Method of characteristics

The method of characteristics is a method of solving partial differential equations (PDGL / PDE), which are typically first order and quasi-linear, i.e. equations of the type

${\displaystyle P(x,t,u){\frac {\partial u}{\partial t}}+Q(x,t,u){\frac {\partial u}{\partial x}}=R(x,t,u),}$

for a function with the initial condition . (An equation is called quasi-linear if it is linear in the highest derivative). ${\displaystyle u(x,t)}$ ${\displaystyle u(x,0)=f(x)}$

The basic idea is to trace the PDE back to a system of ordinary differential equations on certain hypersurfaces , so-called characteristics , by means of a suitable coordinate transformation . The PDE can then be solved as an initial value problem in the new system with initial values ​​on the hypersurfaces intersecting the characteristic. Disturbances propagate along the characteristics. The method can also be applied generally to hyperbolic partial differential equations , the prototype of which is the wave equation , and to some other higher order PDEs.

Characteristics play a role in the qualitative discussion of the solution to certain PDEs and in the question of when initial value problems are correctly posed for this PDE .

The method goes back to Joseph-Louis Lagrange (1779, quasi-linear partial differential equations of the first order). It was founded geometrically by Gaspard Monge in 1784 , which Johann Friedrich Pfaff in 1815 and Augustin-Louis Cauchy in 1819 expanded to more than two dimensions.

idea

In order to convert the partial differential equation in a system of ordinary differential equations, the coordinates and two new coordinates and parameterized, i.e., it has equations and . First, the function you are looking for is derived using a chain rule : ${\displaystyle t}$ ${\displaystyle x}$ ${\displaystyle \tau }$ ${\displaystyle \xi }$ ${\displaystyle t=t(\tau ,\xi )}$ ${\displaystyle x=x(\tau ,\xi )}$ ${\displaystyle u(x,t)=u(x(\tau ,\xi ),t(\tau ,\xi ))}$ ${\displaystyle \tau }$

${\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} \tau }}={\frac {\partial u}{\partial t}}{\frac {\mathrm {d} t}{\mathrm {d} \tau }}+{\frac {\partial u}{\partial x}}{\frac {\mathrm {d} x}{\mathrm {d} \tau }}.}$

The above quasi-linear PDE is calculated using the "characteristic equations"

${\displaystyle {\frac {\mathrm {d} t}{\mathrm {d} \tau }}=P(x,t,u)}$,

${\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} \tau }}=Q(x,t,u)}$

to

${\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} \tau }}={\frac {\partial u}{\partial t}}{\frac {\mathrm {d} t}{\mathrm {d} \tau }}+{\frac {\partial u}{\partial x}}{\frac {\mathrm {d} x}{\mathrm {d} \tau }}=R(x,t,u)}$

So a system of ordinary differential equations in the new coordinates, if you still use the parameterizations and on the right side . ${\displaystyle t=t(\tau ,\xi )}$ ${\displaystyle x=x(\tau ,\xi )}$

Geometric interpretation

Geometrically, the procedure can be described as follows. The solution function leads to area equations in the space of coordinates ( integral areas ). Such an integral surface has the normal vector : ${\displaystyle u=u(x,t)}$ ${\displaystyle z=u(x,t)}$ ${\displaystyle (x,t,z)}$

${\displaystyle {\vec {N}}=\left({\frac {\partial u}{\partial x}},{\frac {\partial u}{\partial t}},-1\right)}$

and the PDE says geometrically that the vector field of the characteristics is tangential to the integral surface , because the scalar product of the vector field with the normal vector vanishes: ${\displaystyle {\vec {X}}(x,t,z)=\left(Q(x,t,z),P(x,t,z),R(x,t,z)\right)}$ ${\displaystyle z=u(x,t)}$ ${\displaystyle z=u(x,t)}$ ${\displaystyle {\vec {X}}(x,t,z)}$ ${\displaystyle {\vec {N}}(x,t,z)}$

${\displaystyle {\vec {X}}\cdot {\vec {N}}={\frac {\partial u}{\partial x}}\cdot Q+{\frac {\partial u}{\partial t}}\cdot P+(-1)\cdot R=0}$.

