-didnt-include-nuclear-energy-in-plan-to-get-off-russian-gas.html. |
[20] |
Côté E, Salm S (2022) Risk-adjusted preferences of utility companies and institutional investors for battery storage and green hydrogen investment. Energ Policy 163: 112821. https://doi.org/10.1016/j.enpol.2022.112821 doi: 10.1016/j.enpol.2022.112821
![]() |
[21] |
da Silva César A, da Silva Veras T, Mozer TS, et al. (2019) Hydrogen productive chain in Brazil: An analysis of the competitiveness' drivers. J Clean Prod 207: 751–763. https://doi.org/10.1016/j.jclepro.2018.09.157 doi: 10.1016/j.jclepro.2018.09.157
![]() |
[22] |
da Silva Veras T, Mozer TS, da Silva César A (2017) Hydrogen: trends, production and characterization of the main process worldwide. Int J Hydrogen Energ 42: 2018–2033. https://doi.org/10.1016/j.ijhydene.2016.08.219 doi: 10.1016/j.ijhydene.2016.08.219
![]() |
[23] |
de las Nieves Camacho M, Jurburg D, Tanco M (2022) Hydrogen fuel cell heavy-duty trucks: Review of main research topics. Int J Hydrogen Energ. https://doi.org/10.1016/j.ijhydene.2022.06.271 doi: 10.1016/j.ijhydene.2022.06.271
![]() |
[24] |
Dehhaghi S, Choobchian S, Ghobadian B, et al. (2022) Five-year development plans of renewable energy policies in Iran: a content analysis. Sustainability 14: 1501. https://doi.org/10.3390/su14031501 doi: 10.3390/su14031501
![]() |
[25] | Department of Energy (DOE) (2020) Energy Department Announces Approximately $64M in Funding for 18 Projects to Advance H2@Scale. Available from: https://www.energy.gov/articles/energy-department-announces-approximately-64m-funding-18-projects-advance-h2scale. |
[26] | Department of Energy (DOE) (2021) DOE Establishes New Office of Clean Energy Demonstrations Under the Bipartisan Infrastructure Law. Available from: https://www.energy.gov/articles/doe-establishes-new-office-clean-energy-demonstrations-under-bipartisan-infrastructure-law. |
[27] | Department of Energy (DOE) (2022) DOE National Clean Hydrogen Strategy and Roadmap. Available from: https://www.hydrogen.energy.gov/pdfs/clean-hydrogen-strategy-roadmap.pdf. |
[28] | Department of Energy2 (DOE) (2021) DOE Announces $20 Million to Produce Clean Hydrogen from Nuclear Power. Available from: https://www.energy.gov/articles/doe-announces-20-million-produce-clean-hydrogen-nuclear-power. |
[29] | Department of Energy2 (DOE) (2022) DOE Announces Nearly $25 Million to Study Advanced Clean Hydrogen Technologies for Electricity Generation. Available from: https://www.energy.gov/articles/doe-announces-nearly-25-million-study-advanced-clean-hydrogen-technologies-electricity. |
[30] | Department of Energy3 (DOE) (2021) DOE Announces $52.5 Million to Accelerate Progress in Clean Hydrogen. Available from: https://www.energy.gov/articles/doe-announces-525-million-accelerate-progress-clean-hydrogen. |
[31] | Department of Energy3 (DOE) (2022) DOE Announces First Loan Guarantee for a Clean Energy Project in Nearly a Decade. Available from: https://www.energy.gov/articles/doe-announces-first-loan-guarantee-clean-energy-project-nearly-decade. |
[32] | Department of Energy4 (DOE) (2022) DOE Announces $60 Million to Advance Clean Hydrogen Technologies and Decarbonize Grid. Available from: https://www.energy.gov/articles/doe-announces-60-million-advance-clean-hydrogen-technologies-and-decarbonize-grid. |
