PDE for IVP on R for a transport equation

In summary, the conversation is about a homework problem involving solving for the IVP of a transport equation on the real numbers. The equation is Ut-4Ux=t^2 for t>0, XER and the initial condition is u=cosx for t=0, XER. The person is asking for help and is wondering if the method of characteristics can be used and if the problem can be solved without the right hand side. Chet suggests eliminating the right hand side by considering U = V(x, t) +T(t) where V satisfies the homogeneous equation and the same boundary condition.
  • #1
Robconway
4
0
Hi guys, I'm having trouble with a homework problem:

I will have to solve for the IVP of a transport equation on R:

the equations are:

Ut-4Ux=t^2 for t>0, XER
u=cosx for t=0, XER



I've actually never seen a transportation problem like this and any help would be greatly appreciated, thank you!
 
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  • #2
Have you learned about the method of characteristics? Suppose the right hand side were zero. Would you be able to solve the problem then?

Chet
 
  • #3
To eliminate the RHS, consider U = V(x, t) +T(t) where V satisfies the homogeneous equation and the same boundary condition.
 

FAQ: PDE for IVP on R for a transport equation

What is a PDE for IVP on R for a transport equation?

A PDE (partial differential equation) for IVP (initial value problem) on R (the set of real numbers) for a transport equation is a mathematical equation that describes the evolution of a quantity over time and space, where the initial value is known and the equation involves derivatives with respect to both time and space variables.

What is the purpose of solving a PDE for IVP on R for a transport equation?

The purpose of solving a PDE for IVP on R for a transport equation is to understand the behavior and evolution of a physical system, such as the movement of a fluid or the propagation of a wave, by determining the values of the quantity at different points in time and space.

What are the key components of a PDE for IVP on R for a transport equation?

The key components of a PDE for IVP on R for a transport equation include the dependent variable, which represents the quantity being studied, the independent variables of time and space, and the partial derivatives that describe the rate of change of the dependent variable with respect to the independent variables.

What methods can be used to solve a PDE for IVP on R for a transport equation?

There are several methods that can be used to solve a PDE for IVP on R for a transport equation, including analytical methods such as separation of variables and numerical methods such as finite difference or finite element methods.

How do initial and boundary conditions affect the solution of a PDE for IVP on R for a transport equation?

Initial and boundary conditions provide additional information about the system being studied and are necessary to obtain a unique solution to the PDE for IVP on R for a transport equation. The initial conditions specify the value of the dependent variable at the starting point, while the boundary conditions define the behavior of the dependent variable at the boundaries of the system.

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