Partial Differential Equations

• A partial differential equation (PDE) is an equation with an unknown function of two or more variables and certain of its partial derivatives.
• Example: $\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 0$, $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = u$
• The order of a PDE relates to the highest derivative that appears in the equation.
• Example: $\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 0$ (first order), $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ (second order)
• A PDE is linear if it is linear in the unknown function and its derivatives.
• Example: $\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 0$ (linear), $u \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 0$ (nonlinear)
• A solution of a PDE is a function that satisfies the equation.
• Example: $u(x, y) = f(x - y)$ is a solution of $\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 0$ for any differentiable function $f$.

PDEs Examples

• The heat equation: $\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}$
• $u(x, t)$ represents the temperature at position $x$ and time $t$ in the heat equation.
• $k$ is a constant that depends on the material properties of the body relating to the heat equation.
• The wave equation: $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$
• $u(x, t)$ represents the displacement of the wave at position $x$ and time $t$ in the wave equation.
• $c$ is the speed of the wave in the wave equation.
• Laplace's equation: $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ or $\nabla^2 u = 0$
• $u(x, y)$ represents the potential function in Laplace's equation.
• $\nabla^2$ stands for the Laplacian operator in Laplace's equation.
• The heat equation describes the flow of heat in a solid body
• The wave equation describes the propagation of waves such as sound or water
• Laplace's equation arises in electrostatics, fluid mechanics, and heat transfer.

Initial and Boundary Conditions

• To obtain a unique solution to a PDE, it is usually necessary to specify additional conditions such as initial and boundary conditions.

• Initial conditions specify the value of the unknown function and its derivatives at a particular time, usually $t = 0$.

• Example: $u(x, 0) = f(x)$, $\frac{\partial u}{\partial t}(x, 0) = g(x)$

• Boundary conditions specify the value of the unknown function or its derivatives on the boundary of the domain.

• Example: $u(0, t) = h(t)$, $u(L, t) = k(t)$

Classification of Second-Order Linear PDEs

• The second-order linear PDE in two independent variables general form is: $A \frac{\partial^2 u}{\partial x^2} + B \frac{\partial^2 u}{\partial x \partial y} + C \frac{\partial^2 u}{\partial y^2} + D \frac{\partial u}{\partial x} + E \frac{\partial u}{\partial y} + F u = G$
• $A, B, C, D, E, F$, and $G$ are functions of $x$ and $y$
• Elliptic: $B^2 - 4AC < 0$
• Parabolic: $B^2 - 4AC = 0$
• Hyperbolic: $B^2 - 4AC > 0$
• Laplace's equation: $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ (elliptic)
• Heat equation: $\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}$ (parabolic)
• Wave equation: $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$ (hyperbolic)

An introduction to partial differential equations (PDEs). Covers the order and linearity of PDEs. Examples of PDEs and their solutions are provided.