# Burgers Equation Question - cannot satisfy initial conditions

• zapz
In summary, the conversation discusses using characteristics to solve Burgers equation with an initial condition of u(x,0)=x^2 on the half line x≥0. The solution process involves finding the function u=f(x-ut) and then using the initial condition to determine the correct sign solution. The conversation ends with the realization that taking the limit as t approaches 0 resolves any issues with the initial condition.
zapz
Burgers Equation Question -- cannot satisfy initial conditions

## Homework Statement

Use characteristics to solve $$u_t+uu_x=0$$ on half line x≥0 with $$u(x,0)=x^2$$

NA

## The Attempt at a Solution

I think I have an issue with the initial condition. So solving via characteristics gives:

$$\frac{dx}{dt}=u \Rightarrow x=ut+x_0 \Rightarrow u=f(x-ut)$$. Then we plug in initial values and get:

$$u(x,0)=f(x)=x^2 \Rightarrow u=(x-ut)^2 \Rightarrow u^2t^2-u(1+2xt)+x^2=0$$

$$u=\dfrac{(1+2xt) \pm \sqrt{(1+2xt)^2+4x^2t^2}}{2t^2}$$

Now here I have a problem. So, in order to pic which sign solution to use, I need to use the initial condition. However, at t=0, I have an obvious problem. If anyone could offer help, or whether there is a typo in the problem, that would be much appreciated!

zapz said:
at t=0, I have an obvious problem.
What is the limit as t→0+?

1 person
haruspex said:
What is the limit as t→0+?

Thank you for a quick reply! I just did the limit out and that resolved my issue. Thank you very much!

## What is Burgers Equation and why is it important in science?

Burgers Equation is a fundamental equation in fluid mechanics that describes the motion of a fluid in a pipe. It is important because it allows scientists to study the behavior of fluids in a variety of situations, from the movement of air and water to the flow of blood in our bodies.

## What does it mean when Burgers Equation cannot satisfy initial conditions?

When Burgers Equation cannot satisfy initial conditions, it means that the equation does not accurately describe the behavior of the fluid in a given situation. This can happen when the initial conditions do not match the physical properties of the fluid, or when the equation itself is not appropriate for the situation being studied.

## What factors can affect the solution of Burgers Equation?

There are several factors that can affect the solution of Burgers Equation, including the initial conditions, the physical properties of the fluid, and the boundary conditions. In addition, the type of fluid and the type of flow being studied can also play a role in determining the solution.

## How can scientists use Burgers Equation to make predictions or solve real-world problems?

Scientists can use Burgers Equation to make predictions about the behavior of fluids in different situations. By solving the equation and analyzing the results, they can gain insights into the movement of fluids in a variety of contexts, from predicting the spread of pollutants in a river to understanding the flow of air around an airplane wing. This information can then be used to solve real-world problems and improve our understanding of the world around us.

## Are there any limitations to using Burgers Equation in scientific research?

Like any mathematical model, Burgers Equation has its limitations. It is a simplified version of the Navier-Stokes equations, which describe the motion of fluids in more detail. Additionally, Burgers Equation assumes that the flow is one-dimensional and the fluid is incompressible, which may not always be the case in real-world situations. Therefore, it is important for scientists to carefully consider the assumptions and limitations of Burgers Equation when using it in their research.

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