Question

In lectures, we have discussed first order quasilinear PDEs. That is, PDEs of the general form A(x, y, u) ∂u(x, y) ∂x + B(x, y, u) ∂u(x, y) ∂y = C(x, y, u), (1) for some A, B and C. To solve such PDEs we first find characteristics, curves in the solution space (x, y, u) parametrically given by (x(τ ), y(τ ), u(τ )), which satisfy dx dτ = A(x, y, u), dy dτ = b(x, y, u), du dτ = C(x, y, u). (2) We find solutions to these equations in the form f(x, y, u) = C1 and g(x, y, u) = C2 where C1 and C2 are arbitrary constants. The independent functions f and g are then used to write the general solution to Equation (1) f(x, y, u(x, y)) = F [g(x, y, u(x, y))] , (3) where F is a sufficiently smooth function (that is, you can expect in this question that its derivative exists everywhere). 1. [12 marks] Verify that (3) for any suitable F and for any f and g as described above is actually a solution to the PDE (1). That is, you should show that given (2) which describe the functions f and g and the solution (3), then Equation (1) is always satisfied. HINT: This is not as simple as it sounds. You should first attempt to differentiate f(x, y, u) = C1 and g(x, y, u) = C2 by τ and differentiate the solution (3) first with respect to x and then with respect to y and use the resultant simultaneous equations to deduce (1).

Answer 1

F is a sufficiently smooth function (that is, you can expect in this question that its acutually a solution to the PDE. That is, you should show that given