Pub Date : 2021-04-01DOI: 10.22034/CMDE.2021.42195.1824
S. Zubova, Abdulftah Hosni Mohamad
We consider a first-order partial differential equation with constant irreversible coefficients in a Banach space in the regular case. The equation is split into equations in subspaces, in which non-degenerate subsystems are obtained. We obtain an analytical solution of each system with Showalter-type conditions. Finally, an example is given to illustrate the theoretical results.
{"title":"Analytical solution for descriptor system in partial differential equations","authors":"S. Zubova, Abdulftah Hosni Mohamad","doi":"10.22034/CMDE.2021.42195.1824","DOIUrl":"https://doi.org/10.22034/CMDE.2021.42195.1824","url":null,"abstract":"We consider a first-order partial differential equation with constant irreversible coefficients in a Banach space in the regular case. The equation is split into equations in subspaces, in which non-degenerate subsystems are obtained. We obtain an analytical solution of each system with Showalter-type conditions. Finally, an example is given to illustrate the theoretical results.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45747450","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 : 2021-03-22DOI: 10.22034/CMDE.2021.40834.1775
M. Molaei, F. D. Saei, M. Javidi, Y. Mahmoudi
In this paper, new analytical solutions for a class of conformable fractional differential equations (CFDE) and some more results about Laplace transform introduced by Abdeljawad cite{abdeljawad2015conformable} are investigated. The Laplace transform method is developed to get the exact solution of conformable fractional differential equations. The aim of this paper is to convert the conformable fractional differential equations into ordinary differential equations (ODE), this is done by using the fractional Laplace transformation of $(alpha+beta)$ order.
{"title":"New analytical methods for solving a class of conformable fractional differential equations by fractional Laplace transform","authors":"M. Molaei, F. D. Saei, M. Javidi, Y. Mahmoudi","doi":"10.22034/CMDE.2021.40834.1775","DOIUrl":"https://doi.org/10.22034/CMDE.2021.40834.1775","url":null,"abstract":"In this paper, new analytical solutions for a class of conformable fractional differential equations (CFDE) and some more results about Laplace transform introduced by Abdeljawad cite{abdeljawad2015conformable} are investigated. The Laplace transform method is developed to get the exact solution of conformable fractional differential equations. The aim of this paper is to convert the conformable fractional differential equations into ordinary differential equations (ODE), this is done by using the fractional Laplace transformation of $(alpha+beta)$ order.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42707162","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 : 2021-03-21DOI: 10.22034/CMDE.2021.40523.1766
M. Bagheri, A. Khani
In this article, a new version of the trial equation method is suggested. This method allows new exact solutions of the nonlinear partial differential equations. The developed method is applied to unstable nonlinear fractional-order Schr¨odinger equation in fractional time derivative form of order. Some exact solutions of the fractional-order fractional PDE are attained by employing the new powerful expansion approach using by beta-fractional derivatives which are used to get many solitary wave solutions by changing various parameters. New exact solutions are expressed with rational hyperbolic function solutions, rational trigonometric function solutions, 1-soliton solutions, dark soliton solitons, and rational function solutions. We can say that the unstable nonlinear Schr¨odinger equation exists I different dynamical behaviors. In addition, the physical behaviors of these new exact solution are given with two and three dimensional graphs.
{"title":"Dynamics of combined soliton solutions of unstable nonlinear fractional-order Schrodinger equation by beta-fractional derivative","authors":"M. Bagheri, A. Khani","doi":"10.22034/CMDE.2021.40523.1766","DOIUrl":"https://doi.org/10.22034/CMDE.2021.40523.1766","url":null,"abstract":"In this article, a new version of the trial equation method is suggested. This method allows new exact solutions of the nonlinear partial differential equations. The developed method is applied to unstable nonlinear fractional-order Schr¨odinger equation in fractional time derivative form of order. Some exact solutions of the fractional-order fractional PDE are attained by employing the new powerful expansion approach using by beta-fractional derivatives which are used to get many solitary wave solutions by changing various parameters. New exact solutions are expressed with rational hyperbolic function solutions, rational trigonometric function solutions, 1-soliton solutions, dark soliton solitons, and rational function solutions. We can say that the unstable nonlinear Schr¨odinger equation exists I different dynamical behaviors. In addition, the physical behaviors of these new exact solution are given with two and three dimensional graphs.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46713356","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 : 2021-03-21DOI: 10.22034/CMDE.2021.37944.1675
S. J. Ghevariya
In this paper, the Projected Differential Transform Method (PDTM) has been used to solve the Black Scholes differential equation for powered Modified Log Payoff (ML-Payoff) functions, $max {S^klnbig(frac{S}{K}big),0}$ and $max{S^klnbig(frac{K}{S}big),0}, (kin mathbb{R^{+}}cup {0})$. It is the generalization of Black Scholes model for ML-Payoff functions. It can be seen that values from PDTM is quite accurate to the closed form solutions.
