Partial Differential Equations (PDEs)

Introduction to Partial Differential Equations (PDEs)

• A PDE involves an unknown function with two or more variables and its partial derivatives.
• The general form of a PDE is represented as F(x, y, u, ux, uy, uxx, uyy, uxy,...) = 0, where x and y are independent variables, u = u(x, y) is the dependent variable, and ux, uy, uxx, uyy, uxy are partial derivatives.
• The order of a PDE is determined by the highest order derivative in the equation.
  • First-order PDE example: ∂u/∂x + ∂u/∂y = u
  • Second-order PDE example: ∂²u/∂x² + ∂²u/∂y² = 0
• PDEs can be classified as linear if the dependent variable and its derivatives appear linearly, otherwise, they are nonlinear.
  • Linear PDE: A(x, y)ux + B(x, y)uy + C(x, y)u = f(x, y)
  • Nonlinear PDE: uxuy + u = 0

Examples of PDEs

• Wave Equation: ∂²u/∂t² = c² ∂²u/∂x²
• Heat Equation: ∂u/∂t = α ∂²u/∂x²
• Laplace Equation: ∂²u/∂x² + ∂²u/∂y² = 0
• Poisson Equation: ∂²u/∂x² + ∂²u/∂y² = f(x, y)

Classification of Second-Order Linear PDEs

• The general form of a second-order linear PDE in two independent variables is A(x, y)uxx + B(x, y)uxy + C(x, y)uyy + D(x, y)ux + E(x, y)uy + F(x, y)u = G(x, y).
• The discriminant Δ = B² - 4AC is used to classify the PDE.

PDE Classification based on the Discriminant

• Hyperbolic: Δ > 0
• Parabolic: Δ = 0
• Elliptic: Δ < 0

PDE Type Examples

• Wave Equation: utt = c² uxx is Hyperbolic because Δ = 4c² > 0.
• Heat Equation: ut = α uxx is Parabolic because Δ = 0.
• Laplace Equation: uxx + uyy = 0 is Elliptic because Δ = -4 < 0.

Boundary Conditions Types

• Dirichlet Boundary Condition: Specifies the value of the solution on the boundary as u(x, y) = f(x, y).
• Neumann Boundary Condition: Specifies the normal derivative of the solution on the boundary as ∂u/∂n = g(x, y), where n is the normal direction.
• Robin Boundary Condition: A combination of Dirichlet and Neumann conditions, given by αu + β(∂u/∂n) = h(x, y) on the boundary.

Boundary Conditions Example

• For the heat equation ut = α uxx on the interval [0, L]:
  • u(0, t) = T1 represents a Dirichlet boundary condition.
  • (∂u/∂x)(L, t) = 0 represents a Neumann boundary condition, indicating an insulated boundary.

Solution Techniques for PDEs

• Separation of Variables: Assumes the solution can be written as a product of functions, each depending on one independent variable, such as u(x, t) = X(x)T(t).
  • This method involves substituting the product into the PDE and separating the variables to obtain ordinary differential equations (ODEs).

Heat Equation Solution Example via Separation of Variables

• For the heat equation ut = α uxx with boundary conditions u(0, t) = 0 and u(L, t) = 0:
  • Assume a solution: u(x, t) = X(x)T(t).
  • Substitute into the heat equation: X(x)T'(t) = α X''(x)T(t).
  • Separate variables to get: (T'(t))/(αT(t)) = (X''(x))/(X(x)) = -λ, where λ is a separation constant.
  • Solve the ODEs: T'(t) + α λ T(t) = 0, giving T(t) = A e^(-αλt), and X''(x) + λ X(x) = 0.
  • Apply boundary conditions to find X(x).

Other Solution Methods

• Fourier Transforms: Useful for problems on infinite domains.
• Laplace Transforms: Useful for problems with time-dependent boundary conditions.
• Numerical Methods: Includes finite difference, finite element, and spectral methods, used for complex problems without analytical solutions.