mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the method of characteristics is a technique for solving

partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...

. Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any

hyperbolic partial differential equation In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n-1 derivatives. More precisely, the Cauchy problem can be ...

. The method is to reduce a partial differential equation to a family of

ordinary differential equations In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...

along which the solution can be integrated from some initial data given on a suitable

hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidea ...

Characteristics of first-order partial differential equation

For a first-order PDE (

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to h ...

), the method of characteristics discovers curves (called characteristic curves or just characteristics) along which the PDE becomes an

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...

(ODE). Once the ODE is found, it can be solved along the characteristic curves and transformed into a solution for the original PDE. For the sake of simplicity, we confine our attention to the case of a function of two independent variables ''x'' and ''y'' for the moment. Consider a quasilinear PDE of the form Suppose that a solution ''z'' is known, and consider the surface graph ''z'' = ''z''(''x'',''y'') in R³. A

normal vector In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve ...

to this surface is given by :

\left(\frac(x,y),\frac(x,y),-1\right).\,

As a result, equation () is equivalent to the geometrical statement that the vector field :

(a(x,y,z),b(x,y,z),c(x,y,z))\,

is tangent to the surface ''z'' = ''z''(''x'',''y'') at every point, for the dot product of this vector field with the above normal vector is zero. In other words, the graph of the solution must be a union of

integral curve In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations. Name Integral curves are known by various other names, depending on the nature and interpret ...

s of this vector field. These integral curves are called the characteristic curves of the original partial differential equation and are given by the ''

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia :

\begin
\frac&=&a(x,y,z),\\
\frac&=&b(x,y,z),\\
\frac&=&c(x,y,z).
\end

A parametrization invariant form of the ''Lagrange–Charpit equations'' is: :

\frac = \frac = \frac .

Linear and quasilinear cases

Consider now a PDE of the form :

\sum_^n a_i(x_1,\dots,x_n,u) \frac=c(x_1,\dots,x_n,u).

For this PDE to be

linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...

, the coefficients ''a''_''i'' may be functions of the spatial variables only, and independent of ''u''. For it to be quasilinear, ''a''_''i'' may also depend on the value of the function, but not on any derivatives. The distinction between these two cases is inessential for the discussion here. For a linear or quasilinear PDE, the characteristic curves are given parametrically by :

(x_1,\dots,x_n,u) = (x_1(s),\dots,x_n(s),u(s))

u(\mathbf(s)) = U(s)

such that the following system of ODEs is satisfied Equations () and () give the characteristics of the PDE.

Proof for quasilinear Case

In the quasilinear case, the use of the method of characteristics is justified by Grönwall's inequality. The above equation may be written as

\mathbf(\mathbf,u) \cdot \nabla u(\mathbf) = c(\mathbf,u)

We must distinguish between the solutions to the ODE and the solutions to the PDE, which we do not know are equal ''a priori.'' Letting capital letters be the solutions to the ODE we find

\mathbf'(s) = \mathbf(\mathbf(s),U(s))

U'(s) = c(\mathbf(s), U(s))

Examining

\Delta(s) = , u(\mathbf(s)) - U(s), ^2

, we find, upon differentiating that

\Delta'(s) = 2\big(u(\mathbf(s)) - U(s)\big) \Big(\mathbf'(s)\cdot \nabla u(\mathbf(s)) - U'(s)\Big)

which is the same as

\Delta'(s) = 2\big(u(\mathbf(s)) - U(s)\big) \Big(\mathbf(\mathbf(s),U(s))\cdot \nabla u(\mathbf(s)) - c(\mathbf(s),U(s))\Big)

We cannot conclude the above is 0 as we would like, since the PDE only guarantees us that this relationship is satisfied for

u(\mathbf)

\mathbf(\mathbf,u) \cdot \nabla u(\mathbf) = c(\mathbf,u)

, and we do not yet know that

U(s) = u(\mathbf(s))

. However, we can see that

\Delta'(s) = 2\big(u(\mathbf(s)) - U(s)\big) \Big(\mathbf(\mathbf(s),U(s))\cdot \nabla u(\mathbf(s)) - c(\mathbf(s),U(s))-\big(\mathbf(\mathbf(s),u(\mathbf(s))) \cdot \nabla u(\mathbf(s)) - c(\mathbf(s),u(\mathbf(s)))\big)\Big)

since by the PDE, the last term is 0. This equals

\Delta'(s) = 2\big(u(\mathbf(s)) - U(s)\big) \Big(\big(\mathbf(\mathbf(s),U(s))-\mathbf(\mathbf(s),u(\mathbf(s)))\big)\cdot \nabla u(\mathbf(s)) - \big(c(\mathbf(s),U(s))-c(\mathbf(s),u(\mathbf(s)))\big)\Big)

