Unit - 3

Numerical solutions of Ordinary Differential Equations

3.1 First order differential equation by and Runge – Kutta method (Fourth order)

Consider the differential equation of first order

Let  be the first interval.

A second order Runge Kutta formula  

Where

Rewrite as     

A fourth order Runge Kutta formula:

Where 

Example1: Use Runge Kutta method to find y when x=1.2  in step of h=0.1 given that

Given equation

Here

Also

By  Runge Kutta formula for first  interval

Again

A fourth  order Runge Kutta formula:

To  find y at 

A fourth  order Runge Kutta formula:

Example 2: Apply Runge Kutta fourth order method to find an approximate value of  y for  x=0.2   in step of 0.1,  if

Given  equation 

Here 

Also

By  Runge Kutta formula for first  interval

A fourth  order Runge Kutta formula:

Again

A fourth  order Runge Kutta formula:

Example 3: Using Runge Kutta method of fourth order,  solve

Given equation 

Here 

Also

By  Runge Kutta formula for first  interval

         )

A fourth  order Runge Kutta formula:

Hence at x  = 0.2 then y  = 1.196

To find the  value of y at x=0.4. In this case

A fourth  order Runge Kutta formula:

Hence at  x  = 0.4 then  y=1.37527

3.2 Simultaneous first order differential equation by Picard’s method and Runge – Kutta method (Fourth order)

Picard’s method:

The general first order differential equation

….(1)

With the initial condition …(2)

In general, the solution of first order differential equation in one of the two forms:

a)     A series for y in terms of power of x, from which the value of y can be obtained by direct solution.

b)    A set of tabulated values of x and y.

The case (a) is solved by Taylor’s Series or Picard method whereas case (b) is solved by Euler’s, Runge Kutta Methods etc.

Picard’s method-

Let us suppose the first order equation-

It is required to find out that particular solution of equation (1) which assumes the value when ,

Now integrate (1) between limits, we get-

This is equivalent to equation (1),

For it contains the not-known y under the integral sign,

As a first approximation to the solution, put in f(x, y) and integrate (2),

For second approximation-

Similarly-

And so on.

Example: Find the value of y for x = 0.1 by using Picard’s method, given that-

Sol.

We have-

For first approximation, we put y = 1, then-

Second approximation-

We find it very hard to integrate.

Hence we use the first approximation and take x = 0.1 in (1)

Example: Obtain the picard’s second approximation for the given initial value problem-

Find y(1).

Sol.

The first approximation will be-

Replace y by , we get-

The second approximation is-

The third approximation-

It is very difficult to solve the integration-

This is the disadvantage  of the method.

Now we get from the second approximation-

At x = 1-

Simultaneous equation using Runge Kutta method of 2 orders:

  The second order differential equation

Let then the above equation reduces to first order simultaneous differential equation

Then

This can be solved as we discuss above by Runge Kutta Method. Here for and for .               

A fourth  order Runge Kutta formula:

Where      

Example1: Using Runge Kutta method of order four , solve   to find

Given second order differential equation is

Let   then above equation reduces to

     Or

   (say)

Or .

By Runge Kutta Method we have

A fourth  order Runge Kutta formula:

Example 2: Using Runge Kutta method, solve

   for correct to four decimal places with initial condition .

Given second order differential equation is

Let   then above equation reduces to

Or

(say)

Or .

By Runge Kutta Method we have

A fourth  order Runge Kutta formula:

And

.

Example 3: Solve the differential equations

       for

Using four order Runge Kutta method with initial conditions

Given differential equation are

Let

And

Also

By Runge Kutta Method we have

A fourth  order Runge Kutta formula:

And  

 .

Key takeaways-

Runge-kutta methods-

A second order Runge Kutta formula

Where

A fourth order Runge Kutta formula:

Where 

References:

1. E. Kreyszig, “Advanced Engineering Mathematics”, John Wiley & Sons, 2006.
2. P. G. Hoel, S. C. Port And C. J. Stone, “Introduction To Probability Theory”, Universal Book Stall, 2003.
3. S. Ross, “A First Course in Probability”, Pearson Education India, 2002.
4. W. Feller, “An Introduction To Probability Theory and Its Applications”, Vol. 1, Wiley, 1968.
5. N.P. Bali and M. Goyal, “A Text Book of Engineering Mathematics”, Laxmi Publications, 2010.
6. B.S. Grewal, “Higher Engineering Mathematics”, Khanna Publishers, 2000.

Unit - 3

Numerical solutions of Ordinary Differential Equations

3.1 First order differential equation by and Runge – Kutta method (Fourth order)

Consider the differential equation of first order

Let  be the first interval.