The solutions of the PDE are integral curves of the vector field (in the subspace of x, t these are the characteristics). In a parametric representation of the integral curve with parameters , the following equations result: ${\displaystyle {\vec {X}}(x,t,z)}$ ${\displaystyle \tau }$

${\displaystyle {\vec {X}}=\left(Q,P,R\right)=\left({\frac {\mathrm {d} x}{\mathrm {d} \tau }},{\frac {\mathrm {d} t}{\mathrm {d} \tau }},{\frac {\mathrm {d} z}{\mathrm {d} \tau }}\right)}$

for the characteristics or (Lagrange-Charpit equations):

${\displaystyle {\frac {\mathrm {d} x}{Q}}={\frac {\mathrm {d} t}{P}}={\frac {\mathrm {d} z}{R}}}$

Examples

Simple transport equation

A simple transport equation is given, a simple example of a type of 1st order PDEs that describe a temporal-spatial flow ( e.g. advection , transport of chemicals in a liquid):

${\displaystyle u_{t}+c\cdot u_{x}=0}$

with the initial condition , and the real constant . The usual index notation or was used here for the partial derivatives of after or . Deriving by and coefficient comparison provides a system of ordinary differential equations: ${\displaystyle u(x,t=0)=f(x)}$ ${\displaystyle t\geq 0,x\in \mathbb {R} }$ ${\displaystyle c\in \mathbb {R} }$ ${\displaystyle u}$ ${\displaystyle t}$ ${\displaystyle x}$ ${\displaystyle u_{t}}$ ${\displaystyle u_{x}}$ ${\displaystyle u}$ ${\displaystyle \tau }$

${\displaystyle {\frac {\mathrm {d} t}{\mathrm {d} \tau }}=P(x,t,u)=1}$

${\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} \tau }}=Q(x,t,u)=c}$

${\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} \tau }}=R(x,t,u)=0}$

as well as the initial conditions . ${\displaystyle t(\tau =0)=0\,,\,x(\tau =0)=\xi \,,\,u(\tau =0)=f(\xi )}$

Since the equations are completely decoupled from each other, the solution is very simple:

${\displaystyle t=\tau }$

${\displaystyle x=c\tau +\xi }$

${\displaystyle u=f(\xi )}$.

From this follows immediately and with it the solution of the transport equation in the old coordinates: ${\displaystyle \xi =x-ct}$

${\displaystyle u(x,t)=f(x-ct)}$.

${\displaystyle x=ct+\xi }$are the equations of the characteristics. The value of on the x-axis at defines the value of along the characteristic straight line with gradient for all times, which is expressed mathematically in the form of the solution . Along the characteristic does not change what is precisely expressed by the differential equation along the characteristic . ${\displaystyle u}$ ${\displaystyle t=0}$ ${\displaystyle u}$ ${\displaystyle c}$ ${\displaystyle u(x,t)=f(x-ct)=f(\xi )}$ ${\displaystyle u}$ ${\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} \tau }}={\frac {\mathrm {d} u}{\mathrm {d} t}}=0}$ ${\displaystyle x=ct+\xi }$

Generalized transport equation

Consider a more general transport equation with variable coefficients:

${\displaystyle P(x,t){\frac {\partial u}{\partial t}}+Q(x,t){\frac {\partial u}{\partial x}}+R(x,t)u=0}$

with the initial condition . ${\displaystyle u(x,0)=f(x)}$

A new variable is introduced so that the PDE is reduced to curves for an ordinary differential equation. This will be ${\displaystyle \tau }$ ${\displaystyle x(\tau ),t(\tau )}$ ${\displaystyle \tau >0}$

${\displaystyle {\frac {\mathrm {d} t}{\mathrm {d} \tau }}=P(x(\tau ),t(\tau ))}$

${\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} \tau }}=Q(x(\tau ),t(\tau ))}$

chosen (the characteristics equations ) such that:

${\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} \tau }}={\frac {\partial u}{\partial x}}{\frac {\mathrm {d} x}{\mathrm {d} \tau }}+{\frac {\partial u}{\partial t}}{\frac {\mathrm {d} t}{\mathrm {d} \tau }}=Q(x,t){\frac {\partial u}{\partial x}}+P(x,t){\frac {\partial u}{\partial t}}=-R(x,t)u}$

The PDE then becomes an ordinary differential equation:

${\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} \tau }}+R(x(\tau ),t(\tau ))u=0}$

The second coordinate of the coordinate transformation is and the function values ​​u along the curves are given by the initial values ​​in . ${\displaystyle \xi =x(\tau =0)}$ ${\displaystyle \tau >0}$ ${\displaystyle \xi }$

For example, consider the equation:

