In this work, we discuss the existence and uniqueness of solution for a boundary value problem for the Langevin equation and inclusion with the Hilfer fractional derivative. First of all, we give some definitions, theorems, and lemmas that are necessary for the understanding of the manuscript. Second of all, we give our first existence result, based on Krasnoselskii’s fixed point, and to deal with the uniqueness result, we use Banach’s contraction principle. Third of all, in the inclusion case, to obtain the existence result, we use the Leray–Schauder alternative. Last but not least, we give an illustrative example.
{"title":"Boundary Value Problem for the Langevin Equation and Inclusion with the Hilfer Fractional Derivative","authors":"K. Hilal, A. Kajouni, Hamid Lmou","doi":"10.1155/2022/3386198","DOIUrl":"https://doi.org/10.1155/2022/3386198","url":null,"abstract":"In this work, we discuss the existence and uniqueness of solution for a boundary value problem for the Langevin equation and inclusion with the Hilfer fractional derivative. First of all, we give some definitions, theorems, and lemmas that are necessary for the understanding of the manuscript. Second of all, we give our first existence result, based on Krasnoselskii’s fixed point, and to deal with the uniqueness result, we use Banach’s contraction principle. Third of all, in the inclusion case, to obtain the existence result, we use the Leray–Schauder alternative. Last but not least, we give an illustrative example.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43605770","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study some reaction-diffusion models of interactive dynamics of the wild and sterile mosquitoes. The well-posedness of the concerned model is proved. The stability of the steady states is discussed. Numerical simulations are presented to illustrate our theoretical results.
{"title":"Dynamics of Mosquito Population Models with Spatial Diffusion","authors":"U. Traoré","doi":"10.1155/2021/9034274","DOIUrl":"https://doi.org/10.1155/2021/9034274","url":null,"abstract":"In this paper, we study some reaction-diffusion models of interactive dynamics of the wild and sterile mosquitoes. The well-posedness of the concerned model is proved. The stability of the steady states is discussed. Numerical simulations are presented to illustrate our theoretical results.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49596479","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
<jats:p>In this paper, the efficient combined method based on the homotopy perturbation Sadik transform method (HPSTM) is applied to solve the physical and functional equations containing the Caputo–Prabhakar fractional derivative. The mathematical model of this equation of order <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mi>μ</mi> <mo>∈</mo> <mfenced open="(" close="]" separators="|"> <mrow> <mn>0,1</mn> </mrow> </mfenced> </math> </jats:inline-formula> with <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mi>λ</mi> <mo>∈</mo> <msup> <mrow> <mi>ℤ</mi> </mrow> <mrow> <mo>+</mo> </mrow> </msup> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <mi>σ</mi> <mo>∈</mo> <msup> <mrow> <mi>ℝ</mi> </mrow> <mrow> <mo>+</mo> </mrow> </msup> </math> </jats:inline-formula> is presented as follows: <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <mmultiscripts> <mrow> <msubsup> <mstyle displaystyle="true"> <mi mathvariant="fraktur">D</mi> </mstyle> <mi>t</mi> <mi>μ</mi> </msubsup> </mrow> <mprescripts /> <none /> <mi>C</mi> </mmultiscripts> <mi>u</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mi>x</mi> <mo>,</mo> <mi>t</mi> </mrow> </mfenced> <mo>+</mo> <mi>θ</mi> <msup> <mrow> <mi>u</mi> </mrow
本文提出了一种基于同伦微扰Sadik变换方法的高效组合方法 （HPSTM）用于求解包含Caputo–Prabhakar分数导数的物理和函数方程。μ∈0,1阶方程的数学模型ℤ + , θ，σ∈ℝ + 表示如下：D tμC u x，t+θuλx，t u x x，t−σu x x tx、t=0时，其中对于λ=1，θ=1，σ=1s和λ=2、θ=3，σ=1时，将方程分别转化为等宽方程和修正的等宽方程。我们用来求解这个方程的分析方法是基于同伦微扰方法和Sadik变换的结合。本文讨论了收敛性和误差分析。给出了三个实例的分析结果图，以表明该数值方法的适用性。并与其它方法进行了比较。
