Learn & Review: Introduction to Partial Differential Equation

Introduction to Partial Differential Equations (PDEs)

This video series introduces Partial Differential Equations (PDEs), covering their modeling of engineering problems, famous examples like the wave and heat equations, solution techniques such as separation of variables, eigenvalues, eigenfunctions, Fourier analysis, Bessel functions, and numerical solutions.

1. Ordinary Differential Equations (ODEs) vs. Partial Differential Equations (PDEs)

• ODEs:

  • Involve a function of one independent variable (e.g., time, t).
  • Describe how a function evolves with respect to that single variable.
  • Involve ordinary derivatives (e.g., d^2u/dt^2).
  • Example: Describing the deflection of a mass-spring-damper system.
  • Require initial conditions for a unique solution (e.g., u(0) and u'(0) for a second-order ODE).

• PDEs:

  • Involve a function of two or more independent variables (e.g., time t and space x, or spatial variables x, y, z).
  • Describe phenomena that vary in both time and space, or across multiple spatial dimensions.
  • Involve partial derivatives (e.g., ∂u/∂x, ∂^2u/∂t^2).
  • Example: Describing the deflection of a continuous string, which depends on both position (x) and time (t).

2. Key Concepts and Definitions

• Partial Derivative: A derivative of a function with multiple variables with respect to one of those variables, treating the others as constants.
• Order of a PDE: The order of the highest partial derivative present in the equation.
• Notation:
  • Subscript Notation: u_x denotes ∂u/∂x, u_xy denotes ∂^2u/∂x∂y.
  • Dot and Prime Notation: Commonly used for time (t) and space (x) derivatives.
    • Dots (ü) denote derivatives with respect to time (u_tt).
    • Primes (u'') denote derivatives with respect to space (u_xx).

3. Classification of PDEs

Similar to ODEs, PDEs can be classified based on several criteria:

• Linearity:
  • A PDE is linear if the dependent variable u and its partial derivatives appear only to the first power, and their coefficients are functions of the independent variables only (not u).
  • Example: a(x,y) u_x + b(x,y) u_y + c(x,y) u = d(x,y)
  • Quasi-linear: Coefficients can depend on u and its derivatives, but in a specific way.
  • Non-linear: If a PDE is neither linear nor quasi-linear.
• Homogeneity:
  • A PDE is homogeneous if the term corresponding to the forcing function (or the right-hand side of the equation) is zero.
  • Otherwise, it is non-homogeneous.
• Order: As defined above, based on the highest derivative.

4. Examples of Famous PDEs

The following are common examples of linear, second-order PDEs:

• 1D Wave Equation: u_tt = c^2 u_xx (describes vibrations of a string)
• 1D Heat Equation: u_t = c^2 u_xx (describes heat diffusion)
• 2D Laplace Equation: u_xx + u_yy = 0 (describes steady-state phenomena)
• 2D Poisson Equation: u_xx + u_yy = f(x,y) (Laplace equation with a forcing function)
• 2D Wave Equation: u_tt = c^2 (u_xx + u_yy) (describes vibrations of a 2D surface)
• 3D Laplace Equation: u_xx + u_yy + u_zz = 0

5. Solving PDEs: Challenges and Techniques

• Multiple Solutions: A single PDE can have infinitely many solutions. For example, the 2D Laplace equation can be satisfied by x^2 - y^2, e^x cos(y), and ln(x^2 + y^2), among others.
• Reducing the Solution Space: To find a unique solution, initial conditions (ICs) and boundary conditions (BCs) are required, similar to ODEs.
• Superposition Principle: For linear homogeneous PDEs, if u1 and u2 are solutions, then any linear combination c1*u1 + c2*u2 is also a solution. This principle is crucial for constructing general solutions.
• Analytical Solutions: For simpler PDEs, analytical techniques can be used.
  • Example: Solving u_xy = -u_x analytically involves integrating with respect to x and then y, introducing arbitrary functions of the other variable at each step. The general solution found was u(x,y) = e^(-y) * f(x) + g(y), where f(x) and g(y) are arbitrary functions of x and y respectively.
• Limitations: Analytical methods are not always practical or tractable for complex PDEs. More formal methods, like separation of variables, are often required.

6. Next Steps

The next lecture will delve into the 1D wave equation and introduce the separation of variables technique for solving more complex linear PDEs.