A second order Runge Kutta formula  

Where

Rewrite as     

A fourth order Runge Kutta formula:

Where 

Example1: Use Runge Kutta method to find y when x=1.2  in step of h=0.1 given that

Given equation

Here

Also

By  Runge Kutta formula for first  interval

Again

A fourth  order Runge Kutta formula:

To  find y at 

A fourth  order Runge Kutta formula:

Example 2: Apply Runge Kutta fourth order method to find an approximate value of  y for  x=0.2   in step of 0.1,  if

Given  equation 

Here 

Also

By  Runge Kutta formula for first  interval

A fourth  order Runge Kutta formula:

Again

A fourth  order Runge Kutta formula:

Example 3: Using Runge Kutta method of fourth order,  solve

Given equation 

Here 

Also

By  Runge Kutta formula for first  interval

         )

A fourth  order Runge Kutta formula:

Hence at x  = 0.2 then y  = 1.196

To find the  value of y at x=0.4. In this case

A fourth  order Runge Kutta formula:

Hence at  x  = 0.4 then  y=1.37527

3.2 Simultaneous first order differential equation by Picard’s method and Runge – Kutta method (Fourth order)

Picard’s method:

The general first order differential equation

….(1)

With the initial condition …(2)

In general, the solution of first order differential equation in one of the two forms:

a)     A series for y in terms of power of x, from which the value of y can be obtained by direct solution.

b)    A set of tabulated values of x and y.

The case (a) is solved by Taylor’s Series or Picard method whereas case (b) is solved by Euler’s, Runge Kutta Methods etc.

Picard’s method-

Let us suppose the first order equation-

It is required to find out that particular solution of equation (1) which assumes the value when ,

Now integrate (1) between limits, we get-

This is equivalent to equation (1),

For it contains the not-known y under the integral sign,

As a first approximation to the solution, put in f(x, y) and integrate (2),

For second approximation-

Similarly-

And so on.

Example: Find the value of y for x = 0.1 by using Picard’s method, given that-

Sol.

We have-

For first approximation, we put y = 1, then-

Second approximation-

We find it very hard to integrate.

Hence we use the first approximation and take x = 0.1 in (1)

Example: Obtain the picard’s second approximation for the given initial value problem-

Find y(1).

Sol.

The first approximation will be-

Replace y by , we get-

The second approximation is-

The third approximation-

It is very difficult to solve the integration-

This is the disadvantage  of the method.

Now we get from the second approximation-

At x = 1-

Simultaneous equation using Runge Kutta method of 2 orders:

  The second order differential equation

Let then the above equation reduces to first order simultaneous differential equation

Then

This can be solved as we discuss above by Runge Kutta Method. Here for and for .               

A fourth  order Runge Kutta formula:

Where      

Example1: Using Runge Kutta method of order four , solve   to find

Given second order differential equation is

Let   then above equation reduces to

     Or

   (say)

Or .

By Runge Kutta Method we have

A fourth  order Runge Kutta formula:

Example 2: Using Runge Kutta method, solve

   for correct to four decimal places with initial condition .

Given second order differential equation is

Let   then above equation reduces to

Or

(say)

Or .

By Runge Kutta Method we have

A fourth  order Runge Kutta formula:

And

.

Example 3: Solve the differential equations

       for

Using four order Runge Kutta method with initial conditions

Given differential equation are

Let

And

Also

By Runge Kutta Method we have

A fourth  order Runge Kutta formula:

And  

 .

Key takeaways-

Runge-kutta methods-

A second order Runge Kutta formula

Where

A fourth order Runge Kutta formula:

Where 

References:

1. E. Kreyszig, “Advanced Engineering Mathematics”, John Wiley & Sons, 2006.
2. P. G. Hoel, S. C. Port And C. J. Stone, “Introduction To Probability Theory”, Universal Book Stall, 2003.
3. S. Ross, “A First Course in Probability”, Pearson Education India, 2002.
4. W. Feller, “An Introduction To Probability Theory and Its Applications”, Vol. 1, Wiley, 1968.
5. N.P. Bali and M. Goyal, “A Text Book of Engineering Mathematics”, Laxmi Publications, 2010.
6. B.S. Grewal, “Higher Engineering Mathematics”, Khanna Publishers, 2000.

