Does a Unique Solution Exist for a PDE with Specific Boundary Conditions?

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The discussion centers on the uniqueness of solutions for the partial differential equation (PDE) Ux - Uy = 0 with the boundary condition U(x,0) = f(x) for x in [0,1]. It is established that a unique solution does not exist at the point (5,1) due to the limitation of the function f(x) being defined only within the interval [0,1]. The concept of characteristic lines, specifically x + y = C, is crucial in understanding the solution's behavior, as any solution can be expressed as F(x+y), where F is an arbitrary function. The inability to extend f(x) beyond the defined interval leads to multiple potential functions that could satisfy the boundary condition, thus negating uniqueness.

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If I have a PDE like Ux-Uy=0 and U(x,0)=f(x) when x in [0,1]. Then is there an uniqueness solution exist at point (5,1)?
How can I explain it using characteristics lines?

Thanks
 
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t.t.h8701 said:
If I have a PDE like Ux-Uy=0 and U(x,0)=f(x) when x in [0,1]. Then is there an uniqueness solution exist at point (5,1)?
How can I explain it using characteristics lines?

Thanks
If you only know f(x) between 0 and 1, you are going to have a problem extending the solution to x= 5!

The "characteristic lines" are of the form x+ y= C and any solution to this equation is of the form F(x+y) where F is an arbitrary function of one variable. Since you require that U(x,0)= F(x+0)= F(x)= f(x) for x between 0 and 1, you will need to take F(x) to be f(x) between 0 and 1 but that does not define it for x+ y= 5+ 1= 6. Consider any number of functions "f(x)" which are identical between 0 and 1 but differ outside that interval.
 

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