6.1 BASIC CONCEPTS | unit 6 applications of partial differential equations pde

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Unit – 6

Applications of Partial Differential Equations (PDE)

6.1 BASIC CONCEPTS:

Differential equation in which partial derivatives one involved, are called partial differential equation (PDE).

The order of the PDE is the highest order of partial derivatives presenting it.

We know that,

Partial differential equation of order one obtained by eliminating one arbitrary function .

Having two order obtained by eliminating two arbitrary function and .

6.2 MODELLING OF VIBRATING STRING:

While deriving a differential equation corresponding to a string problem we assume the following assumptions.

The string is perfectly plastic and doesn’t offer any resistance to bending.

The mass of the string per unit length is constant.

The tension is so large.

Consider the forces acting on a small portion of string. Let T1 and T2 be the tensions at end P and Q.

The horizontal components of T1 and T2 must be constant.

(1)

and vertical components of and are and

By using (1), we can divide this by

We know that, and

Let

This is called one-dimensional wave equation.

6.3 WAVE EQUATION

Solution of Wave Equation by method of separation of variable:

The vibration of an elastic string are governed by are dimensional wave equation

where is the deflection of the sting.

Since the string is fixed at the ends and .

We have two boundary conditions,

and for all t

We have two initial conditions:

1. If represents the vibrations of a string of length fixed at both ends. Find the solution with boundary conditions.

and initial conditions.

Solution:

Given

The most general solutions is given by

Applying conditions (1),

To apply conditions (iii) we first obtain

Here,

The most general solution will be

(1)

Applying condition (ii)

and

Solution (1) becomes

Combining all these solution we get

Applying condition (iv)

This is Fourier Half Range she series

For

This is Fourier Half Range she series

For

Substituting in equation (2) we get

2. A tightly stretched string with fixed end points and is initially in a position given by If it is released from rest from this positon. Find the displacement y at any distance x from one end at any time t.

Solution:

The differential equation satisfied by is

The initial and boundary conditions are given by

(i)

(ii)

(iii)

(iv)

The most general solution is given by

From condition (i) we get,

From condition (iii) we get,

The most general solution will become

(1)

From Condition (ii)

Equation (1) becomes

Combining all these solutions we get,

(2)

Applying condition (iv)

By using

Comparing we get,

Substitute in equation (2) we get,

6.4 ONE AND TWO DIMENSIONAL HEAT FLOW EQUATIONS

One dimensional Heat flow by method of separation of variables

We have to obtain solution of P.D.E.

The most general solution is

1. Solve: subject to condition

(i)

(ii)

(iii) is odd

(iv) for

Solution:

We have

Step 1: The most general Heat equation is

(1)

Step 2: Use first B.C

Step 3: Equation (1) becomes

(2)

Step 4: Use 2nd B.C

Step 5: Equation (2) becomes

Step 6: Take addition of all solutions and

(3)

Step 7: Use I.C

Step 8: To find we have,

Put in eqn (3) we get,

2. Solve : subject to :

Solution:

We have

Step 1: The most general heart equation is

….. (1)

Step 2: Use B.C

Step 3: Equation (1) becomes

(2)

Step 4: Use 2nd B.C

Step 5: Equation (2) becomes

Step 6: Take addition of all solutions and

(3)

Step 7: Use I.C.

Step 8: To find

Put in Equation (3) we get,

Two dimensional Heat flow by method of separation of variables:

Two dimensional Heat flow also called Laplace’s equation in two dimensions.

The equation is

The most general solution is given by

Use this Equation when

1. Solve the equation with conditions

(i) when for all .

(ii) when for all values of .

(iii) when for all values of .

(iv) when for

Solution:

Here in condition (i), when for all values of , for all z is given

Step 1:

The most general solution is

(1)

Step 2:

Now condition (i) that must remain finite as , this is possible only if

Step 3:

Apply condition (ii)

Step 4: The equation (1) becomes

Step 5:

By condition (iii)

Step 6:

Combining all these solution we get

Applying condition (iv) we have

Which is half range since series

The complete solution is

2. A rectangular plate with insulated surfaces is 10 cm wide and so long compared to its width that may be considered infinite in length without introducing an appreciable error. If the temperature along short edge is given

while the two long and as well as the other short edge are kept at 0C. Find steady state temperature

Solution:

We have to solve the P.D.E

Subject to the conditions

(i)

(ii)

(iii)

(iv)

Step 1: The most general solution is

(1)

Step 2:

By (i) condition,

Step 3:

By (ii) condition

The equation (1) becomes

Step 4: By (iii) condition

Step 5:

Applying condition (iv), we have

By comparing we get,

The answer is

6.5 Applications of P.D.E to general problem:

1. Solve by method separation of variables

where

Solution:

Given

Let the solution of this equation be S

Solving

Integrate both sides

By solving

Integrating both sides,

When ,

(given)

By comparing

Also

Extra Practice Problems:

1. An infinity long uniform metal plate is enclosed between lines and for . The temperature is zero along the edges and at infinity. If the edge is kept at a constant temperature . Find the temperature distribution .

Solution:

We have to solve P.D.E.

Subject to the boundary condition.

In condition (iii), when is given

(i)

(ii)

(iii)

(iv)

In condition (iii), when is given

The most general solution is given by

Now condition (iii) u(x , y) must remain as

By condition (i)

By condition (ii)

Apply condition (iv) we have

Which is represented by half range Fourier fine series

For in

The complete solution is

2. Solve the equation where satisfies the following conditions

(i)

(ii)

(iii)

(iv) is finite

Ans: we have

Solution:

We have

Step 1: The most general solution is

Step 2: Use 1st B.C

step 3: Equation (1) becomes

Step 4: Use 2nd B.C

Step 5: Equation (2) becomes

Step 6: Take addition of all solutions and

Step 7: Use I.C

Step 8: To find we have

3. A string is stretched and fastened to two points apart. Motion is started by displacing by the string in form from which it is released at time t=0 find the displacement from one end.