[33] | Donoghue NM, Thompson P, Arora T (2022) UK Government Publishes Hydrogen Investment Roadmap. Available from: https://www.bakermckenzie.com/en/insight/publications/2022/04/uk-hydrogen-investment-roadmap. |
[34] | Energy Efficiency and Renewable Energy (EERE) (2020) Energy Department Announces $33 Million to Advance Hydrogen and Fuel Cell R & D and the H2@Scale Vision. Available from: https://www.energy.gov/eere/articles/energy-department-announces-33-million-advance-hydrogen-and-fuel-cell-rd-and-h2scale. |
[35] | Energy Efficiency and Renewable Energy (EERE) (2021) DOE Announces Nearly $8 Million for National Laboratory H2@Scale Projects to Help Reach Hydrogen Shot Goals. |
[36] | Energy Efficiency and Renewable Energy (EERE) (2022) Clean Hydrogen Production Standard. Available from: https://www.energy.gov/eere/fuelcells/articles/clean-hydrogen-production-standard. |
[37] | Erbach G, Jensen L (2021) EU hydrogen policy: Hydrogen as an energy carrier for a climate-neutral economy. Available from: https://policycommons.net/artifacts/1426785/eu-hydrogen-policy/2041311/. |
[38] | European Commission (2019) Energy and the Green Deal: A clean energy transition. Available from: https://commission.europa.eu/strategy-and-policy/priorities-2019-2024/european-green-deal/energy-and-green-deal_en |
[39] | European Commission (2020) A hydrogen strategy for a climate-neutral Europe, EPRS: European Parliamentary Research Service. Available from: https://ec.europa.eu/energy/sites/ener/files/hydrogen_strategy.pdf. |
[40] | European Commission (2022) EU Emissions Trading System (EU ETS). Available from: https://ec.europa.eu/clima/eu-action/eu-emissions-trading-system-eu-ets_en. |
[41] | European Commission2 (2022) Key actions of the EU Hydrogen Strategy. Available from: https://energy.ec.europa.eu/topics/energy-systems-integration/hydrogen/key-actions-eu-hydrogen-strategy_en. |
[42] | European Commission3 (2022) What is the Innovation Fund? Available from: https://competition-policy.ec.europa.eu/state-aid/legislation/modernisation/ipcei_en. |
[43] | European Commission4 (2022) What is the Innovation Fund? Available from: https://climate.ec.europa.eu/eu-action/funding-climate-action/innovation-fund/what-innovation-fund_en. |
[44] | European Commission5 (2022) REPowerEU: Joint European action for more affordable, secure and sustainable energy? Available from: https://ec.europa.eu/commission/presscorner/detail/en/ip_22_1511. |
[45] | European Commission6 (2022) Commission awards over €1 billion to innovative projects for the EU climate transition. Available from: https://ec.europa.eu/commission/presscorner/detail/en/IP_22_2163. |
[46] | European Commission7 (2022) EU Taxonomy Compass. Database. Available from: https://ec.europa.eu/sustainable-finance-taxonomy/home. |
[47] |
Falcone PM, Morone P, Sica E (2018) Greening of the financial system and fuelling a sustainability transition: A discursive approach to assess landscape pressures on the Italian financial system. Technol Forecast Soc 127: 23–37. https://doi.org/10.1016/j.techfore.2017.05.020 doi: 10.1016/j.techfore.2017.05.020
![]() |
[48] |
Falcone PM, Sica E (2019) Assessing the opportunities and challenges of green finance in Italy: An analysis of the biomass production sector. Sustainability 11: 517. https://doi.org/10.3390/su11020517 doi: 10.3390/su11020517
![]() |
[49] |
Fankhauser S, Jotzo F (2018) Economic growth and development with low‐carbon energy. Wires Clim Change 9: e495. https://doi.org/10.1002/wcc.495 doi: 10.1002/wcc.495
