Pub Date : 2020-08-01DOI: 10.22034/CMDE.2020.30174.1446
Z. Monfared, Z. Dadi, Z. Afsharnezhad
The area of discontinuous dynamical systems is almost a young research area, and the enthusiasm and necessity for analysing these systems have been growing. On the other hand, chaos appears in a rather wide class of discontinuous systems. One of the most important properties of chaos is sensitive dependence on initial conditions. Also, the most effective way to diagnosis chaotic systems is defining Lyapunov exponents of these systems. In addition, defining and calculating Lyapunov exponents for all discontinuous systems are real challenges. This paper is devoted to define Lyapunov exponents for discontinuous dynamical systems of Filippov type in order to investigate chaos for these systems.
{"title":"Lyapunov exponents for discontinuous dynamical systems of Filippov type","authors":"Z. Monfared, Z. Dadi, Z. Afsharnezhad","doi":"10.22034/CMDE.2020.30174.1446","DOIUrl":"https://doi.org/10.22034/CMDE.2020.30174.1446","url":null,"abstract":"The area of discontinuous dynamical systems is almost a young research area, and the enthusiasm and necessity for analysing these systems have been growing. On the other hand, chaos appears in a rather wide class of discontinuous systems. One of the most important properties of chaos is sensitive dependence on initial conditions. Also, the most effective way to diagnosis chaotic systems is defining Lyapunov exponents of these systems. In addition, defining and calculating Lyapunov exponents for all discontinuous systems are real challenges. This paper is devoted to define Lyapunov exponents for discontinuous dynamical systems of Filippov type in order to investigate chaos for these systems.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44131785","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}
Pub Date : 2020-08-01DOI: 10.22034/CMDE.2020.27993.1382
J. Biazar, Roxana Asayesh
This paper is devoted to applying the sixth-order compact finite difference approach to the Helmholtz equation. Instead of using matrix inversion, a discrete sinusoidal transform is used as a quick solver to solve the discretized system resulted from the compact finite difference method. Through this way, the computational costs of the method with large numbers of nodes are greatly reduced. The efficiency and accuracy of the scheme are investigated by solving some illustrative examples, having the exact solutions.
{"title":"An efficient high-order compact finite difference method for the Helmholtz equation","authors":"J. Biazar, Roxana Asayesh","doi":"10.22034/CMDE.2020.27993.1382","DOIUrl":"https://doi.org/10.22034/CMDE.2020.27993.1382","url":null,"abstract":"This paper is devoted to applying the sixth-order compact finite difference approach to the Helmholtz equation. Instead of using matrix inversion, a discrete sinusoidal transform is used as a quick solver to solve the discretized system resulted from the compact finite difference method. Through this way, the computational costs of the method with large numbers of nodes are greatly reduced. The efficiency and accuracy of the scheme are investigated by solving some illustrative examples, having the exact solutions.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44200745","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}
Pub Date : 2020-08-01DOI: 10.22034/CMDE.2020.32382.1505
K. Ali, R. Nuruddeen, A. Yıldırım
In this paper, we analytically study the newly developed (2+1)-dimensional BenjaminOno equation by Wazwaz and propose its (3+1)-dimensional version. For this purpose, we successfully employed the modified extended tanh expansion method to construct certain hyperbolic, periodic and complex solitary wave structures simulated with the aid of symbolic computation using Mathematica. Also, we have depicted graphically the constructed solutions.
{"title":"On the new extensions to the Benjamin-Ono equation","authors":"K. Ali, R. Nuruddeen, A. Yıldırım","doi":"10.22034/CMDE.2020.32382.1505","DOIUrl":"https://doi.org/10.22034/CMDE.2020.32382.1505","url":null,"abstract":"In this paper, we analytically study the newly developed (2+1)-dimensional BenjaminOno equation by Wazwaz and propose its (3+1)-dimensional version. For this purpose, we successfully employed the modified extended tanh expansion method to construct certain hyperbolic, periodic and complex solitary wave structures simulated with the aid of symbolic computation using Mathematica. Also, we have depicted graphically the constructed solutions.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48914475","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}
Pub Date : 2020-08-01DOI: 10.22034/CMDE.2020.32139.1502
O. F. Rouz
This article examines asymptotic mean-square stability analysis of stochastic linear theta (SLT) scheme for n-dimensional stochastic delay differential equations (SDDEs). We impose some conditions on drift and diffusion terms, which admit that the diffusion coefficient can be highly nonlinear and does not necessarily satisfy a linear growth or global Lipschitz condition. We prove that the proposed scheme is asymptotically mean square stable if the employed stepsize is smaller than a given and easily computable upper bound. In particular, based on our investigation in the case θ ∈[ 1/2 , 1], the stepsize is arbitrary. Eventually, numerical examples are given to demonstrate the effectiveness of our work.
