ES 111 Mathematical Methods in the Earth Sciences
Lecture Outline 18 - Weds 26th Nov ’08
Higher-Order Partial Differential Equations
The last thing we are going to mention is higher-order PDE’s. These are almost invariably
found in real physical problems, and we can only touch on them here. We will simply look at one
technique which can simplify their solution; taking solutions further involves techniques such as
Fourier analysis not covered in this course.
The Laplacian
Recall how we describe heat conduction. In one dimension, heat conduction is governed by the
equation
∂T
∂t =κ∂2T
∂x2
where κis the thermal diffusivity and T(x, t) is a temperature field varying in xand time only.
In 2 dimensions, the analogous equation may be written
∂T
∂t =κ ∂2T
∂x2+∂2T
∂z2!=κ∇2T
where now Tis a function of x, z and tand ∇2is called the Laplacian operator, also referred to
as ”del-squared”. The Laplacian can be written for 2,3 . . . dimensions; it has a slightly different
explicit form in curvilinear coordinate systems.
The reason we use the Laplacian is that it is a useful shorthand, and that it works irrespective
of which coordinate system we adopt. So ∂T
∂t =κ∇2Tis a true statement irrespective of whether
we are using 2D, 3D, spherical, cylindrical etc. etc. Note, however, that the explicit form of the
Laplacian does change depending on what coordinate system you’re in.
Another place where the Laplacian arises is in (not surprisingly) Laplace’s equation:
∂2f
∂x2+∂2f
∂y2+∂2f
∂z2=∇2f= 0
Here fis a scalar quantity f(x, y, z) and this equation is written in Cartesian coordinates. This
equation is fundamental to the description of gravity and charge, among other phenomena.
Separation of Variables
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