How Do You Solve a First-Order Inhomogeneous PDE?

[mcl4]

Homework Statement

Assume u_t+cu_x = xt, u(x,0) = f(x) for t>0. Find a formula for u(x,t) in terms of f, x, t, and c.

The Attempt at a Solution

I don't really follow what the professor is doing in class, and his office hours and the textbook weren't much more help, so the only thing I know about PDE's is what I've read online. That said:

$$\frac{du}{dr}$$ = $$\frac{dx}{dr}$$u_x+$$\frac{dt}{dr}$$u_t

$$\frac{dt}{dr}$$=1
t=r
$$\frac{dx}{dt}$$=c
x=ct+c'
x₀=c'
x=ct+x₀

$$\frac{du}{dr}$$=xt=ct²+x₀t
$$\int$$du=$$\int$$(ct²+x₀t)dr
u(x,t) = ct²r+x₀tr+c''
u(x₀,0) = 0+0+c'' = f(x₀)
u(x,t) = ct²r+x₀tr+f(x₀)
= ct³+(x-ct)t²+f(x-ct)
= xt²+f(x-ct)

but when I calculate u_t and u_x and substitute into the original equation I do not get xt.

Any pointers would be much appreciated!

 

[mighty2000]

Thank you for your post. I understand that PDEs can be difficult to grasp, but I will try my best to explain the steps in finding a formula for u(x,t) in terms of f, x, t, and c.

Firstly, let's start with the equation given: ut+cux = xt. This is a first-order linear PDE in two variables, u and x. To solve this, we will use the method of characteristics.

The Method of Characteristics:

1. Determine the characteristic equations:
We start by finding the characteristic equations, which are given by: dx/dt = c and du/dt = x.

2. Solve the characteristic equations:
Integrating the first equation with respect to t, we get x = ct + k, where k is a constant of integration. Similarly, integrating the second equation with respect to t, we get u = x^2/2 + f(k), where f(k) is a function of k.

3. Find the value of k:
To find the value of k, we use the initial condition given: u(x,0) = f(x). Substituting t = 0 and u = f(x) into the expression for u above, we get f(x) = x^2/2 + f(k). This implies that f(k) = 0, and hence k = x. Therefore, our characteristic equations become x = ct + x and u = x^2/2.

4. Substitute the values of x and u into the original equation:
Substituting x = ct + x and u = x^2/2 into the original equation, we get ut + cux = ct^2 + cx. Rearranging this, we get u(x,t) = ct^2 + cx - f(x).

5. Substitute the value of x into the expression for u:
Finally, substitute the value of x = ct + x into the expression for u, we get u(x,t) = ct^2 + cx - f(ct + x).

This is the formula for u(x,t) in terms of f, x, t, and c. I hope this helps to clarify the steps in solving this PDE. If you have any further questions, please do not hesitate to ask. Good luck with your studies!