practise problems for solving first order PDEs using method of characteristics

by Nasser Abbasi

Problem 2

Solve

Solution

Seek solution where constant,hence

Compare to (1) we see that or and , but since then , and this has solution but hence

Now at , the solution is but this solution is valid any where on this characteristic line and not just when . hence

But from (2), hence

Problem 3

Solve

Solution

Seek solution where constant,hence

Compare to (1) we see that or and , but since then , and this has solution

but hence Hence At ,

Now we are told the solution at is , or but this solution is valid any where on this characteristic line and not just when . hence

Replace the value of obtained in (2) we obtain

Hence

Problem 4

Solve

Solution

Seek solution where constant,hence

Compare to (1) we see that or and , but since then hence we need to solve

but hence Hence

At ,

Now we are told the solution at is , or but this solution is valid any where on this characteristic line and not just when . hence

Replace the value of obtained in (2) we obtain

Hence

To avoid a solution which blow up, we need , hence for example, and will not give a valid solution. so all region in plane in which is not a valid region to apply this solution at.

The solution breaks down along this line in the plane

To see it in 3D, here is the solution that includes the above line, and we see that the solution below the line and the above the line are not continuous across it. ( I think there is a name to this phenomena that I remember reading about sometime, may be related to shockwaves but do not now know how this would happen in reality)

Problem 5

Solution

Nonhomogeneous pde first order.

(TO DO)