Module 5 B

Numerical method – 2

5.1 ORDINARY DIFFERENTIAL EQUATIONS:

5.2 TAYLOR’S SERIES

Solve.

For y(0.1) correct to four and places of decimal using Taylor series method.

Ans. The Taylor's series for y (x) about is given by

…

Here

And so on using the value in Taylor's series

To get y(0.1) correct the four place of decimal it is found that the term up to are to be taken and other neglected.

Thus y(0.1) = 0.9138

Model questions

Apply Taylor series method to find y(0.2) from given that y(0))=1

5.3 EULER AND MODIFIED EULER’S METHODS.

Euler’s method:-

Given compute y (0.02) by Euler’s method taking h=0.01.

Ans. We have

Here

Apply Euler’s method

Modified Euler's method

But which occurs in the right hand side of given equation cannot be calculated since is unknown so first we calculate  from Euler's first formula

Or

Predicator formula

Corrector formula

Using modified Euler's method solved the equation for x=1.2 correct to 3 decimal places.

Ans. Given

By predictor formula

By corrector formula

Again, apply character formula

Once again applying the character formula

Since takes up to three place of decimal we get y(1.2)=2.2332

5.4 RUNGE- KUTTA METHOD OF FOURTH ORDER FOR SOLVING FIRST AND SECOND ORDER EQUATIONS.

Let equation be

Then,

→  Using Runge kutta method of fourth order determine y (0.1) and y(0.2) correct to four decimal place given that where y(0)=2 and h=0.1.

Ans.

To find y(0.1) we have

For

5.5 MILNE’S AND ADAM’S PREDICATOR-CORRECTOR METHODS.

Predictor formula

Corrector formula

Example. Apply Milne’s method to find solution of differential equation  in interval

In step of h=0.1 it is given that

Ans.

Milne’s Predictor formula

Corrector Formula

Adam’s Bashtorth method

Predictor Formula

Corrector formula

Using Adam’s Bashforth method find y(1.4), given y(1.4) given ,

Ans. Given,

Adam’s Predictor formula

Corrector formula

=2.57494

5.6 PARTIAL DIFFERENTIAL EQUATIONS:

Partial Differential Equation

Forward difference approximately

Backward difference equation

Laplace Equation :- Elliptic type is called Laplace equation.

Poisson Equation :-  

In this case standard fire point formula is of the form.

Solve the Poisson equation

For the square mesh of the figure given below with u(x,y)=0 on the boundary and mesh length=1

Ans   Here h=1

The standard five-point formula for the given equation is

For

Equation 1 becomes

=8(-1)(1)

         ---2

For equation 1 becomes

For equation 1 becomes

Putting in 1 we get

Putting for in 2 we get

Heat Equation

Bender Schmidt method :-

And

Crank Nicholson Method

Wave Equation

Where  

TEXTBOOKS/REFERENCES:

1. P. KANDASAMY, K. THILAGAVATHY, K. GUNAVATHI, NUMERICAL METHODS, S. CHAND & COMPANY, 2ND EDITION, REPRINT 2012.
2. S.S. SASTRY, INTRODUCTORY METHODS OF NUMERICAL ANALYSIS, PHI, 4TH EDITION, 2005.
3. ERWIN KREYSZIG, ADVANCED ENGINEERING MATHEMATICS, 9TH EDITION, JOHN WILEY & SONS, 2006.
4. B.S. GREWAL, HIGHER ENGINEERING MATHEMATICS, KHANNA PUBLISHERS, 35TH EDITION, 2010.