Help with PDE (first partial derivatives)

[tpm]

Homework Statement

I would like to know how to solve the PDE

$$\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}+\frac{\partial f}{\partial z}=g(x,y,z)$$

* f is the unknown function * g(x,y,z) is a known and smooth function

Homework Equations

$$divf=g$$

The Attempt at a Solution

- I have tried using 'Divergence theorem' but got no further info abut f.

- If g(x,y,z)= g(x)+g(y)+g(z) i can provide a solution othewise i have no
idea ..

 

[mighty2000]

Thank you for your question. The PDE you have presented is a first-order linear partial differential equation, also known as an advection equation. The general form of this equation is:

a(x,y,z) \frac{\partial f}{\partial x} + b(x,y,z) \frac{\partial f}{\partial y} + c(x,y,z) \frac{\partial f}{\partial z} = g(x,y,z)

where a(x,y,z), b(x,y,z), and c(x,y,z) are known functions and g(x,y,z) is a known and smooth function. In your case, a(x,y,z) = b(x,y,z) = c(x,y,z) = 1, and g(x,y,z) = g(x,y,z).

To solve this PDE, we can use the method of characteristics. This involves finding a set of curves, known as characteristics, along which the PDE reduces to an ordinary differential equation. The solution to this ODE can then be used to find the solution to the original PDE.

To find the characteristics, we can set up the following system of ODEs:

\frac{dx}{dt} = a(x,y,z), \frac{dy}{dt} = b(x,y,z), \frac{dz}{dt} = c(x,y,z)

Solving this system of ODEs will give us the equations of the characteristics:

x = x(t), y = y(t), z = z(t)

Now, we can substitute these equations into the original PDE to obtain:

\frac{d}{dt} f(x(t), y(t), z(t)) = g(x(t), y(t), z(t))

This is an ODE that can be solved using standard techniques, such as separation of variables or integrating factors. Once we have the solution, we can substitute back into the equations of the characteristics to obtain the solution to the original PDE.

In the case where g(x,y,z) = g(x) + g(y) + g(z), we can use the method of separation of variables to solve the ODE and obtain a solution for f(x,y,z). However, for more complicated g(x,y,z) functions, we may need to use numerical methods to obtain a solution.

I hope this helps to answer your question. If you have any further questions, please do not hesitate to ask. Good luck with your studies