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.
Highlights
We set up a simple graphical method for the real roots of the equation F x 0 illustrated by four transcendental equations
Symbolic Computation is a modern area of research of interdisciplinary character that is placed in the common area of action of mathematics and the sciences of computation
The 1970s and 1980s have seen the development, of environments that place a greater emphasis on computation with mathematical objects in an implicit or symbolic form
Summary
Symbolic Computation is a modern area of research of interdisciplinary character that is placed in the common area of action of mathematics and the sciences of computation. Symbolic computing is concerned with the representation and manipulation of information in a symbolic form. It is based on defining objects not as numerical quantities but as entities that have certain mathematical properties. The representation of mathematical objects in a symbolic rather than numeric computational form has existed since the early days of computer science [1]. The 1970s and 1980s have seen the development, of environments that place a greater emphasis on computation with mathematical objects in an implicit or symbolic form. Symbolic and graphical computations utilized with nowadays existing symbols used for manipulating digital computer programs, they are definitely invaluable for obtaining solutions with any desired accuracy [3,4]. The above mentioned advantages motivated us introducing graphical and symbolic developments to a class of slightly perturbed equations that would have some applications in the future
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