Spectral methods on the semi-infinite line

Scientific computing has an important role in applied mathematics. Many problems that occur in physics and engineering can be modelled by linear or nonlinear differential equations. The main topic of this thesis is the solution of Blasius and Lane-Emden type equations which are nonlinear ordinary differential equations on a semi-infinite interval. The Blasius equation is a third-order nonlinear ordinary differential equation. The Lane-Emden type equations have been considered by many mathematicians. An orthogonal Laguerre basis is proposed to provide an effective and simple way to improve the solution by spectral methods. Through comparisons among the exact solutions of Horedt [54] and the series solutions of Wazwaz [84], Liao [61], and Ramos [76], and the current work, it is shown that the present work provides an effective approach for Lane-Emden type equations; also it is confirmed by the numerical results that this approach has exponentially convergence rate. In the Blasius equation, the second derivative at zero is an important point of the function, so we have computed It and compared the result with other well-known methods and show that the present solution is accurate.