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An Introduction to Partial Differential Equations

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Definition Of Partial Differential Equations
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Definition Of Partial Differential Equations

This section presents some basic definitions regarding Partial Differential Equations.

Partial Derivatives

Note
The partial derivative of a function f with respect to the variable x is usually denoted by

$\frac{\partial f}{\partial x}\;,\;f_x$

A partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary).

For example if $\inline f(x,y)=ye^x$ , then

$\frac{\partial f}{\partial x}=ye^x$

$\frac{\partial f}{\partial y}=e^x$

Partial Differential Equations

Note
If a function $\inline u=u(x,y)$ is solution for

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

then $\inline u$ is a harmonic function.

Partial differential equations (PDE) are a type of differential equation, i.e., a relation involving an unknown function (or functions) of several independent variables and their partial derivatives with respect to those variables. Partial differential equations are used to formulate, and thus aid the solution of, problems involving functions of several variables; such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, and elasticity.

For example

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

where $\inline u=u(x,y)$ is a two variable function, is a partial differential equation .

Example:

Example - Simple example

Problem

$z = (x - A)^2 + (y - B)^2$

Workings

Therefore

$\frac{\partial z}{\partial x} = 2(x - A)\;\;\;\;and\;\;\;\;\frac{\partial z}{\partial y} = 2(y - B)$

Note : This equation is of the first order although the equation from which it is derived has two arbitrary constants.

The Solution Of Partial Differential Equations

It will be clear from these examples that the methods used for the solution of ordinary differential equations will not apply to Partial Differential Equations without considerable modification. A general discussion of partial differential equations is both difficult and lengthy. The objective in the following examples is to show some of the substitutions which may be used in the solution of the types of equation which occur in Scientific and engineering applications.

Example:

Example - Exponential Solution

Problem

The following linear equation gives the conduction of heat in one direction.

$\frac{\partial ^2z}{\partial x^2} = \frac{1}{a^2}\;\frac{\partial z}{\partial t}$

Workings

In the treatment of ordinary linear equations it was found that the use of exponentials was useful. This suggests $\displaystyle z = e^{mx+nt}$ as a trial solution. Substituting in the differential equation:

$m^2\;e^{mx+nt} = \frac{1}{a^2}\;ne^{mx+nt}$

Solution

Which is true if $\displaystyle n = m^2\,a^2$

Thus $\displaystyle e^{mx+m^2a^2t}$ is a solution. Changing the sign of m gives $\displaystyle e^{-mx+m^2a^2t}$ which is also a solution.