Series solution to the Thomas–Fermi equation

An adaptive algorithm for the Thomas–Fermi equation

“…Table 6 shows the comparison between our results from Table 5 and those obtained by the Adomian decomposition method combined with Padé approximation. Comparison of our results with those obtained by the homotopy analysis method [13,15,16] is presented in Table 7. The values for u (0) using (31) and (32) with 1000 mesh points are much better than all other Table 1 Truncated free boundary x and iteration number K Table 7 Comparison between our approach and the homotopy analysis method in [13,15,16] for u (0) N u (0) via (31) and (32) Padé Yao [16] K h a n [ 15] L i a o [ 13] results except for that obtained with the 30-th order diagonal homotopy-Padé approximation in [16].…”

“…The following tests are obtained with x [0] = 10. At each iteration for x , Newton's method requires about 3-10 iterations to solve (15) to the desired accuracy. The converged value of the free boundary x is obtained about in 10-25 iterations for all cases reported below.…”

“…Andrianov and Awrejcewicz [12] applied the two-point Padé and quasifractional approximants to achieve a uniform analytical solution. Recently, in [13][14][15][16], explicit series solutions were achieved using the homotopy analysis method combined with the diagonal Padé approximation. Romas [17] applied the piecewiseadaptive homotopy perturbation method by means of the Padé approximation to solve the problem.…”

An Explicit Analytic Solution to the Thomas-Fermi Equation by the Improved Differential Transform Method

“…It was shown that, unlike the previous works, our approach is free of integration and requires simple arithmetic operations together with dierentiation. Compared to a number of well-known results in the literature [5,6,8,9], the proposed method was revealed to be highly accurate and ecient. It can be concluded that the IDTM, thanks to its mentioned merits, holds the promise to become of interest in analytical treatment of nonlinear equations of various types in the future.…”

On the rational second kind Chebyshev pseudospectral method for the solution of the Thomas–Fermi equation over an infinite interval

“…This method provided a convenient way to control the convergence of approximation series and adjusted convergence regions when necessary, which was a fundamental qualitative difference in analysis between the homotopy analysis method and all other reported analytic techniques. Khan [32] used the homotopy analysis method with a new and better transformation which improved the results in comparison with Liao's work. In [33], the quasilinearization approach was applied for solving Eq.…”