Solutions of time-dependent Emden–Fowler type equations by homotopy-perturbation method

In this Letter, we apply the homotopy-perturbation method (HPM) to obtain approximate analytical solutions of the time-dependent Emden– Fowler type equations. We also present a reliable new algorithm based on HPM to overcome the difficulty of the singular point at x = 0. The analysis is accompanied by some linear and nonlinear time-dependent singular initial value problems. The results prove that HPM is very effective and simple

Solving SIVPs of Lane–Emden–Fowler Type Using a Pair of Optimized Nyström Methods with a Variable Step Size.

[EN]This research article introduces an efficient method for integrating Lane–Emden–Fowler equations of second-order singular initial value problems (SIVPs) using a pair of hybrid block methods with a variable step-size mode. The method pairs an optimized Nyström technique with a set of formulas applied at the initial step to circumvent the singularity at the beginning of the interval. The variable step-size formulation is implemented using an embedded-type approach, resulting in an efficient technique that outperforms its counterpart methods that used fixed step-size implementation. The numerical simulations confirm the better performance of the variable step-size implementation

A New Homotopy Perturbation Method for Solving Systems of Nonlinear Equations of Emden-Fowler Type

In this work, we apply the new homotopy perturbation method (NHPM) to get accurate results for solving systems of nonlinear equations of Emden–Fowler type, we indicate that our method (NHPM) is equivalent  to the variational iteration method (VIM) with a specific convex. Four examples  are given  to illustrate our proposed methods. The method is easy to carry out and gives very accurate solutions for solving linear and nonlinear differential equations

Solving third-order Lane–Emden–Fowler equations using a variable stepsize formulation of a pair of block methods.

[EN]This manuscript presents a variable stepsize formulation of a pair of block methods to efficiently solve third-order IVP models of Lane–Emden–Fowler type equations. The main method is obtained considering two intermediate points. This method combines an appropriate set of formulas for dealing with the singularity at the left endpoint . The proposed method is implemented in variable step-size mode to ensure that the truncation error is kept within a specified tolerance. The results of the numerical experiments confirm the good performance of the variable step-size implementation presented in this paper

Numerical investigation of Differential Biological-Models via GA-Kansa Method Inclusive Genetic Strategy

In this paper, we use Kansa method for solving the system of differential equations in the area of biology. One of the challenges in Kansa method is picking out an optimum value for Shape parameter in Radial Basis Function to achieve the best result of the method because there are not any available analytical approaches for obtaining optimum Shape parameter. For this reason, we design a genetic algorithm to detect a close optimum Shape parameter. The experimental results show that this strategy is efficient in the systems of differential models in biology such as HIV and Influenza. Furthermore, we prove that using Pseudo-Combination formula for crossover in genetic strategy leads to convergence in the nearly best selection of Shape parameter.Comment: 42 figures, 23 page