Solving Partial Differential Equations

A PDE of order $n$ has at most $n$ arbitrary functions in its general solution. The general solution is written as linear combination of the $n$ functions.

Boundary Condition

An additional requirement imposed on the solution of a partial differential equation. Defines a condition fo the boundary of the domain in which the PDE is posed.

Suppose a PDE is defined on a region $D$ with boundary $\partial D$. A boundary condition prescribes a relation of the form:

$$B\big(x, u(x), \nabla u(x)\big) = 0, \qquad x \in \partial D$$

Initial Condition

A constraint that specifies the value of the solution at the initial time on the entire spatial domain for a time-dependent PDE.

Direct Integration

Used when variables can be integrated directly with respect to one variable.

Separation of Variables

This method assumes a product solution $u = X(x)Y(y)$. A PDE is converted into two ODEs.

Used in Laplace, heat, and wave equations.

Two Dimensional Heat Flow

Assumptions when formulating a mathematical model:

• Thermal conductivity of the metal is uniform.
• The plate is sufficiently thin (no heat flow in z-axis).
• The temperature distribution is in a steady state.
  If not, eigenvalues and eigenfunctions are used to model transient states. And they are not covered in this module.

The model in Cartesian coordinates:

$$\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0$$

The final solution in Cartesian coordinates:

$$T(x,y)=\sum_{n=1}^\infty A_n \sin\left(\frac{n\pi x}{a}\right) \sinh\left(\frac{n\pi y}{a}\right).$$

With $BC\theta(x,b)=100$, Fourier sine series gives constants.

The model in polar coordinates:

$$r^2 \frac{\partial^2 T}{\partial r^2} + r\frac{\partial T}{\partial r} + \frac{\partial^2 T}{\partial \theta^2} = 0$$

Here:

• $T$ - temperature
• $x,y$ - spatial coordinates
• $r, \theta$ - polar coordinates

Temperatures of the edges of the plate are the boundary conditions.

One Dimensional Heat Flow

Laws of heat flows:

• The amount of heat in a body is proportional to its mass and temperature.
• The rate of heat flow through a plane surface is proportional to the area and the rate of change of temperature with respect to the perpendicular distance.

$$\frac{\partial^2 T}{\partial x^2}=\frac1k\frac{\partial T}{\partial t}$$

Here $k \gt 0$.

Boundary conditions:

• $T(0,t)=0$
• $T(l,t)=0$
• $T(x,0)=f(x)$

General solution:

$$T(x,t)=\sum_{r=1}^\infty B_r \sin\left(\frac{r\pi x}{l}\right) \exp \left(\frac{r^2\pi^2 k t}{l^2} \right)$$ $$B_r=\frac{2}{l}\int_0^l f(x)\sin\left(\frac{r\pi x}{l}\right)\,\text{d}x$$

Heat

One-Dimensional Wave Equation

$$\frac{\partial^2 y}{\partial t^2}=c^2\frac{\partial^2 y}{\partial x^2}$$

Boundary conditions:

• $y(0,t)=0$
• $y(L,t)=0$
• $y(x,0)=f(x)$
• $y_t(x,0)=0$

$$y(x,t)=\sum_{n=1}^\infty A_n \sin\left(\frac{n\pi x}{L}\right) \cos\left(\frac{n\pi c t}{L}\right)$$ $$A_n=\frac{2}{L}\int_0^L f(x)\sin\left(\frac{n\pi x}{L}\right)\,dx$$