Pub Date : 2021-01-02DOI: 10.22034/CMDE.2020.27563.1369
A. Beiranvand, K. Ivaz, H. Beiranvand
This paper introduces a novel method for estimation of the parameters of the constant elasticity of variance model. To do this, the likelihood function will be constructed based on the approximate density function. Then, to estimate the parameters, some optimization algorithms will be applied
{"title":"A new methodology to estimate constant elasticity of variance","authors":"A. Beiranvand, K. Ivaz, H. Beiranvand","doi":"10.22034/CMDE.2020.27563.1369","DOIUrl":"https://doi.org/10.22034/CMDE.2020.27563.1369","url":null,"abstract":"This paper introduces a novel method for estimation of the parameters of the constant elasticity of variance model. To do this, the likelihood function will be constructed based on the approximate density function. Then, to estimate the parameters, some optimization algorithms will be applied","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":"67985517","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.2020.36633.1633
B. Sepehrian, Z. Shamohammadi
A radial basis functions (RBFs) method for solving nonlinear time- and space-fractional Fokker-Planck equation is presented. The time-fractional derivative is Caputo type and the space-fractional derivative is Caputo or Riemann-Liouville type. The Caputo and Riemann-Liouville fractional derivatives of RBFs are utilized for approximating the spatial fractional derivatives of the unknown function. Also, in each time step the time-fractional derivative is approximated by the high order formulas introduced in cite{CaoLiChen} and, a collocation method is applied. The centers of RBFs are chosen as suitable collocation points. Thus, in each time step, the computations of fractional Fokker-Planck equation are reduced to a nonlinear system of algebraic equations. Two numerical examples are included to demonstrate the applicability, accuracy and stability of the method. Numerical experiments show that the experimental order of convergence is $4-alpha$, $alpha$ is the order of time derivative.
{"title":"Radial basis functions method for nonlinear time- and space-fractional Fokker-Planck equation","authors":"B. Sepehrian, Z. Shamohammadi","doi":"10.22034/CMDE.2020.36633.1633","DOIUrl":"https://doi.org/10.22034/CMDE.2020.36633.1633","url":null,"abstract":"A radial basis functions (RBFs) method for solving nonlinear time- and space-fractional Fokker-Planck equation is presented. The time-fractional derivative is Caputo type and the space-fractional derivative is Caputo or Riemann-Liouville type. The Caputo and Riemann-Liouville fractional derivatives of RBFs are utilized for approximating the spatial fractional derivatives of the unknown function. Also, in each time step the time-fractional derivative is approximated by the high order formulas introduced in cite{CaoLiChen} and, a collocation method is applied. The centers of RBFs are chosen as suitable collocation points. Thus, in each time step, the computations of fractional Fokker-Planck equation are reduced to a nonlinear system of algebraic equations. Two numerical examples are included to demonstrate the applicability, accuracy and stability of the method. Numerical experiments show that the experimental order of convergence is $4-alpha$, $alpha$ is the order of time derivative.","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":"41832318","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 : 2021-01-01DOI: 10.22034/CMDE.2020.37349.1653
Noora Habibi, Ali Mesforush
In this work, a new two-grid method presented for the elliptic partial differential equations is generalized to the time-dependent linear parabolic partial differential equations. The new two-grid waveform relaxation method uses the numerical method of lines, replacing any spatial derivative by a discrete formula, obtained here by the finite element method. A convergence analysis in terms of the spectral radius of the corresponding two-grid waveform relaxation operator is also developed. Moreover, the efficiency of the presented method and its analysis are tested, applying the two-dimensional heat equation.
