Partial Differential Equations

• Partial Differential Equations (PDEs) involve unknown functions with multiple variables and their partial derivatives.
• A PDE's general form is represented as: $F(x, y, u, u_x, u_y, u_{xx}, u_{xy}, u_{yy},...) = 0$, where x and y are independent variables and u = u(x, y) is the dependent variable.
• Examples of PDEs include the heat equation, wave equation, and Laplace's equation.
• A solution to a PDE satisfies the equation within a specific region.
• Unique solutions to PDEs require initial conditions (system state at a specific time) and boundary conditions (system state at region boundaries).
• PDEs can be linear (unknown function and derivatives appear linearly) or nonlinear.
• The order of a PDE is determined by its highest-order derivative.

Heat Equation

• The heat equation is represented as: $\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}$
• Consider a one-dimensional rod of length L, where u(x, t) represents the temperature at position x and time t.

Assumptions for Deriving the Heat Equation

• The rod is homogeneous and isotropic.
• Heat flows only in the x-direction.
• No internal heat sources are present.

Derivation Steps

• Heat flux ($\phi$) is proportional to the temperature gradient: $\phi = -k \frac{\partial u}{\partial x}$, where k is thermal conductivity.
• Heat balance involves the rate of heat entering and leaving a small segment of the rod.
• Heat accumulation within the segment is given by: $\rho c \Delta x \frac{\partial u}{\partial t}$, where $\rho$ is density and c is specific heat.
• Heat Balance Equation: $\rho c \Delta x \frac{\partial u}{\partial t} = -k \frac{\partial u}{\partial x}(x + \Delta x, t) + k \frac{\partial u}{\partial x}(x, t)$
• The heat equation is: $\frac{\partial u}{\partial t} = K \frac{\partial^2 u}{\partial x^2}$, where $K = \frac{k}{\rho c}$ is the thermal diffusivity.

Boundary Conditions for the Heat Equation

• Dirichlet: Temperature specified at the boundary, such as $u(0, t) = T_1$, $u(L, t) = T_2$.
• Neumann: Heat flux specified at the boundary, such as $\frac{\partial u}{\partial x}(0, t) = \phi_1$, $\frac{\partial u}{\partial x}(L, t) = \phi_2$.
• Robin: Combination of temperature and heat flux, such as $h u(0, t) + k \frac{\partial u}{\partial x}(0, t) = 0$.
• The initial temperature distribution is specified as: $u(x, 0) = f(x)$.

Solving the Heat Equation Using Separation of Variables

• Assume a solution of the form: $u(x, t) = X(x)T(t)$.
• Substituting into the heat equation yields: $X(x)T'(t) = K X''(x)T(t)$.
• Divide by $X(x)T(t)$: $\frac{T'(t)}{K T(t)} = \frac{X''(x)}{X(x)} = -\lambda$, where $\lambda$ is a separation constant.