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The simplest equation method is employed to construct some new exact closed-form solutions of the general Prandtl's boundary layer equation for two-dimensional flow with vanishing or uniform mainstream velocity. We obtain solutions for the case when the simplest equation is the Bernoulli equation or the Riccati equation. Prandtl's boundary layer equation arises in the study of various physical models of fluid dynamics. Thus finding the exact solutions of this equation is of great importance and interest.

Many scientific and engineering problems and phenomena are modeled by nonlinear differential equations. Therefore, the study of nonlinear differential equations has been an active area of research from the past few decades. Considerable attention has been devoted to the construction of exact solutions of nonlinear equations because of their important role in the study of nonlinear physical models. For nonlinear differential equations, we do not have the freedom to compute exact (closed-form) solutions and for analytical work we have to rely on some approximate analytical or numerical techniques which may be helpful for us to understand the complex physical phenomena involved. The exact solutions of the nonlinear differential equations are of great interest and physically more important. These exact solutions, if reported, facilitate the verification of complex numerical codes and are also helpful in a stability analysis for solving special nonlinear problems. In recent years, much attention has been devoted to the development of several powerful and useful methods for finding exact analytical solutions of nonlinear differential equations. These methods include the powerful Lie group method [

Prandtl [

The simplest equation method is a powerful mathematical tool for finding exact solutions of nonlinear ordinary differential equations. It has been developed by Kudryashov [

Prandtl's boundary layer equation for the stream function

Here

By the use of Lie group theoretic method of infinitesimal transformations [

Equation (

Here we present a brief description of the simplest equation method for solving nonlinear ordinary differential equations.

We first consider a general form of a nonlinear ordinary differential equation:

The basic idea of the simplest equation method consists in expanding the solutions of the previous ordinary differential equation in a finite series:

In this paper we use the Bernoulli and Riccati equations as the simplest equations. These equations are well-known nonlinear ODEs whose solutions can be expressed in terms of elementary functions.

For the Bernoulli equation

For the Riccati equation

One of the main steps in using the simplest equation method is to determine the positive number

By the substitution of (

By solving the algebraic equations obtained in Step

In this section, we employ the simplest equation method and obtain exact closed-form solutions of Prandtl's boundary layer equation (

The balancing procedure yields

By the substitution of (

Likewise, if we take

The balancing procedure yields

By the insertion of (

By taking

In this study, we have utilized the method of simplest equation for obtaining exact closed-form solutions of the well-known Prandtl's boundary layer equation for two-dimensional flow with uniform mainstream velocity. As the simplest equations, we have used the Bernoulli and Riccati equations. Prandtl's boundary layer equations arise in various physical models of fluid dynamics and thus the exact solutions obtained may be very useful and significant for the explanation of some practical physical models dealing with Prandtl's boundary layer theory. We have also verified that the solutions obtained here are indeed the solutions of Prandtl's boundary layer equation.

T. Aziz and A. Fatima would like to thank the School of Computational and Applied Mathematics and the Financial Aid and Scholarship Office, University of the Witwatersrand, for financial support and research grant.