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Perhaps the simplest hyperbolic partial differential equation imaginable is the so-called scalar advection or one-way wave equation in (1+1)D, defined by
The (1+1)D Advection Equation
where is a real constant .
It is complemented by the initial condition
Here, the solution is assumed continuously differentiable (though it need not be), and is defined over the entire -axis, and for . The solution is simply
That is, the initial data travels to the left or right (depending on the sign of ) with speed . Despite its simplicity, it is often used as a model for numerical schemes .