Partial Differential Equations

Overview of PDEsSecond-Order PDEs
First-Order PDEs

This notebook is about finding analytical solutions of partial differential equations (PDEs). If you are interested in numeric solutions of PDEs, then the numeric PDEModels Overview is a good starting point.

Overview of PDEs

A partial differential equation (PDE) is a relationship between an unknown function and its derivatives with respect to the variables .

Here is an example of a PDE:

PDEs occur naturally in applications; they model the rate of change of a physical quantity with respect to both space variables and time variables. At this stage of development, DSolve typically only works with PDEs having two independent variables.

The order of a PDE is the order of the highest derivative that occurs in it. The previous equation is a first-order PDE.

A function is a solution to a given PDE if and its derivatives satisfy the equation.

Here is one solution to the previous equation:
This verifies the solution:

Here are some well-known examples of PDEs (clicking a link in the table will bring up the relevant examples). DSolve gives symbolic solutions to equations of all these types, with certain restrictions, particularly for second-order PDEs.

name of equation
general form
classification
transport equation with constantlinear first-order PDE
Burgers' equationquasilinear first-order PDE
eikonal equationnonlinear first-order PDE
Laplace's equationelliptic linear second-order PDE
wave equation where is the speed of lighthyperbolic linear second-order PDE
heat equation where is the thermal diffusivityparabolic linear second-order PDE

Recall that the general solutions to PDEs involve arbitrary functions rather than arbitrary constants. The reason for this can be seen from the following example.

The partial derivative with respect to y does not appear in this example, so an arbitrary function C[1][y] can be added to the solution, since the partial derivative of C[1][y] with respect to x is 0:

If there are several arbitrary functions in the solution, they are labeled as C[1], C[2], and so on.

First-Order PDEs

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 :
This verifies that the solution is correct:
Particular solutions of the homogeneous PDE are obtained by specifying the function C[1]:
Here is a plot of the surface for this particular solution:

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

In this transport equation, for convenience:

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:
Here is a linear homogeneous PDE with variable coefficients:
This verifies the solution:
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:

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

This PDE is quasi-linear because of the term on the right-hand side:
This verifies the solution:
Burgers’ equation is an important example of a quasi-linear PDE.
It can be written using the notation introduced earlier:

The term makes this equation quasi-linear.

This solves the equation:
This verifies the solution to Burgers' equation:

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.

Nonlinear PDEs

The general first-order nonlinear PDE for an unknown function is given by

Here is a function of , , and .

The term "nonlinear" refers to the fact that is a nonlinear function of and . For instance, the eikonal equation involves a quadratic expression in and .

The general solution to a first-order linear or quasi-linear PDE involves an arbitrary function. If the PDE is nonlinear, a very useful solution is given by the complete integral. This is a function of u(x,y,C[1],C[2]), where C[1] and C[2] are independent parameters and u satisfies the PDE for all values of (C[1],C[2]) in an open subset of the plane. The complete integral can be used to find a general solution for the PDE as well as to solve initial value problems for it.

Here is a simple nonlinear PDE:
The complete integral depends on the parameters C[1] and C[2]. Since DSolve returns a general solution for linear and quasi-linear PDEs, a warning message appears before a complete integral is returned:
This verifies the solution:

If the values of C[1] and C[2] are fixed, the previous solution represents a plane in three dimensions. Thus, the complete integral for this PDE is a two-parameter family of planes, each of which is a solution surface for the equation.

Next, the envelope of a one-parameter family of surfaces is a surface that touches each member of the family. If the complete integral is restricted to a one-parameter family of planes, for example by setting C[2]=5C[1], the envelope of this family is also a solution to the PDE called a general integral.

This finds the envelope of the one-parameter family given by setting C[2]=5C[1] in the complete integral for the preceding PDE p*q==1:

Like nonlinear ODEs, some nonlinear PDEs also have a singular solution (or singular integral) that is obtained by constructing the envelope of the entire two-parameter family of surfaces represented by the complete integral.

Here is an example of such a construction, (equation 6.4.13, page 429 of [K00]):

Thus, the singular integral for this PDE is a plane parallel to the - plane.

