Understanding the Method of Characteristics for First-Order PDEs

The method of characteristics is a powerful analytical tool used to solve first-order partial differential equations (PDEs). This technique involves transforming the PDE into a set of ordinary differential equations (ODEs), which can often be solved more easily. In this article, we will explore the fundamental principles behind the method of characteristics and illustrate how it can be applied to solve a specific first-order linear PDE.

Introduction to the Method of Characteristics

The method of characteristics is based on the idea of integrating the PDE along a specific path, known as a characteristic curve. These curves are selected such that the PDE reduces to a set of ODEs, which are easier to solve. Essentially, the method transforms the original PDE into a system of ODEs by parameterizing the solution using a single parameter, typically denoted as (t).

Parameterized Equation and Multivariate Chain Rule

Consider a first-order linear PDE of the form:

[ f(x, y, t) frac{partial u}{partial x} g(x, y, t) frac{partial u}{partial y} h(x, y, t) frac{partial u}{partial t} w(x, y, z) ]

We aim to find a parameterized equation of the form:

[ u(x(x_0, y_0, t_0), y(x_0, y_0, t_0), t) U(x_0, y_0, t_0, t) ]

Here, ( (x_0, y_0, t_0) ) are initial conditions, and ( U ) is a function of these initial conditions and the parameter ( t ). The ODEs for the characteristic curve can be obtained by applying the multivariate chain rule:

[ frac{dx}{dt} frac{partial x}{partial x_0} frac{d x_0}{dt} frac{partial x}{partial y_0} frac{d y_0}{dt} frac{partial x}{partial t_0} frac{d t_0}{dt} ]

Using this, we can write:

[ frac{dx}{dt} f(x(x_0, y_0, t_0), y(x_0, y_0, t_0), t) ]

Similarly, we have:

[ frac{dy}{dt} g(x(x_0, y_0, t_0), y(x_0, y_0, t_0), t) ]

[ frac{dt}{dt} h(x(x_0, y_0, t_0), y(x_0, y_0, t_0), t) ]

And the ODE for the solution ( U ) is:

[ frac{dU}{dt} w(x(x_0, y_0, t_0), y(x_0, y_0, t_0), t) ]

Solving a Specific First-Order Linear PDE

Let's consider the following first-order linear PDE:

[ y frac{partial u}{partial x} - 2xy frac{partial u}{partial y} 2xu ]

And the initial condition:

[ u(0, y) y^3 ]

First, we write the system of ODEs by applying the method of characteristics:

[ frac{dx}{dt} 1 ]

[ frac{dy}{dt} -2xy ]

[ frac{du}{dt} 2xu ]

Solving the first ODE: [ x(t) x_0 t ]

Using the chain rule to eliminate ( x ) from the second ODE, we get:

[ frac{d(y(x_0 t))}{dt} -2(y(x_0 t))(x_0 t) ]

Integrating both sides, we have:

[ y(x_0 t) y_0 e^{- t^2 / 2 - x_0 t} ]

Substituting the expression for ( y ) back into the third ODE, we obtain:

[ frac{du}{dt} 2(x_0 t)u ]

Integrating both sides, we find:

[ u(x_0 t, y_0 e^{- t^2 / 2 - x_0 t}) y_0^3 e^{- t^2 - 2 x_0 t} ]

Now, substituting ( t 0 ) to implement the initial condition ( u(0, y_0) y_0^3 ), we get:

[ u(x_0, y_0 e^{- x_0^2 / 2}) y_0^3 ]

Finally, substituting back ( y_0 y e^{x_0^2 / 2} ), the solution to the PDE is:

[ u(x, y) y e^{- x^2 / 4} ]

N-Independent Variables and Quasi-Linear PDEs

The method of characteristics can also be extended to PDEs with ( n ) independent variables. For a first-order quasi-linear PDE of the form:

[ sum_{i1}^n a_i u_{x_1 dots x_n} b u_{x_1 dots x_n} ]

the differential of a function with ( n ) independent variables is:

[ sum_{i1}^n dx_i u_{x_1 dots x_n} du ]

Equating these, we get a system of ( n-1 ) equations and ( du ). Solving these ODEs, we obtain a system of functions ( phi_1, dots, phi_n ) that define the relationship between the PDE and the differential. Any combination of these functions should generate a solution to the equation that is:

[ f(phi_1, dots, phi_n) 0 ]

for any function ( f ).

Conclusion

The method of characteristics is an essential tool for solving first-order PDEs. By parameterizing the solution and transforming the PDE into a system of ODEs, the method simplifies the problem significantly. This article has provided a detailed explanation of the method and illustrated its application with examples. Understanding this method is crucial for anyone dealing with first-order PDEs in mathematics, physics, and engineering.