Method of Characteristics

The Method of Characteristics is a technique for solving first-order partial differential equations (partial differential equations) by reducing them to a system of ordinary differential equations (ODEs) along characteristic curves.

The Method of Characteristics focuses on the case of two independent variables, .

Along a characteristic curve in the plane, the partial differential equation can be integrated directly to give the solution .

To use the Method of Characteristics, the region on which the partial differential equation is solved must be covered by characteristic curves.

For a partial differential equation of the form , the characteristic equations are::

The element of arc length squared along a characteristic curve is::.

Using the arc length parameterisation, we can write:: and .

Along a characteristic curve, the solution satisfies::, from which can be found by integration.


Homogeneous Case

A partial differential equation is homogeneous when the function is equal to zero.

For homogeneous equations, , which means the solution is constant along each characteristic curve.

In the homogeneous case, the characteristic equations reduce to::.

For homogeneous partial differential equations, the general solution has the form::, where is a parameter labeling each characteristic curve, and is determined by boundary conditions.

In the Method of Characteristics, boundary conditions are used to fix the form of the function itself, not just constant values.


Example: Solving

Given: with along .

Step 1: Find the characteristic curves. Since , write , or equivalently .

Step 2: The ODE is not separable. Notice that the right-hand side contains and only in combinations of , so substitute to make it separable.

With , we have .

Substitute into the characteristic ODE:

Simplify:

Step 3: Separate variables:

Recognize that , so multiply both sides by :

Step 4: Integrate both sides:

Exponentiate both sides:

Set (a parameter labeling each characteristic):

Substitute :

Multiply through by :

Step 5: The general solution for the homogeneous partial differential equation is:

Step 6: Apply the boundary condition along .

Substitute into :

On the boundary, .

So,

Set , then:

Step 7: The solution to the partial differential equation is: