Partial differential equations (PDEs) are the most common method by which we model physical problems in engineering. Finite element methods are one of many ways of solving PDEs. This handout reviews the basics of PDEs and discusses some of the classes of PDEs in brief. The contents are based on Partial Differential Equations in Mechanics volumes 1 and 2 by A.P.S. Selvadurai and Nonlinear Finite Elements of Continua and Structures by T. Belytschko, W.K. Liu, and B. Moran.
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A PDE is a relationship between an unknown function of several variables and its partial derivatives.
Let u(x_{1},x_{2}, x_{3},t) be an unknown function. The independent variables are x_{1}, x_{2}, x_{3}, and t. We usually write
and say that u is the dependent variable.
Partial derivatives are denoted by expressions such as
Some examples of partial differential equations are
An example of a system of partial differential equations is
In expanded form this system of equations is
It is often more convenient to write PDEs in vector notation or index notation.
The order of a PDE is determined by the highest derivative in the equation. For example,
A linear PDE is one that of the first degree of its field variable and partial derivatives. For example,
The above equations can also be written in operator notation as
Let L be a linear operator. Then an linear partial differential equation can be written in the form
If f(x_{1},x_{2}, x_{3},t) = 0, the PDE is called homogeneous.
We usually come across threetypes of secondorder PDEs in mechanics. These are classified as elliptic, hyperbolic, and parabolic.
The equations of elasticity (without inertial terms) are elliptic PDEs. Hyperbolic PDEs describe wave propagation phenomena. The heat conduction equation is an example of a parabolic PDE.
Each type of PDE has certain characteristics that help determine if a particular finite element approach is appropriate to the problem being described by the PDE. Interestingly, just knowing the type of PDE can give us insight into how smooth the solution is, how fast information propagates, and the effect of initial and boundary conditions.
Suppose we have a secondorder PDE of the form
Then, the PDE is called elliptic if
An example is
The PDE is called hyperbolic if
An example is
The PDE is called parabolic if
An example is
Appears in almost every field of physics.
Appears in almost every field of physics.
