Abstract: Singular initial value problems, linear and nonlinear, homogeneous and nonhomogeneous, are investigated by using Taylor series method. The solutions are constructed in the form of a convergent series. A general formula is established. The approach is illustrated with few examples.
INTRODUCTION
In recent years, the studies of singular initial value problems in the second-order Ordinary Differential Equations (ODEs) have attracted the attention of many mathematicians and physicists. One of the equations describing this type is the Lane-Emden-type equations formulated as.
| (1) |
On the other hand, studies have been carried out on another class of singular initial value problems of the form
| (2) |
where A and B are constants, f (x, y) is a continuous real valued function and g (x) ∈ c [0,1]. Equation 2 differs from the classical Lane-Emden-type Eq. 1 for the function f (x, y) and the inhomogeneous term g (x).
Equation 1 with specializing f (y) was used to model several
phenomena in mathematical Physics and astrophysics such as the theory of stellar
structure, the thermal behavior of a spherical cloud of gas, isothermal gas
spheres and theory of thermionic currents Chandrasekhar (1976) and Davis (1962).
Due to the significant applications of Lane-Emden-type equations in the scientific
community, various forms of f (y) have been investigated in many research works.
A discussion of the formulation of these models and the physical structure of
the solutions can be found in Chandrasekhar (1976), Davis (1962), Shawagfeh
(1993), Adomian (1986) and Wazwaz (2001). Most algorithms currently in use for
handling the Lane-Emden-type problems are based on either series solution or
perturbation techniques. Wazwaz (2001) has given a general study to construct
exact and series solution to Lane-Emden-type equations by employing the Adomian
decomposition method. Moreover, a generalization was developed in Wazwaz (2001)
by replacing the coefficient 2/x of
It is important to note that (2), with boundary conditions, has attracted many mathematicians and has been studied from various points of view. Russell and Shampine (1975) have investigated (2) for the liner function f (x, y) = ky+h (x) and have proved that a unique solution exists if h (x) ∈ C [0,1] and –∞4<k≤ π2. Three-point difference methods of second order have been used in Russell and Shampine (1975). Moreover, three-point difference method of second order have been also used by Chawal and Katt I(1984), Chawla et al. (1986) and Iyengar and Jain (1987). However, Jain and Jain (1989) derived three-point difference method of four and six order to solve this problem. The numerical results obtained in Jain and Jain (1989) demonstrate o (h4) and o (h6) convergence of the method. Recently, El- sayed (1999) used a multi-integral method to investigate the nonlinear problem (2) with two boundary conditions.
The Eq. 1 can not have a taylor series expansion directly over the interval in which a solution is desired. For example, if
then
THE METHOD
In theory, the infinite Taylor series can be used to evaluate a function, given its derivative function and its value at some point, consider the nonlinear first-order
ODE:
The Taylor series for y (x) at x = x0 is
We solving the Eq. 2 by the following steps :
| (a) | (3) |
| (b) |
| (c) |
| (d) |
| (e) |
substituting (e) the initial condition and values found in (d) we obtain the solution.
NUMERICAL ILLUSTRATIONS
Example 1: We consider the linear singular initial value problem:
| (4) |
By (3), Eq. 4 becomes
substituting the initial condition and the values find, we obtain the solution y = x2+x3.
Example 2: We consider the linear singular initial value problem
by Taylor series method with x = 0 we have
substituting the initial condition and the values find, we obtain the solution
and in the closed form y (x) = ex2.
GENERALIZATION
A generalization of the Lane-Emaden- like Eq. 1 has been
studied by Wazwaz (2001). We replace the standard coefficient of
| (5) |
With initial conditions
can be formulated. It is convenient to consider a modification to the approach presented before in order to enable us to handle (5). It is convenient to introduce the derivative of order k. From Eq. 5,
we can get the higher derivatives for y in the following
| (6) |
Hence x = 0, y (0) = A,
By taylor series method with x = 0. We have
substituting the initial condition and the values find, we obtain the solution.
To illustrate the generalization discussed above, we discuss this example:
Example 3. We consider the linear initial value problem:
| (7) |
By (6), Eq. 7 becomes
substituting the initial condition and the values find we obtain the solution y (x) = x4-x3.
CONCLUSION
In the discussion it was shown that, with the proper use of the taylor series method, it is possible to obtain an analytic solution to a class of singular initial value problems, homogeneous
or inhomogeneous. The difficulty in using a taylor series method directly to
this type of equations, due to the existence of singular point at x = 0, is
overcome here. The class of singular equations was generalized, by changing
the coefficient of