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Now, we prove that the series (11) converges to the solution of the IVP. The proof is similar to the proof of the Banach fixed-point theorem. From our numerica simulations, we establish the fact that for two successive Ym and y+, we have the relation Hel < 1. Here |].| Ym denotes the usual sup norm. This means that there is constant c with 0 <c < 1, such that ||yn+1|| <c|lym||. We illustrate this fact in the next table, where severa calculations are shown. The calculations are done for different values of the parameter a. The first row of the table shows the values of a and the corresponding intervals.

Table 1 Now, we prove that the series (11) converges to the solution of the IVP. The proof is similar to the proof of the Banach fixed-point theorem. From our numerica simulations, we establish the fact that for two successive Ym and y+, we have the relation Hel < 1. Here |].| Ym denotes the usual sup norm. This means that there is constant c with 0 <c < 1, such that ||yn+1|| <c|lym||. We illustrate this fact in the next table, where severa calculations are shown. The calculations are done for different values of the parameter a. The first row of the table shows the values of a and the corresponding intervals.

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starting from an initial guess to the actual solution of the problem. Homotopy Analysis Method (HAM) does not even require a small parameter in the model. Moreover, it transforms a non-linear problem into a recursive sequence of linear problems which are comparitvely easy to solve. Adding the solutions of linear problems provides an
starting from an initial guess to the actual solution of the problem. Homotopy Analysis Method (HAM) does not even require a small parameter in the model. Moreover, it transforms a non-linear problem into a recursive sequence of linear problems which are comparitvely easy to solve. Adding the solutions of linear problems provides an
Fig. 3: Numerical comparison with the 20'" order HAM series solution for differenent values of auxiliary parameter i when a = 1 in (a), when a = 5 in (b) and when a = 10 in (c).
Fig. 3: Numerical comparison with the 20'" order HAM series solution for differenent values of auxiliary parameter i when a = 1 in (a), when a = 5 in (b) and when a = 10 in (c).
Fig. 4: h-curve analysis for the 20’" order HAM series solution with a = | in (a), with a = 5 in (b) and with a = 10 in (c).
Fig. 4: h-curve analysis for the 20’" order HAM series solution with a = | in (a), with a = 5 in (b) and with a = 10 in (c).
Fig. 5: Numerical comparison with 30" order power, its Pade approximant and 20'" order HAM series solution when a = | in (a), when a = 5 in (b) and when a = 10 in (c). (c) solution comparison for Taylor, Pade and HAM when a = 10
Fig. 5: Numerical comparison with 30" order power, its Pade approximant and 20'" order HAM series solution when a = | in (a), when a = 5 in (b) and when a = 10 in (c). (c) solution comparison for Taylor, Pade and HAM when a = 10
F. Abbas et al.: Approximate solutions to Lane-Emden equation...
F. Abbas et al.: Approximate solutions to Lane-Emden equation...
Theorem 3.3. For yo = a, the deformation derivative ym associated with homotopy H> are obtained recursively as solutions of the initial value problem 3.5 Convergence Interval Analysis for Different HAM Algorithms
Theorem 3.3. For yo = a, the deformation derivative ym associated with homotopy H> are obtained recursively as solutions of the initial value problem 3.5 Convergence Interval Analysis for Different HAM Algorithms
Thus, for the mth-deformation derivative, we obtain with the initial conditions in (13) the recursive linear initial value problem. Appl. Math. Inf. Sci. 13, No. 2, 143-152 (2019) / www.naturalspublishing.com/Journals.asp
Thus, for the mth-deformation derivative, we obtain with the initial conditions in (13) the recursive linear initial value problem. Appl. Math. Inf. Sci. 13, No. 2, 143-152 (2019) / www.naturalspublishing.com/Journals.asp
F. Abbas et al.: Approximate solutions to Lane-Emden equation...
F. Abbas et al.: Approximate solutions to Lane-Emden equation...
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