Pub Date : 2021-01-05DOI: 10.22034/CMDE.2020.41130.1784
Elnaz Aryani, A. Babaei, Ali Valinejad
This paper deals with the numerical solution of nonlinear fractional stochastic integro-differential equations with the n-dimensional Wiener process. A new computational method is employed to approximate the solution of the considered problem. This technique is based on the modified hat functions, the Caputo derivative and a suitable numerical integration rule. Error estimate of the method is investigated in detail. At the end, illustrative examples are included to demonstrate the validity and effectiveness of the presented approach.
{"title":"A numerical technique for solving nonlinear fractional stochastic integro-differential equations with n-dimensional Wiener process","authors":"Elnaz Aryani, A. Babaei, Ali Valinejad","doi":"10.22034/CMDE.2020.41130.1784","DOIUrl":"https://doi.org/10.22034/CMDE.2020.41130.1784","url":null,"abstract":"This paper deals with the numerical solution of nonlinear fractional stochastic integro-differential equations with the n-dimensional Wiener process. A new computational method is employed to approximate the solution of the considered problem. This technique is based on the modified hat functions, the Caputo derivative and a suitable numerical integration rule. Error estimate of the method is investigated in detail. At the end, illustrative examples are included to demonstrate the validity and effectiveness of the presented approach.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45862645","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 : 2021-01-05DOI: 10.22034/CMDE.2020.38964.1708
Marzieh Dehghani-Madiseh
Matrix functions play important roles in various branches of science and engineering. In numerical computations and physical measurements there are several sources of error which significantly affect the main results obtained from solving the problems. This effect also influences the matrix computations. In this paper, we propose some approaches to enclose the matrix functions. We then present some analytical arguments to ensure that the obtained enclosures contain the exact result. Numerical experiments are given to illustrate the performance and effectiveness of the proposed approaches.
{"title":"Bounding error of calculating the matrix functions","authors":"Marzieh Dehghani-Madiseh","doi":"10.22034/CMDE.2020.38964.1708","DOIUrl":"https://doi.org/10.22034/CMDE.2020.38964.1708","url":null,"abstract":"Matrix functions play important roles in various branches of science and engineering. In numerical computations and physical measurements there are several sources of error which significantly affect the main results obtained from solving the problems. This effect also influences the matrix computations. In this paper, we propose some approaches to enclose the matrix functions. We then present some analytical arguments to ensure that the obtained enclosures contain the exact result. Numerical experiments are given to illustrate the performance and effectiveness of the proposed approaches.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48637753","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 : 2021-01-03DOI: 10.22034/CMDE.2020.41028.1780
Khuddush Mahammad, K. R. Prasad, P. Veeraiah
In this paper, we consider the iterative system of singular Rimean-Liouville fractional order boundary value problems with RiemannStieltjes integral boundary conditions involving increasing homeomorphism and positive homomorphism operator(IHPHO). By using Krasnoselskii’s cone fixed point theorem in a Banach space, we derive sufficent conditions for the existence of infinite number of nonnegative solutions. The sufficient conditions are also derived for the existence of unique nonnegative solution to the addressed problem by fixed point theorem in a complete metric space. As an application, we present an example to illustrate the main results.
{"title":"An infinite number of nonnegative solutions for iterative system of singular fractional order boundary value problems","authors":"Khuddush Mahammad, K. R. Prasad, P. Veeraiah","doi":"10.22034/CMDE.2020.41028.1780","DOIUrl":"https://doi.org/10.22034/CMDE.2020.41028.1780","url":null,"abstract":"In this paper, we consider the iterative system of singular Rimean-Liouville fractional order boundary value problems with RiemannStieltjes integral boundary conditions involving increasing homeomorphism and positive homomorphism operator(IHPHO). By using Krasnoselskii’s cone fixed point theorem in a Banach space, we derive sufficent conditions for the existence of infinite number of nonnegative solutions. The sufficient conditions are also derived for the existence of unique nonnegative solution to the addressed problem by fixed point theorem in a complete metric space. As an application, we present an example to illustrate the main results.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44387796","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 : 2021-01-03DOI: 10.22034/CMDE.2020.41761.1805
Zhen Wang
In this paper, we present an efficient numerical method to solve a one-dimensional time-fractional convection equation whose solution has a certain weak regularity at the starting time, where the time-fractional derivative in the Caputo sense with order in (0,1) is discretized by the L1 finite difference method on non-uniform meshes and the spatial derivative by the discontinuous Galerkin (DG) finite element method. The stability and convergence of the method are analyzed. Numerical experiments are provided to confirm the theoretical results.