{"title":"Preserving asymptotic mean-square stability of stochastic theta scheme for systems of stochastic delay differential equations","authors":"O. F. Rouz","doi":"10.22034/CMDE.2020.32139.1502","DOIUrl":"https://doi.org/10.22034/CMDE.2020.32139.1502","url":null,"abstract":"This article examines asymptotic mean-square stability analysis of stochastic linear theta (SLT) scheme for n-dimensional stochastic delay differential equations (SDDEs). We impose some conditions on drift and diffusion terms, which admit that the diffusion coefficient can be highly nonlinear and does not necessarily satisfy a linear growth or global Lipschitz condition. We prove that the proposed scheme is asymptotically mean square stable if the employed stepsize is smaller than a given and easily computable upper bound. In particular, based on our investigation in the case θ ∈[ 1/2 , 1], the stepsize is arbitrary. Eventually, numerical examples are given to demonstrate the effectiveness of our work.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49454342","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}
Pub Date : 2020-08-01DOI: 10.22034/CMDE.2020.28604.1395
Z. Eskandari, M. Dahaghin
In this paper we present two class of third derivative multistep methods (TDMMs) that have good stability properties. Stability analysis of this method is examined and our numerical results are compared with the results of the existing method.
{"title":"Stability analysis of third derivative multi-step methods for stiff initial value problems","authors":"Z. Eskandari, M. Dahaghin","doi":"10.22034/CMDE.2020.28604.1395","DOIUrl":"https://doi.org/10.22034/CMDE.2020.28604.1395","url":null,"abstract":"In this paper we present two class of third derivative multistep methods (TDMMs) that have good stability properties. Stability analysis of this method is examined and our numerical results are compared with the results of the existing method.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47474866","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}
Pub Date : 2020-08-01DOI: 10.22034/CMDE.2020.28975.1407
A. Motamednezhad, F. Khajevand
The Lie symmetry method for differential equations is applied to study the exact solutions of the acoustic PDE. This study is based on two methods: Kudryashov and direct method for reduction's process. By using the symmetry operators some exact solutions are found with their graphs are plotted.
{"title":"Symmetry analysis and exact solutions of acoustic equation","authors":"A. Motamednezhad, F. Khajevand","doi":"10.22034/CMDE.2020.28975.1407","DOIUrl":"https://doi.org/10.22034/CMDE.2020.28975.1407","url":null,"abstract":"The Lie symmetry method for differential equations is applied to study the exact solutions of the acoustic PDE. This study is based on two methods: Kudryashov and direct method for reduction's process. By using the symmetry operators some exact solutions are found with their graphs are plotted.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43626858","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}
Pub Date : 2020-08-01DOI: 10.22034/CMDE.2020.26327.1337
Zahra Poursepahi Samian, M. Yaghouti
This paper is concerned with the reflected forward-backward stochastic differential equations with continuous monotone coefficients. Using the continuity approach, we prove that there exists at least one solution for the reflected forward-backward stochastic differential equations. The distinct character of our result is that the coefficient of the reflected forward SDEs contains the solution variable of the reflected BSDEs.
{"title":"Some Results on Reflected Forward-Backward Stochastic differential equations","authors":"Zahra Poursepahi Samian, M. Yaghouti","doi":"10.22034/CMDE.2020.26327.1337","DOIUrl":"https://doi.org/10.22034/CMDE.2020.26327.1337","url":null,"abstract":"This paper is concerned with the reflected forward-backward stochastic differential equations with continuous monotone coefficients. Using the continuity approach, we prove that there exists at least one solution for the reflected forward-backward stochastic differential equations. The distinct character of our result is that the coefficient of the reflected forward SDEs contains the solution variable of the reflected BSDEs.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42628634","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}
Pub Date : 2020-08-01DOI: 10.22034/CMDE.2020.28609.1396
B. Sepehrian, M. Radpoor
In this study, we solve the Fokker-Planck equation by a compact finite difference method. By the finite difference method the computation of Fokker-Planck equation is reduced to a system of ordinary differential equations. Two different methods, boundary value method and cubic $C^1$-spline collocation method, for solving the resulting system are proposed. Both methods have fourth order accuracy in time variable. By the boundary value method some pointwise approximate solutions are only obtained. But, $C^1$-spline method gives a closed form approximation in each space step, too. Illustrative examples are included to demonstrate the validity and efficiency of the methods. A comparison is made with existing results.
