This paper discusses some sufficient conditions for oscillatory behavior of even-order half-linear neutral differential equation. An example is given to illustrate the main result.
讨论了偶阶半线性中立型微分方程振动性的几个充分条件。给出了一个例子来说明主要结果。
{"title":"Oscillatory Behavior of Even-Order Half-Linear Neutral Differential Equations","authors":"S. Sangeetha, S. Thamilvanan, E. Thandapani","doi":"10.1155/2022/3352789","DOIUrl":"https://doi.org/10.1155/2022/3352789","url":null,"abstract":"This paper discusses some sufficient conditions for oscillatory behavior of even-order half-linear neutral differential equation. An example is given to illustrate the main result.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2022-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47471958","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}
Convection, diffusion, and reaction mechanisms are characteristics of transient mass-transfer phenomena that occur in natural and industrial systems. In this article, we contemplate a passive scalar transport governed by the convection-diffusion-reaction (CDR) equation in 2D flow. The efficiency of solving computationally partial differential equations can be illustrated by using a precise numerical method that yields remarkable precision at a low cost. The accuracy and computational efficiency of two second-order finite difference methods were investigated. The results were compared to a finite volume technique, which has a memory advantage and conserves mass, momentum, and energy even on coarse grids. For various diffusion coefficient values, numerical simulation of unsteady CDR equation are also performed. The techniques were examined for consistency and convergence. The effectiveness and accuracy of these approaches for solving CDR equations are demonstrated by simulation results. Efficiency is measured using L 2 and L ∞ , and the estimated results are compared to the corresponding analytical solution.
{"title":"Finite Volume Method for a Time-Dependent Convection-Diffusion-Reaction Equation with Small Parameters","authors":"Uzair Ahmed, D. Mashat, D. Maturi","doi":"10.1155/2022/3476309","DOIUrl":"https://doi.org/10.1155/2022/3476309","url":null,"abstract":"Convection, diffusion, and reaction mechanisms are characteristics of transient mass-transfer phenomena that occur in natural and industrial systems. In this article, we contemplate a passive scalar transport governed by the convection-diffusion-reaction (CDR) equation in 2D flow. The efficiency of solving computationally partial differential equations can be illustrated by using a precise numerical method that yields remarkable precision at a low cost. The accuracy and computational efficiency of two second-order finite difference methods were investigated. The results were compared to a finite volume technique, which has a memory advantage and conserves mass, momentum, and energy even on coarse grids. For various diffusion coefficient values, numerical simulation of unsteady CDR equation are also performed. The techniques were examined for consistency and convergence. The effectiveness and accuracy of these approaches for solving CDR equations are demonstrated by simulation results. Efficiency is measured using \u0000 \u0000 \u0000 \u0000 L\u0000 \u0000 \u0000 2\u0000 \u0000 \u0000 \u0000 and \u0000 \u0000 \u0000 \u0000 L\u0000 \u0000 \u0000 ∞\u0000 \u0000 \u0000 \u0000 , and the estimated results are compared to the corresponding analytical solution.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2022-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46237404","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, the Bernstein collocation method (BCM) is used for the first time to solve the nonlinear magnetohydrodynamics (MHD) Jeffery–Hamel arterial blood flow issue. The flow model described by nonlinear partial differential equations is first transformed to a third-order one-dimensional equation. By using the Bernstein collocation method, the problem is transformed into a nonlinear system of algebraic equations. The residual correction procedure is used to estimate the error; it is simple to use and can be used even when the exact solution is unknown. In addition, the corrected Bernstein solution can be found. As a consequence, the solution is estimated using a numerical approach based on Bernstein polynomials, and the findings are verified by the 4th-order Runge–Kutta results. Comparison with the homotopy perturbation method shows that the present method gives much higher accuracy. The accuracy and efficiency of the proposed method were supported by the analysis of variance (ANOVA) and 95% of confidence on interval error. Finally, the results revealed that the MHD Jeffery–Hamel flow is directly proportional to the product of the angle between the plates α and Reynolds number Re .
{"title":"Bernstein Collocation Method for Solving MHD Jeffery–Hamel Blood Flow Problem with Error Estimations","authors":"A. Bataineh, O. Isik, I. Hashim","doi":"10.1155/2022/9123178","DOIUrl":"https://doi.org/10.1155/2022/9123178","url":null,"abstract":"In this paper, the Bernstein collocation method (BCM) is used for the first time to solve the nonlinear magnetohydrodynamics (MHD) Jeffery–Hamel arterial blood flow issue. The flow model described by nonlinear partial differential equations is first transformed to a third-order one-dimensional equation. By using the Bernstein collocation method, the problem is transformed into a nonlinear system of algebraic equations. The residual correction procedure is used to estimate the error; it is simple to use and can be used even when the exact solution is unknown. In addition, the corrected Bernstein solution can be found. As a consequence, the solution is estimated using a numerical approach based on Bernstein polynomials, and the findings are verified by the 4th-order Runge–Kutta results. Comparison with the homotopy perturbation method shows that the present method gives much higher accuracy. The accuracy and efficiency of the proposed method were supported by the analysis of variance (ANOVA) and 95% of confidence on interval error. Finally, the results revealed that the MHD Jeffery–Hamel flow is directly proportional to the product of the angle between the plates \u0000 \u0000 α\u0000 \u0000 and Reynolds number \u0000 \u0000 Re\u0000 \u0000 .","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2022-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49623648","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}
G. Sumitha, R. Kodeeswaran, S. Noeiaghdam, S. Balamuralitharan, V. Govindan
In this paper, we establish the solution of the fourth-order nonlinear homogeneous neutral functional difference equation. Moreover, we study the new oscillation criteria have been established which generalize some of the existing results of the fourth-order nonlinear homogeneous neutral functional difference equation in the literature. Likewise, a few models are given to represent the significance of the primary outcomes.
{"title":"Oscillation of Fourth-Order Nonlinear Homogeneous Neutral Difference Equation","authors":"G. Sumitha, R. Kodeeswaran, S. Noeiaghdam, S. Balamuralitharan, V. Govindan","doi":"10.1155/2022/2406736","DOIUrl":"https://doi.org/10.1155/2022/2406736","url":null,"abstract":"In this paper, we establish the solution of the fourth-order nonlinear homogeneous neutral functional difference equation. Moreover, we study the new oscillation criteria have been established which generalize some of the existing results of the fourth-order nonlinear homogeneous neutral functional difference equation in the literature. Likewise, a few models are given to represent the significance of the primary outcomes.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41937534","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 work, we prove the existence and uniqueness of mild solution of the fractional conformable Cauchy problem with nonlocal condition. We obtained these results by applying the fixed point theorems precisely to the fixed point theorem of Krasnoselskii and Banach’s fixed point theorem. At the end, we provide application.
{"title":"Existence of Solution for a Conformable Fractional Cauchy Problem with Nonlocal Condition","authors":"K. Hilal, A. Kajouni, Najat Chefnaj","doi":"10.1155/2022/6468278","DOIUrl":"https://doi.org/10.1155/2022/6468278","url":null,"abstract":"In this work, we prove the existence and uniqueness of mild solution of the fractional conformable Cauchy problem with nonlocal condition. We obtained these results by applying the fixed point theorems precisely to the fixed point theorem of Krasnoselskii and Banach’s fixed point theorem. At the end, we provide application.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2022-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47059637","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 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":null,"pages":null},"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":null,"pages":null},"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}
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 with is presented as follows:
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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
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