Solving PDE by Separation of Variables

Recap

• Last video: Started solving PDE by separation of variables.
• Showed constant equated on both sides must be negative.

Negative Constant Representation

• Write the constant as -λ² (since squaring any real number gives positive, and its negative is always negative).
• Results in two ordinary differential equations (ODEs): one in terms of T, the other in terms of X.

Solving ODE for T

• dT/dt = -λ² * T
• Solution: T(t) = A * exp(-λ² * t) (decaying exponential).*

Solving ODE for X

• d²X/dx² + λ² * X = 0
• Solution: X(x) = B * cos(λx) + C * sin(λx)
• Instead of hyperbolic sines and cosines (since λ² is negative).*

General Solution of PDE

• u(x,t) = (A exp(-λ² t))(B cos(λx) + C sin(λx))
• Simplify: Combine constants AB into one new constant, so solution simplifies to a form involving only two constants.

Applying Boundary Conditions

1. Boundary Condition at x = 0:
   • u(0, t) = 0
   • Result: C1 = 0
2. Boundary Condition at x = L:
   • u(L, t) = 0
   • Result: sin(λL) = 0;
   • Condition: λL = nπ which gives λ = nπ/L

Infinite Solutions Set

• Multiple values of n provide an infinite set of solutions.
• General solution: u(x,t) = Σ (from n=1 to ∞) [An exp(-λ² t) sin(λx)]
• Coefficients An are determined by initial conditions.

Sturm-Liouville Theorem

• Relation to Sturm-Liouville problems (second-order ODEs with specific boundary conditions).
• Solutions to ODEs related to specific eigenvalues.
  • Eigenfunctions satisfy orthogonality relation.

Applying Initial Condition

• u(x, 0) = φ(x);
• Leads to finding coefficients An:
  • Multiply by sin(λMx) and integrate.
  • Use of orthogonality: Simplifies to find An.
  • Result: An = (2/L) ∫ (from 0 to L) [sin(λMx) φ(x) dx]

Final General Solution

• General solution combining orthogonal functions and exponential decay: u(x,t) = Σ (from n=1 to ∞) [(2/L) ∫ (from 0 to L) [sin(λnx) φ(x) dx] exp(-λ²n t) sin(λnx)]

Next Topics

• Solving PDEs with non-homogeneous boundary conditions (requires different techniques).