
Citation: B. Ravit, K. Cooper, B. Buckley, I. Yang, A. Deshpande. Organic compounds associated with microplastic pollutants in New Jersey, U.S.A. surface waters[J]. AIMS Environmental Science, 2019, 6(6): 445-459. doi: 10.3934/environsci.2019.6.445
[1] | Debao Yan . Existence results of fractional differential equations with nonlocal double-integral boundary conditions. Mathematical Biosciences and Engineering, 2023, 20(3): 4437-4454. doi: 10.3934/mbe.2023206 |
[2] | Abdon Atangana, Jyoti Mishra . Analysis of nonlinear ordinary differential equations with the generalized Mittag-Leffler kernel. Mathematical Biosciences and Engineering, 2023, 20(11): 19763-19780. doi: 10.3934/mbe.2023875 |
[3] | Allaberen Ashyralyev, Evren Hincal, Bilgen Kaymakamzade . Crank-Nicholson difference scheme for the system of nonlinear parabolic equations observing epidemic models with general nonlinear incidence rate. Mathematical Biosciences and Engineering, 2021, 18(6): 8883-8904. doi: 10.3934/mbe.2021438 |
[4] | Sebastian Builes, Jhoana P. Romero-Leiton, Leon A. Valencia . Deterministic, stochastic and fractional mathematical approaches applied to AMR. Mathematical Biosciences and Engineering, 2025, 22(2): 389-414. doi: 10.3934/mbe.2025015 |
[5] | Hardik Joshi, Brajesh Kumar Jha, Mehmet Yavuz . Modelling and analysis of fractional-order vaccination model for control of COVID-19 outbreak using real data. Mathematical Biosciences and Engineering, 2023, 20(1): 213-240. doi: 10.3934/mbe.2023010 |
[6] | Barbara Łupińska, Ewa Schmeidel . Analysis of some Katugampola fractional differential equations with fractional boundary conditions. Mathematical Biosciences and Engineering, 2021, 18(6): 7269-7279. doi: 10.3934/mbe.2021359 |
[7] | Jian Huang, Zhongdi Cen, Aimin Xu . An efficient numerical method for a time-fractional telegraph equation. Mathematical Biosciences and Engineering, 2022, 19(5): 4672-4689. doi: 10.3934/mbe.2022217 |
[8] | Yingying Xu, Chunhe Song, Chu Wang . Few-shot bearing fault detection based on multi-dimensional convolution and attention mechanism. Mathematical Biosciences and Engineering, 2024, 21(4): 4886-4907. doi: 10.3934/mbe.2024216 |
[9] | H. M. Srivastava, Khaled M. Saad, J. F. Gómez-Aguilar, Abdulrhman A. Almadiy . Some new mathematical models of the fractional-order system of human immune against IAV infection. Mathematical Biosciences and Engineering, 2020, 17(5): 4942-4969. doi: 10.3934/mbe.2020268 |
[10] | Guodong Li, Ying Zhang, Yajuan Guan, Wenjie Li . Stability analysis of multi-point boundary conditions for fractional differential equation with non-instantaneous integral impulse. Mathematical Biosciences and Engineering, 2023, 20(4): 7020-7041. doi: 10.3934/mbe.2023303 |
Fractional calculus is a main branch of mathematics that can be considered as the generalisation of integration and differentiation to arbitrary orders. This hypothesis begins with the assumptions of L. Euler (1730) and G. W. Leibniz (1695). Fractional differential equations (FDEs) have lately gained attention and publicity due to their realistic and accurate computations [1,2,3,4,5,6,7]. There are various types of fractional derivatives, including Riemann–Liouville, Caputo, Grü nwald–Letnikov, Weyl, Marchaud, and Atangana. This topic's history can be found in [8,9,10,11]. Undoubtedly, fractional calculus applies to mathematical models of different phenomena, sometimes more effectively than ordinary calculus [12,13]. As a result, it can illustrate a wide range of dynamical and engineering models with greater precision. Applications have been developed and investigated in a variety of scientific and engineering fields over the last few decades, including bioengineering [14], mechanics [15], optics [16], physics [17], mathematical biology, electrical power systems [18,19,20] and signal processing [21,22,23].