Unit - 3

Numerical solutions of Ordinary Differential Equations

Unit - 3

Numerical solutions of Ordinary Differential Equations

Unit - 3

Numerical solutions of Ordinary Differential Equations

Unit - 3

Numerical solutions of Ordinary Differential Equations

Unit - 3

Numerical solutions of Ordinary Differential Equations

Unit - 3

Numerical solutions of Ordinary Differential Equations

3.1 First order differential equation by and Runge – Kutta method (Fourth order)

Consider the differential equation of first order

Let  be the first interval.

A second order Runge Kutta formula  

Where

Rewrite as     

A fourth order Runge Kutta formula:

Where 

Example1: Use Runge Kutta method to find y when x=1.2  in step of h=0.1 given that

Given equation

Here

Also

By  Runge Kutta formula for first  interval

Again

A fourth  order Runge Kutta formula:

To  find y at 

A fourth  order Runge Kutta formula:

Example 2: Apply Runge Kutta fourth order method to find an approximate value of  y for  x=0.2   in step of 0.1,  if

Given  equation 

Here 

Also

By  Runge Kutta formula for first  interval

A fourth  order Runge Kutta formula:

Again

A fourth  order Runge Kutta formula:

Example 3: Using Runge Kutta method of fourth order,  solve

Given equation 

Here 

Also

By  Runge Kutta formula for first  interval

         )

A fourth  order Runge Kutta formula:

Hence at x  = 0.2 then y  = 1.196

To find the  value of y at x=0.4. In this case

A fourth  order Runge Kutta formula:

Hence at  x  = 0.4 then  y=1.37527

3.2 Simultaneous first order differential equation by Picard’s method and Runge – Kutta method (Fourth order)

Picard’s method:

The general first order differential equation

….(1)

With the initial condition …(2)

In general, the solution of first order differential equation in one of the two forms:

a)     A series for y in terms of power of x, from which the value of y can be obtained by direct solution.

b)    A set of tabulated values of x and y.

The case (a) is solved by Taylor’s Series or Picard method whereas case (b) is solved by Euler’s, Runge Kutta Methods etc.

Picard’s method-

Let us suppose the first order equation-

It is required to find out that particular solution of equation (1) which assumes the value when ,

Now integrate (1) between limits, we get-

This is equivalent to equation (1),

For it contains the not-known y under the integral sign,

As a first approximation to the solution, put in f(x, y) and integrate (2),

For second approximation-

Similarly-

And so on.

Example: Find the value of y for x = 0.1 by using Picard’s method, given that-

Sol.

We have-

For first approximation, we put y = 1, then-

Second approximation-

We find it very hard to integrate.

Hence we use the first approximation and take x = 0.1 in (1)

Example: Obtain the picard’s second approximation for the given initial value problem-

Find y(1).

Sol.

The first approximation will be-

Replace y by , we get-

The second approximation is-

The third approximation-

It is very difficult to solve the integration-

This is the disadvantage  of the method.

Now we get from the second approximation-

At x = 1-

Simultaneous equation using Runge Kutta method of 2 orders:

  The second order differential equation

Let then the above equation reduces to first order simultaneous differential equation

Then

This can be solved as we discuss above by Runge Kutta Method. Here for and for .               

A fourth  order Runge Kutta formula:

Where      

Example1: Using Runge Kutta method of order four , solve   to find

Given second order differential equation is

Let   then above equation reduces to

     Or

   (say)

Or .

By Runge Kutta Method we have

A fourth  order Runge Kutta formula:

Example 2: Using Runge Kutta method, solve

   for correct to four decimal places with initial condition .

Given second order differential equation is

Let   then above equation reduces to

Or

(say)

Or .

By Runge Kutta Method we have

A fourth  order Runge Kutta formula:

And

.

Example 3: Solve the differential equations

       for

Using four order Runge Kutta method with initial conditions

Given differential equation are

Let

And

Also

By Runge Kutta Method we have

A fourth  order Runge Kutta formula:

And  

 .

Key takeaways-

Runge-kutta methods-

A second order Runge Kutta formula

Where

A fourth order Runge Kutta formula:

Where 

References:

1. E. Kreyszig, “Advanced Engineering Mathematics”, John Wiley & Sons, 2006.
2. P. G. Hoel, S. C. Port And C. J. Stone, “Introduction To Probability Theory”, Universal Book Stall, 2003.
3. S. Ross, “A First Course in Probability”, Pearson Education India, 2002.
4. W. Feller, “An Introduction To Probability Theory and Its Applications”, Vol. 1, Wiley, 1968.
5. N.P. Bali and M. Goyal, “A Text Book of Engineering Mathematics”, Laxmi Publications, 2010.
6. B.S. Grewal, “Higher Engineering Mathematics”, Khanna Publishers, 2000.