![]() |
[50] | Faure A, Okullo SJ, Pahle M (2020) Price and quantity policies to improve the EU-ETS: which is best? Technical Report. |
[51] | Fossil Energy and Carbon Management (FECM) (2022) DOE Invests $2.4 Million for Next-Generation Energy Storage Technologies. Available from: https://www.energy.gov/fecm/articles/doe-invests-24-million-next-generation-energy-storage-technologies. |
[52] | Fossil Energy and Carbon Management2 (FECM) (2022) U.S. Department of Energy Announces $28 Million to Develop Clean Hydrogen. Available from: https://www.energy.gov/fecm/articles/us-department-energy-announces-28-million-develop-clean-hydrogen?utm_medium = email & utm_source = govdelivery. |
[53] |
Gordon JA, Balta-Ozkan N, Nabavi SA (2023) Socio-technical barriers to domestic hydrogen futures: Repurposing pipelines, policies, and public perceptions. Appl Energ 336: 120850. https://doi.org/10.1016/j.apenergy.2023.120850 doi: 10.1016/j.apenergy.2023.120850
![]() |
[54] | Greenstone M, Nath I (2019) Do renewable portfolio standards deliver? University of Chicago, Becker Friedman Institute for Economics Working Paper 62. |
[55] |
Haas R, Resch G, Panzer C, et al. (2011) Efficiency and effectiveness of promotion systems for electricity generation from renewable energy sources–Lessons from EU countries. Energy 36: 2186–2193. https://doi.org/10.1016/j.energy.2010.06.028 doi: 10.1016/j.energy.2010.06.028
![]() |
[56] |
Hafeznia H, Aslani A, Anwar S, et al. (2017) Analysis of the effectiveness of national renewable energy policies: A case of photovoltaic policies. Renew Sust Energ Rev 79: 669–680. https://doi.org/10.1016/j.rser.2017.05.033 doi: 10.1016/j.rser.2017.05.033
![]() |
[57] | Holloway S, Vincent CJ, Kirk K (2006) Industrial carbon dioxide emissions and carbon dioxide storage potential in the UK. Nottingham, UK, British Geological Survey, 60. Available from: https://www.energy.gov/eere/articles/doe-announces-nearly-8-million-national-laboratory-h2scale-projects-help-reach. |
[58] | Hoogland O, Eklund L, Dahl V (2022) The fiscal implications of the clean energy transition, Directorate-General for Energy. Publications Office of the European Union. Available from: https://data.europa.eu/doi/10.2833/941143. |
[59] |
Howlett M, Leong C (2022) Policy volatility and the propensity of policies to fail: dealing with uncertainty, maliciousness and compliance in public policy-making. Int J Public Policy 16: 236–252. https://doi.org/10.1504/IJPP.2022.127431 doi: 10.1504/IJPP.2022.127431
![]() |
[60] |
Hu J, Harmsen R, Crijns-Graus W, et al. (2018) Identifying barriers to large-scale integration of variable renewable electricity into the electricity market: A literature review of market design. Renew Sust Energ Rev 81: 2181–2195. https://doi.org/10.1016/j.rser.2017.06.028 doi: 10.1016/j.rser.2017.06.028
![]() |
[61] | HyResource (2022) A collaborative knowledge sharing resource supporting the development of Australia's hydrogen industry. Available from: https://research.csiro.au/hyresource/policy/international/european-commission/. |
[62] | IEA (2022) Hydrogen Projects Database. Available from: https://www.iea.org/data-and-statistics/data-product/hydrogen-projects-database. |
[63] | IEA1 (2021) Hydrogen. International Energy Agency, Paris. Available from: https://www.iea.org/reports/hydrogen. |
[64] | IEA2 (2021) Global Hydrogen Review 2021. IEA, Paris. Available from: https://www.iea.org/reports/global-hydrogen-review-2021. |
