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9 Solving Partial Differential Equations
the equation is known as the heat equation. We remark that the temperature in a fluid
is influenced not only by diffusion, but also by the flow of the liquid. If present,
the latter effect requires an extra term in the equation (known as an advection or
convection term).
The term g is known as the source term and represents generation, or loss, of heat
(by some mechanism) within the body. For diffusive transport, g models injection
or extraction of the substance.
We should also mention that the diffusion equation may appear after simplifying
more complicated PDEs. For example, flow of a viscous fluid between two flat and
parallel plates is described by a one-dimensional diffusion equation, where u then
is the fluid velocity.
A PDE is solved in some domain Ω in space and for a time interval [0, T ]. The
solution of the equation is not unique unless we also prescribe initial and boundary
conditions. The type and number of such conditions depend on the type of equation.
For the diffusion equation, we need one initial condition, u(x, 0), stating what u
is when the process starts. In addition, the diffusion equation needs one boundary
condition at each point of the boundary ∂Ω of Ω. This condition can either be that
u is known or that we know the normal derivative, ∇u · n = ∂u/∂n (n denotes an
outward unit normal to ∂Ω).
9.1 Example: Temperature Development in a Rod
Let us look at a specific application and how the diffusion equation with initial
and boundary conditions then appears. We consider the evolution of temperature
in a one-dimensional medium, more precisely a long rod, where the surface of
the rod is covered by an insulating material. The heat can then not escape from
the surface, which means that the temperature distribution will only depend on
a coordinate along the rod, x, and time t. At one end of the rod, x = L,
we also assume that the surface is insulated, but at the other end, x = 0, we
assume that we have some device for controlling the temperature of the medium.
Here, a function s(t) tells what the temperature is in time. We therefore have
a boundary condition u(0, t) = s(t). At the other insulated end, x = L, heat
cannot escape, which is expressed by the boundary condition ∂u(L, t)/∂x = 0.
The surface along the rod is also insulated and hence subject to the same boundary
condition (here generalized to ∂u/∂n = 0 at the curved surface). However, since
we have reduced the problem to one dimension, we do not need this physical
boundary condition in our mathematical model. In one dimension, we can set
Ω = [0, L].
To summarize, the PDE with initial and boundary conditions reads
∂u(x, t)
∂t
= β
∂ 2 u(x, t)
∂x 2 + g(x, t),
x ∈ (0, L) ,t ∈ (0, T ],
(9.1)
u(0, t) = s(t),
t ∈ (0, T ],
(9.2)
9 Solving Partial Differential Equations
the equation is known as the heat equation. We remark that the temperature in a fluid
is influenced not only by diffusion, but also by the flow of the liquid. If present,
the latter effect requires an extra term in the equation (known as an advection or
convection term).
The term g is known as the source term and represents generation, or loss, of heat
(by some mechanism) within the body. For diffusive transport, g models injection
or extraction of the substance.
We should also mention that the diffusion equation may appear after simplifying
more complicated PDEs. For example, flow of a viscous fluid between two flat and
parallel plates is described by a one-dimensional diffusion equation, where u then
is the fluid velocity.
A PDE is solved in some domain Ω in space and for a time interval [0, T ]. The
solution of the equation is not unique unless we also prescribe initial and boundary
conditions. The type and number of such conditions depend on the type of equation.
For the diffusion equation, we need one initial condition, u(x, 0), stating what u
is when the process starts. In addition, the diffusion equation needs one boundary
condition at each point of the boundary ∂Ω of Ω. This condition can either be that
u is known or that we know the normal derivative, ∇u · n = ∂u/∂n (n denotes an
outward unit normal to ∂Ω).
9.1 Example: Temperature Development in a Rod
Let us look at a specific application and how the diffusion equation with initial
and boundary conditions then appears. We consider the evolution of temperature
in a one-dimensional medium, more precisely a long rod, where the surface of
the rod is covered by an insulating material. The heat can then not escape from
the surface, which means that the temperature distribution will only depend on
a coordinate along the rod, x, and time t. At one end of the rod, x = L,
we also assume that the surface is insulated, but at the other end, x = 0, we
assume that we have some device for controlling the temperature of the medium.
Here, a function s(t) tells what the temperature is in time. We therefore have
a boundary condition u(0, t) = s(t). At the other insulated end, x = L, heat
cannot escape, which is expressed by the boundary condition ∂u(L, t)/∂x = 0.
The surface along the rod is also insulated and hence subject to the same boundary
condition (here generalized to ∂u/∂n = 0 at the curved surface). However, since
we have reduced the problem to one dimension, we do not need this physical
boundary condition in our mathematical model. In one dimension, we can set
Ω = [0, L].
To summarize, the PDE with initial and boundary conditions reads
∂u(x, t)
∂t
= β
∂ 2 u(x, t)
∂x 2 + g(x, t),
x ∈ (0, L) ,t ∈ (0, T ],
(9.1)
u(0, t) = s(t),
t ∈ (0, T ],
(9.2)