{"title":"PDTM Approach to Solve Black Scholes Equation for Powered ML-Payoff Function","authors":"S. J. Ghevariya","doi":"10.22034/CMDE.2021.37944.1675","DOIUrl":"https://doi.org/10.22034/CMDE.2021.37944.1675","url":null,"abstract":"In this paper, the Projected Differential Transform Method (PDTM) has been used to solve the Black Scholes differential equation for powered Modified Log Payoff (ML-Payoff) functions, $max {S^klnbig(frac{S}{K}big),0}$ and $max{S^klnbig(frac{K}{S}big),0}, (kin mathbb{R^{+}}cup {0})$. It is the generalization of Black Scholes model for ML-Payoff functions. It can be seen that values from PDTM is quite accurate to the closed form solutions.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43266007","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 : 2021-03-21DOI: 10.22034/CMDE.2021.40223.1757
A. Golbabai, N. Safaei, Mahboubeh Molavi‐Arabshahi
In the current paper, for the economic growth model, an efficient numerical approach on arbitrary collocation points is described according to Radial Basis Functions (RBFs) interpolation to approximate the solutions of optimal control problem. The proposed method is based on parametrizing the solutions with any arbitrary global RBF and transforming the optimal control problem into a constrained optimization problem using arbitrary collocation points. The superiority of the method is its flexibility to select between different RBF functions for the interpolation and also parametrization an extensive range of arbitrary nodes. The Lagrange multipliers method is employed to convert the constrained optimization problem into a system of algebraic equations. Numerical results approve the accuracy and performance of the presented method for solving optimal control problems in the economic growth model.
{"title":"Numerical solution of optimal control problem for economic growth model using RBF collocation method","authors":"A. Golbabai, N. Safaei, Mahboubeh Molavi‐Arabshahi","doi":"10.22034/CMDE.2021.40223.1757","DOIUrl":"https://doi.org/10.22034/CMDE.2021.40223.1757","url":null,"abstract":"In the current paper, for the economic growth model, an efficient numerical approach on arbitrary collocation points is described according to Radial Basis Functions (RBFs) interpolation to approximate the solutions of optimal control problem. The proposed method is based on parametrizing the solutions with any arbitrary global RBF and transforming the optimal control problem into a constrained optimization problem using arbitrary collocation points. The superiority of the method is its flexibility to select between different RBF functions for the interpolation and also parametrization an extensive range of arbitrary nodes. The Lagrange multipliers method is employed to convert the constrained optimization problem into a system of algebraic equations. Numerical results approve the accuracy and performance of the presented method for solving optimal control problems in the economic growth model.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43009278","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 : 2021-02-22DOI: 10.22034/CMDE.2021.37458.1669
W. Adel, A. Yıldırım
In this work, a direct computational method has been developed for solving the thermal analysis of porous fins with a rectangular cross-section with the aid of Chebyshev polynomials. The method transforms the nonlinear differential equation into a system of nonlinear algebraic equations and then solved using a novel technique. The solution of the system gives the unknown Chebyshev coefficients. An algorithm for solving this nonlinear system is presented. The results are obtained for different values of the variables and a comparison with other methods is made to demonstrate the effectiveness of the method.
{"title":"Studying the Thermal Analysis of Rectangular Cross Section Porous Fin: A Numerical Approach","authors":"W. Adel, A. Yıldırım","doi":"10.22034/CMDE.2021.37458.1669","DOIUrl":"https://doi.org/10.22034/CMDE.2021.37458.1669","url":null,"abstract":"In this work, a direct computational method has been developed for solving the thermal analysis of porous fins with a rectangular cross-section with the aid of Chebyshev polynomials. The method transforms the nonlinear differential equation into a system of nonlinear algebraic equations and then solved using a novel technique. The solution of the system gives the unknown Chebyshev coefficients. An algorithm for solving this nonlinear system is presented. The results are obtained for different values of the variables and a comparison with other methods is made to demonstrate the effectiveness of the method.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45619256","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 : 2021-02-22DOI: 10.22034/CMDE.2021.41486.1799
Marvin Hoti
In this manuscript a center manifold reduction of the flow of a non-hyperbolic equilibrium point on a planar dynamical system with the Caputo derivative is proposed. The stability of the non-hyperbolic equilibrium point is shown to be locally asymptotically stable, under suitable conditions, by using the fractional Lyapunov direct method.