[65] | IEA2 (2022) Policies database, 2022, IEA. Available from: https://www.iea.org/policies. |
[66] | IRENA (2022) World Energy Transitions Outlook 2022: 1.5℃ Pathway, International Renewable Energy Agency, Abu Dhabi. |
[67] |
Iribarren D, Martín-Gamboa M, Navas-Anguita Z, et al. (2020) Influence of climate change externalities on the sustainability-oriented prioritisation of prospective energy scenarios. Energy 196: 117179. https://doi.org/10.1016/j.energy.2020.117179 doi: 10.1016/j.energy.2020.117179
![]() |
[68] |
Jafari H, Safarzadeh S, Azad-Farsani E (2022) Effects of governmental policies on energy-efficiency improvement of hydrogen fuel cell cars: A game-theoretic approach. Energy 254: 124394. https://doi.org/10.1016/j.energy.2022.124394 doi: 10.1016/j.energy.2022.124394
![]() |
[69] |
Jones J, Genovese A, Tob-Ogu A (2020) Hydrogen vehicles in urban logistics: A total cost of ownership analysis and some policy implications. Renew Sust Energ Rev 119: 109595. https://doi.org/10.1016/j.rser.2019.109595 doi: 10.1016/j.rser.2019.109595
![]() |
[70] |
Krozer Y (2019) Financing of the global shift to renewable energy and energy efficiency. Green Financ 1: 264–278. https://doi.org/10.3934/GF.2019.3.264 doi: 10.3934/GF.2019.3.264
![]() |
[71] | Kurmayer N (2021) €2 billion 'Clean Hydrogen Partnership' signals move away from hydrogen cars. Euractiv. Available from: https://www.euractiv.com/section/energy/news/e2-billion-clean-hydrogen-partnership-another-move-away-from-hydrogen-cars/. |
[72] |
Lezama F, Soares J, Hernandez-Leal P, et al. (2018) Local energy markets: Paving the path toward fully transactive energy systems. IEEE T Power Syst 34: 4081–4088. https://doi.org/10.1109/TPWRS.2018.2833959 doi: 10.1109/TPWRS.2018.2833959
![]() |
[73] |
Mahajan D, Tan K, Venkatesh T, et al. (2022) Hydrogen Blending in Gas Pipeline Networks—A Review. Energies 15: 3582. https://doi.org/10.3390/en15103582 doi: 10.3390/en15103582
![]() |
[74] | Maisonneuve C (2022) European Energy Sovereignty: Putting an End to the Stigma of Nuclear Power. Institut Montaigne. Available from: https://www.institutmontaigne.org/en/blog/european-energy-sovereignty-putting-end-stigma-nuclear-power. |
[75] |
Marques AC, Fuinhas JA (2012) Are public policies towards renewables successful? Evidence from European countries. Renew Energ 44: 109–118. https://doi.org/10.1016/j.renene.2012.01.007 doi: 10.1016/j.renene.2012.01.007
![]() |
[76] | McQueen S, Stanford J, Satyapal S, et al. (2020) Department of energy hydrogen program plan (No. DOE/EE-2128). US Department of Energy (USDOE), Washington DC (United States). https://doi.org/10.2172/1721803 |
[77] |
Moore J, Shabani B (2016) A critical study of stationary energy storage policies in Australia in an international context: the role of hydrogen and battery technologies. Energies 9: 674. https://doi.org/10.3390/en9090674 doi: 10.3390/en9090674
![]() |
[78] | OECD (2023) Economic Outlook, Interim Report March 2023: A Fragile Recovery. Available from: https://www.oecd-ilibrary.org/sites/d14d49eb-en/index.html?itemId = /content/publication/d14d49eb-en. |
[79] |
Oliveira AM, Beswick RR, Yan Y (2021) A green hydrogen economy for a renewable energy society. Curr Opin Chem Eng 33: 100701. https://doi.org/10.1016/j.coche.2021.100701 doi: 10.1016/j.coche.2021.100701
![]() |
[80] |
