A more complicated operator would be the heat operator. We get the heat operator from a slight rewrite of the heat equation without sources. The heat operator is,.
XVIII. Partial differential operators - classification and adjoints.
Now, what we really want to define here is not an operator but instead a linear operator. A linear operator is any operator that satisfies,. You might want to verify for yourself that the derivative and integral operators we gave above are also linear operators. In fact, in the process of showing that the heat operator is a linear operator we actually showed as well that the first order and second order partial derivative operators are also linear.
Partial differential equations and operators Fundamental solutions and semigroups: Part I
The next term we need to define is a linear equation. A linear equation is an equation in the form,. The first two from this list are of course the heat equation and the wave equation. The fourth equation was just made up to give a more complicated example.
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The main issue that allows these to be linear equations is the fact that the operator in each is linear. Now just to be complete here are a couple of examples of nonlinear partial differential equations.
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Lewy , An example of a smooth linear partial differential equation without solution. Annals of Math.
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