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Chapter 5: Q5.3-27E (page 261)

Generalized Blasius Equation. H. Blasius, in his study of the laminar flow of a fluid, encountered an equation of the form y'''+yy''=(y')2-1. Use the Runge鈥揔utta algorithm for systems with h = 0.1 to approximate the solution that satisfies the initial conditions y(0)=0,y'(0)=0,y''(0)=1.32824. Sketch this solution on the interval [0, 2].

Short Answer

Expert verified

The result can get by the Runge-Kutta method, and the result is y(2)=1.6001.

Step by step solution

01

Transform the equation

Here the equation is y'''+yy''=(y')2-1.

The system can be written as:

x1=yx2=y'=x'1x3=y''=x'2

The transform equation is:

x'1=x2x'2=x3x'3=-x1x3+x22-1

The initial conditions are:

x1(0)=y(0)=0x2(0)=y'(0)=0x3(0)=y''(0)=1.32824

02

Apply the Runge-Kutta method

For h=0.1

t

Y

T

Y

0

0

1.1

0.599

0.1

0.00647

1.2

0.69515

0.2

0.0252

1.3

0.79515

0.3

0.0553

1.4

0.89926

0.4

0.0957

1.5

1.0072

0.5

0.1456

1.6

1.1189

0.6

0.20407

1.7

1.234

0.7

0.27032

1.8

1.3526

0.8

0.34363

1.9

1.4747

0.9

0.4233

2

1.6001

1

0.50882

03

Graph

Therefore, the value of y(2)=1.6001.

Thus, this is the required result.

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