Park C, Lim S, Shin J, et al. (2022) How much hydrogen should be supplied in the transportation market? Focusing on hydrogen fuel cell vehicle demand in South Korea: Hydrogen demand and fuel cell vehicles in South Korea. Technol Forecast Soc Change 181: 121750. https://doi.org/10.1016/j.techfore.2022.121750 doi: 10.1016/j.techfore.2022.121750
![]() |
[81] | Parkes R (2022) 'From niche to scale' | EU launches €3bn European Hydrogen Bank with a bang but keeps quiet about the details. Recharge, 14 September 2022. Available from: https://www.rechargenews.com/energy-transition/from-niche-to-scale-eu-launches-3bn-european-hydrogen-bank-with-a-bang-but-keeps-quiet-about-the-details/2-1-1299131. |
[82] | Parkes R2 (2022) Biden invokes wartime legislation to ramp up US hydrogen electrolyser production, but what will this mean in practice? Available from: https://www.rechargenews.com/energy-transition/biden-invokes-wartime-legislation-to-ramp-up-us-hydrogen-electrolyser-production-but-what-will-this-mean-in-practice-/2-1-1235045. |
[83] |
Polzin F, Migendt M, Täube FA, et al. (2015) Public policy influence on renewable energy investments—A panel data study across OECD countries. Energ policy 80: 98-111. https://doi.org/10.1016/j.enpol.2015.01.026 doi: 10.1016/j.enpol.2015.01.026
![]() |
[84] |
Prăvălie R, Bandoc G (2018) Nuclear energy: Between global electricity demand, worldwide decarbonisation imperativeness, and planetary environmental implications. J Environ Manage 209: 81–92. https://doi.org/10.1016/j.jenvman.2017.12.043 doi: 10.1016/j.jenvman.2017.12.043
![]() |
[85] | REPowerEU Plan (2018) Communication from the Commission to the European Parliament, the European Council, the Council, the European Economic and Social Committee and the Committee of the Regions. |
[86] |
Rodríguez MLÁ, Flores JJA, Vera JVA, et al. (2022) The regulatory framework of the hydrogen market in Mexico: a look at energy governance. Int J Hydrogen Energ 47: 29986–29998. https://doi.org/10.1016/j.ijhydene.2022.05.168 doi: 10.1016/j.ijhydene.2022.05.168
![]() |
[87] |
Roy J, Ghosh D, Ghosh A, et al. (2013) Fiscal instruments: crucial role in financing low carbon transition in energy systems. Curr Opin Environ Sust 5: 261–269. https://doi.org/10.1016/j.cosust.2013.05.003 doi: 10.1016/j.cosust.2013.05.003
![]() |
[88] | S & P Global Commodity insights (2022) UK's gas grid ready for 20% hydrogen blend from 2023: network companies. Electric Power. Energy Transition. Natural Gas. Available from: https://www.spglobal.com/commodityinsights/en/market-insights/latest-news/electric-power/011422-uks-gas-grid-ready-for-20-hydrogen-blend-from-2023-network-companies. |
[89] |
Sasanpour S, Cao KK, Gils HC, et al. (2021) Strategic policy targets and the contribution of hydrogen in a 100% renewable European power system. Energ Rep 7: 4595–4608. https://doi.org/10.1016/j.egyr.2021.07.005 doi: 10.1016/j.egyr.2021.07.005
![]() |
[90] |
Shin J, Hwang WS, Choi H (2019) Can hydrogen fuel vehicles be a sustainable alternative on vehicle market? Comparison of electric and hydrogen fuel cell vehicles. Technol Forecast Soc 143: 239–248. https://doi.org/10.1016/j.techfore.2019.02.001 doi: 10.1016/j.techfore.2019.02.001
![]() |
[91] | Surana K, Anadon LD (2015) Public policy and financial resource mobilization for wind energy in developing countries: A comparison of approaches and outcomes in China and IndiaGlob. Environ. Chang, 35: 340–359. https://doi.org/10.1016/j.gloenvcha.2015.10.001 |