{"title":"Non-uniform L1/DG method for one-dimensional time-fractional convection equation","authors":"Zhen Wang","doi":"10.22034/CMDE.2020.41761.1805","DOIUrl":"https://doi.org/10.22034/CMDE.2020.41761.1805","url":null,"abstract":"In this paper, we present an efficient numerical method to solve a one-dimensional time-fractional convection equation whose solution has a certain weak regularity at the starting time, where the time-fractional derivative in the Caputo sense with order in (0,1) is discretized by the L1 finite difference method on non-uniform meshes and the spatial derivative by the discontinuous Galerkin (DG) finite element method. The stability and convergence of the method are analyzed. Numerical experiments are provided to confirm the theoretical results.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44163359","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 : 2021-01-03DOI: 10.22034/CMDE.2020.41145.1785
Zehra Pinar
The simplified phenomenological model of long-crested shallow-water wave propagations is considered without/with the Coriolis effect. Symmetry analysis is taken into consideration to obtain exact solutions. Both classical wave transformation and transformations are obtained with symmetries and solvable equations are kept thanks to these transformations. Additionally, the exact solutions are obtained via various methods which are ansatz based methods. The obtained results have a major role in the literature so that the considered equation is seen in a large scale of applications in the area of geophysical.
{"title":"The Symmetry Analysis and Analytical Studies of the Rotational Green-Naghdi (R-GN) Equation","authors":"Zehra Pinar","doi":"10.22034/CMDE.2020.41145.1785","DOIUrl":"https://doi.org/10.22034/CMDE.2020.41145.1785","url":null,"abstract":"The simplified phenomenological model of long-crested shallow-water wave propagations is considered without/with the Coriolis effect. Symmetry analysis is taken into consideration to obtain exact solutions. Both classical wave transformation and transformations are obtained with symmetries and solvable equations are kept thanks to these transformations. Additionally, the exact solutions are obtained via various methods which are ansatz based methods. The obtained results have a major role in the literature so that the considered equation is seen in a large scale of applications in the area of geophysical.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43341043","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 : 2021-01-03DOI: 10.22034/CMDE.2020.30193.1447
Anas Rachid, M. Bahaj, R. Fakhar
Error estimates for element schemes for time-dependent for convection-diffusion-reaction equations with memory are derived and stated. For the spatially discrete scheme, optimal order error estimates in $L^{2},$ $H^{1}, $ and $W^{1,p }$ norms for $2leq p
导出并说明了具有记忆的对流扩散反应方程的含时单元格式的误差估计。对于空间离散格式，获得了$2leq p
{"title":"Finite Volume Element Approximation For Time-dependent Convection-Diffusion-ReactionEquations With Memory","authors":"Anas Rachid, M. Bahaj, R. Fakhar","doi":"10.22034/CMDE.2020.30193.1447","DOIUrl":"https://doi.org/10.22034/CMDE.2020.30193.1447","url":null,"abstract":"Error estimates for element schemes for time-dependent for convection-diffusion-reaction equations with memory are derived and stated. For the spatially discrete scheme, optimal order error estimates in $L^{2},$ $H^{1}, $ and $W^{1,p }$ norms for $2leq p <infty ,$ are obtained. Inthis paper, we also study the lumped mass modification. Based on the Crank-Nicolson method, a time discretization scheme is discussed and related error estimates are derived.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45589537","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 : 2021-01-03DOI: 10.22034/CMDE.2020.30802.1460
M. H. Akrami
This paper is devoted to study dynamical behaviours of the fractional-order Bazykin-Berezovskaya model and its discretization. The fractional derivative has been described in the Caputo sense. We show that the discretized system, exhibits more complicated dynamical behaviours than its corresponding fractional-order model. Specially, in the discretized model Neimark-Sacker and flip bifurcations and also chaos phenomena will happen. In the final part, some numerical simulation verify the analytical results.
