scholarly journals Numerical Solution of Nonlinear Integro-Differential Equations with Initial Conditions by Bernstein Operational Matrix of Derivative

2013 ◽  
Vol 02 (02) ◽  
pp. 141-149 ◽  
Author(s):  
Behrooz Basirat ◽  
Mohammad Amin Shahdadi
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Initial Conditions ◽  
Operational Matrix

scholarly journals Numerical Approach Based on Two-Dimensional Fractional-Order Legendre Functions for Solving Fractional Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-12
Author(s):  
Qingxue Huang ◽  
Fuqiang Zhao ◽  
Jiaquan Xie ◽  
Lifeng Ma ◽  
Jianmei Wang ◽  
...  
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Fractional Differential Equations ◽  
Operational Matrix ◽  
Numerical Approach ◽  
Principal Characteristic ◽  
Legendre Functions ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Fractional Differential

In this paper, a robust, effective, and accurate numerical approach is proposed to obtain the numerical solution of fractional differential equations. The principal characteristic of the approach is the new orthogonal functions based on shifted Legendre polynomials to the fractional calculus. Also the fractional differential operational matrix is driven. Then the matrix with the Tau method is utilized to transform this problem into a system of linear algebraic equations. By solving the linear algebraic equations, the numerical solution is obtained. The approach is tested via some examples. It is shown that the FLF yields better results. Finally, error analysis shows that the algorithm is convergent.

A method for the estimation of errors propagated in the numerical solution of a system of ordinary differential equations

1966 ◽  
Vol 25 ◽  
pp. 281-287 ◽  
Author(s):  
P. E. Zadunaisky
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Difference Method ◽  
Initial Conditions ◽  
Current Theory ◽  
Numerical Process

Let bex′=f(t,x) a system of ordinary differential equations, with initial conditionsx(a) =s, which is integrated numerically by a finite difference method of orderpand constant steph.To estimate the truncation and round-off errors accumulated during the numerical process it is established a method based on the current theory of the asymptotic behaviour (whenh→0) of errors. This method should avoid the main difficulties that arise when the results of the theory must be applied to practical cases. The method has been successfully tested and applied to estimate the errors accumulated in a numerical computation of planetary perturbations on the orbit of a comet.

Near-circular satellite orbits in an oblate, diurnally varying atmosphere

1983 ◽  
Vol 389 (1796) ◽  
pp. 153-170 ◽  
Keyword(s):  
Gravitational Field ◽  
Differential Equations ◽  
Diurnal Variation ◽  
Numerical Solution ◽  
Initial Conditions ◽  
Satellite Orbits ◽  
Atmospheric Parameters ◽  
Orbital Elements ◽  
Air Drag ◽  
Circular Satellite

The influence of air drag and the geopotential on near-circular satellite orbits, eccentricity e < 0.01, is considered. A model of the atmosphere is adopted that allows for oblateness, and in which the density behaviour approximates to the observed diurnal variation. Differential equations governing the variation in e and the argument of perigee ω are derived by combining the effects of air drag with those of the Earth’s gravitational field. These are solved numerically using initial conditions obtained from a series of computed orbits of the satellite 1963-27 A. The behaviour of the orbital elements predicted by the numerical solution is compared with the observed elements to test the developed theory, and to obtain values of atmospheric parameters at heights near 400 km.

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