${\displaystyle u_{t}+c\,u_{x}+a\,u=0}$

with , then again the characteristics as in example 1 result, but solutions result from the third equation . So here one does not have constant solutions along the characteristic, as was the case in the previous example, but an exponential decay with time. ${\displaystyle u(x,0)=u(x_{0},0)=f(x_{0})=K}$ ${\displaystyle P=1,Q=c,R=a,}$ ${\displaystyle x-ct=x_{0}}$ ${\displaystyle u_{t}+au=0}$ ${\displaystyle u(x,t)=f(x-ct)\exp {(-at)}=f(x_{0})\exp {(-at)}=K\exp {(-at)}}$

As another example, I'll

${\displaystyle u_{t}+x\,u_{x}=0}$

considered, with . Here is and one does not have straight lines as characteristics, but rather . The function value is constant along the characteristics, so that it turns out to be a solution ${\displaystyle u(x,0)=f(x)}$ ${\displaystyle P=1,Q=x,R=0,}$ ${\displaystyle x=x_{0}\exp {(t)}}$

${\displaystyle u(x,t)=f(x_{0})=f(x\,\exp {(-t)})}$

results.

Burgers equation

→ Main article : Burgers equation

Another example are the laws of conservation of form that occur in physics

${\displaystyle u_{t}+F_{x}(u(x,t))=0}$,

For example the Burgers equation in the case of vanishing viscosity (non-viscous Burgers equation):

${\displaystyle F={\frac {1}{2}}u^{2}}$

and thus

${\displaystyle u_{t}+u\,u_{x}=0}$

with the initial condition . Here's , the equation is nonlinear. The characteristics are , that is to say straight lines, but which have a variable slope that depends on the function value along the characteristics. The solution is formally similar to the example of the simple transport equation and is constant along the characteristic, where the following applies . ${\displaystyle u(x,0)=f(x)}$ ${\displaystyle P=1,Q=u,R=0}$ ${\displaystyle x=x_{0}+u\,t=x_{0}+f(x_{0})\,t}$ ${\displaystyle u(x,t)=f(x-ut)=f(x_{0})}$ ${\displaystyle u_{t}=0}$

The Burgers equation is often used as a model system of nonlinear hydrodynamic equations. The novelty in this case is that the characteristics can intersect because of the variable slope. At the intersection, the solution becomes ambiguous and a clear solution to the problem no longer exists. A discontinuity forms, a shock wave front for characteristics converging in the direction of advancing time , and a dilution front for diverging characteristics. However, one can avoid the collapse of classical solutions by considering weak solutions ( distributions ), whereby entropy conditions are used to select the physically correct solution. In the case of Burgers' equation, the shock wave has a speed which corresponds to the mean value of the function values ​​u on the right and left of the shock front.

Wave equation

→ Main article : wave equation

The wave equation is the prototype of a linear hyperbolic partial differential equation of the 2nd order:

${\displaystyle {\frac {\partial ^{2}u}{{\partial x}^{2}}}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}u}{{\partial t}^{2}}}=0}$

with a constant . Transforming to new variables , with which the wave equation: ${\displaystyle c}$ ${\displaystyle w=x+ct}$ ${\displaystyle v=x-ct}$

${\displaystyle {\left({\frac {\partial }{\partial w}}+{\frac {\partial }{\partial v}}\right)}^{2}u={\left({\frac {\partial }{\partial w}}-{\frac {\partial }{\partial v}}\right)}^{2}u}$

transformed, from which:

${\displaystyle {\frac {\partial ^{2}}{\partial w\partial v}}u=0}$

or

${\displaystyle \left({\frac {\partial }{\partial x}}+{\frac {1}{c}}{\frac {\partial }{\partial t}}\right)\cdot \left({\frac {\partial }{\partial x}}-{\frac {1}{c}}{\frac {\partial }{\partial t}}\right)u=0}$

follows, so or . ${\displaystyle u(w,v)=u_{1}(w)+u_{2}(v)}$ ${\displaystyle u=u_{1}(x+ct)+u_{2}(x-ct)}$

The equations of the characteristics are or with a constant . ${\displaystyle w=\mathrm {const} ,v=\mathrm {const} }$ ${\displaystyle x=x_{0}\pm ct}$ ${\displaystyle x_{0}}$

General partial differential equation of the 2nd order

The general 2nd order partial differential equation is given by:

${\displaystyle Au_{tt}+2Bu_{tx}+Cu_{xx}+Du_{x}+Eu_{t}+Fu=0}$

partial derivatives are indicated here by indices.

