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Numerical solution of Abel equation using operational matrix method with Chebyshev polynomials

ASIAN-EUROPEAN JOURNAL OF MATHEMATICS 2017年第3期10卷 1750053-1-1750053-11页

作者： Ozturk, Yalcin Gulsu, Mustafa Mugla Sitki Kocman Univ Ula Ali Kocman Vocat Sch Mugla Turkey Mugla Sitki Kocman Univ Fac Sci Dept Math Mugla Turkey

In this paper, we present a numerical scheme for solving the Abel equation. The approach is based on the shifted Chebyshev polynomials together with operational method. We reduce the problem to a set of nonlinear algebraic equations using operational matrix method. In addition, convergence analysis of the method is presented. Some numerical examples are given to demonstrate the validity and applicability of the method. The only a small number of Chebyshev polynomials is needed to obtain a satisfactory result.

关键词： Approximation by polynomials physical problem Abel equation nonlinear differential equation operational matrix method Chebyshev polynomial

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Numerical solution of Riccati equation using operational matrix method with Chebyshev polynomials

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ASIAN-EUROPEAN JOURNAL OF MATHEMATICS 2015年第2期8卷 1550020-1-1550020-11页

作者： Ozturk, Yalcin Gulsu, Mustafa Mugla Univ Fac Sci Dept Math Mugla Turkey

In this paper, we present numerical technique for solving the Riccati equation by using operational matrix method with Chebyshev polynomials. The method consists of expanding the required approximate solution as truncated Chebyshev series. Using operational matrix method, we reduce the problem to a set of algebraic equations. Some numerical examples are given to demonstrate the validity and applicability of the method. The method is easy to implement and produces very accurate results.

关键词： Riccati equation nonlinear differential equation operational matrix method Chebyshev polynomial approximate solution

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Simulating variable-order fractional Brownian motion and solving nonlinear stochastic differential equations

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MATHEMATICAL methodS IN THE APPLIED SCIENCES 2024年第11期47卷 8471-8489页

作者： Samadyar, Nasrin Ordokhani, Yadollah Alzahra Univ Fac Math Sci Dept Math Tehran Iran

Stochastic differential equations (SDEs) are very useful in modeling many problems in biology, economic data, turbulence, and medicine. Fractional Brownian motion (fBm) and variable-order fractional Brownian motion (vofBm) are suitable alternatives to standard Brownian motion (sBm) for describing and modeling many phenomena, since the increments of these processes are dependent of the past and for H > 1/2 these increments have the property of long-term dependence. Classical mathematical techniques such as Ito's calculus do not work for stochastic computations on fBm and vofBm due to they are not semi-Martingale for H(xi) not equal 1/2. Therefore, solving these equations is much more difficult than solving SDEs with sBm. On the other hand, these equations do not have an analytical solution, so we have to use numerical methods to find their solution. In this paper, a computational approach based on hybrid of block-pulse and parabolic functions (HBPFs) has been introduced for simulating vofBm and solving a modern class of SDEs. The mechanism of this approach is based on stochastic and fractional integration operational matrices, which transform the intended problem to a nonlinear system of algebraic equations. Thus, the complexity of solving the mentioned problem is reduced significantly. Also, convergence analysis of the expressed method has been theoretically examined. Finally, the accuracy and efficiency of the proposed algorithm have been experimentally investigated through some test problems and comparison of obtained results with results of previous papers. High accurate numerical results are obtained by using a small number of basic functions. Therefore, this method deals with smaller matrices and vectors, which is one of the most important advantage of our suggested method. Also, presenting an applicable procedure to construct vofBm is another innovation of this work. To gain this aim, at first, discretized sBm is generated via fundamental features of this proc

关键词： error analysis operational matrix method piecewise function stochastic differential equations variable-order fractional Brownian motion

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Approximating a New Class of Stochastic Differential Equations via operational Matrices of Bernoulli Polynomials

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IRANIAN JOURNAL OF SCIENCE 2023年第5-6期47卷 1633-1647页

作者： Ordokhani, Yadollah Samadyar, Nasrin Alzahra Univ Fac Math Sci Dept Math Tehran Iran

