scholarly journals An Operator-Based Scheme for the Numerical Integration of FDEs

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1372
Author(s):  
Inga Timofejeva ◽  
Zenonas Navickas ◽  
Tadas Telksnys ◽  
Romas Marcinkevicius ◽  
Minvydas Ragulskis
Keyword(s):  
Power Series ◽  
Numerical Integration ◽  
Analytic Solution ◽  
Selection Strategy ◽  
Integration Step ◽  
Adaptive Integration ◽  
Fractional Differential ◽  
Generalized Differential ◽  
Infinite Power ◽  
Step Selection

An operator-based scheme for the numerical integration of fractional differential equations is presented in this paper. The generalized differential operator is used to construct the analytic solution to the corresponding characteristic ordinary differential equation in the form of an infinite power series. The approximate numerical solution is constructed by truncating the power series, and by changing the point of the expansion. The developed adaptive integration step selection strategy is based on the controlled error of approximation induced by the truncation. Computational experiments are used to demonstrate the efficacy of the proposed scheme.

Approximations of boundary crossing probabilities for a Brownian motion

1999 ◽  
Vol 36 (4) ◽  
pp. 1019-1030 ◽  
Author(s):  
Alex Novikov ◽  
Volf Frishling ◽  
Nino Kordzakhia
Keyword(s):  
Brownian Motion ◽  
Power Series ◽  
Numerical Integration ◽  
Piecewise Linear ◽  
Boundary Crossing ◽  
Square Root ◽  
Girsanov Transformation ◽  
Piecewise Linear Approximations ◽  
Infinite Power ◽  
Crossing Probabilities

Using the Girsanov transformation we derive estimates for the accuracy of piecewise approximations for one-sided and two-sided boundary crossing probabilities. We demonstrate that piecewise linear approximations can be calculated using repeated numerical integration. As an illustrative example we consider the case of one-sided and two-sided square-root boundaries for which we also present analytical representations in a form of infinite power series.

Approximations of boundary crossing probabilities for a Brownian motion

1999 ◽  
Vol 36 (04) ◽  
pp. 1019-1030 ◽  
Author(s):  
Alex Novikov ◽  
Volf Frishling ◽  
Nino Kordzakhia
Keyword(s):  
Brownian Motion ◽  
Power Series ◽  
Numerical Integration ◽  
Piecewise Linear ◽  
Boundary Crossing ◽  
Square Root ◽  
Girsanov Transformation ◽  
Piecewise Linear Approximations ◽  
Infinite Power ◽  
Crossing Probabilities

Using the Girsanov transformation we derive estimates for the accuracy of piecewise approximations for one-sided and two-sided boundary crossing probabilities. We demonstrate that piecewise linear approximations can be calculated using repeated numerical integration. As an illustrative example we consider the case of one-sided and two-sided square-root boundaries for which we also present analytical representations in a form of infinite power series.

scholarly journals The Use of Integrals in Numerical Integrations of theN-Body Problem

1971 ◽  
Vol 10 ◽  
pp. 40-51
Author(s):  
Paul E. Nacozy
Keyword(s):  
Angular Momentum ◽  
Numerical Integration ◽  
Degrees Of Freedom ◽  
Body Problem ◽  
Center Of Mass ◽  
Integration Step ◽  
Gravitational System ◽  
Systems Of Differential Equations ◽  
Numerical Integrations ◽  
Original Number

AbstractThe numerical integration of systems of differential equations that possess integrals is often approached by using the integrals to reduce the number of degrees of freedom or by using the integrals as a partial check on the resulting solution, retaining the original number of degrees of freedom.Another use of the integrals is presented here. If the integrals have not been used to reduce the system, the solution of a numerical integration may be constrained to remain on the integral surfaces by a method that applies corrections to the solution at each integration step. The corrections are determined by using linearized forms of the integrals in a least-squares procedure.The results of an application of the method to numerical integrations of a gravitational system of 25-bodies are given. It is shown that by using the method to satisfy exactly the integrals of energy, angular momentum, and center of mass, a solution is obtained that is more accurate while using less time of calculation than if the integrals are not satisfied exactly. The relative accuracy is ascertained by forward and backward integrations of both the corrected and uncorrected solutions and by comparison with more accurate integrations using reduced step-sizes.

