Solutions of Frational Emden-Fowler Equations by Homotopy Analysis Method

Abstract

I S S N 2 3 4 7 -1 9 2 1 V o l u m e 1 3 N u m b e r 4 J o u r n a l o f A d v a n c e i n M a t h e m a t i c s 7042 | P a g e J a n u a r y 2018, h t t p s : / / c i r w o r l d . c o m / SOLUTIONS OF FRATIONAL EMDEN-FOWLER EQUATIONS BY HOMOTOPY ANALYSIS METHOD Keyword:fractional Emden-Fowler type equations, homotopy analysis method, Caputo fractional derivative 1.IntroductionConsider the following fractional Emden-Fowler type equations: The purpose of the present work is to use the homotopy analysis method(HAM) [3,4]to obtain the approximate analytical solutions of the Emden-Fowler type equations. The HAM is an analytical approach to get the series solution of linear or nonlinear fractional differential equations. The method provides great freedom to choose base function to approximate the linear and nonlinear problems. Main advantage of the method is that one can construct a continuous mapping of an initial guess approximation to the exact solution of the given problem through an auxiliary linear operator and to ensure the convergence of the series solution an auxiliary parameter is used. Recently Liao [3,4] has claimed that the difference from the other analytical methods is that one can ensure the convergence of series solution by means of choosing a proper value of convergence-control parameter.

High‐Performance n‐ and p‐Type Field‐Effect Transistors Based on Hybridly Surface‐Passivated Colloidal PbS Nanosheets

“…As recently shown, by increasing the amount of chloroalkane in the synthesis (the number of Cl − ions on the crystal), the lateral shape of the nanosheets is transformed to stripes . Field‐effect transistors based on these stripes show even higher performances (compared to the squared sheets), since the gate has a more effective influence on the channel conductivity, resulting from the reduced width of the quasi‐1D crystal …”

Solutions of the heat-conduction model described by fractional Emden-Fowler type equation

“…When 1 α = , eq. (1) was adopted to describe the heat-conduction model of the human head [7]. The singularity behavior that occurs at the origin is the main difficulty in the analysis of eqs.…”

“…(1) and (2). Recently, several researchers discussed such initial value problems [7][8][9][10][11]. Various numerical methods, such as, were applied to this kind of the initial value problems, see [12][13][14][15][16][17][18][19][20] and the cited references therein.…”

“…Various numerical methods, such as, were applied to this kind of the initial value problems, see [12][13][14][15][16][17][18][19][20] and the cited references therein. The purpose of the present work is to use the homotopy analysis method (HAM) [7,12] to obtain the approximate analytical solutions of eq. (1).…”