In the current paper, we introduce an efficient methodology to solve nonlinear stochastic differential equations (SDEs) driven by variable order fractional Brownian motion (vofBm) with appropriate initial condition. SDEs have many applications throughout pure mathematics and are used to model various behaviors of stochastic models such as stock prices, random growth models or physical systems that are subjected to thermal fluctuations. The mechanism of our proposed approach, which is based on Bernoulli polynomials operational matrices, is that it first transforms the problem under consideration into a nonlinear stochastic integral equation (SIE) driven by vofBm by using given initial condition and integrating from both sides of nonlinear SDE over the interval [0, t], where t is an element of [0, T]. Then, operational matrices of integration (omi) based on Bernoulli polynomials are utilized to significantly reduce the complexity of solving obtained SIE through converting such SIE into a nonlinear system of algebraic equations. Error analysis and convergence of suggested technique are also analyzed under some mild conditions. It is concluded that by increasing (n) over cap, the approximate solution more accurately estimates the exact solution, where <^>n denotes the number of elements in the used Bernoulli vector. Finally, some test problems are included to emphasize that the introduced idea is accurate, efficient and applicable. One of the most important innovations of this paper is the numerical simulation of vofBm, which is done in two steps. In the first step, standard Brownian motion (sBm) is constructed via spline interpolation scheme and then block-pulse and hat functions are used in the second step to simulate vofBm. Moreover, the obtained numerical results are also compared with achieved results from Chebyshev cardinal wavelets (Ccw) method to confirm the superiority of the presented method respect to the previous methods. This paper finishes with presenting

关键词： Stochastic differential equations Variable order fractional Brownian motion Bernoulli polynomials operational matrix method Error analysis

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A numerical solution of a non-classical Stefan problem with space-dependent thermal conductivity, variable latent heat and Robin boundary condition

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JOURNAL OF THERMAL ANALYSIS AND CALORIMETRY 2022年第24期147卷 14649-14657页

作者： Kumar, Abhishek Rajeev Gomez-Aguilar, J. F. Banaras Hindu Univ Dept Math Sci Indian Inst Technol Varanasi 221005 Uttar Pradesh India CONACyT Tecnol Nacl Mexico CENIDET Interior Internado Palmira S-N Col Palmira Cuernavaca 62490 Morelos Mexico

In this article, we have considered a non-classical Stefan problem that involves the space-dependent thermal conductivity, variable latent heat and Robin boundary condition at the fixed boundary. The solution of the problem is obtained by using the operational matrix method based on Genocchi polynomials and the collocation scheme. To show the accuracy, we have compared the results obtained by the Genocchi operational matrix method and a finite difference scheme with the exact solution for a particular case of the considered Stefan problem. It is found that the proposed Genocchi operational matrix method is more accurate than the finite difference scheme. Moreover, it is shown that the proposed algorithm rapidly converges as the degree of Genocchi polynomials increases. We have also analyzed the effects of the parameters alpha and delta on the temperature distribution and the evolution of moving phase front in the case of the considered non-classical Stefan problem. It is observed that the phase change processes become fast as the parameters alpha or/and delta increase.

关键词： Non-classical Stefan problem operational matrix method Space-dependent thermal conductivity Robin boundary condition Variable latent heat

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Numerical solution method for multi-term variable order fractional differential equations by shifted Chebyshev polynomials of the third kind

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ALEXANDRIA ENGINEERING JOURNAL 2022年第7期61卷 5145-5153页

作者： Tural-Polat, Sadiye Nergis Dincel, Arzu Turan Yildiz Tech Univ Dept Elect & Commun Engn Davutpasa Campus TR-34220 Istanbul Turkey Yildiz Tech Univ Dept Engn Math Davutpasa Campus TR-34220 Istanbul Turkey

Multi-term variable-order fractional differential equations (VO-FDEs) are considered to be one of the tools to illustrate the behavior of transient-regime real-life phenomena precisely. The analytical solutions of VO-FDEs are generally very hard to obtain. Therefore, in this study, an approximation method for the solution of multi-term VO-FDEs is proposed using shifted Chebyshev polynomials of the third kind (SCP3). The method employs the operational matrices of SCP3 to proximate the VO-FDEs with a system of algebraic equations. The solution of which also yields the approximate result for the multi-term VO-FDE. An error analysis is also included. The SCP3 method is examined on several examples demonstrating the precision and prowess of the method. (c) 2021 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://***/ licenses/by-nc-nd/4.0/).

