A numerical solution of a singular boundary value problem arising in boundary layer theory

Equation (10) can be changed to the following equivalent form

(13)

subject to the boundary conditions

(14)

(15)

In this paper, the numerical solution of Eq. (13) with boundary conditions (14, 15) is based on the the finite difference method. The interval is divided into N subintervals with step size , and define for . Let denotes the values of for . Let , the finite difference formulation of Eq. (13) writes as

(16)

for . The boundary condition (14) corresponds to

(17)

And the discretization of boundary condition (15) reads as

(18)

The discretization formulation (16–18) is a nonlinear equation system, so Newton iteration method is recommended to solve
approximate solutions. We now proceed to describe the iterative process for the solution
of the nonlinear system (16–18). Let , and

(19)

where

(20)

and

(21)

for .

The solving Eqs. (16–18) is equivalent to solving the system described by

(22)

Newton’s iteration method is recommended to solve nonlinear system (22). Given and initial values , the k–th Newton’s iterates can be obtained by solving system (22). Newton’s method for the solution of Eq. (22) proceeds to yield subsequent iterates for w as

(23)

where satisfies the equation

(24)

The iterative process described by Eqs. (23, 24) may be repeated in succession until for some prescribed error tolerance .

The algorithm is then given as:

Step 1. Input the values , number of subintervals N and stopping condition

Step 2. Initialize ,, step size and ,

Step 3. Compute by Eqs. (23, 24);

Step 4. Repeat through step 3 until is satisfied.