{"title":"Solving the Fokker-Planck equation via the compact finite difference method","authors":"B. Sepehrian, M. Radpoor","doi":"10.22034/CMDE.2020.28609.1396","DOIUrl":"https://doi.org/10.22034/CMDE.2020.28609.1396","url":null,"abstract":"In this study, we solve the Fokker-Planck equation by a compact finite difference method. By the finite difference method the computation of Fokker-Planck equation is reduced to a system of ordinary differential equations. Two different methods, boundary value method and cubic $C^1$-spline collocation method, for solving the resulting system are proposed. Both methods have fourth order accuracy in time variable. By the boundary value method some pointwise approximate solutions are only obtained. But, $C^1$-spline method gives a closed form approximation in each space step, too. Illustrative examples are included to demonstrate the validity and efficiency of the methods. A comparison is made with existing results.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44235702","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}
Pub Date : 2020-08-01DOI: 10.22034/CMDE.2020.32623.1515
Sedigheh Sabermahani, Y. Ordokhani
This manuscript is devoted to present an efficient numerical method for finding numerical solution of Volterra-Fredholm integro-differential equations of fractional-order. The technique is based on the M"{u}ntz-Legendre polynomials and Petrov-Galerkin method. A new Riemann-Liouville operational matrix for M"{u}ntz-Legendre polynomials is proposed using Laplace transform. Employing this operational matrix and Petrov-Galerkin method, the problem transforms to a system of algebraic equations. Next, we solve this system by applying any iterative method. An estimation of the error is proposed. The efficiency and accuracy of the present scheme is illustrated using several examples.
{"title":"A new operational matrix of Muntz-Legendre polynomials and Petrov-Galerkin method for solving fractional Volterra-Fredholm integro-differential equations","authors":"Sedigheh Sabermahani, Y. Ordokhani","doi":"10.22034/CMDE.2020.32623.1515","DOIUrl":"https://doi.org/10.22034/CMDE.2020.32623.1515","url":null,"abstract":"This manuscript is devoted to present an efficient numerical method for finding numerical solution of Volterra-Fredholm integro-differential equations of fractional-order. The technique is based on the M\"{u}ntz-Legendre polynomials and Petrov-Galerkin method. A new Riemann-Liouville operational matrix for M\"{u}ntz-Legendre polynomials is proposed using Laplace transform. Employing this operational matrix and Petrov-Galerkin method, the problem transforms to a system of algebraic equations. Next, we solve this system by applying any iterative method. An estimation of the error is proposed. The efficiency and accuracy of the present scheme is illustrated using several examples.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42340528","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}
Pub Date : 2020-08-01DOI: 10.22034/CMDE.2020.31155.1468
M. Esmaeilzadeh, H. Najafi, H. Aminikhah
This article is concerned with using a finite difference method, namely the theta-methods, to solve the diffusion-convection equation with piecewise constant arguments.The stability of this scheme is also obtained. Since there are not many published results on the numerical solution of this sort of differential equation and because of the importance of the above equation in the physics and engineering sciences, we have decided to study and present a stable numerical solution for the above mentioned problem. At the end of article some experiments are done to demonstrate the stability of the scheme. We also draw the figures for the numerical and analytical solutions which confirm ou results.The numerical solutions have also been compared with analytical solutions.
{"title":"A numerical scheme for diffusion-convection equation with piecewise constant arguments","authors":"M. Esmaeilzadeh, H. Najafi, H. Aminikhah","doi":"10.22034/CMDE.2020.31155.1468","DOIUrl":"https://doi.org/10.22034/CMDE.2020.31155.1468","url":null,"abstract":"This article is concerned with using a finite difference method, namely the theta-methods, to solve the diffusion-convection equation with piecewise constant arguments.The stability of this scheme is also obtained. Since there are not many published results on the numerical solution of this sort of differential equation and because of the importance of the above equation in the physics and engineering sciences, we have decided to study and present a stable numerical solution for the above mentioned problem. At the end of article some experiments are done to demonstrate the stability of the scheme. We also draw the figures for the numerical and analytical solutions which confirm ou results.The numerical solutions have also been compared with analytical solutions.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42005167","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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