One of the definitions of fractional derivatives is Caputo-Fabrizo, which adds a new dimension in the study of FDEs. The new derivative's feature is that it has a nonsingular kernel, which is made from a combination of an ordinary derivative with an exponential function, but it has the same supplementary motivating properties with various scales as in the Riemann-Liouville fractional derivatives and Caputo. The Caputo-Fabrizio fractional derivative has been used to solve real-world problems in numerous areas of mathematical modelling for example, numerical solutions for groundwater pollution, the movement of waves on the surface of shallow water modelling [24], RLC circuit modelling [25], and heat transfer modelling [26,27] were discussed.
Rach (1987), Bellomo and Sarafyan (1987) first compared the Adomian Decomposition method (ADM) [28,29,30,31,32] to the Picard method on a variety of examples. These methods have many benefits: they effectively work with various types of linear and nonlinear equations and also provide an analytic solution for all of these equations with no linearization or discretization. These methods are more realistic compared with other numerical methods as each technique is used to solve a specific type of equations, on the other hand ADM and Picard are useful for many types of equations. In the numerical examples provided, we compare ADM and Picard solutions of multidimentional fractional order equations with Caputo-Fabrizio.
The fractional derivative of Caputo-Fabrizio for the function x(t) is defined as [33]
CFDα0x(t)=B(α)1−α∫t0dds(x(s)) e−α1−α(t−s)ds, | (1.1) |
and its corresponding fractional integral is
CFIαx(t)=1−αB(α)x(t)+αB(α)∫t0x (s)ds, 0<α<1, | (1.2) |
where x(t) be continuous and differentiable on [0, T]. Also, in the above definition, the function B(α)>0 is a normalized function which satisfy the condition B(0)=B(1)=0. The relation between the Caputo–Fabrizio fractional derivate and its corresponding integral is given by
(CFIα0)(CFDα0f(t))=f(t)−f(a). | (1.3) |
In this section, we will introduce a multidimentional FDE subject to the initial condition. Let α∈(0,1], 0<α1<α2<...,αm<1, and m is integer real number,
CFDx=f(t,x,CFDα1x,CFDα2x,...,CFDαmx,) ,x(0)=c0, | (2.1) |
where x=x(t),t∈J=[0,T],T∈R+,x∈C(J).
To facilitate the equation and make it easy for the calculation, we let x(t)=c0+X(t) so Eq (2.1) can be witten as
CFDαX=f(t,c0+X,CFDα1X,CFDα2X,...,CFDαmX), X(0)=0. | (2.2) |
the algorithm depends on converting the initial condition from a constant c0 to 0.
Let CFDαX=y(t) then X=CFIαy, so we have
CFDαiX= CFIα−αi CFDαX= CFIα−αiy, i=1,2,...,m. | (2.3) |
Substituting in Eq (2.2) we obtain
y=f(t,c0+ CFIαy, CFIα−α1y,..., CFIα−αmy). | (2.4) |
Assume f satisfies Lipschtiz condition with Lipschtiz constant L given by,
|f(t,y0,y1,...,ym)|−|f(t,z0,z1,...,zm)|≤Lm∑i=0|yi−zi|, | (2.5) |
which implies
|f(t,c0+CFIαy,CFIα−α1y,..,CFIα−αmy)−f(t,c0+CFIαz,CFIα−α1z,..,CFIα−αmz)|≤Lm∑i=0| CFIα−αiy− CFIα−αiz|. | (2.6) |
The solution algorithm of Eq (2.4) using ADM is,
y0(t)=a(t)yn+1(t)=An(t), j⩾0. | (2.7) |
where a(t) pocesses all free terms in Eq (2.4) and An are the Adomian polynomials of the nonlinear term which takes the form [34]
An=f(Sn)−n−1∑i=0Ai, | (2.8) |
where f(Sn)=∑ni=0Ai. Later, this accelerated formula of Adomian polynomial will be used in convergence analysis and error estimation. The solution of Eq (2.4) can be written in the form,
y(t)=∞∑i=0yi(t). | (2.9) |
lastly, the solution of the Eq (2.4) takes the form
x(t)=c0+X(t)=c0+ CFIαy(t). | (2.10) |
At which we convert the parameter to the initial form y to x in Eq (2.10), so we have the solution of the original Eq (2.1).
Define a mapping F:E→E where E=(C[J],‖⋅‖) is a Banach space of all continuous functions on J with the norm ‖x‖= maxtϵJx(t).