关键词： Multi-term fractional differ-ential equations Variable-order fractional differential equations Shifted Chebyshev polyno-mials of the third kind operational matrix method

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An operational method for solutions of Riccati type differential equations with functional arguments

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JOURNAL OF TAIBAH UNIVERSITY FOR SCIENCE 2020年第1期14卷 661-669页

作者： Yuzbasi, Suayip Akdeniz Univ Fac Sci Dept Math TR-07058 Antalya Turkey

In this article, an operational matrix approach is presented to solve the Riccati type differential equations with functional arguments. These equations are encountered in Mathematical Physics. The method is based on the least-squares approximation and the operational matrices of integration and product. By obtaining the operation matrices for each term of the problem, the method converts the problem to a system of nonlinear algebraic equations. The roots of last system are used in determination of unknown function. Error analysis is made. Numerical applications are given to show efficiency of the method and also the comparisons are made with other methods from literature. In applications of the method, it is observed from the applications that the suggested method gives effective results.

关键词： Riccati differential equations functional differential equations operational matrix method error analysis

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Numerical Solutions of Multi-order Fractional Antiperiodic Boundary Value Problems

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IRANIAN JOURNAL OF SCIENCE AND TECHNOLOGY TRANSACTION A-SCIENCE 2020年第6期44卷 1839-1851页

作者： Choi, HuiChol Jong, KumSong Jon, KyongSon Sin, YongSim Kim Il Sung Univ Fac Math Pyongyang North Korea Northeast Normal Univ Key Lab Appl Stat MOE Sch Math & Stat Changchun Peoples R China

In this paper, we propose the numerical scheme for solving nonlinear higher multi-order Caputo fractional differential equations with integral and antiperiodic boundary conditions, establish the existence of an approximate solution to our problem, and prove the convergence of our numerical method. The operational matrix of fractional integration based on hat functions is used to solve the given problem. Also, we present a numerical example to illustrate our main results.

关键词： Multi-order fractional differential equation operational matrix method Antiperiodic boundary condition Integral boundary condition Hat functions Convergence analysis

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Jacobi collocation method for the approximate solution of some fractional-order Riccati differential equations with variable coefficients

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PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS 2019年 523卷 1130-1149页

作者： Singh, Harendra Srivastava, H. M. Postgrad Coll Dept Math Ghazipur 233001 Uttar Pradesh India Univ Victoria Dept Math & Stat Victoria BC V8W 3R4 Canada China Med Univ China Med Univ Hosp Dept Med Res Taichung 40402 Taiwan

This paper presents a computational method for the approximate solution of arbitrary-order non-linear fractional Riccati differential equations with variable coefficients. Proposed computational method is a combination of the operational matrix of integration method and the collocation method associated with the Jacobi polynomials. Convergence analysis of the proposed method is provided. Numerical results for different fractional orders of the Riccati differential equations are discussed. Figures and tables are used to show the numerical results derived from the proposed computational method for particular cases of Jacobi polynomials such as the Legendre polynomials, the Chebyshev polynomials of the second kind, the Chebyshev polynomials of the third kind, the Chebyshev polynomial of the fourth kind, and the Gegenbauer (or ultraspherical) polynomials. Numerical results from the proposed methods are compared from those derived by using the existing analytical and numerical methods. It is observed that the results from the proposed method are more accurate. Maximum absolute error and the root-mean square error tables are given for all five kinds of polynomials for comparison purposes. (C) 2019 Elsevier B.V. All rights reserved.

关键词： Fractional-order Riccati differential equations Jacobi polynomials Collocation method operational matrix method Convergence analysis Error analysis

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A numerical approach for solving weakly singular partial integro-differential equations via two-dimensional-orthonormal Bernstein polynomials with the convergence analysis

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NUMERICAL methodS FOR PARTIAL DIFFERENTIAL EQUATIONS 2019年第2期35卷 615-637页

作者： Mirzaee, Farshid Alipour, Sahar Samadyar, Nasrin Malayer Univ Fac Math Sci & Stat POB 65719-95863 Malayer Iran

In this paper, we develop an efficient matrix method based on two-dimensional orthonormal Bernstein polynomials (2D-OBPs) to provide approximate solution of linear and nonlinear weakly singular partial integro-differential equations (PIDEs). First, we approximate all functions involved in the considerable problem via 2D-OBPs. Then, by using the operational matrices of integration, differentiation, and product, the solution of Volterra singular PIDEs is transformed to the solution of a linear or nonlinear system of algebraic equations which can be solved via some suitable numerical methods. With a small number of bases, we can find a reasonable approximate solution. Moreover, we establish some useful theorems for discussing convergence analysis and obtaining an error estimate associated with the proposed method. Finally, we solve some illustrative examples by employing the presented method to show the validity, efficiency, high accuracy, and applicability of the proposed technique.

关键词： error analysis operational matrix method singular integral equation two-dimensional orthonormal Bernstein polynomials

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