Tutorial-1

Department of Mathematics
MAL630: Partial Diﬀerential Equations
Tutorial 1: Cauchy Problem for the Quasilinear Equations and the method of characteristics
Course Coordinator: V. V. K. Srinivas Kumar.
The below exercise problems are copied from the text book titled Partial Diﬀerential Equations
by Robert McOwen ( I have purchased the original version of the same. )
�
�
1. Show that if z = u(x, y) is an integral surface of V = a, b, c containing a point P, then
the surface contains the characteristic curve χ passing through P. (Assume the vector
ﬁeld V is C 1 .)
2. If S1 and S2 are two graphs [i.e. Si is given by z = ui (x, y), i = 1, 2] that are integral
�
�
surfaces of V = a, b, c and intersect in a curve χ, show that χ is a characteristic curve.
3. If Γ is a characteristic curve of V, show that there is an inﬁnite number of solutions u of
a(x, y, u)ux + b(x, y, u)uy = c(x, y, u) containing Γ in the graph of u.
4. Solve the given initial value problem and determine the values of x and y for which it
exists:
(a) xux +uy = y, u(x, 0) = x2 (b) ux −2uy = u, u(0, y) = y (c) y −1 ux +uy = u2 , u(x, 1) = x2 .
5. Solve the given initial value problem and determine the values of x, y and z for which it
exists:
(a) xux + yuy + uz = u, u(x, y, 0) = h(x, y) (b) ux + uy + zuz = u3 , u(x, y, 1) = h(x, y).
6. Solve the given initial value problem and determine the values of x and y for which it
exists:
√
(a) ux + u2 uy = 1, u(x, 0) = 1 (b) ux + uuy = 0, u(x, 0) = x2 + 1.
7. Find a general solution of the following equations:
(a) (x + u)ux + (y + u)uy = 0, (b)(x2 + 3y 2 + 3u2 )ux − 2xyuy + 2xu = 0.
√
8. Consider the equation ux +uy = u. Derive the general solution u(x, y) = (x+f (x−y))2 /4.
Observe that the trivial solution u(x, y) = 0 is not covered by the general solution.
9. Consider the equation y 2 ux + xuy = sin(u2 ).
(a) Describe all projected characteristic curves in the xy−plane.
(b) For the solution u of the initial value problem with u(x, 0) = x, determine the values
of ux , uy , uxx , uxy , uyy on the x−axis.

Download Report