9
Solving Partial Differential Equations
We now turn to the solving of differential equations in which the solution is a
function that depends on several independent variables. One such equation is called
a partial differential equation (PDE, plural: PDEs).
The subject of PDEs is enormous. At the same time, it is very important, since
so many phenomena in nature and technology find their mathematical formulation
through such equations. Knowing how to solve at least some PDEs is therefore of
great importance to engineers. In an introductory book like this, nowhere near full
justice to the subject can be made. However, we still find it valuable to give the
reader a glimpse of the topic by presenting a few basic and general methods that we
will apply to a very common type of PDE.
We shall focus on one of the most widely encountered partial differential
equations: the diffusion equation, which in one dimension looks like
∂u
∂t
= β
∂ 2 u
∂x 2 + g .
The multi-dimensional counterpart is often written as
∂u
∂t
= β∇
2 u + g .
We shall restrict the attention here to the one-dimensional case.
The unknown in the diffusion equation is a function u(x, t) of space and time.
The physical significance of u depends on what type of process that is described
by the diffusion equation. For example, u is the concentration of a substance if
the diffusion equation models transport of this substance by diffusion. Diffusion
processes are of particular relevance at the microscopic level in biology, e.g.,
diffusive transport of certain ion types in a cell caused by molecular collisions.
There is also diffusion of atoms in a solid, for instance, and diffusion of ink in a
glass of water.
One very popular application of the diffusion equation is for heat transport
in solid bodies. Then u is the temperature, and the equation predicts how the
temperature evolves in space and time within the solid body. For such applications,
© The Author(s) 2020
S. Linge, H. P. Langtangen, Programming for Computations - Python,
Texts in Computational Science and Engineering 15,
https://doi.org/10.1007/978-3-030-16877-3_9
287
Solving Partial Differential Equations
We now turn to the solving of differential equations in which the solution is a
function that depends on several independent variables. One such equation is called
a partial differential equation (PDE, plural: PDEs).
The subject of PDEs is enormous. At the same time, it is very important, since
so many phenomena in nature and technology find their mathematical formulation
through such equations. Knowing how to solve at least some PDEs is therefore of
great importance to engineers. In an introductory book like this, nowhere near full
justice to the subject can be made. However, we still find it valuable to give the
reader a glimpse of the topic by presenting a few basic and general methods that we
will apply to a very common type of PDE.
We shall focus on one of the most widely encountered partial differential
equations: the diffusion equation, which in one dimension looks like
∂u
∂t
= β
∂ 2 u
∂x 2 + g .
The multi-dimensional counterpart is often written as
∂u
∂t
= β∇
2 u + g .
We shall restrict the attention here to the one-dimensional case.
The unknown in the diffusion equation is a function u(x, t) of space and time.
The physical significance of u depends on what type of process that is described
by the diffusion equation. For example, u is the concentration of a substance if
the diffusion equation models transport of this substance by diffusion. Diffusion
processes are of particular relevance at the microscopic level in biology, e.g.,
diffusive transport of certain ion types in a cell caused by molecular collisions.
There is also diffusion of atoms in a solid, for instance, and diffusion of ink in a
glass of water.
One very popular application of the diffusion equation is for heat transport
in solid bodies. Then u is the temperature, and the equation predicts how the
temperature evolves in space and time within the solid body. For such applications,
© The Author(s) 2020
S. Linge, H. P. Langtangen, Programming for Computations - Python,
Texts in Computational Science and Engineering 15,
https://doi.org/10.1007/978-3-030-16877-3_9
287