Partial Differential Equations

• Partial Differential Equations (PDEs) are differential equations involving multivariable functions and their partial derivatives.

Order of PDE

• The order is determined by the highest order derivative present in the equation.

Linear PDE

• Defined as: $\qquad \sum_{i=1}^{n} a_i(x) \frac{\partial u}{\partial x_i} + b(x)u = f(x)$
• $a_i$ and $b$ are functions of $x$ only.

Examples of PDEs

First Order Linear PDE

• $\qquad A(x, y) \frac{\partial u}{\partial x} + B(x, y) \frac{\partial u}{\partial y} + C(x, y)u = D(x, y)$

Second Order Linear PDE

• $\qquad A \frac{\partial^2 u}{\partial x^2} + B \frac{\partial^2 u}{\partial x \partial y} + C \frac{\partial^2 u}{\partial y^2} + D \frac{\partial u}{\partial x} + E \frac{\partial u}{\partial y} + Fu = G$
• A through G are functions of $(x,y)$.
  • Elliptic: $B^2 - 4AC < 0$
  • Parabolic: $B^2 - 4AC = 0$
  • Hyperbolic: $B^2 - 4AC > 0$

Key Equations in PDEs

Heat/Diffusion Equation

• $\qquad \frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}$
  • $u(x, t)$ represents temperature or concentration.
  • $t$ is time.
  • $x$ is position.
  • $k$ is thermal diffusivity.

Wave Equation

• $\qquad \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$
  • $u(x, t)$ is the wave's displacement.
  • $t$ is time.
  • $x$ is position.
  • $c$ is the wave speed.

Laplace's Equation

• $\qquad \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$
  • $u(x, y)$ denotes steady-state temperature or electric potential.

Methods to Solve PDEs

Separation of Variables

• Assume a solution as a product of single-variable functions.
• Substitute into the PDE to separate variables.
• Solve the resulting Ordinary Differential Equations (ODEs).
• Apply boundary conditions to find integration constants.

Fourier Transform

• Apply the Fourier transform to the PDE.
• Solve the resulting algebraic equation.
• Apply the inverse Fourier transform to obtain the solution.

Laplace Transform

• Apply the Laplace transform to the PDE.
• Solve the resulting algebraic equation.
• Apply the inverse Laplace transform to obtain the solution.

Numerical Methods for PDEs

• Finite Difference Method
• Finite Element Method
• Finite Volume Method

Boundary Conditions

Dirichlet Boundary Condition

• The solution's value is specified on the boundary: $\qquad u(x, t) = f(x)$

Neumann Boundary Condition

• The normal derivative of the solution is specified on the boundary: $\qquad \frac{\partial u}{\partial n} = g(x)$

Robin Boundary Condition

• A linear combination of the solution and its normal derivative is specified on the boundary: $\qquad \alpha u + \beta \frac{\partial u}{\partial n} = h(x)$

Applications of PDEs

• PDEs have various real-world applications.
• Including Heat Transfer
• Fluid Dynamics
• Electromagnetism
• Quantum Mechanics
• Finance

Additional Notes on PDEs

Superposition Principle

• If $u_1$ and $u_2$ are solutions of a linear PDE, then $c_1u_1 + c_2u_2$ is also a solution, where $c_1$ and $c_2$ are constants.

Well-Posedness

• A PDE is well-posed if it has a unique solution that depends continuously on the initial and boundary conditions.

Ill-Posedness

• A PDE is ill-posed if it does not have a unique solution or if the solution does not depend continuously on the initial and boundary conditions.