In this paper, Homotopy Analysis method with Genetic Algorithm is presented and used to obtain an analytical solution for the time-dependent Emden-Fowler type of equations and wave-type equation with singular behavior...In this paper, Homotopy Analysis method with Genetic Algorithm is presented and used to obtain an analytical solution for the time-dependent Emden-Fowler type of equations and wave-type equation with singular behavior at x = 0. The advantage of this single global method employed to present a reliable framework is utilized to overcome the singularity behavior at the point x = 0 for both models. The method is demonstrated for a variety of problems in one and higher dimensional spaces where approximate-exact solutions are obtained. The results obtained in all cases show the reliability and the efficiency of this method.展开更多
Finite Element Method (FEM), based on p and h versions approach, and the Adomians decomposition algorithm (ADM) are introduced for solving the Emden-Fowler Equation. A number of special cases of p and h versions of FE...Finite Element Method (FEM), based on p and h versions approach, and the Adomians decomposition algorithm (ADM) are introduced for solving the Emden-Fowler Equation. A number of special cases of p and h versions of FEM are introduced. Several iterated forms of the ADM are considered also. To demonstrate the efficiency of both methods, the numerical solutions of different examples are compared for both methods with the analytical solutions. It is observed that the results obtained by FEM are quite satisfactory and more accurate than ADM. Moreover, the FEM method is applicable for a wide range of classes including the singularity cases with the given special treatments by the FEM. Comparing the results with the existing true solutions shows that the FEM approach is highly accurate and converges rapidly.展开更多
文摘In this paper, Homotopy Analysis method with Genetic Algorithm is presented and used to obtain an analytical solution for the time-dependent Emden-Fowler type of equations and wave-type equation with singular behavior at x = 0. The advantage of this single global method employed to present a reliable framework is utilized to overcome the singularity behavior at the point x = 0 for both models. The method is demonstrated for a variety of problems in one and higher dimensional spaces where approximate-exact solutions are obtained. The results obtained in all cases show the reliability and the efficiency of this method.
文摘Finite Element Method (FEM), based on p and h versions approach, and the Adomians decomposition algorithm (ADM) are introduced for solving the Emden-Fowler Equation. A number of special cases of p and h versions of FEM are introduced. Several iterated forms of the ADM are considered also. To demonstrate the efficiency of both methods, the numerical solutions of different examples are compared for both methods with the analytical solutions. It is observed that the results obtained by FEM are quite satisfactory and more accurate than ADM. Moreover, the FEM method is applicable for a wide range of classes including the singularity cases with the given special treatments by the FEM. Comparing the results with the existing true solutions shows that the FEM approach is highly accurate and converges rapidly.