Looking at the matrix

${\displaystyle M={\begin{pmatrix}A&B\\B&C\end{pmatrix}}}$

of the coefficients of the highest derivatives, the equations are elliptic for , parabolic for, and hyperbolic for . ${\displaystyle \mathrm {det} \,(M)=AC-B^{2}>0}$ ${\displaystyle \mathrm {det} \,(M)=0}$ ${\displaystyle \det \,(M)<0}$

In addition to the PDE, the following applies to any curve:

${\displaystyle du_{t}=u_{tt}dt+u_{tx}dx}$

${\displaystyle du_{x}=u_{xt}dt+u_{xx}dx}$

That's three linear equations for the second derivative . So that this can be clearly determined from the values ​​of assumed to be known, the following must apply to the determinant ${\displaystyle u_{tt},u_{tx},u_{xx}}$ ${\displaystyle u,u_{t},u_{x}}$

${\displaystyle \Delta =\mathrm {det} {\begin{pmatrix}A&2B&C\\dt&dx&0\\0&dt&dx\end{pmatrix}}\neq 0}$

For some curves, the characteristics of the PDE (the name comes from Gaspard Monge ), this does not apply, there the following applies : ${\displaystyle \Delta =0}$

${\displaystyle A{(dx)}^{2}-2Bdxdt+C{(dt)}^{2}=0}$

or

${\displaystyle x_{t}={\frac {B\pm {\sqrt {(B^{2}-AC)}}}{A}}}$

The initial value problem can only be solved uniquely if the curves on which the initial values ​​are specified are not tangential to the characteristics. That is the statement of the Cauchy-Kovalevskaya theorem for the so-called non-characteristic Cauchy problem. Since it is under the root sign , it follows that hyperbolic equations have two sets of characteristics, parabolic one and elliptical none at all. ${\displaystyle -\mathrm {det} M}$

The characteristics can also be viewed geometrically as curves in two dimensions , the normal vectors of which the equation ${\displaystyle (x,t)}$ ${\displaystyle {\vec {n}}}$

${\displaystyle {\vec {n}}M{\vec {n}}^{T}=0}$

meet (this is equivalent to the tangential vectors of the curves).

There , then applies ${\displaystyle {\vec {n}}\sim (u_{t},u_{x})}$

${\displaystyle u_{t}^{2}A+2Bu_{t}u_{x}+Cu_{x}^{2}=0}$

If a principal axis transformation is carried out to diagonalize the quadratic equation , only in the case of the hyperbolic equation, i.e. the eigenvalues have opposite signs, a form is obtained which, as in the above example of the wave equation, is reduced to first order equations with two characteristics by means of variable transformation can.

For example, for the wave equation:

${\displaystyle M={\begin{pmatrix}-{\frac {1}{c^{2}}}&0\\0&1\end{pmatrix}}}$

and the normal vectors are perpendicular to the associated characteristics or . ${\displaystyle {\vec {n}}=(-c,1),{\vec {n}}=(c,1)}$ ${\displaystyle x+ct}$ ${\displaystyle x-ct}$

An example of an equation in which all three types of PDE appear is the Euler-Tricomi equation or Tricomi equation :

${\displaystyle u_{tt}-tu_{xx}=0}$

for that which is hyperbolic for positive , for parabolic and for negative elliptical. Correspondingly, it has no characteristics for negative ones, for one that branches out and has the characteristic equation there , that is, characteristics . ${\displaystyle \mathrm {det} \,(M)=AC-B^{2}=-t}$ ${\displaystyle t}$ ${\displaystyle t=0}$ ${\displaystyle t}$ ${\displaystyle t}$ ${\displaystyle t\geq 0}$ ${\displaystyle t>0}$ ${\displaystyle dx^{2}-t\,dt^{2}=0}$ ${\displaystyle 3x\pm 2t^{\frac {3}{2}}=\mathrm {const} }$

literature

• Fritz John : Partial Differential Equations, 4th edition, Springer Verlag 1982
• Eberhard Zeidler u. a .: Teubner Taschenbuch der Mathematik , Part 1, 1996, pp. 533ff, 488f

Web links

• Scott Sarra, The Method of Characteristics with applications to Conservation Laws, 2002

Individual evidence

1. ↑ Helmut Fischer, Helmut Kaul, Mathematik für Physiker, 3rd edition, Teubner 2008, p. 198. The characteristic method is dealt with in Paragraph 7 (p. 172ff).
2. ^ Fritz John: Partial Differential Equations , 4th edition, Springer Verlag 1982, p. 9.
3. ^ Discussion according to Arnold Sommerfeld Partial Differential Equations in Physics , Academic Press 1949, p. 36f