Theorem 3.1. Equation (2.4) has a unique solution whenever 0<ϕ<1 where ϕ=L(∑mi=0[(α−αi)(T−1)]+1B(α−αi)).
Proof. First, we define the mapping F:E→E as
Fy=f(t,c0+ CFIαy, CFIα−α1y,..., CFIα−αmy). |
Let y and z∈E are two different solutions of Eq (2.4). Then
Fy−Fz=f(t,c0+CFIαy,CFIα−α1y,..,CFIα−αmy)−f(t,c0+CFIαz,CFIα−α1z,...,CFIα−αmz) |
which implies that
|Fy−Fz|=|f(t,c0+ CFIαy, CFIα−α1y,..., CFIα−αmy)−f(t,c0+ CFIαz, CFIα−α1z,..., CFIα−αmz)|≤Lm∑i=0| CFIα−αiy− CFIα−αiz|≤Lm∑i=0|1−(α−αi)B(α−αi)(y−z)+α−αiB(α−αi)∫t0(y−z)ds|‖Fy−Fz‖≤Lm∑i=01−(α−αi)B(α−αi)maxtϵJ|y−z|+α−αiB(α−αi)maxtϵJ|y−z|∫t0ds≤Lm∑i=01−(α−αi)B(α−αi)‖y−z‖+α−αiB(α−αi)‖y−z‖T≤L‖y−z‖(m∑i=01−(α−αi)B(α−αi)+α−αiB(α−αi)T)≤L‖y−z‖(m∑i=0[(α−αi)(T−1)]+1B(α−αi))≤ϕ‖y−z‖. |
under the condition 0<ϕ<1, the mapping F is contraction and hence there exists a unique solution y∈C[J] for the problem Eq (2.4) and this completes the proof.
Theorem 3.2. The series solution of the problem Eq (2.4)converges if |y1(t)|<c and c isfinite.
Proof. Define a sequence {Sp} such that Sp=∑pi=0yi(t) is the sequence of partial sums from the series solution ∑∞i=0yi(t), we have
f(t,c0+ CFIαy, CFIα−α1y,..., CFIα−αmy)=∞∑i=0Ai, |
So
f(t,c0+ CFIαSp, CFIα−α1Sp,..., CFIα−αmSp)=p∑i=0Ai, |
From Eq (2.7) we have
∞∑i=0yi(t)=a(t)+∞∑i=0Ai−1 |
let Sp,Sq be two arbitrary sums with p⩾q. Now, we are going to prove that {Sp} is a Caushy sequence in this Banach space. We have
Sp=p∑i=0yi(t)=a(t)+p∑i=0Ai−1,Sq=q∑i=0yi(t)=a(t)+q∑i=0Ai−1. |
Sp−Sq=p∑i=0Ai−1−q∑i=0Ai−1=p∑i=q+1Ai−1=p−1∑i=qAi−1=f(t,c0+ CFIαSp−1, CFIα−α1Sp−1,..., CFIα−αmSp−1)−f(t,c0+ CFIαSq−1, CFIα−α1Sq−1,..., CFIα−αmSq−1) |
|Sp−Sq|=|f(t,c0+ CFIαSp−1, CFIα−α1Sp−1,..., CFIα−αmSp−1)−f(t,c0+ CFIαSq−1, CFIα−α1Sq−1,..., CFIα−αmSq−1)|≤Lm∑i=0| CFIα−αiSp−1− CFIα−αiSq−1|≤Lm∑i=0|1−(α−αi)B(α−αi)(Sp−1−Sq−1)+α−αiB(α−αi)∫t0(Sp−1−Sq−1)ds|≤Lm∑i=01−(α−αi)B(α−αi)|Sp−1−Sq−1|+α−αiB(α−αi)∫t0|Sp−1−Sq−1|ds |
‖Sp−Sq‖≤Lm∑i=01−(α−αi)B(α−αi)maxtϵJ|Sp−1−Sq−1|+α−αiB(α−αi)maxtϵJ|Sp−1−Sq−1|∫t0ds≤L‖Sp−Sq‖m∑i=0(1−(α−αi)B(α−αi)+α−αiB(α−αi)T)≤L‖Sp−Sq‖(m∑i=0[(α−αi)(T−1)]+1B(α−αi))≤ϕ‖Sp−Sq‖ |
let p=q+1 then,
‖Sq+1−Sq‖≤ϕ‖Sq−Sq−1‖≤ϕ2‖Sq−1−Sq−2‖≤...≤ϕq‖S1−S0‖ |
From the triangle inequality we have
‖Sp−Sq‖≤‖Sq+1−Sq‖+‖Sq+2−Sq+1‖+...‖Sp−Sp−1‖≤[ϕq+ϕq+1+...+ϕp−1]‖S1−S0‖≤ϕq[1+ϕ+...+ϕp−q+1]‖S1−S0‖≤ϕq[1−ϕp−q1−ϕ]‖y1(t)‖ |
Since 0<ϕ<1,p⩾q then (1−ϕp−q)≤1. Consequently
‖Sp−Sq‖≤ϕq1−ϕ‖y1(t)‖≤ϕq1−ϕmax∀tϵJ|y1(t)| | (3.1) |
but |y1(t)|<∞ and as q→∞ then, ‖Sp−Sq‖→0 and hence, {Sp} is a Caushy sequence in this Banach space then the proof is complete.
