Unit4 - Subjective Questions

Definition:
A Partial Differential Equation (PDE) is an equation involving an unknown function of two or more independent variables and its partial derivatives with respect to those variables.

Order:
The order of a PDE is the order of the highest partial derivative occurring in the equation.

Degree:
The degree of a PDE is the power of the highest order derivative occurring in the equation, provided the equation is free from radicals and fractions regarding the derivatives.

Example:
Consider the equation:

• Order: 2 (due to )
• Degree: 3 (the power of the highest order derivative)

Given equation:

Differentiating partially with respect to :

Differentiating partially with respect to :

Substituting values from and back into equation :

Result:

Given:

Let . Then .

Differentiate w.r.t :

Differentiate w.r.t :

Divide by to eliminate :

Result:

The classification is based on the discriminant :

1. Hyperbolic PDE:
   If .
   Example: The Wave Equation ().

2. Parabolic PDE:
   If .
   Example: The Heat Equation ().

3. Elliptic PDE:
   If .
   Example: The Laplace Equation ().

The Method of Separation of Variables is a technique used to solve linear PDEs with boundary conditions. The steps are:

1. Assumption: Assume the solution can be written as a product of functions, each depending on only one independent variable. For example: .
2. Substitution: Substitute this product and its derivatives into the original PDE.
3. Separation: Rearrange the terms so that terms involving are on one side and terms involving are on the other.
4. Constant Equating: Since the two sides depend on different independent variables, they must both equal a common constant (separation constant, usually denoted as , , or ).
5. ODE Solution: This breaks the PDE into separate Ordinary Differential Equations (ODEs).
6. General Solution: Solve the ODEs and combine the results to find the general solution of the PDE.
7. Boundary Conditions: Apply boundary and initial conditions to determine arbitrary constants.

Let .
Substitute into PDE:

Divide by :

Solve for X:

Solve for Y:

General Solution:

Apply Condition :

Comparing coefficients:
and .

Substitute and back:

Consider a tightly stretched elastic string of length between two fixed points.

Assumptions:

1. The string is perfectly flexible.
2. The tension is constant.
3. Deflection is small.

Derivation:
Consider a small element of the string between and . Let be mass per unit length.

Vertical force at :
Vertical force at :

Resultant Force = .
For small angles, .

Force

By Newton's 2nd Law ():

Dividing by and taking limits as :

Let .
Final Equation:

Let . The separated equation is:

There are three possibilities for :

1. (say ):
   The solution involves exponential terms (). These represent non-oscillatory solutions, which contradicts the physical nature of a vibrating string returning to equilibrium.

2. :
   The solution is linear (). This implies a straight line, which cannot represent a vibrating mode satisfying fixed boundary conditions ( at ends) appropriately for all time.

3. (say ):
   The solution involves trigonometric terms (). This represents periodic oscillatory motion, which physically models vibrations.

Conclusion: We choose to obtain a periodic, bounded solution suitable for wave motion.

For a string of length fixed at and , the boundary conditions are:

Using the Method of Separation of Variables with constant , the general solution for is:

Applying .
Applying . Since (trivial), .

The corresponding time part is:

The General Solution is the summation of all normal modes:

Wave Equation:
Boundary Conditions: .
Initial Conditions:

1. Velocity (Rest)
2. Displacement

General Solution for zero initial velocity:

Apply Initial Displacement:
At :

Using identity :

Comparing coefficients:

• For :
• For :
• All other

Final Solution:

Consider a homogeneous rod of conducting material of length , density , specific heat , and thermal conductivity .

Let be the temperature at position at time .
Consider an element between and with cross-sectional area .

1. Heat Inflow at per second:
   

2. Heat Outflow at per second:
   

3. Net Heat Gain:
   

4. Rate of Heat Absorption (by material):
   Mass = . Heat Capacity = Mass Specific Heat.
   

Equating Net Gain to Absorption:

Let (Thermal Diffusivity).
Final Equation:

Let . The separated equation is .

Case 1: (say )


Solution: (Temperature approaches as , usually rejected).

Case 2:


Solution: (Steady State Solution).

Case 3: (say )


Solution: (Transient Solution, decays with time).

Using Separation of Variables with constant :

1. Apply Boundary Conditions:

• .
• . Since , .
  

2. General Solution Series:

3. Apply Initial Condition:

This is a Fourier Half-Range Sine Series. The coefficient is given by:

Steady State Condition:
It is the state where the temperature at any point in the body does not change with time. Mathematically, .

Heat Equation reduces to:

Solution:
Integrating twice:

Applying Boundary Conditions:

1. At , .
2. At , .

Final Temperature Distribution:

Step 1: Initial State (Steady State)
Ends at $0$ and $100$. .
.
.
Initial temp distribution: .

Step 2: Transient State (Heat Equation)
New BCs: .
Initial condition: .

General solution for zero BCs:

Find using Fourier Sine series for :

Final Solution:

Equation:

Or using the Laplacian operator: .

Physical Significance:

1. Steady State Heat Flow: It represents the temperature distribution in a two-dimensional plate (lamina) in a steady state (temperature does not change with time).
2. Electrostatics: It represents the electric potential in a charge-free region.
3. Fluid Dynamics: It represents the velocity potential for an irrotational, incompressible fluid flow.

Let .
Substitute into PDE: .
Divide by : .

Possible Solutions:

1. (say ):
   
   
   

2. (say ):
   
   
   

3. :
   
   
   

The choice of solution depends on the specific boundary conditions given (e.g., if periodic in , use case 2 for ).

Problem Setup:
Laplace Eq: .
BCs:

3. (Implied by "infinite length")
4. (Triangle function given)

Selection of Solution:
Since at , needs trigonometric terms (). Since as , needs decaying exponential.

Apply BCs:
.
.

Series Solution:

Find :

Split integral $0$ to $5$ () and $5$ to $10$ (). Evaluate to find coefficients.

1. Wave Equation:

• Equation:
• Classification: Hyperbolic ().
• Nature: Solutions represent wave propagation. They are oscillatory in time (involve sine/cosine for time).
• Phenomena: Vibrating string, sound waves, electromagnetic waves.
• Reversibility: Time-reversible (invariant under ).

2. Heat Equation:

• Equation:
• Classification: Parabolic ().
• Nature: Solutions represent diffusion/smoothing. They decay exponentially in time.
• Phenomena: Heat conduction, diffusion of particles.
• Reversibility: Irreversible (cannot predict past state from future state easily due to smoothing).

Principle of Superposition:
If are solutions of a linear homogeneous Partial Differential Equation, then any linear combination

where are constants, is also a solution of that equation.

Application:
When solving Boundary Value Problems using Separation of Variables, we often obtain an infinite sequence of specific solutions (eigenfunctions) that satisfy the homogeneous boundary conditions. To satisfy the final non-homogeneous initial condition (like an initial shape ), we construct an infinite series:

The coefficients are then determined using Fourier series techniques to match the initial condition.