Solving a First Order Linear PDE

In summary, the conversation is about a first order linear partial differential equation and its solution, which is δ(z, t) = √[μ z]/[/ρg t]. The person is asking for a step-by-step solution and expresses gratitude for any help.
  • #1
sagigirl
1
0
Good day. I was wondering if you could help me solve this first order linear partial differential equation:

[∂δ]/[/∂t] = [ρg]/[/μ] δ^2 [∂δ]/[/∂z].

The solution for this is:

δ(z, t) = √[μ z]/[/ρg t].

I don't really understand how the PDE became like this. If you could show the step-by-step solution, I would really, gladly appreciate it. Thank you :)
 
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  • #2
sagigirl said:
Good day. I was wondering if you could help me solve this first order linear partial differential equation:

[∂δ]/[/∂t] = [ρg]/[/μ] δ^2 [∂δ]/[/∂z].

The solution for this is:

δ(z, t) = √[μ z]/[/ρg t].

I don't really understand how the PDE became like this. If you could show the step-by-step solution, I would really, gladly appreciate it. Thank you :)

Welcome to the PF.

Is this for schoolwork?
 

FAQ: Solving a First Order Linear PDE

What is a first order linear PDE?

A first order linear partial differential equation (PDE) is a mathematical equation that involves an unknown function of two or more variables and its partial derivatives. It is called linear because the unknown function and its derivatives appear to the first power only, and there are no products or powers of the unknown function.

How do you solve a first order linear PDE?

To solve a first order linear PDE, you need to follow a specific method called the method of characteristics. This involves finding a set of curves, called characteristics, along which the PDE reduces to an ordinary differential equation (ODE). The solution to the ODE can then be used to find the solution to the original PDE.

What are the initial and boundary conditions for a first order linear PDE?

The initial conditions for a first order linear PDE are specified at a particular point in the domain of the unknown function. These conditions give the value of the function at that point and are used to determine the constants of integration in the solution. The boundary conditions, on the other hand, are specified on the boundary of the domain and are used to determine the form of the solution.

Can you give an example of a first order linear PDE?

One example of a first order linear PDE is the heat equation, which is used to model the flow of heat in a medium. It is given by the equation ∂u/∂t = k∂²u/∂x², where u is the temperature, t is time, and k is the thermal diffusivity of the medium. This equation satisfies the criteria for a first order linear PDE as the unknown function u and its derivatives appear to the first power.

What are some applications of solving first order linear PDEs?

First order linear PDEs have many applications in fields such as physics, engineering, and economics. They are used to model and analyze various physical processes, such as heat transfer, fluid flow, and diffusion. They are also used in financial modeling to study the behavior of stock prices and interest rates. Additionally, first order linear PDEs are an important tool in the development of numerical methods for solving more complex PDEs.

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