Theorem 3.3. The maximum absolute truncated error Eq (2.4)is estimated to be maxtϵJ|y(t)−∑qi=0yi(t)|≤ϕq1−ϕmaxtϵJ|y1(t)|
Proof. From the convergence theorm inequality (Eq 3.1) we have
‖Sp−Sq‖≤ϕq1−ϕmaxtϵJ|y1(t)| |
but, Sp=∑pi=0yi(t) as p→∞ then, Sp→y(t) so,
‖y(t)−Sq‖≤ϕq1−ϕmaxtϵJ|y1(t)| |
so, the maximum absolute truncated error in the interval J is,
maxtϵJ|y(t)−q∑i=0yi(t)|≤ϕq1−ϕmaxtϵJ|y1(t)| | (3.2) |
and this completes the proof.
In this part, we introduce several numerical examples with unkown exact solution and we will use inequality (Eq 3.2) to estimate the maximum absolute truncated error.
Example 4.1. Application of linear FDE
CFDx(t)+2aCFD1/2x(t)+bx(t)=0, x(0)=1. | (4.1) |
A Basset problem in fluid dynamics is a classical problem which is used to study the unsteady movement of an accelerating particle in a viscous fluid under the action of the gravity [36]
Set
X(t)=x(t)−1 |
Equation (4.1) will be
CFDX(t)+2aCFD1/2X(t)+bX(t)=0, X(0)=0. | (4.2) |
Appling Eq (2.3) to Eq (4.2), and using initial condition, also we take a = 1, b = 1/2,
y=−12−2I1/2y−12I y | (4.3) |
Appling ADM to Eq (4.3), we find the solution algorithm become
y0(t)=−12,yi(t)=−2 CFI1/2yi−1−12 CFI yi−1, i≥1. | (4.4) |
Appling Picard solution to Eq (4.2), we find the solution algorithm become
y0(t)=−12,yi(t)=−12−2 CFI1/2yi−1−12 CFI yi−1, i≥1. | (4.5) |
From Eq (4.4), the solution using ADM is given by y(t)=Limq→∞qi=0yi(t) while from Eq (4.5), the solution using Picard technique is given by y(t)=Limi→∞yi(t). Lately, the solution of the original problem Eq (4.2), is
x(t)=1+ CFI y(t). |
One the same processor (q = 20), the time consumed using ADM is 0.037 seconds, while the time consumed using Picard is 7.955 seconds.
Figure 1 gives a comparison between ADM and Picard solution of Ex. 4.1.
Example 4.2. Consider the following nonlinear FDE [35]
CFD1/2x=8t3/23√π−t7/44Γ(114)−t44+18 CFD1/4x+14x2, x(0)=0. | (4.6) |
Appling Eq (2.3) to Eq (4.6), and using initial condition,
y=8t3/23√π−t7/44Γ(114)−t44+18 CFI1/4y+14(CFI1/2y)2. | (4.7) |
Appling ADM to Eq (4.7), we find the solution algorithm will be become
y0(t)=8t3/23√π−t7/44Γ(114)−t44,yi(t)=18 CFI1/4yi−1+14(Ai−1), i≥1. | (4.8) |
at which Ai are Adomian polynomial of the nonliner term (CFI1/2y)2.