[92] |
Talebian H, Herrera OE, Mérida W (2021) Policy effectiveness on emissions and cost reduction for hydrogen supply chains: The case for British Columbia. Int J Hydrogen Energ 46: 998–1011. https://doi.org/10.1016/j.ijhydene.2020.09.190 doi: 10.1016/j.ijhydene.2020.09.190
![]() |
[93] | UK Government (2020) The ten point plan for a green industrial revolution. Department for Business, Energy & Industrial Strategy. Available from: https://www.gov.uk/government/publications/the-ten-point-plan-for-a-green-industrial-revolution. |
[94] | UK Government (2021) UK Hydrogen Strategy. Department for Business, Energy & Industrial Strategy. Available from: https://www.gov.uk/government/publications/the-ten-point-plan-for-a-green-industrial-revolution. |
[95] | UK Government (2022) Hydrogen Strategy update to the market: July 2022. Department for Business Energy & Industrial Strategy. Available from: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/1092555/hydrogen-strategy-update-to-the-market-july-2022.pdf. |
[96] | UK Government2 (2021) Net Zero Strategy: Build Back Greener. Available from: https://www.gov.uk/government/publications/net-zero-strategy. |
[97] | UK Government2 (2022) Hydrogen BECCS Innovation Programme: successful projects. Available from: https://www.gov.uk/government/publications/hydrogen-beccs-innovation-programme-successful-projects. |
[98] | UK Government3 (2021) Industrial Fuel Switching competition Phase 1: feasibility studies. Available from: https://www.gov.uk/government/publications/industrial-fuel-switching-competition. |
[99] | UK Government3 (2022) Government unveils investment for energy technologies of the future. Available from: https://www.gov.uk/government/news/government-unveils-investment-for-energy-technologies-of-the-future. |
[100] | UK Government4 (2021) Tees Valley multi-modal hydrogen transport hub. Available from: https://www.gov.uk/government/publications/tees-valley-multi-modal-hydrogen-transport-hub. |
[101] | UK Government4 (2022) UK Low Carbon Hydrogen Standard: emissions reporting and sustainability criteria. Available from: https://www.gov.uk/government/publications/uk-low-carbon-hydrogen-standard-emissions-reporting-and-sustainability-criteria. |
[102] | UK Government5 (2021) Rail centre and green energy funding in Budget boost for Wales. Available from: https://www.gov.uk/government/news/rail-centre-and-green-energy-funding-in-budget-boost-for-wales. |
[103] |
Van de Graaf T, Overland I, Scholten D, et al. (2020) The new oil? The geopolitics and international governance of hydrogen. Energ Res Soc Sci 70: 101667. https://doi.org/10.1016/j.erss.2020.101667 doi: 10.1016/j.erss.2020.101667
![]() |
[104] |
Van Renssen S (2020) The hydrogen solution? Nat Clim Change 10: 799–801. https://doi.org/10.1038/s41558-020-0891-0 doi: 10.1038/s41558-020-0891-0
![]() |
[105] |
Willner M, Perino G (2022) Beyond control: Policy incoherence of the EU emissions trading system. Politics Gov10: 256–264. https://doi.org/10.17645/pag.v10i1.4797 doi: 10.17645/pag.v10i2.5183
![]() |
[106] |
Ye F, Paulson N, Khanna M (2022) Are renewable energy policies effective to promote technological change? The role of induced technological risk. J Environ Econ Manag 114: 102665. https://doi.org/10.1016/j.jeem.2022.102665 doi: 10.1016/j.jeem.2022.102665
![]() |
[107] |
