# Domain of Dependence of 1D First-order Linear Convection

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 January 10, 2018, 23:37 Domain of Dependence of 1D First-order Linear Convection #1 New Member   abcd efgh ijkl Join Date: Oct 2015 Posts: 26 Rep Power: 10 I know about the domain of dependence of 2nd order hyperbolic equations. It is defined using characteristics and initial/boundary data. But how do we define domain of dependence for 1st order hyperbolic equation? There will be only one characteristic. Thanks

January 11, 2018, 03:25
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Filippo Maria Denaro
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Quote:
 Originally Posted by alibaig1991 I know about the domain of dependence of 2nd order hyperbolic equations. It is defined using characteristics and initial/boundary data. But how do we define domain of dependence for 1st order hyperbolic equation? There will be only one characteristic. Thanks

Well, a first order PDE is classified by definition only hyperbolic. As example for a 2D case, being a and b real functions of (x1,x2).

a*df/dx1+b*df/dx2=0 -> df/dx1+(b/a)*df/dx2=0

so that you can make at system with the total differential

df = dx1*df/dx1+dx2*df/dx2 =dx1*(df/dx1+dx2/dx1*df/dx2)

and see that on the curves of the plane (x1,x2) dx2/dx1=b/a you get df=0.
A classical example is x1=t, x2=x, a=1, b=u=const.
The curve dx/dt=u is a straight line in the (x,t) plane. That means the function at a point Xp and at a time T is equal to the value of the initial data f(Xp-u*T,0). Thus, you can trace the domain of dependence at the time t=0.

 Tags mathematical analysis, pde