By the triangle inequality, we have

, \Delta'(s),  \leq 2\big, u(\mathbf(s)) - U(s)\big,  \Big(\big\, \mathbf(\mathbf(s),U(s))-\mathbf(\mathbf(s),u(\mathbf(s)))\big\,  \ \, \nabla u(\mathbf(s))\,  + \big, c(\mathbf(s),U(s))-c(\mathbf(s),u(\mathbf(s)))\big, \Big)

Assuming

\mathbf,c

are at least

C^1

, we can bound this for small times. Choose a neighborhood

\Omega

around

\mathbf(0), U(0)

small enough such that

\mathbf,c

are locally Lipschitz. By continuity,

(\mathbf(s),U(s))

will remain in

\Omega

for small enough

s

. Since

U(0) = u(\mathbf(0))

, we also have that

(\mathbf(s), u(\mathbf(s)))

will be in

\Omega

for small enough

s

by continuity. So,

(\mathbf(s),U(s)) \in \Omega

and

(\mathbf(s), u(\mathbf(s))) \in \Omega

for

s \in,s_0

. Additionally,

\, \nabla u(\mathbf(s))\,  \leq M

for some

M \in \R

for

s \in,s_0

by compactness. From this, we find the above is bounded as

, \Delta'(s),  \leq C, u(\mathbf(s)) - U(s), ^2 = C , \Delta(s),

for some

C \in \mathbb

. It is a straightforward application of Grönwall's Inequality to show that since

\Delta(0) = 0

we have

\Delta(s) = 0

for as long as this inequality holds. We have some interval

[0, \epsilon)

such that

u(X(s)) = U(s)

in this interval. Choose the largest

\epsilon

such that this is true. Then, by continuity,

U(\epsilon) = u(\mathbf(\epsilon))

. Provided the ODE still has a solution in some interval after

\epsilon

, we can repeat the argument above to find that

u(X(s)) = U(s)

in a larger interval. Thus, so long as the ODE has a solution, we have

u(X(s)) = U(s)

Fully nonlinear case

Consider the partial differential equation where the variables ''p''_i are shorthand for the partial derivatives :

p_i = \frac.

Let (''x''_i(''s''),''u''(''s''),''p''_i(''s'')) be a curve in R²ⁿ⁺¹. Suppose that ''u'' is any solution, and that :

u(s) = u(x_1(s),\dots,x_n(s)).

Along a solution, differentiating () with respect to ''s'' gives :

\sum_i(F_ + F_u p_i)\dot_i + \sum_i F_\dot_i = 0

\dot - \sum_i p_i \dot_i = 0

\sum_i (\dot_i dp_i - \dot_i dx_i)= 0.

The second equation follows from applying the chain rule to a solution ''u'', and the third follows by taking an exterior derivative of the relation

du - \sum_i p_i \, dx_i = 0

. Manipulating these equations gives :

\dot_i=\lambda F_,\quad\dot_i=-\lambda(F_+F_up_i),\quad \dot=\lambda\sum_i p_iF_

where λ is a constant. Writing these equations more symmetrically, one obtains the Lagrange–Charpit equations for the characteristic :

\frac=-\frac=\frac.

Geometrically, the method of characteristics in the fully nonlinear case can be interpreted as requiring that the

Monge cone In the mathematical theory of partial differential equations (PDE), the Monge cone is a geometrical object associated with a first-order equation. It is named for Gaspard Monge. In two dimensions, let :F(x,y,u,u_x,u_y) = 0\qquad\qquad (1) be a PDE ...

of the differential equation should everywhere be tangent to the graph of the solution. The second order partial differential equation is solved with Charpit method .

Example

As an example, consider the

advection equation In the field of physics, engineering, and earth sciences, advection is the transport of a substance or quantity by bulk motion of a fluid. The properties of that substance are carried with it. Generally the majority of the advected substance is al ...

