Pub Date : 2021-05-01DOI: 10.22034/CMDE.2021.44264.1871
Omid Farkhonderooz, D. Ahmadian
In this paper, we are interested in construction of an explicit third-order stochastic Runge–Kutta (SRK3) schemes for the weak approximation of stochastic differential equations (SDEs) with the general diffusion coefficient b(t, x). To this aim, we use the Itˆo-Taylor method and compare them with the stochastic expansion of the approximation. In this way, the authors encountered with a large number of equations and could find to derive four families for SRK3 schemes. Also we investigate the mean-square stability (MS-stability) properties of SRK3 schemes for a linear SDE. Finally, the proposed families are implemented on some examples to illustrate convergence results.
{"title":"Mean-square stability of a constructed Third-order stochastic Runge--Kutta schemes for general stochastic differential equations","authors":"Omid Farkhonderooz, D. Ahmadian","doi":"10.22034/CMDE.2021.44264.1871","DOIUrl":"https://doi.org/10.22034/CMDE.2021.44264.1871","url":null,"abstract":"In this paper, we are interested in construction of an explicit third-order stochastic Runge–Kutta (SRK3) schemes for the weak approximation of stochastic differential equations (SDEs) with the general diffusion coefficient b(t, x). To this aim, we use the Itˆo-Taylor method and compare them with the stochastic expansion of the approximation. In this way, the authors encountered with a large number of equations and could find to derive four families for SRK3 schemes. Also we investigate the mean-square stability (MS-stability) properties of SRK3 schemes for a linear SDE. Finally, the proposed families are implemented on some examples to illustrate convergence results.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41251151","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 : 2021-05-01DOI: 10.22034/CMDE.2021.38146.1680
A. Mohamed
In this paper, we evaluate the approximate numerical solution for the Volterra-Fredholm integral equation (V-FIE) using the shifted Jacobi collocation (SJC) method. This method depends on the operational matrices. We present some properties of the shifted Jacobi polynomials. These properties together with the shifted Jacobi polynomials transform the Volterra-Fredholm integral equation into a system of algebraic equations in the expansion coefficients of the solution. We discuss the convergence and error analysis of the shifted Jacobi polynomials in detail. The efficiency of this method is verified through numerical examples and compared with others.
{"title":"Shifted Jacobi collocation method for Volterra-Fredholm integral equation","authors":"A. Mohamed","doi":"10.22034/CMDE.2021.38146.1680","DOIUrl":"https://doi.org/10.22034/CMDE.2021.38146.1680","url":null,"abstract":"In this paper, we evaluate the approximate numerical solution for the Volterra-Fredholm integral equation (V-FIE) using the shifted Jacobi collocation (SJC) method. This method depends on the operational matrices. We present some properties of the shifted Jacobi polynomials. These properties together with the shifted Jacobi polynomials transform the Volterra-Fredholm integral equation into a system of algebraic equations in the expansion coefficients of the solution. We discuss the convergence and error analysis of the shifted Jacobi polynomials in detail. The efficiency of this method is verified through numerical examples and compared with others.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47951412","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 : 2021-05-01DOI: 10.22034/CMDE.2021.41430.1795
M. Tahmasebi, F. Mahmoudi
In this paper, we propose an exponential Euler method to approximate the solution of a stochastic functional differential equation driven by weighted fractional Brownian motion B{a,b} under some assumptions on a and b. We obtain also the convergence rate of the method to the true solution after proving an L2 -maximal bound for the stochastic ntegrals in this case.
{"title":"The Convergence of exponential Euler method for weighted fractional stochastic equations","authors":"M. Tahmasebi, F. Mahmoudi","doi":"10.22034/CMDE.2021.41430.1795","DOIUrl":"https://doi.org/10.22034/CMDE.2021.41430.1795","url":null,"abstract":"In this paper, we propose an exponential Euler method to approximate the solution of a stochastic functional differential equation driven by weighted fractional Brownian motion B{a,b} under some assumptions on a and b. We obtain also the convergence rate of the method to the true solution after proving an L2 -maximal bound for the stochastic ntegrals in this case.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46293098","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 : 2021-05-01DOI: 10.22034/CMDE.2021.39419.1728
S. A. Alavi, A. Haghighi, A. Yari, F. Soltanian
This paper presents a numerical method for solving a class of fractional optimal control problems (FOCPs) based on numerical polynomial approximation. The fractional derivative in the dynamic system is described in the Caputo sense. We used the approach in order to approximate the state and control functions by the Mott polynomials (M-polynomials). We introduced the operational matrix of fractional Riemann-Liouville integration and apply it to approximate the fractional derivative of the basis. We investigated the convergence of the new method and some examples are included to demonstrate the validity and applicability of the proposed method.
{"title":"A Numerical Method For Solving Fractional Optimal Control Problems Using The Operational Matrix Of Mott Polynomials","authors":"S. A. Alavi, A. Haghighi, A. Yari, F. Soltanian","doi":"10.22034/CMDE.2021.39419.1728","DOIUrl":"https://doi.org/10.22034/CMDE.2021.39419.1728","url":null,"abstract":"This paper presents a numerical method for solving a class of fractional optimal control problems (FOCPs) based on numerical polynomial approximation. The fractional derivative in the dynamic system is described in the Caputo sense. We used the approach in order to approximate the state and control functions by the Mott polynomials (M-polynomials). We introduced the operational matrix of fractional Riemann-Liouville integration and apply it to approximate the fractional derivative of the basis. We investigated the convergence of the new method and some examples are included to demonstrate the validity and applicability of the proposed method.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41685651","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 : 2021-04-24DOI: 10.22034/CMDE.2021.37369.1654
Mahboubeh Aalaei, M. Manteqipour
Abstract. In this paper, a new adaptive Monte Carlo algorithm is proposed to solve systems of linear algebraic equations (SLAEs). The corresponding properties of the algorithm and its advantages over the conventional and previous adaptive Monte Carlo algorithms are discussed and theoretical results are established to justify the convergence of the algorithm. Furthermore, the algorithm is used to solve the SLAEs obtained from finite difference method for the problem of European and American options pricing. Numerical tests are performed on examples with matrices of different size and on SLAEs coming from option pricing problems. Comparisons with standard numerical and stochastic algorithms are also done which demonstrate the computational efficiency of the proposed algorithm.
{"title":"An adaptive Monte Carlo algorithm for European and American options","authors":"Mahboubeh Aalaei, M. Manteqipour","doi":"10.22034/CMDE.2021.37369.1654","DOIUrl":"https://doi.org/10.22034/CMDE.2021.37369.1654","url":null,"abstract":"Abstract. In this paper, a new adaptive Monte Carlo algorithm is proposed to solve systems of linear algebraic equations (SLAEs). The corresponding properties of the algorithm and its advantages over the conventional and previous adaptive Monte Carlo algorithms are discussed and theoretical results are established to justify the convergence of the algorithm. Furthermore, the algorithm is used to solve the SLAEs obtained from finite difference method for the problem of European and American options pricing. Numerical tests are performed on examples with matrices of different size and on SLAEs coming from option pricing problems. Comparisons with standard numerical and stochastic algorithms are also done which demonstrate the computational efficiency of the proposed algorithm.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49346688","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 : 2021-04-24DOI: 10.22034/CMDE.2021.42640.1834
E. Dastranj, Hossein Sahebi Fard
In this paper, European options with transaction cost under some Black-Scholes markets are priced. In fact stochastic analysis and Lie group analysis are applied to find exact solutions for European options pricing under considered markets. In the sequel, using the finite difference method, numerical solutions are presented as well. Finally European options pricing are presented in four maturity times under some Black-Scholes models equipped with the gold asset as underlying asset. For this, the daily gold world price has been followed from Jan 1, 2016 to Jan 1, 2019 and the results of the profit and loss of options under the considered models indicate that call options prices prevent arbitrage opportunity but put options create it.