Appling Picard solution to Eq (4.7), we find the the solution algorithm become
y0(t)=8t3/23√π−t7/44Γ(114)−t44,yi(t)=y0(t)+18 CFI1/4yi−1+14(CFI1/2yi−1)2, i≥1. | (4.9) |
From Eq (4.8), the solution using ADM is given by y(t)=Limq→∞qi=0yi(t) while from Eq (4.9), the solution using Picard technique is given by y(t)=Limi→∞yi(t). Finally, the solution of the original problem Eq (4.7), is.
x(t)= CFI1/2y. |
One the same processor (q = 2), the time consumed using ADM is 65.13 seconds, while the time consumed using Picard is 544.787 seconds.
Table 1 showed the maximum absolute truncated error of of ADM solution (using Theorem 3.3) at different values of m (when t = 0:5; N = 2):
q | max. absolute error |
2 | 0.114548 |
5 | 0.099186 |
10 | 0.004363 |
Figure 2 gives a comparison between ADM and Picard solution of Ex. 4.2.
Example 4.3. Consider the following nonlinear FDE [35]
CFDαx=3t2−128125πt5+110(CFD1/2x)2,x(0)=0. | (4.10) |
Appling Eq (2.3) to Eq (4.10), and using initial condition,
y=3t2−128125πt5+110(CFI1/2y)2 | (4.11) |
Appling ADM to Eq (4.11), we find the solution algorithm become
y0(t)=3t2−128125πt5,yi(t)=110(Ai−1), i≥1 | (4.12) |
at which Ai are Adomian polynomial of the nonliner term (CFI1/2y)2.
Then appling Picard solution to Eq (4.11), we find the solution algorithm become
y0(t)=3t2−128125πt5,yi(t)=y0(t)+110(CFI1/2yi−1)2, i≥1. | (4.13) |
From Eq (4.12), the solution using ADM is given by y(t)=Limq→∞qi=0yi(t) while from Eq (4.13), the solution is y(t)=Limi→∞yi(t). Finally, the solution of the original problem Eq (4.11), is
x(t)=CFIy(t). |
One the same processor (q = 4), the time consumed using ADM is 2.09 seconds, while the time consumed using Picard is 44.725 seconds.
Table 2 showed the maximum absolute truncated error of of ADM solution (using Theorem 3.3) at different values of m (when t = 0:5; N = 4):
q | max. absolute error |
2 | 0.00222433 |
5 | 0.0000326908 |
10 | 2.88273*10−8 |
Figure 3 gives a comparison between ADM and Picard solution of Ex. 4.3 with α=1.
Example 4.4. Consider the following nonlinear FDE [35]
CFDαx=t2+12 CFDα1x+14 CFDα2x+16 CFDα3x+18x4,x(0)=0. | (4.14) |
Appling Eq (2.3) to Eq (4.10), and using initial condition,
y=t2+12(CFIα−α1y)+14(CFIα−α2y)+16(CFIα−α3y)+18(CFIαy)4, | (4.15) |
Appling ADM to Eq (4.15), we find the solution algorithm become
y0(t)=t2,yi(t)=12(CFIα−α1y)+14(CFIα−α2y)+16(CFIα−α3y)+18Ai−1, i≥1 | (4.16) |
where Ai are Adomian polynomial of the nonliner term (CFIαy)4.
Then appling Picard solution to Eq (4.15), we find the solution algorithm become
y0(t)=t2,yi(t)=t2+12(CFIα−α1yi−1)+14(CFIα−α2yi−1)+16(CFIα−α3yi−1)+18(CFIαyi−1)4 i≥1. | (4.17) |
From Eq (4.16), the solution using ADM is given by y(t)=Limq→∞qi=0yi(t) while from Eq (4.17), the solution using Picard technique is y(t)=Limi→∞yi(t). Finally, the solution of the original problem Eq (4.14), is
x(t)=CFIαy(t). |
One the same processor (q = 3), the time consumed using ADM is 0.437 seconds, while the time consumed using Picard is (16.816) seconds. Figure 4 shows a comparison between ADM and Picard solution of Ex. 4.4 atα=0.7,α1=0.1,α2=0.3,α3=0.5.
