# Linear and Quasi-Linear PDEs

First-order PDEs are usually classified as linear, quasi-linear, or nonlinear. The first two types are discussed in this tutorial.

A first-order PDE for an unknown function is said to be *linear* if it can be expressed in the form

The PDE is said to be *quasilinear* if it can be expressed in the form

A PDE which is neither linear nor quasi-linear is said to be *nonlinear*.

For convenience, the symbols , , and are used throughout this tutorial to denote the unknown function and its partial derivatives.

Here is a linear homogeneous first-order PDE with constant coefficients.

The equation is linear because the left-hand side is a linear polynomial in , , and . Since there is no term free of , , or , the PDE is also homogeneous.

As mentioned earlier, the general solution contains an arbitrary function

C[1] of the argument

.

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This verifies that the solution is correct.

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Particular solutions of the homogeneous PDE are obtained by specifying the function

C[1].

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Here is a plot of the surface for this particular solution.

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The *transport equation* is a good example of a linear first-order homogeneous PDE with constant coefficients.

In this transport equation,

for convenience.

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Note that the solution to the transport equation is constant on any straight line of the form in the plane. These straight lines are called the *base characteristic curves*. The equation defines a plane in three dimensions. The intersections of these planes with the solution surface are called *characteristic curves*. Since the characteristic curves are solutions to a system of ODEs, the problem of solving the PDE is reduced to that of solving a system of ODEs for , , and , where is a parameter along the characteristic curves. These ODEs are called characteristic ODEs.

The solution to an inhomogeneous PDE has two components: the general solution to the homogeneous PDE and a particular solution to the inhomogeneous PDE.

This is a linear inhomogeneous PDE of the first order.

The first part of the solution,

, is the particular solution to the inhomogeneous PDE. The rest of the solution is the general solution to the homogeneous equation.

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Here is a linear homogeneous PDE with variable coefficients.

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This verifies the solution.

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Here is a linear inhomogeneous PDE with variable coefficients.

The solution is once again composed of the general solution to the homogeneous PDE and a particular solution,

Sin[x], to the inhomogeneous PDE.

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Now consider some examples of *first-order quasi-linear PDEs*.

This PDE is quasi-linear because of the term

on the right-hand side.

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This verifies the solution.

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It can be written using the notation introduced earlier.

The term makes this equation quasi-linear.

This solves the equation.

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This verifies the solution to Burgers' equation.

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A practical consequence of quasi-linearity is the appearance of shocks and steepening and breaking of solutions. Thus, although the procedures for finding general solutions to linear and quasi-linear PDEs are quite similar, there are sharp differences in the nature of the solutions.