Abstract
A numerical method for the solution of the Falkner–Skan equation, which is a nonlinear differential equation, is presented. The method has been derived by truncating the semi-infinite domain of the problem to a finite domain and then expanding the required approximate solution as the elements of the Chebyshev series. Using matrix representation of a function and their derivatives, the problem is reduced to a system of algebraic equations in a simple way. From the computational point of view, the results are in excellent agreement with those presented in published works.
Highlights
Ordinary differential equations are important tools in solving real-world problems
E Falkner–Skan equation arises in the study of laminar boundary layers exhibiting similarity. e similarity solutions of the two-dimensional incompressible laminar boundary layer equations are well-known as the Falkner–Skan equation. e F–S equation is a one-dimensional third-order nonlinear two-point boundaryvalue problem which has no closed-form solution. e problem is given by f′′′ + β0ff′′ + β1 − f′2 0, 0 ≤ η < ∞, (1)
Using matrix notation of the Chebyshev series, the nonlinear differential equation converts into a system of algebraic equations which can be solved. e importance of the present method arises from its simplicity and the fact that it does not require to guess the value of f′′(0)
Summary
Ordinary differential equations are important tools in solving real-world problems. Several natural phenomena are modelled by ODEs that have been used in several fields, such as physics, engineering, and biology [1,2,3,4,5,6,7].e Falkner–Skan equation arises in the study of laminar boundary layers exhibiting similarity. e similarity solutions of the two-dimensional incompressible laminar boundary layer equations are well-known as the Falkner–Skan equation. e F–S equation is a one-dimensional third-order nonlinear two-point boundaryvalue problem which has no closed-form solution. e problem is given by f′′′ + β0ff′′ + β1 − f′2 0, 0 ≤ η < ∞, (1) (2) lim f′(η) 1. η⟶∞e solution of (1) and (2) is characterized by f′′(0) α.e numerical treatment of this problem was addressed by many authors, namely, Lakestani [8], Parand et al [9], ElNady and Abd Rabbo [10, 11], Cebeci and Keller [12], Na [13], Asaithambi [14], Asaithambi [15], Elgazery [16], and Ganapol [17]. ese techniques have mainly used shooting algorithms or invariant imbedding. e Chebshev collocation matrix method [18] has been presented the numerical solution of nonlinear differential equations. e method in [18] transforms the nonlinear differential equation into the matrix equation, which corresponds to a system of nonlinear algebraic equations with unknown Chebyshev coefficients, via Chebyshev collocation points. e method in [19] transforms the nonlinear differential equation into the system of nonlinear algebraic equations with unknownJournal of Engineering shifted Chebyshev coefficients, via Chebyshev–Gauss collocation points. e solution of this system yields the Chebyshev coefficients of the solution function. e method is valid for both initial-value and boundary-value problems.e purpose of this paper is to develop an efficient method based on the Chebyshev series which is much more straightforward and simpler than the other existing algorithms. E Chebshev collocation matrix method [18] has been presented the numerical solution of nonlinear differential equations. Any continuous function f(ξ) in the interval 0 ≤ ξ ≤ 1 and its derivatives can be written in a matrix form of the Chebyshev series as follows: f(ξ) Tr[I]ai, r 0, 1, 2, .
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