The Caputo-Fabrizo fractional deivative has a nonsingular kernel, and consequently, this definition is appropriate in solving nonlinear multidimensional FDE [37,38]. Since the selected numerical problems have an unkown exact solution, the formula (3.2) can be used to estimate the maximum absolute truncated error. By comparing the time taken on the same processor (i7-2670QM), it was found that the time consumed by ADM is much smaller compared with the Picard technique. Furthermore Picard gives a more accurate solution than ADM at the same interval with the same number of terms.
The authors declare there is no conflict of interest.
[1] |
Auta HS, Emenike CU, Fauziah SH (2017) Distribution and importance of microplastics in the marine environment: A review of the sources, fate, effects, and potential solutions. Environ Int 102: 165-176. doi: 10.1016/j.envint.2017.02.013
![]() |
[2] |
Horton AA, Walton A, Spurgeon DJ, et al. (2017) Review: Microplastics in freshwater and terrestrial systems: Evaluating the current understanding to identify knowledge gaps and future research priorities. Sci Total Environ 586: 127-141. doi: 10.1016/j.scitotenv.2017.01.190
![]() |
[3] |
Eerkes-Medrano D, Thomson RC, Aldridge DC (2015) Microplastics in freshwater systems: A review of the emerging threats, identification of knowledge gaps and prioritization of research needs. Water Res 75: 63-82. doi: 10.1016/j.watres.2015.02.012
![]() |
[4] |
Estahbanati S, Fahrenfeld NL (2016) Influence of wastewater treatment plant discharges on microplastic concentrations in surface water. Chemosphere 162: 277-284. doi: 10.1016/j.chemosphere.2016.07.083
![]() |
[5] |
Mason SA, Garneau D, Sutton R, et al. (2016) Microplastic pollution is widely detected in US municipal wastewater treatment plant effluent. Environ Pollut 218: 1045-1054. doi: 10.1016/j.envpol.2016.08.056
![]() |
[6] |
Ravit B, Cooper K, Moreno G, et al. (2017) Microplastics in urban New Jersey freshwaters: distribution, chemical identification, and biological effects. AIMS Environ Sci 4: 809-826. doi: 10.3934/environsci.2017.6.809
![]() |
[7] |
Carr SA (2017) Sources and dispersive modes of micro-fibers in the environment. Int Environ Assess Manage 13: 466-469. doi: 10.1002/ieam.1916
![]() |
[8] |
Dris R, Gasperi J, Rocher V, et al. (2015) Microplastic contamination in an urban area: a case study in Greater Paris. Environ Chem 12: 592-599. doi: 10.1071/EN14167
![]() |
[9] |
Rochman CM, Kross SM, Armstrong JB, et al. (2015) Scientific evidence supports a ban on microbeads. Environ Sci Technol 49: 10759-10761. doi: 10.1021/acs.est.5b03909
![]() |
[10] |
Rochman CM (2018) Microplastics research - from sink to source. Science 360: 28-29. doi: 10.1126/science.aar7734
![]() |
[11] | Farrell P, Nelson K (2013) Trophic level transfer of microplastic: Mytilus edulis (L.) to Carcinus maenas (L.). Environ Pollut 177: 1-3. |
[12] |
Lusher AL, McHugh M, Tompson RC (2013) Occurrence of microplastics in the gastrointestinal track of pelagic and demersal fish from the English Channel. Mar Pollut Bull 67: 94-99. doi: 10.1016/j.marpolbul.2012.11.028
![]() |
[13] |
Sanchez W, Bender C, Porcher JM (2014) Wild gudgeons (Gobio gobio) from French rivers are contaminated by microplastics: Preliminary study and first evidence. Environ Res 128: 98-100. doi: 10.1016/j.envres.2013.11.004
![]() |
[14] |
Browne MA, Dissanayake Galloway TS, Lowe DM, et al. (2008) Ingested microscopic plastic translocates to the circulatory system of the mussel, Mytilus edulis (L). Environ Sci Technol 42: 5026-5031. doi: 10.1021/es800249a
![]() |
[15] |
Van Cauwenberghe L, Janssen CR (2014) Microplastics in bivalves cultured for human consumption. Environ Pollut 193: 65-70. doi: 10.1016/j.envpol.2014.06.010