{"title":"Exact solutions and numerical simulation for Bakstein-Howison model","authors":"E. Dastranj, Hossein Sahebi Fard","doi":"10.22034/CMDE.2021.42640.1834","DOIUrl":"https://doi.org/10.22034/CMDE.2021.42640.1834","url":null,"abstract":"In this paper, European options with transaction cost under some Black-Scholes markets are priced. In fact stochastic analysis and Lie group analysis are applied to find exact solutions for European options pricing under considered markets. In the sequel, using the finite difference method, numerical solutions are presented as well. Finally European options pricing are presented in four maturity times under some Black-Scholes models equipped with the gold asset as underlying asset. For this, the daily gold world price has been followed from Jan 1, 2016 to Jan 1, 2019 and the results of the profit and loss of options under the considered models indicate that call options prices prevent arbitrage opportunity but put options create it.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49035716","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 : 2021-04-24DOI: 10.22034/CMDE.2021.39351.1725
M. Azizi, M. Amirfakhrian, M. A. Araghi
This paper presents a numerical fuzzy indirect method based on the fuzzy basis functions technique to solve an optimal control problem governed by Poisson's differential equation. The considered problem may or may not be accompanied by a control box constraint. The first-order necessary optimality conditions have been derived, which may contain a variational inequality in function space. In the presented method, the obtained optimality conditions have been discretized using fuzzy basis functions and a system of equations introduced as the discretized optimality conditions. The derived system mostly contains some nonsmooth equations and conventional system solvers fail to solve it. A fuzzy-system-based semi-smooth Newton method has also been introduced to deal with the obtained system. Solving optimality systems by the presented method gets us unknown fuzzy quantities on the state and control fuzzy expansions. Finally, some test problems have been studied to demonstrate the efficiency and accuracy of the presented fuzzy numerical technique.
{"title":"Application of fuzzy systems on the numerical solution of the elliptic PDE-constrained optimal control problems","authors":"M. Azizi, M. Amirfakhrian, M. A. Araghi","doi":"10.22034/CMDE.2021.39351.1725","DOIUrl":"https://doi.org/10.22034/CMDE.2021.39351.1725","url":null,"abstract":"This paper presents a numerical fuzzy indirect method based on the fuzzy basis functions technique to solve an optimal control problem governed by Poisson's differential equation. The considered problem may or may not be accompanied by a control box constraint. The first-order necessary optimality conditions have been derived, which may contain a variational inequality in function space. In the presented method, the obtained optimality conditions have been discretized using fuzzy basis functions and a system of equations introduced as the discretized optimality conditions. The derived system mostly contains some nonsmooth equations and conventional system solvers fail to solve it. A fuzzy-system-based semi-smooth Newton method has also been introduced to deal with the obtained system. Solving optimality systems by the presented method gets us unknown fuzzy quantities on the state and control fuzzy expansions. Finally, some test problems have been studied to demonstrate the efficiency and accuracy of the presented fuzzy numerical technique.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47978083","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 : 2021-04-11DOI: 10.22034/CMDE.2021.41690.1815
S. Mockary, A. Vahidi, E. Babolian
The Riesz fractional advection-diffusion is a result of the mechanics of chaotic dynamics. It's of preponderant importance to solve this equation numerically. Moreover, the utilization of Chebyshev polynomials as a base in several mathematical equations shows the exponential rate of convergence. To this approach, we transform the interval of state space into the interval [-1,1] * [-1,1] Then, we use the operational matrix to discretize fractional operators. Applying the resulting discretization, we obtain a linear system of equations, which leads to the numerical solution. Examples show the effectiveness of the method.