{"title":"Analytical Solutions for the Equal Width Equations Containing Generalized Fractional Derivative Using the Efficient Combined Method","authors":"M. Derakhshan","doi":"10.1155/2021/7066398","DOIUrl":"https://doi.org/10.1155/2021/7066398","url":null,"abstract":"<jats:p>In this paper, the efficient combined method based on the homotopy perturbation Sadik transform method (HPSTM) is applied to solve the physical and functional equations containing the Caputo–Prabhakar fractional derivative. The mathematical model of this equation of order <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\">\u0000 <mi>μ</mi>\u0000 <mo>∈</mo>\u0000 <mfenced open=\"(\" close=\"]\" separators=\"|\">\u0000 <mrow>\u0000 <mn>0,1</mn>\u0000 </mrow>\u0000 </mfenced>\u0000 </math>\u0000 </jats:inline-formula> with <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M2\">\u0000 <mi>λ</mi>\u0000 <mo>∈</mo>\u0000 <msup>\u0000 <mrow>\u0000 <mi>ℤ</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mo>+</mo>\u0000 </mrow>\u0000 </msup>\u0000 <mo>,</mo>\u0000 <mi>θ</mi>\u0000 <mo>,</mo>\u0000 <mi>σ</mi>\u0000 <mo>∈</mo>\u0000 <msup>\u0000 <mrow>\u0000 <mi>ℝ</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mo>+</mo>\u0000 </mrow>\u0000 </msup>\u0000 </math>\u0000 </jats:inline-formula> is presented as follows: <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M3\">\u0000 <mmultiscripts>\u0000 <mrow>\u0000 <msubsup>\u0000 <mstyle displaystyle=\"true\">\u0000 <mi mathvariant=\"fraktur\">D</mi>\u0000 </mstyle>\u0000 <mi>t</mi>\u0000 <mi>μ</mi>\u0000 </msubsup>\u0000 </mrow>\u0000 <mprescripts />\u0000 <none />\u0000 <mi>C</mi>\u0000 </mmultiscripts>\u0000 <mi>u</mi>\u0000 <mfenced open=\"(\" close=\")\" separators=\"|\">\u0000 <mrow>\u0000 <mi>x</mi>\u0000 <mo>,</mo>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 </mfenced>\u0000 <mo>+</mo>\u0000 <mi>θ</mi>\u0000 <msup>\u0000 <mrow>\u0000 <mi>u</mi>\u0000 </mrow","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46073124","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we consider the reconstruction of heat field in one-dimensional quasiperiodic media with an unknown source from the interior measurement. The innovation of this paper is solving the inverse problem by means of two different homotopy iteration processes. The first kind of homotopy iteration process is not convergent. For the second kind of homotopy iteration process, a convergent result is proved. Based on the uniqueness of this inverse problem and convergence results of the second kind of homotopy iteration process with exact data, the results of two numerical examples show that the proposed method is efficient, and the error of the inversion solution r t is given.
{"title":"The Method Based on Series Solution for Identifying an Unknown Source Coefficient on the Temperature Field in the Quasiperiodic Media","authors":"Bingxian Wang, C. Bai, M. Xu, L. Zhang","doi":"10.1155/2021/2893299","DOIUrl":"https://doi.org/10.1155/2021/2893299","url":null,"abstract":"In this paper, we consider the reconstruction of heat field in one-dimensional quasiperiodic media with an unknown source from the interior measurement. The innovation of this paper is solving the inverse problem by means of two different homotopy iteration processes. The first kind of homotopy iteration process is not convergent. For the second kind of homotopy iteration process, a convergent result is proved. Based on the uniqueness of this inverse problem and convergence results of the second kind of homotopy iteration process with exact data, the results of two numerical examples show that the proposed method is efficient, and the error of the inversion solution \u0000 \u0000 r\u0000 \u0000 \u0000 t\u0000 \u0000 \u0000 \u0000 is given.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45283393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The cube-root truly nonlinear oscillator and the inverse cube-root truly nonlinear oscillator are the most meaningful and classical nonlinear ordinary differential equations on behalf of its various applications in science and engineering. Especially, the oscillators are used widely in the study of elastic force, structural dynamics, and elliptic curve cryptography. In this paper, we have applied modified Mickens extended iteration method to solve the cube-root truly nonlinear oscillator, the inverse cube-root truly nonlinear oscillator, and the equation of pendulum. Comparison is made among iteration method, harmonic balance method, He’s amplitude-frequency formulation, He’s homotopy perturbation method, improved harmonic balance method, and homotopy perturbation method. After comparison, we analyze that modified Mickens extended iteration method is more accurate, effective, easy, and straightforward. Also, the comparison of the obtained analytical solutions with the numerical results represented an extraordinary accuracy. The percentage error for the fourth approximate frequency of cube-root truly nonlinear oscillator is 0.006 and the percentage error for the fourth approximate frequency of inverse cube-root truly nonlinear oscillator is 0.12.