{"title":"Dynamical behaviours of Bazykin-Berezovskaya model with fractional-order and its discretization","authors":"M. H. Akrami","doi":"10.22034/CMDE.2020.30802.1460","DOIUrl":"https://doi.org/10.22034/CMDE.2020.30802.1460","url":null,"abstract":"This paper is devoted to study dynamical behaviours of the fractional-order Bazykin-Berezovskaya model and its discretization. The fractional derivative has been described in the Caputo sense. We show that the discretized system, exhibits more complicated dynamical behaviours than its corresponding fractional-order model. Specially, in the discretized model Neimark-Sacker and flip bifurcations and also chaos phenomena will happen. In the final part, some numerical simulation verify the analytical results.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48332087","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 : 2021-01-02DOI: 10.22034/CMDE.2020.39020.1710
A. D. Nezhad, M. Moghaddam
One of the aims of this article is to investigate the solvability and unsolvability conditions for fractional cohomological equation $ psi^{alpha} f=g $, on $ mathbb{T}^n $. We prove that if $ f $ is not analytic, then fractional integro-differential equation $ I_t^{1-alpha} D_x^{alpha}u(x,t)+i I_x^{1-alpha} D_t^{alpha}u(x,t)=f(t) $ has no solution in $ C^1(B) $ with $0< alpha leq 1$. We also obtain solutions for the space-time fractional heat equations on $ mathbb{S}^1 $ and $ mathbb{T}^n $. At the end of this article, there are examples of fractional partial differential equations and a fractional integral equation together with their solutions.
{"title":"Toward a new understanding of cohomological method for fractional partial differential equations","authors":"A. D. Nezhad, M. Moghaddam","doi":"10.22034/CMDE.2020.39020.1710","DOIUrl":"https://doi.org/10.22034/CMDE.2020.39020.1710","url":null,"abstract":"One of the aims of this article is to investigate the solvability and unsolvability conditions for fractional cohomological equation $ psi^{alpha} f=g $, on $ mathbb{T}^n $. We prove that if $ f $ is not analytic, then fractional integro-differential equation $ I_t^{1-alpha} D_x^{alpha}u(x,t)+i I_x^{1-alpha} D_t^{alpha}u(x,t)=f(t) $ has no solution in $ C^1(B) $ with $0< alpha leq 1$. We also obtain solutions for the space-time fractional heat equations on $ mathbb{S}^1 $ and $ mathbb{T}^n $. At the end of this article, there are examples of fractional partial differential equations and a fractional integral equation together with their solutions.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43070857","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 : 2021-01-02DOI: 10.22034/CMDE.2020.24117.1294
F. Topal, Buse Eralp
In this study, existence of positive solutions are considered for second order boundary value problems on any time scales even in the case when y =0 may also be a solution.
{"title":"Second Order Boundary Value Problems of Nonsingular Type on Time Scales","authors":"F. Topal, Buse Eralp","doi":"10.22034/CMDE.2020.24117.1294","DOIUrl":"https://doi.org/10.22034/CMDE.2020.24117.1294","url":null,"abstract":"In this study, existence of positive solutions are considered for second order boundary value problems on any time scales even in the case when y =0 may also be a solution.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47412146","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 : 2021-01-02DOI: 10.22034/CMDE.2020.37139.1646
Santosh Kumar, Khursheed Alam
In the present study, we propose an effective nonlinear anisotropic diffusion-based hyperbolic parabolic model for image denoising and edge detection. The hyperbolic-parabolic model employs a second-order PDEs and have a second-time derivative to time t. This approach is very effective to preserve sharper edges and better-denoised images of noisy images. Our model is well-posed and it has a unique weak solution under certain conditions, which is obtained by using an iterative finite difference explicit scheme. The results are obtained in terms of peak signal to noise ratio (PSNR) as a metric, using an explicit scheme with forward-backward diffusivities.
{"title":"PDE-based hyperbolic-parabolic model for image denoising with forward-backward diffusivity","authors":"Santosh Kumar, Khursheed Alam","doi":"10.22034/CMDE.2020.37139.1646","DOIUrl":"https://doi.org/10.22034/CMDE.2020.37139.1646","url":null,"abstract":"In the present study, we propose an effective nonlinear anisotropic diffusion-based hyperbolic parabolic model for image denoising and edge detection. The hyperbolic-parabolic model employs a second-order PDEs and have a second-time derivative to time t. This approach is very effective to preserve sharper edges and better-denoised images of noisy images. Our model is well-posed and it has a unique weak solution under certain conditions, which is obtained by using an iterative finite difference explicit scheme. The results are obtained in terms of peak signal to noise ratio (PSNR) as a metric, using an explicit scheme with forward-backward diffusivities.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49134427","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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