Zahedi R, Zahedi A, Ahmadi A (2022) Strategic study for renewable energy policy, optimizations and sustainability in Iran. Sustainability 14: 2418. https://doi.org/10.3390/su14042418 doi: 10.3390/su14042418
![]() |
![]() |
![]() |
1. | Zhenghui Li, Qinyang Lai, Jiajia He, Does digital technology enhance the global value chain position?, 2024, 24, 22148450, 856, 10.1016/j.bir.2024.04.016 |
\varrho | \left\vert \frac{\Phi _{ES}-\Phi _{ADM}}{\Phi _{ES}}\right\vert | \left\vert \frac{\Phi _{ES}-\Phi _{PM}}{\Phi _{ES}}\right\vert |
0.1 | 2.48379 \times 10^{-9} | 2.19628 \times 10^{-8} |
0.2 | 2.47045 \times 10^{-7} | 7.0281 \times 10^{-7} |
0.3 | 2.04146 \times 10^{-6} | 5.33709 \times 10^{-6} |
0.4 | 6.32111 \times 10^{-6} | 0.0000224935 |
0.5 | 5.3704 \times 10^{-6} | 0.0000686815 |
0.6 | 0.0000360447 | 0.000171175 |
0.7 | 0.000215013 | 0.000371487 |
0.8 | 0.000750286 | 0.000731218 |
0.9 | 0.0020904 | 0.00134688 |
1 | 0.00511876 | 0.00240123 |
1.1 | 0.0115464 | 0.00436134 |
1.2 | 0.024631 | 0.00870185 |
1.3 | 0.0503602 | 0.0201514 |
1.4 | 0.0990462 | 0.0519358 |
1.5 | 0.186618 | 0.132278 |
\varrho | \left\vert \frac{\chi _{ES}-\chi _{ADM}}{\chi _{ES}}\right\vert | \left\vert \frac{\chi _{ES}-\chi _{PM}}{\chi _{ES}}\right\vert |
0.1 | 1.0964*10 ^{{-6}} | 0.0000724208 |
0.2 | 0.0000268761 | 0.00060454 |
0.3 | 0.000179457 | 0.00212539 |
0.4 | 0.000701695 | 0.0052377 |
0.5 | 0.00204302 | 0.0106117 |
0.6 | 0.00493079 | 0.0189737 |
0.7 | 0.0104472 | 0.0310892 |
0.8 | 0.0201116 | 0.0477376 |
0.9 | 0.0359674 | 0.0696801 |
1 | 0.0606745 | 0.0976178 |
IVP | Initial value problem |
ADM | Adomian decomposition method |
PM | Picard method |
FDEs | Fractional differential equations |
HDE | Hybrid differential equation |
CQIE | Chandrasekhar quadratic integral equation |
RLFI | Riemann--Liouville fractional integral |
MNC | Measure of noncompactness |
BS | Banach space |
CS | Cauchy sequence |
ARE | Absolute relative error |
\varrho | \left\vert \frac{\Phi _{ES}-\Phi _{ADM}}{\Phi _{ES}}\right\vert | \left\vert \frac{\Phi _{ES}-\Phi _{PM}}{\Phi _{ES}}\right\vert |
0.1 | 2.48379 \times 10^{-9} | 2.19628 \times 10^{-8} |
0.2 | 2.47045 \times 10^{-7} | 7.0281 \times 10^{-7} |
0.3 | 2.04146 \times 10^{-6} | 5.33709 \times 10^{-6} |
0.4 | 6.32111 \times 10^{-6} | 0.0000224935 |
0.5 | 5.3704 \times 10^{-6} | 0.0000686815 |
0.6 | 0.0000360447 | 0.000171175 |
0.7 | 0.000215013 | 0.000371487 |
0.8 | 0.000750286 | 0.000731218 |
0.9 | 0.0020904 | 0.00134688 |
1 | 0.00511876 | 0.00240123 |
1.1 | 0.0115464 | 0.00436134 |
1.2 | 0.024631 | 0.00870185 |
1.3 | 0.0503602 | 0.0201514 |
1.4 | 0.0990462 | 0.0519358 |
1.5 | 0.186618 | 0.132278 |
\varrho | \left\vert \frac{\chi _{ES}-\chi _{ADM}}{\chi _{ES}}\right\vert | \left\vert \frac{\chi _{ES}-\chi _{PM}}{\chi _{ES}}\right\vert |
0.1 | 1.0964*10 ^{{-6}} | 0.0000724208 |
0.2 | 0.0000268761 | 0.00060454 |
0.3 | 0.000179457 | 0.00212539 |
0.4 | 0.000701695 | 0.0052377 |
0.5 | 0.00204302 | 0.0106117 |
0.6 | 0.00493079 | 0.0189737 |
0.7 | 0.0104472 | 0.0310892 |
0.8 | 0.0201116 | 0.0477376 |
0.9 | 0.0359674 | 0.0696801 |
1 | 0.0606745 | 0.0976178 |
IVP | Initial value problem |
ADM | Adomian decomposition method |
PM | Picard method |
FDEs | Fractional differential equations |
HDE | Hybrid differential equation |
CQIE | Chandrasekhar quadratic integral equation |
RLFI | Riemann--Liouville fractional integral |
MNC | Measure of noncompactness |
BS | Banach space |
CS | Cauchy sequence |
ARE | Absolute relative error |