{"title":"Extending a new two-grid waveform relaxation on a spatial finite element discretization","authors":"Noora Habibi, Ali Mesforush","doi":"10.22034/CMDE.2020.37349.1653","DOIUrl":"https://doi.org/10.22034/CMDE.2020.37349.1653","url":null,"abstract":"In this work, a new two-grid method presented for the elliptic partial differential equations is generalized to the time-dependent linear parabolic partial differential equations. The new two-grid waveform relaxation method uses the numerical method of lines, replacing any spatial derivative by a discrete formula, obtained here by the finite element method. A convergence analysis in terms of the spectral radius of the corresponding two-grid waveform relaxation operator is also developed. Moreover, the efficiency of the presented method and its analysis are tested, applying the two-dimensional heat equation.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67987587","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 : 2021-01-01DOI: 10.22034/CMDE.2020.38135.1678
S. Alfaqeih, E. Mısırlı
In the present article, we utilize the Conformable Double Laplace Transform Method (CDLTM) to get the exact solutions of a wide class of Conformable fractional differential in mathematical physics. The results obtained show that the proposed method is efficient, reliable and easy to be implemented on related linear problems in applied mathematics and physics. Moreover, the (CDLTM) has a small computational size as compared to other methods.
{"title":"Conformable Double Laplace Transform Method for Solving Conformable Fractional Partial Differential Equations","authors":"S. Alfaqeih, E. Mısırlı","doi":"10.22034/CMDE.2020.38135.1678","DOIUrl":"https://doi.org/10.22034/CMDE.2020.38135.1678","url":null,"abstract":"In the present article, we utilize the Conformable Double Laplace Transform Method (CDLTM) to get the exact solutions of a wide class of Conformable fractional differential in mathematical physics. The results obtained show that the proposed method is efficient, reliable and easy to be implemented on related linear problems in applied mathematics and physics. Moreover, the (CDLTM) has a small computational size as compared to other methods.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67987644","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 : 2021-01-01DOI: 10.22034/CMDE.2020.31728.1482
Nigar Ali, G. Zaman
A double time delayed- HIV-1 infection model with optimal controls functions is taken into account. The proposed model consists of double time delays and the following five compartments: uninfected CD4+ T cells, infected cells, double infected cells, human immunodeficiency virus and recombinant virus. The optimal control functions are introduced into the model. Then, the existence and uniqueness results for the optimal control pair are established. The optimality of system is derived and then solved numerically using a forward and backward difference scheme. The role of objective functional is to minimize the the density of infected cells; (ii) minimize free virus particles number; and (iii) maximize healthy cells density in blood
{"title":"Optimal control of double delayed HIV-1 infection model of fighting a virus with another virus","authors":"Nigar Ali, G. Zaman","doi":"10.22034/CMDE.2020.31728.1482","DOIUrl":"https://doi.org/10.22034/CMDE.2020.31728.1482","url":null,"abstract":"A double time delayed- HIV-1 infection model with optimal controls functions is taken into account. The proposed model consists of double time delays and the following five compartments: uninfected CD4+ T cells, infected cells, double infected cells, human immunodeficiency virus and recombinant virus. The optimal control functions are introduced into the model. Then, the existence and uniqueness results for the optimal control pair are established. The optimality of system is derived and then solved numerically using a forward and backward difference scheme. The role of objective functional is to minimize the the density of infected cells; (ii) minimize free virus particles number; and (iii) maximize healthy cells density in blood","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67986310","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 : 2021-01-01DOI: 10.22034/CMDE.2020.35574.1610
Manjit Singh
The nonlinear variable coefficient Zakharov-Kuznetsov (Vc-ZK) equation is derived using reductive perturbation technique for ion-acoustic solitary waves in magnetized three-component dusty plasma having negatively charged dust particles, isothermal ions, and electrons. The equation is investigated for generalized symmetries using a recently proposed compatibility method. Some more general symmetries are obtained and group invariant solutions are also constructed for these symmetries. Besides this, the equation is also investigated for nontrivial local conservation laws
{"title":"Generalized symmetries and conservation laws of (3+1)-dimensional variable coefficient Zakharov-Kuznetsov equation","authors":"Manjit Singh","doi":"10.22034/CMDE.2020.35574.1610","DOIUrl":"https://doi.org/10.22034/CMDE.2020.35574.1610","url":null,"abstract":"The nonlinear variable coefficient Zakharov-Kuznetsov (Vc-ZK) equation is derived using reductive perturbation technique for ion-acoustic solitary waves in magnetized three-component dusty plasma having negatively charged dust particles, isothermal ions, and electrons. The equation is investigated for generalized symmetries using a recently proposed compatibility method. Some more general symmetries are obtained and group invariant solutions are also constructed for these symmetries. Besides this, the equation is also investigated for nontrivial local conservation laws","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67986592","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 : 2021-01-01DOI: 10.22034/CMDE.2020.39685.1737
T. A. Bullo, G. Duressa, G. Degla
This paper deals with the numerical treatment of singularly perturbed parabolic reaction-diffusion initial boundary value problems. Introducing a fitting parameter into the asymptotic solution and applying average finite difference approximation, a fitted operator finite difference method is developed for solving the problem. To accelerate the rate of convergence of the method, the Richardson extrapolation technique is applied. The consistency and stability of the proposed method have been established very well to ensure the convergence of the method. Numerical experimentation is carried out on some model problems and the results are presented both in tables and graphs. The numerical results are compared with the findings of some methods existing in the literature and found to be more accurate. Generally, the formulated method is consistent, stable, and more accurate than some methods existing in the literature for solving singularly perturbed parabolic reaction-diffusion initial boundary value problems.