(this example assumes familiarity with PDE notation, and solutions to basic ODEs). :

a \frac + \frac = 0

where

a

is constant and

u

is a function of

x

and

t

. We want to transform this linear first-order PDE into an ODE along the appropriate curve; i.e. something of the form :

\fracu(x(s), t(s)) = F(u, x(s), t(s)) ,

where

(x(s),t(s))

is a characteristic line. First, we find :

\fracu(x(s), t(s)) = \frac \frac + \frac \frac

by the chain rule. Now, if we set

\frac = a

and

\frac = 1

we get :

a \frac + \frac

which is the left hand side of the PDE we started with. Thus :

\fracu = a \frac + \frac = 0.

So, along the characteristic line

(x(s), t(s))

, the original PDE becomes the ODE

u_s = F(u, x(s), t(s)) = 0

. That is to say that along the characteristics, the solution is constant. Thus,

u(x_s, t_s) = u(x_0, 0)

where

(x_s, t_s)\,

and

(x_0, 0)

lie on the same characteristic. Therefore, to determine the general solution, it is enough to find the characteristics by solving the characteristic system of ODEs: *

\frac = 1

, letting

t(0)=0

we know

t=s

, *

\frac = a

, letting

x(0)=x_0

we know

x=as+x_0=at+x_0

, *

\frac = 0

, letting

u(0)=f(x_0)

we know

u(x(t), t)=f(x_0)=f(x-at)

. In this case, the characteristic lines are straight lines with slope

a

, and the value of

u

remains constant along any characteristic line.

Characteristics of linear differential operators

Let ''X'' be a

differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...

and ''P'' a linear

differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...

P : C^\infty(X) \to C^\infty(X)

of order ''k''. In a local coordinate system ''x''^''i'', :

P = \sum_ P^(x)\frac

in which ''α'' denotes a

multi-index Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices. ...

. The principal

symbol A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different conc ...

of ''P'', denoted ''σ''_''P'', is the function on the

cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. Th ...

T^∗''X'' defined in these local coordinates by :

\sigma_P(x,\xi) = \sum_ P^\alpha(x)\xi_\alpha

where the ''ξ''_''i'' are the fiber coordinates on the cotangent bundle induced by the coordinate differentials ''dx''^''i''. Although this is defined using a particular coordinate system, the transformation law relating the ''ξ''_''i'' and the ''x''^''i'' ensures that ''σ''_''P'' is a well-defined function on the cotangent bundle. The function ''σ''_''P'' is

homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...

of degree ''k'' in the ''ξ'' variable. The zeros of ''σ''_''P'', away from the zero section of T^∗''X'', are the characteristics of ''P''. A hypersurface of ''X'' defined by the equation ''F''(''x'') = ''c'' is called a characteristic hypersurface at ''x'' if :

\sigma_P(x,dF(x)) = 0.

Invariantly, a characteristic hypersurface is a hypersurface whose

conormal bundle In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion). Definition Riemannian manifold Let (M,g) be a Rieman ...

is in the characteristic set of ''P''.

Qualitative analysis of characteristics

Characteristics are also a powerful tool for gaining qualitative insight into a PDE. One can use the crossings of the characteristics to find

shock wave In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a me ...

s for potential flow in a compressible fluid. Intuitively, we can think of each characteristic line implying a solution to

u

along itself. Thus, when two characteristics cross, the function becomes multi-valued resulting in a non-physical solution. Physically, this contradiction is removed by the formation of a shock wave, a tangential discontinuity or a weak discontinuity and can result in non-potential flow, violating the initial assumptions. Characteristics may fail to cover part of the domain of the PDE. This is called a

rarefaction Rarefaction is the reduction of an item's density, the opposite of compression. Like compression, which can travel in waves ( sound waves, for instance), rarefaction waves also exist in nature. A common rarefaction wave is the area of low relat ...

, and indicates the solution typically exists only in a weak, i.e.

integral equation In mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; u(x_1,x_2,x_3,...,x_n) ...

, sense. The direction of the characteristic lines indicates the flow of values through the solution, as the example above demonstrates. This kind of knowledge is useful when solving PDEs numerically as it can indicate which

finite difference A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for t ...

scheme is best for the problem.

Notes

References

* * * * * . *

External links

Prof. Scott Sarra tutorial on Method of Characteristics

{{Numerical PDE Partial differential equations Hyperbolic partial differential equations

Characteristics of first-order partial differential equation

Linear and quasilinear cases

Proof for quasilinear Case

Fully nonlinear case

Example

Characteristics of linear differential operators

Qualitative analysis of characteristics

See also

Notes

References

External links