In this section, we will investigate Lane-Emden type equations with exponential and logarithmic nonlinearities which occur in the stellar structure theory (see [6, 11, 28–30]).
Theorem 2 If , , and is the differential transformation of the function , then
(5)
where
Proof From the definition of the transformation,
Put
(6)
By differentiation with respect to x, we get
(7)
From (6) and Theorem 1, we obtain
Comparing the terms with the same power of , we have
From this, we get
(8)
Replacing by k and by i, it follows
The proof is complete. □
Example 1 Consider the isothermal gas spheres equation in the case that the temperature remains constant (see [6, 12, 15, 31, 32]) which is described by the Lane-Emden type equation
(9)
with initial conditions .
From initial conditions and Theorem 1, we get
Multiplying equation (9) by x and using Theorem 1, we obtain
From here, we obtain the recurrence relation
(11)
Replacing by k, we get
(12)
Using (5), we can write recurrence relation (12) in the form
(13)
Since
then we can write relation (13) in the form
(14)
It is obvious that
Then recurrence relation (14) gives
Using the inverse transformation rule, we obtain an approximate solution of equation (6) in the form
Batiha [30], Gupta [28], Rafig et al. [29], Yildirim et al. [32], Parand et al. [31] obtained the same result by the variation iteration method, the homotopy perturbation method and the Hermite functions collocation method but using symbolic calculations as integral iterative functionals and solving differential equations of the second order.
Example 2 Now, we consider a more general type of equation (9)
(15)
with initial conditions .
Put
(16)
Multiplying equation (15) by x and using Theorem 1, we obtain the recurrence relation
(17)
Replacing by k, we get
(18)
In the cases of a linear combination of several nonlinearities, it is better to solve such type of equations as follows.
From initial conditions and Theorem 2, we have
Relation (18) yields
Then
Following the same procedure, , for can be solved as follows:
Using the inverse transformation rule, we obtain the solution of equation (6) in the form
In the limit case , we can observe that the series solution obtained by the differential transformation method converges to the series expansion of the closed form solution
Equation (15) with the coefficient instead of has been solved by Chowdhury and Hashim [33] using the homotopy perturbation method and Adomian [18, 19] using the Adomian decomposition method. They obtained a closed form solution as well but with the help of many symbolic calculations.
Equation (15) has been also investigated by Yiǧider et al. [34] using the differential transformation method. They obtained only the series solution (not in the closed form)
because they came out only from linear approximations of exponential nonlinearities.
Now, we derive the differential transformation of the certain logarithmic nonlinearity of the Lane-Emden type equation which occurs in the stellar structure theory and the thermionic current theory (see [11, 13]).
Theorem 3 If , and is the differential transformation of the function , then
Proof Put
(20)
It is obvious that . Now, we use the following identity:
(21)
If we apply the inverse transformation, we obtain from (20) and Theorem 1
Comparing the terms with the same power of , we have
(22)
From (22) we get
(23)
Replacing by i in the second sum on the right-hand side of identity (9) and considering the fact that , we obtain
The proof is complete. □
Example 3 Consider the following Lane-Emden type equation:
(24)
with initial conditions , .
Multiplying equation (24) by x and using Theorem 1, we obtain
Thus,
From here it follows
(26)
Replacing by k relation (26) gives
(27)
and from here we get
(28)
Using (19) for , we can write relation (27) in the form
which in view of (28) implies
(29)
From initial conditions and relation (27), we get
Then recurrence relation (29) yields
Continuing in this way, we obtain
Hence, the solution of the equation has the following form:
If , then the series solution converges to the series expansion of the closed form solution
as obtained by Chowdhury [33] by the homotopy perturbation method and by Wazwaz [19, 20] by the Adomian decomposition method. Disadvantage of both mentioned methods is solving many differential equations of the second order or complicated symbolic calculations of so-called Adomian polynomials.
Parand et al. [31] obtained a series solution of (24) (not in the closed form) using the Hermite functions collocation method.