{"title":"Accelerated fitted operator finite difference method for singularly perturbed parabolic reaction-diffusion problems","authors":"T. A. Bullo, G. Duressa, G. Degla","doi":"10.22034/CMDE.2020.39685.1737","DOIUrl":"https://doi.org/10.22034/CMDE.2020.39685.1737","url":null,"abstract":"This paper deals with the numerical treatment of singularly perturbed parabolic reaction-diffusion initial boundary value problems. Introducing a fitting parameter into the asymptotic solution and applying average finite difference approximation, a fitted operator finite difference method is developed for solving the problem. To accelerate the rate of convergence of the method, the Richardson extrapolation technique is applied. The consistency and stability of the proposed method have been established very well to ensure the convergence of the method. Numerical experimentation is carried out on some model problems and the results are presented both in tables and graphs. The numerical results are compared with the findings of some methods existing in the literature and found to be more accurate. Generally, the formulated method is consistent, stable, and more accurate than some methods existing in the literature for solving singularly perturbed parabolic reaction-diffusion initial boundary value problems.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67987308","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 : 2021-01-01DOI: 10.22034/CMDE.2020.31022.1466
M. F. Aghdaei, H. Adibi
In this paper, we construct exact families of traveling wave (periodic wave, singular wave, singular periodic wave, singular-solitary wave and shock wave) solutions of a well-known equation of nonlinear PDE, the variable coefficients combined HirotaLakshmanan-Porsezian-Daniel (Hirota-LPD) equation with the fourth nonlinearity, which describes an important development, and application of soliton dispersion management experiment in nonlinear optics is considered, and as an achievement, a series of exact traveling wave solutions for the aforementioned equation is formally extracted. This nonlinear equation is solved by using the extended trial equation method (ETEM) and the improved tan(ϕ/2)-expansion method (ITEM). Meanwhile, the mechanical features of some families are explained through offering the physical descriptions. Analytical treatment to find the nonautonomous rogue waves are investigated for the combined Hirota-LPD equation.