{"title":"An Effective Solution of the Cube-Root Truly Nonlinear Oscillator: Extended Iteration Procedure","authors":"B. Haque, M. Hossain","doi":"10.1155/2021/7819209","DOIUrl":"https://doi.org/10.1155/2021/7819209","url":null,"abstract":"The cube-root truly nonlinear oscillator and the inverse cube-root truly nonlinear oscillator are the most meaningful and classical nonlinear ordinary differential equations on behalf of its various applications in science and engineering. Especially, the oscillators are used widely in the study of elastic force, structural dynamics, and elliptic curve cryptography. In this paper, we have applied modified Mickens extended iteration method to solve the cube-root truly nonlinear oscillator, the inverse cube-root truly nonlinear oscillator, and the equation of pendulum. Comparison is made among iteration method, harmonic balance method, He’s amplitude-frequency formulation, He’s homotopy perturbation method, improved harmonic balance method, and homotopy perturbation method. After comparison, we analyze that modified Mickens extended iteration method is more accurate, effective, easy, and straightforward. Also, the comparison of the obtained analytical solutions with the numerical results represented an extraordinary accuracy. The percentage error for the fourth approximate frequency of cube-root truly nonlinear oscillator is 0.006 and the percentage error for the fourth approximate frequency of inverse cube-root truly nonlinear oscillator is 0.12.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41390758","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Numerical computation for the class of singularly perturbed delay parabolic reaction diffusion equations with integral boundary condition has been considered. A parameter-uniform numerical method is constructed via the nonstandard finite difference method for the spatial direction, and the backward Euler method for the resulting system of initial value problems in temporal direction is used. The integral boundary condition is treated using numerical integration techniques. Maximum absolute errors and the rate of convergence for different values of perturbation parameter ε and mesh sizes are tabulated for two model examples. The proposed method is shown to be parameter-uniformly convergent.
{"title":"Parameter-Uniform Numerical Scheme for Singularly Perturbed Delay Parabolic Reaction Diffusion Equations with Integral Boundary Condition","authors":"Wakjira Tolassa Gobena, G. Duressa","doi":"10.1155/2021/9993644","DOIUrl":"https://doi.org/10.1155/2021/9993644","url":null,"abstract":"Numerical computation for the class of singularly perturbed delay parabolic reaction diffusion equations with integral boundary condition has been considered. A parameter-uniform numerical method is constructed via the nonstandard finite difference method for the spatial direction, and the backward Euler method for the resulting system of initial value problems in temporal direction is used. The integral boundary condition is treated using numerical integration techniques. Maximum absolute errors and the rate of convergence for different values of perturbation parameter \u0000 \u0000 ε\u0000 \u0000 and mesh sizes are tabulated for two model examples. The proposed method is shown to be parameter-uniformly convergent.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45274169","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present in this paper a numerical study of the validity limit of the optics geometrical approximation in comparison with a differential method which is established according to rigorous formalisms based on the electromagnetic theory. The precedent studies show that this method is adopted to the study of diffraction by periodic rough surfaces. For periods much larger than the wavelength, the mechanism is analog to what happens in a cavity where a ray is trapped and undergoes a large number of reflections. For gratings with a period much smaller than the wavelength, the roughness essentially behaves as a transition layer with a gradient of the optical index. Such a layer reduces the reflection there by increasing the absorption. The code has been implemented for TE polarization. We determine by the two methods such as differential method and the optics geometrical approximation the emissivity of gold and tungsten cylindrical surfaces with a sinusoidal profile, for a wavelength equal to 0.55 microns. The obtained results for a fixed height of the grating allowed us to delimit the validity domain of the optic geometrical approximation for the treated cases. The emissivity calculated by the differential method and that given on the basis of the homogenization theory are satisfactory when the period is much smaller than the wavelength.
{"title":"Electric Transverse Emissivity of Sinusoidal Surfaces Determined by a Differential Method: Comparison with Approximation of Geometric Optics","authors":"Taoufik Ghabara","doi":"10.1155/2021/1506485","DOIUrl":"https://doi.org/10.1155/2021/1506485","url":null,"abstract":"We present in this paper a numerical study of the validity limit of the optics geometrical approximation in comparison with a differential method which is established according to rigorous formalisms based on the electromagnetic theory. The precedent studies show that this method is adopted to the study of diffraction by periodic rough surfaces. For periods much larger than the wavelength, the mechanism is analog to what happens in a cavity where a ray is trapped and undergoes a large number of reflections. For gratings with a period much smaller than the wavelength, the roughness essentially behaves as a transition layer with a gradient of the optical index. Such a layer reduces the reflection there by increasing the absorption. The code has been implemented for TE polarization. We determine by the two methods such as differential method and the optics geometrical approximation the emissivity of gold and tungsten cylindrical surfaces with a sinusoidal profile, for a wavelength equal to 0.55 microns. The obtained results for a fixed height of the grating allowed us to delimit the validity domain of the optic geometrical approximation for the treated cases. The emissivity calculated by the differential method and that given on the basis of the homogenization theory are satisfactory when the period is much smaller than the wavelength.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46817935","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study the global bifurcation of infinity of a class of nonlinear eigenvalue problems for fourth-order ordinary differential equations with nondifferentiable nonlinearity. We prove the existence of two families of unbounded continuance of solutions bifurcating at infinity and corresponding to the usual nodal properties near bifurcation intervals.