![]() |
[16] |
Leslie HA, Brandsma SH, van Velzen MJM, et al. (2017) Microplastics en route: Field measurements in the Dutch river delta and Amsterdam canals, wastewater treatment plants, North Sea sediments and biota. Environ Int 101: 133-142. doi: 10.1016/j.envint.2017.01.018
![]() |
[17] | Zubris KAV, Richards BK (2005) Synthetic fibers as an indicator of land application of sludge. Environ Pollut138: 201-211. |
[18] |
McCormick A, Hoellein TJ, Mason SA, et al. (2014) Microplastic is an abundant and distinct microbial habitat in an urban river. Environ Sci Technol 48: 11863-11871. doi: 10.1021/es503610r
![]() |
[19] |
Focazio MJ, Kolpin DW, Barnes KK, et al. (2008). A national reconnaissance for pharmaceuticals and other organic wastewater contaminants in the United States - II) Untreated drinking water sources. Sci Total Environ 402: 201-216. doi: 10.1016/j.scitotenv.2008.02.021
![]() |
[20] | Rochman CM, Hoh E, Hentschel BT, et al. (2013). Long-term field measurements of sorption of organic contaminants to five types of plastic pellets: implications for plastic marine debris. Environ Sci Technol 47: 1646-1654. |
[21] |
Bakir A, Rowland SJ, Thompson RC (2014). Transport of persistent organic pollutants by microplastics in estuarine conditions. Est Coast Shelf Sci 140: 14-21. doi: 10.1016/j.ecss.2014.01.004
![]() |
[22] | Buckley B, Stiles R (2005) Analytical determination of tentatively identified compounds in drinking water supplies to correspond with the USGS pharmaceutical work. Report to the New Jersey Department of Environmental Protection. |
[23] | Murphy E, Buckley B, Lippencott L, et al. (2003) The characterization of tentatively identified compounds in water samples collected from public water systems in New Jersey. Environ Assess 2003: 1-37. |
[24] |
Wright SL, Thompson RC, Galloway TS (2013) The physical impacts of microplastics on marine organisms: A review. Environ Pollut 178: 483-492. doi: 10.1016/j.envpol.2013.02.031
![]() |
[25] |
Andrady AL (2011) Microplastics in the marine environment. Mar Pollut Bull 62: 1596-1605. doi: 10.1016/j.marpolbul.2011.05.030
![]() |
[26] | Baldwin AK, Corsi SR, De Cicco LA, et al. Organic contaminants in Great Lakes tributaries: Prevalence and potential aquatic toxicity. Sci Total Environ 554: 42-52. |
1. | Eman A. A. Ziada, Salwa El-Morsy, Osama Moaaz, Sameh S. Askar, Ahmad M. Alshamrani, Monica Botros, Solution of the SIR epidemic model of arbitrary orders containing Caputo-Fabrizio, Atangana-Baleanu and Caputo derivatives, 2024, 9, 2473-6988, 18324, 10.3934/math.2024894 | |
2. | H. Salah, M. Anis, C. Cesarano, S. S. Askar, A. M. Alshamrani, E. M. Elabbasy, Fourth-order differential equations with neutral delay: Investigation of monotonic and oscillatory features, 2024, 9, 2473-6988, 34224, 10.3934/math.20241630 | |
3. | Said R. Grace, Gokula N. Chhatria, S. Kaleeswari, Yousef Alnafisah, Osama Moaaz, Forced-Perturbed Fractional Differential Equations of Higher Order: Asymptotic Properties of Non-Oscillatory Solutions, 2024, 9, 2504-3110, 6, 10.3390/fractalfract9010006 | |
4. | A.E. Matouk, Monica Botros, Hidden chaotic attractors and self-excited chaotic attractors in a novel circuit system via Grünwald–Letnikov, Caputo-Fabrizio and Atangana-Baleanu fractional operators, 2025, 116, 11100168, 525, 10.1016/j.aej.2024.12.064 | |
5. | Zahra Barati, Maryam Keshavarzi, Samaneh Mosaferi, Anatomical and micromorphological study of Phalaris (Poaceae) species in Iran, 2025, 68, 1588-4082, 9, 10.14232/abs.2024.1.9-15 |
q | max. absolute error |
2 | 0.114548 |
5 | 0.099186 |
10 | 0.004363 |
q | max. absolute error |
2 | 0.00222433 |
5 | 0.0000326908 |
10 | 2.88273*10−8 |