To summarize, the complete integral for a nonlinear PDE includes a rich variety of solutions.

These remarkable properties account for the usefulness of the complete integral in geometrical optics, dynamics, and other areas of application. Following are various examples of nonlinear PDEs that show different kinds of complete integrals.

Here is the complete integral for the eikonal equation:

This complete integral is a two-parameter family of planes. This type of solution arises whenever the PDE depends explicitly only on and , but not on , , or . For a fixed value of , it is a line in the plane at a distance of C[1] units from the origin that makes an angle of ArcCos[C[2]] with the axis. This is the familiar picture of wave-front propagation from geometrical optics.

This verifies the solution for the eikonal equation:
This is an example of a Clairaut equation ():
Once again, the complete integral is a family of planes:
This verifies the solution:
In the following equation, the variables can be separated; that is, the PDE can be written in the form . Hence, the equation can be integrated easily:
This verifies the solution:
In this example (equation 6.49, page 202 of [K74]), the independent variables and are not explicitly present:
This verifies the solution:

Often a coordinate transformation can be used to cast a given PDE into one of the previous types. The expression for the complete integral will then have the same form as for the standard types. Here are some examples of nonlinear PDEs for which DSolve applies a coordinate transformation to find the complete integral.

This PDE (equation 6.47, page 201 of [K74]) can be reduced to the form using the transformation and :
This PDE (equation 6.93, page 213 of [K74]) can be solved easily in a polar coordinate system, in which the variables are separable:
This equation (equation 6.36, page 196 of [K74]) can be transformed into a linear PDE using a Legendre transformation:
This verifies the solution:

It should be noted that there is no general practical algorithm for finding complete integrals, and that the answers are often available only in implicit form.

The solution to this example (problem 2, page 66 of [S57]) is in implicit form:
The solution can be verified as follows:

Second-Order PDEs

The general form of a linear second-order PDE is

Here , and , , , , , , and are functions of and only—they do not depend on . If , the equation is said to be homogeneous.

The first three terms containing the second derivatives are called the principal part of the PDE. They determine the nature of the general solution to the equation. In fact, the coefficients of the principal part can be used to classify the PDE as follows.

The PDE is said to be elliptic if . The Laplace equation has , , and and is therefore an elliptic PDE.

The PDE is said to be hyperbolic if . The wave equation has , , and and is therefore a hyperbolic PDE.

The PDE is said to be parabolic if . The heat equation has , , and and is therefore a parabolic PDE.

DSolve can find the general solution for a restricted type of homogeneous linear second-order PDEs; namely, equations of the form

Here , , and are constants. Thus, DSolve assumes that the equation has constant coefficients and a vanishing non-principal part.

Following are some examples of the three basic types (elliptic, hyperbolic, and parabolic) and an explanation of their significance.

Here is the general solution for Laplace's equation, an elliptic PDE:

This general solution contains two arbitrary functions, C[1] and C[2]. The arguments of these functions, and , indicate that the solution is constant along the imaginary straight line when C[2]0 and along when C[1]0 . These straight lines are called characteristic curves of the PDE. In general, elliptic PDEs have imaginary characteristic curves.

Here is another elliptic PDE:
Note the imaginary characteristic curves for the equation:
The solution is verified as follows:
This finds the general solution of the wave equation, a hyperbolic PDE. The constant in the wave equation represents the speed of light and is set to 1 here for convenience:

The characteristic lines for the wave equation are and where is an arbitrary constant. Hence the wave equation (or any hyperbolic PDE) has two families of real characteristic curves. If initial conditions are specified for the wave equation, the solution propagates along the characteristic lines. Also, any fixed pair of characteristic lines determine the null cone of an observer sitting at their intersection.

Here is another example of a hyperbolic PDE:
Notice that the equation has two families of real characteristics:
The solution can be verified as follows:
Finally, here is an example of a parabolic PDE:

The equation has only one family of real characteristics, the lines . In fact, any parabolic PDE has only a single family of real characteristics.

The solution can be verified as follows:

The heat equation is parabolic, but it is not considered here because it has a nonvanishing non-principal part, and the algorithm used by DSolve is not applicable in this case.