{"title":"Qualitative Stability Analysis of a Non-Hyperbolic Equilibrium Point of a Caputo Fractional System","authors":"Marvin Hoti","doi":"10.22034/CMDE.2021.41486.1799","DOIUrl":"https://doi.org/10.22034/CMDE.2021.41486.1799","url":null,"abstract":"In this manuscript a center manifold reduction of the flow of a non-hyperbolic equilibrium point on a planar dynamical system with the Caputo derivative is proposed. The stability of the non-hyperbolic equilibrium point is shown to be locally asymptotically stable, under suitable conditions, by using the fractional Lyapunov direct method.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43818454","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 : 2021-02-22DOI: 10.22034/CMDE.2021.34215.1567
Y. Khalili, M. K. Moghadam
In this study, we discuss the inverse problem for the Sturm-Liouville operator with the impulse and with the spectral boundary conditions on the finite interval (0, pi). By taking the Mochizuki-Trooshin's method, we have shown that some information of eigenfunctions at some interior point and parts of two spectra can uniquely determine the potential function q(x) and the boundary conditions.
{"title":"The interior inverse boundary value problem for the impulsive Sturm-Liouville operator with the spectral boundary conditions","authors":"Y. Khalili, M. K. Moghadam","doi":"10.22034/CMDE.2021.34215.1567","DOIUrl":"https://doi.org/10.22034/CMDE.2021.34215.1567","url":null,"abstract":"In this study, we discuss the inverse problem for the Sturm-Liouville operator with the impulse and with the spectral boundary conditions on the finite interval (0, pi). By taking the Mochizuki-Trooshin's method, we have shown that some information of eigenfunctions at some interior point and parts of two spectra can uniquely determine the potential function q(x) and the boundary conditions.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45101887","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 : 2021-02-21DOI: 10.22034/CMDE.2021.41150.1787
J. Rashidinia, Elham Mohmedi
In this paper, A reliable numerical scheme is developed and reviewed in order to obtain approximate solution of time fractional parabolic partial differential equations. The introduced scheme is based on Legendre tau spectral approximation and the time fractional derivative is employed in the Caputo sense. TheL2convergence analysis of the numerical method is analyzed. Numerical results for different examples are examined to verify the accuracy of spectral method and justification the theoretical analysis, and to compare with other existing methods in the literatures
{"title":"Numerical solution for solving fractional parabolic partial differential equations","authors":"J. Rashidinia, Elham Mohmedi","doi":"10.22034/CMDE.2021.41150.1787","DOIUrl":"https://doi.org/10.22034/CMDE.2021.41150.1787","url":null,"abstract":"In this paper, A reliable numerical scheme is developed and reviewed in order to obtain approximate solution of time fractional parabolic partial differential equations. The introduced scheme is based on Legendre tau spectral approximation and the time fractional derivative is employed in the Caputo sense. TheL2convergence analysis of the numerical method is analyzed. Numerical results for different examples are examined to verify the accuracy of spectral method and justification the theoretical analysis, and to compare with other existing methods in the literatures","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45333714","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 : 2021-02-21DOI: 10.22034/CMDE.2021.43439.1854
M. Niknam, H. Kheiri, Nadereh Abdi Sobouhi
This paper proposes an optimal control method for the chaotic attitude of the satellite when it is exposed to external disturbances. When there is no control over the satellite, its chaotic attitude is investigated using Lyapunov exponents (LEs), Poincare diagrams, and bifurcation diagrams. In order to overcome the problem of singularity in the great maneuvers of satellite, we consider the kinematic equations based on quaternion parameters instead of Euler angles, and obtain control functions by using the Pontryagin maximum principle (PMP). These functions are able to reach the satellite attitude to its equilibrium point. Also the asymptotic stability of these control functions is investigated by Lyapunov's stability theorem. Some simulation results are given to visualize the effectiveness and feasibility of the proposed method.
{"title":"Optimal control of satellite attitude and its stability based on quaternion parameters","authors":"M. Niknam, H. Kheiri, Nadereh Abdi Sobouhi","doi":"10.22034/CMDE.2021.43439.1854","DOIUrl":"https://doi.org/10.22034/CMDE.2021.43439.1854","url":null,"abstract":"This paper proposes an optimal control method for the chaotic attitude of the satellite when it is exposed to external disturbances. When there is no control over the satellite, its chaotic attitude is investigated using Lyapunov exponents (LEs), Poincare diagrams, and bifurcation diagrams. In order to overcome the problem of singularity in the great maneuvers of satellite, we consider the kinematic equations based on quaternion parameters instead of Euler angles, and obtain control functions by using the Pontryagin maximum principle (PMP). These functions are able to reach the satellite attitude to its equilibrium point. Also the asymptotic stability of these control functions is investigated by Lyapunov's stability theorem. Some simulation results are given to visualize the effectiveness and feasibility of the proposed method.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43257799","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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