{"title":"Global Bifurcation of Fourth-Order Nonlinear Eigenvalue Problems’ Solution","authors":"Fatma Aydin Akgun","doi":"10.1155/2021/7516324","DOIUrl":"https://doi.org/10.1155/2021/7516324","url":null,"abstract":"In this paper, we study the global bifurcation of infinity of a class of nonlinear eigenvalue problems for fourth-order ordinary differential equations with nondifferentiable nonlinearity. We prove the existence of two families of unbounded continuance of solutions bifurcating at infinity and corresponding to the usual nodal properties near bifurcation intervals.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47499536","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this study, we are going to explore mathematically the dynamics of giving up smoking behavior. For this purpose, we will perform a mathematical analysis of a smoking model and suggest some conditions to control this serious burden on public health. The model under consideration describes the interaction between the potential smokers P , the occasional smokers L , the chain smokers S , the temporarily quit smokers Q T , and the permanently quit smokers Q P . Existence, positivity, and boundedness of the proposed problem solutions are proved. Local stability of the equilibria is established by using Routh–Hurwitz conditions. Moreover, the global stability of the same equilibria is fulfilled through using suitable Lyapunov functionals. In order to study the optimal control of our problem, we will take into account a two controls’ strategy. The first control will represent the government prohibition of smoking in public areas which reduces the contact between nonsmokers and smokers, while the second will symbolize the educational campaigns and the increase of cigarette cost which prevents occasional smokers from becoming chain smokers. The existence of the optimal control pair is discussed, and by using Pontryagin minimum principle, these two optimal controls are characterized. The optimality system is derived and solved numerically using the forward and backward difference approximation. Finally, numerical simulations are performed in order to check the equilibria stability, confirm the theoretical findings, and show the role of optimal strategy in controlling the smoking severity.
{"title":"Mathematical Analysis and Optimal Control of Giving up the Smoking Model","authors":"Omar Khyar, J. Danane, K. Allali","doi":"10.1155/2021/8673020","DOIUrl":"https://doi.org/10.1155/2021/8673020","url":null,"abstract":"In this study, we are going to explore mathematically the dynamics of giving up smoking behavior. For this purpose, we will perform a mathematical analysis of a smoking model and suggest some conditions to control this serious burden on public health. The model under consideration describes the interaction between the potential smokers \u0000 \u0000 \u0000 \u0000 P\u0000 \u0000 \u0000 \u0000 , the occasional smokers \u0000 \u0000 \u0000 \u0000 L\u0000 \u0000 \u0000 \u0000 , the chain smokers \u0000 \u0000 \u0000 \u0000 S\u0000 \u0000 \u0000 \u0000 , the temporarily quit smokers \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 Q\u0000 \u0000 \u0000 T\u0000 \u0000 \u0000 \u0000 \u0000 \u0000 , and the permanently quit smokers \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 Q\u0000 \u0000 \u0000 P\u0000 \u0000 \u0000 \u0000 \u0000 \u0000 . Existence, positivity, and boundedness of the proposed problem solutions are proved. Local stability of the equilibria is established by using Routh–Hurwitz conditions. Moreover, the global stability of the same equilibria is fulfilled through using suitable Lyapunov functionals. In order to study the optimal control of our problem, we will take into account a two controls’ strategy. The first control will represent the government prohibition of smoking in public areas which reduces the contact between nonsmokers and smokers, while the second will symbolize the educational campaigns and the increase of cigarette cost which prevents occasional smokers from becoming chain smokers. The existence of the optimal control pair is discussed, and by using Pontryagin minimum principle, these two optimal controls are characterized. The optimality system is derived and solved numerically using the forward and backward difference approximation. Finally, numerical simulations are performed in order to check the equilibria stability, confirm the theoretical findings, and show the role of optimal strategy in controlling the smoking severity.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43269115","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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