{"title":"An efficient approximate solution of Riesz fractional advection-diffusion equation","authors":"S. Mockary, A. Vahidi, E. Babolian","doi":"10.22034/CMDE.2021.41690.1815","DOIUrl":"https://doi.org/10.22034/CMDE.2021.41690.1815","url":null,"abstract":"The Riesz fractional advection-diffusion is a result of the mechanics of chaotic dynamics. It's of preponderant importance to solve this equation numerically. Moreover, the utilization of Chebyshev polynomials as a base in several mathematical equations shows the exponential rate of convergence. To this approach, we transform the interval of state space into the interval [-1,1] * [-1,1] Then, we use the operational matrix to discretize fractional operators. Applying the resulting discretization, we obtain a linear system of equations, which leads to the numerical solution. Examples show the effectiveness of the method.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43469882","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 : 2021-04-08DOI: 10.22034/CMDE.2021.38468.1692
Reza A. Hejazi, E. Dastranj, Noora Habibi, A. Naderifard
In this paper option pricing is given via stochastic analysis and invariant subspace method. Finally numerical solutions is driven and shown via diagram. The considered model is one of the most well known non-linear time series model in which the switching mechanism is controlled by an unobservable state variable that follows a first-order Markov chain. Some analytical solutions for option pricing are given under our considered model. Then numerical solutions are presented via finite difference method.
{"title":"Stochastic analysis and invariant subspace method for handling option pricing with numerical simulation","authors":"Reza A. Hejazi, E. Dastranj, Noora Habibi, A. Naderifard","doi":"10.22034/CMDE.2021.38468.1692","DOIUrl":"https://doi.org/10.22034/CMDE.2021.38468.1692","url":null,"abstract":" In this paper option pricing is given via stochastic analysis and invariant subspace method. Finally numerical solutions is driven and shown via diagram. The considered model is one of the most well known non-linear time series model in which the switching mechanism is controlled by an unobservable state variable that follows a first-order Markov chain. Some analytical solutions for option pricing are given under our considered model. Then numerical solutions are presented via finite difference method.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48230508","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 : 2021-04-08DOI: 10.22034/CMDE.2021.39707.1740
M. Nemati, M. Shafiee, H. Ebrahimi
The radial basis functions (RBFs) methods were first developed by Kansa to approximate partial differential equations (PDEs). The RBFs method is being truly meshfree becomes quite appealing, owing to the presence of distance function, straight-forward implementation, and ease of programming in higher dimensions. Another considerable advantage is the presence of a tunable free shape parameter, contained in most of the RBFs that control the accuracy of the RBFs method. Here, the solution of the two dimensional system of nonlinear partial differential equations is examined numerically by a Global Radial Basis Functions Collocation Method (GRBFCM). It can work on a set of random or uniform nodes with no need for element connectivity of input data. For the time-dependent partial differential equations, a system of ordinary differential equations (ODEs) is derived from this scheme. Like some other numerical methods, a comparison between numerical results with analytical solutions is implemented confirming the efficiency, accuracy, and simple performance of the suggested method.
{"title":"A meshless technique based on the radial basis functions for solving systems of partial differential equations","authors":"M. Nemati, M. Shafiee, H. Ebrahimi","doi":"10.22034/CMDE.2021.39707.1740","DOIUrl":"https://doi.org/10.22034/CMDE.2021.39707.1740","url":null,"abstract":"The radial basis functions (RBFs) methods were first developed by Kansa to approximate partial differential equations (PDEs). The RBFs method is being truly meshfree becomes quite appealing, owing to the presence of distance function, straight-forward implementation, and ease of programming in higher dimensions. Another considerable advantage is the presence of a tunable free shape parameter, contained in most of the RBFs that control the accuracy of the RBFs method. Here, the solution of the two dimensional system of nonlinear partial differential equations is examined numerically by a Global Radial Basis Functions Collocation Method (GRBFCM). It can work on a set of random or uniform nodes with no need for element connectivity of input data. For the time-dependent partial differential equations, a system of ordinary differential equations (ODEs) is derived from this scheme. Like some other numerical methods, a comparison between numerical results with analytical solutions is implemented confirming the efficiency, accuracy, and simple performance of the suggested method.","PeriodicalId":44352,"journal":{"name":"Computational Methods for Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49509119","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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