A new Reliable Numerical Algorithm Based on the First Kind of Bessel Functions to Solve Prandtl–Blasius Laminar Viscous Flow over a Semi-Infinite Flat Plate

Abstract

In this paper, a new numerical algorithm is introduced to solve the Blasius equation, which is a third-order nonlinear ordinary differential equation arising in the problem of two-dimensional steady state laminar viscous flow over a semi-infinite flat plate. The proposed approach is based on the first kind of Bessel functions collocation method. The first kind of Bessel function is an infinite series, defined on R and is convergent for any x ∈ R. In this work, we solve the problem on semi-infinite domain without any domain truncation, variable transformation basis functions or transformation of the domain of the problem to a finite domain. This method reduces the solution of a nonlinear problem to the solution of a system of nonlinear algebraic equations. To illustrate the reliability of this method, we compare the numerical results of the present method with some well-known results in order to show the applicability and efficiency of our method.

A matrix formulation of the Tau method for the numerical solution of non-linear problems

“…Bessel functions and polynomials are used to solve the more number of problems in physics, engineering, mathematics, and etc., such as Blasius equation, Lane-Emden equations, integro-differential equations of the fractional order, unsteady gas equation, systems of linear Volterra integral equations, high-order linear complex differential equations in circular domains, systems of high-order linear Fredholm integro-differential equations, nolinear Thomas-Fermi on semi-infinite domain, etc. [44,45,46,47,48,49,50,51,52,53,54].…”

A new approach for solving nonlinear Thomas-Fermi equation based on fractional order of rational Bessel functions

“…Bessel functions and polynomials are used to solve the more number of problems in physics, engineering, mathematics, and etc., such as Blasius equation, Lane-Emden equations, integro-differential equations of the fractional order, unsteady gas equation, systems of linear Volterra integral equations, high-order linear complex differential equations in circular domains, systems of high-order linear Fredholm integro-differential equations, etc. [82,83,84,85,86,87,88,89,90,91,92,93,94].…”

Analytic Approximate Solution of the Extended Blasius Equation with Temperature-Dependent Viscosity

“…There is no closed-form exact solution for the equation, but many attempts have been made to find analytical approximate solutions and numerical solutions . Almost all methods and techniques were applied to obtain either numerical or approximate analytical solutions to the equation, Runge-Kutta method [3,11], series approximation and Padé approximant [2,17], numerical integration [3], perturbation [4,8], variational iteration method [5,14], homotopy method [6,23], Adomian decomposition method [9,13], recursive relations [10], finite difference method [12], shooting method [15,16], method of gradient [18], collocation method [19,21], generalized iterative differential quadrature method [20], combined Laplace transform and homotopy perturbation methods [22], numerical transformation methods [24], Haar wavelet approximation and a collocation method [25], optimal auxiliary functions method [26], direct algorithm method [27], quartic B-spline method [28], reproducing kernel method [29], optimal auxiliary functions method [30].…”