Symbolic and Graphical Computations of a Class of Slightly Perturbed Equations

Mohammed A. Sharaf, Abdel-naby S. Saad, Zainab A. Mominkhan

Department of Astronomy, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia.
Department of Astronomy, National Research Institute of Astronomy and Geophysics, Cairo, Egypt.
Department of Mathematics, College of Science for Girls, King Abdulaziz University, Jeddah, Saudi Arabia.

DOI: 10.4236/am.2013.45112                

Abstract

In this paper, a class of slightly perturbed equations of the form F(x)= ξ -x+αΦ(x) will be treated graphically and symbolically, where Φ(x) is an analytic function of x. For graphical developments, we set up a simple graphical method for the real roots of the equation F(x)=0 illustrated by four transcendental equations. In fact, the graphical solution usually provides excellent initial conditions for the iterative solution of the equation. A property avoiding the critical situations between divergent to very slow convergent solutions may exist in the iterative methods in which no good initial condition close to the root is available. For the analytical developments, literal analytical solutions are obtained for the most celebrated slightly perturbed equation which is Kepler’s equation of elliptic orbit. Moreover, the effect of the orbital eccentricity on the rate of convergence of the series is illustrated graphically.

Keywords

Symbolic Computation; Small Perturbations; Graphical Representations; Transcendental Equations

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	E. Fiume, “An Introduction to Scientific, Symbolic, and Graphical Computation,” A K Peters/CRC Press, London, 1995.
[2]	E. Kaltofen, “Challenges of Symbolic Computation,” Jour nal of Symbolic Computation, Vol. 29, 2000, pp. 891-919. doi:10.1006/jsco.2000.0370
[3]	V. A. Brumberg, “Analytical Techniques of Celestial Mechanics,” Springer-Verlag, Berlin, Heidelberg, 1995. doi:10.1007/978-3-642-79454-4
[4]	M. A. Sharaf, A.-N. S. Saad and A. A. Sharaf, “Unified Symbolic Algorithm of Gauss Method for Near-Parabolic Orbits,” Celestial Mechanics and Dynamical Astronomy, Vol. 70, No. 3, 1998, pp. 201-214. doi:10.1023/A:1008339612666
[5]	M. P. Fitzpatrick, “Principles of Celestial Mechanics,” Academic Press, New York and London, 1970.
[6]	R. H. Battin, “An Introduction to the Mathematics and Methods of Astrodynamics,” AIAA, Education Series, Reston, 1999. doi:10.2514/4.861543