scholarly journals New efficiency algorithm for solving fractional order differential-algebraic system

2016 ◽  
Vol 5 (3) ◽  
pp. 152
Author(s):  
Sameer Hasan ◽  
Eman Namah
Keyword(s):  
Exact Solution ◽  
Fractional Order ◽  
Lagrange Multiplier ◽  
Iteration Method ◽  
Algebraic System ◽  
Analytic Solution ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Differential Algebraic System ◽  
Fractional Differential

This work provided the evolution of the algorithm for analytic solution of system of fractional differential-algebraic equations (FDAEs).The algorithm referred to good effective method for combination the Laplace Iteration method with general Lagrange multiplier (LLIM). Through this method we have reached excellent results in comparison with exact solution as we illustrated in our examples.

scholarly journals Solutions of Conformable Fractional-Order SIR Epidemic Model

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Atimad Harir ◽  
Said Malliani ◽  
Lalla Saadia Chandli
Keyword(s):  
Fractional Order ◽  
Epidemic Model ◽  
Analytic Solution ◽  
Nonlinear Problems ◽  
Variational Iteration Method ◽  
Transformation Method ◽  
Conformable Fractional Derivative ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Fractional Differential

In this paper, the conformable fractional-order SIR epidemic model are solved by means of an analytic technique for nonlinear problems, namely, the conformable fractional differential transformation method (CFDTM) and variational iteration method (VIM). These models are nonlinear system of conformable fractional differential equation (CFDE) that has no analytic solution. The VIM is based on conformable fractional derivative and proved. The result revealed that both methods are in agreement and are accurate and efficient for solving systems of OFDE.

scholarly journals Computational Optimization of Residual Power Series Algorithm for Certain Classes of Fuzzy Fractional Differential Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Mohammad Alaroud ◽  
Mohammed Al-Smadi ◽  
Rokiah Rozita Ahmad ◽  
Ummul Khair Salma Din
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Optimization Technique ◽  
Fractional Power ◽  
Taylor Formula ◽  
Residual Power Series ◽  
Fractional Differential ◽  
Residual Power ◽  
Fuzzy Fractional Differential Equations

This paper aims to present a novel optimization technique, the residual power series (RPS), for handling certain classes of fuzzy fractional differential equations of order 1<γ≤2 under strongly generalized differentiability. The proposed technique relies on generalized Taylor formula under Caputo sense aiming at extracting a supportive analytical solution in convergent series form. The RPS algorithm is significant and straightforward tool for creating a fractional power series solution without linearization, limitation on the problem’s nature, sort of classification, or perturbation. Some illustrative examples are provided to demonstrate the feasibility of the RPS scheme. The results obtained show that the scheme is simple and reliable and there is good agreement with exact solution.

scholarly journals Construction of Fractional Power Series Solutions to Fractional Boussinesq Equations Using Residual Power Series Method

2016 ◽  
Vol 2016 ◽  
pp. 1-15 ◽  
Author(s):  
Fei Xu ◽  
Yixian Gao ◽  
Xue Yang ◽  
He Zhang
Keyword(s):  
Power Series ◽  
Fractional Power ◽  
Boussinesq Equations ◽  
Power Series Method ◽  
Residual Power Series Method ◽  
Time Space ◽  
Residual Power Series ◽  
Fractional Differential ◽  
Fractional Power Series ◽  
Residual Power

This paper is aimed at constructing fractional power series (FPS) solutions of time-space fractional Boussinesq equations using residual power series method (RPSM). Firstly we generalize the idea of RPSM to solve any-order time-space fractional differential equations in high-dimensional space with initial value problems inRn. Using RPSM, we can obtain FPS solutions of fourth-, sixth-, and 2nth-order time-space fractional Boussinesq equations inRand fourth-order time-space fractional Boussinesq equations inR2andRn. Finally, by numerical experiments, it is shown that RPSM is a simple, effective, and powerful method for seeking approximate analytic solutions of fractional differential equations.

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