{"title":"Exact solutions of the combined Hirota-LPD equation with variable coefficients","authors":"M. F. Aghdaei, H. Adibi","doi":"10.22034/CMDE.2020.31022.1466","DOIUrl":"https://doi.org/10.22034/CMDE.2020.31022.1466","url":null,"abstract":"In this paper, we construct exact families of traveling wave (periodic wave, singular wave, singular periodic wave, singular-solitary wave and shock wave) solutions of a well-known equation of nonlinear PDE, the variable coefficients combined HirotaLakshmanan-Porsezian-Daniel (Hirota-LPD) equation with the fourth nonlinearity, which describes an important development, and application of soliton dispersion management experiment in nonlinear optics is considered, and as an achievement, a series of exact traveling wave solutions for the aforementioned equation is formally extracted. This nonlinear equation is solved by using the extended trial equation method (ETEM) and the improved tan(ϕ/2)-expansion method (ITEM). Meanwhile, the mechanical features of some families are explained through offering the physical descriptions. Analytical treatment to find the nonautonomous rogue waves are investigated for the combined Hirota-LPD equation.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67985609","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 : 2021-01-01DOI: 10.22034/CMDE.2020.34163.1563
T. Allahviranloo, M. Keshavarz, S. Abbasbandy, M. Modarressi
The cardiovascular system is an extremely intelligent and dynamic system which adjusts its performance depending on the individual's physical and environmental conditions. Some of these physical and environmental conditions may create slight disruptions in the cardiovascular system leading to a variety of diseases. Since prevention has always been preferable to treatment, this paper examined the Instantaneous Pressure-Volume Relation (IPVR) and also the pressure of the artery root. The fuzzy mathematics as a powerful tool is used to evaluate and predict the status of an individual's blood pressure. The arterial pressure is modeled as a first-order fuzzy differential equation and an analytical solution for this equation is obtained and an example show the behavior of the solution. The risk factors using fuzzy rules are assessed, which help diagnose the status of individual's blood pressure. Using the outcome by drawing the individual's attention to these risk factors, the individual's health is improved. Moreover, in this study adaptive neuro-fuzzy inference system (ANFIS) models is evaluated to predict the status of an individual's blood pressure on the basis of the inputs.
{"title":"Analytical Fuzzy Solution of the Ventricular Pressure Equation and Prediction of the Blood Pressure","authors":"T. Allahviranloo, M. Keshavarz, S. Abbasbandy, M. Modarressi","doi":"10.22034/CMDE.2020.34163.1563","DOIUrl":"https://doi.org/10.22034/CMDE.2020.34163.1563","url":null,"abstract":"The cardiovascular system is an extremely intelligent and dynamic system which adjusts its performance depending on the individual's physical and environmental conditions. Some of these physical and environmental conditions may create slight disruptions in the cardiovascular system leading to a variety of diseases. Since prevention has always been preferable to treatment, this paper examined the Instantaneous Pressure-Volume Relation (IPVR) and also the pressure of the artery root. The fuzzy mathematics as a powerful tool is used to evaluate and predict the status of an individual's blood pressure. The arterial pressure is modeled as a first-order fuzzy differential equation and an analytical solution for this equation is obtained and an example show the behavior of the solution. The risk factors using fuzzy rules are assessed, which help diagnose the status of individual's blood pressure. Using the outcome by drawing the individual's attention to these risk factors, the individual's health is improved. Moreover, in this study adaptive neuro-fuzzy inference system (ANFIS) models is evaluated to predict the status of an individual's blood pressure on the basis of the inputs.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67986418","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 : 2021-01-01DOI: 10.22034/CMDE.2020.35895.1621
A. Zamiri, A. Borhanifar, A. Ghannadiasl
In this paper, a Laguerre collocation method is presented in order to obtain numerical solutions for linear and nonlinear Lane-Emden type equations and their initial conditions. The basis of the present method is operational matrices with respect to modified generalized Laguerre polynomials(MGLPs) that transforms the solution of main equation and its initial conditions to the solution of a matrix equation corresponding to the system of algebraic equations with the unknown Laguerre coefficients. By solving this system, coefficients of approximate solution of the main problem will be determined. Implementation of the method is easy and has more accurate results in comparison with results of other methods.
{"title":"Laguerre collocation method for solving Lane-Emden type equations","authors":"A. Zamiri, A. Borhanifar, A. Ghannadiasl","doi":"10.22034/CMDE.2020.35895.1621","DOIUrl":"https://doi.org/10.22034/CMDE.2020.35895.1621","url":null,"abstract":"In this paper, a Laguerre collocation method is presented in order to obtain numerical solutions for linear and nonlinear Lane-Emden type equations and their initial conditions. The basis of the present method is operational matrices with respect to modified generalized Laguerre polynomials(MGLPs) that transforms the solution of main equation and its initial conditions to the solution of a matrix equation corresponding to the system of algebraic equations with the unknown Laguerre coefficients. By solving this system, coefficients of approximate solution of the main problem will be determined. Implementation of the method is easy and has more accurate results in comparison with results of other methods.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67986543","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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