Analyzing PDE BVP for ut + ux = 0 with given boundary condition

• Nicolaus
In summary, the conversation discusses a problem involving a well-posed PDE BVP with the condition of u(t,x) = x on x^2 + y^2 = 1. The proposed solution is u(t,x) = f(x-t) with an initial condition of t(0) = 0 and x(0) = xo. The variable s is used to show that ∂(u(t(s), x(s))/∂s = 0. It is questioned whether this solution is well-posed since f must be a function of (x-t). There is also a question about the accuracy of the given condition x^2 + y^2 = 1.
Nicolaus

Homework Statement

1. ut +ux = 0 subject to u(t,x) = x on x^2 + y^2 = 1

Is this a well-posed PDE BVP?

The Attempt at a Solution

This is an easy one to solve: u(t,x) = f(x-t)
I let t(0) = 0 as an initial condition, and so t=s => x= ts + xo, where x(0) = xo
s is the variable such that ∂(u(t(s), x(s))/∂s = 0
If I let u(t,x) = x = f(x-t), would this not be well-posed since f must be a function of (x-t)?

In the problem setup, is the condition really ## x^2 + y^2 = 1 ##, or did you mean ## x^2 + t^2 = 1 ##?

1. What is a partial differential equation (PDE)?

A partial differential equation (PDE) is a mathematical equation that involves multiple independent variables and their partial derivatives. It is used to describe various physical phenomena in fields such as physics, engineering, and economics.

2. What are the characteristics of a PDE?

The three main characteristics of a PDE are the order, linearity, and type. The order of a PDE is determined by the highest derivative involved, the linearity refers to whether the equation is linear or nonlinear, and the type refers to the type of dependent variable involved (e.g. scalar, vector, or tensor).

3. How are characteristics used to classify PDEs?

Characteristics are used to classify PDEs into different types, including elliptic, parabolic, and hyperbolic equations. Elliptic equations have no time dependence and are used to model steady-state problems, parabolic equations have one time variable and are used to model diffusion and heat transfer, and hyperbolic equations have two time variables and are used to model wave-like phenomena.

4. What are the boundary and initial conditions for a PDE?

Boundary conditions are the conditions that must be satisfied at the boundaries of the domain in which the PDE is being solved. Initial conditions, on the other hand, specify the values of the dependent variable at the initial time or point in the domain. These conditions are necessary to uniquely define a solution to a PDE.

5. How are characteristics used in solving PDEs?

Characteristics play a crucial role in solving PDEs, as they determine the type of equation and the appropriate solution method. For example, elliptic equations can be solved using numerical methods such as finite differences, while hyperbolic equations require methods such as finite volume or finite element methods. The characteristics of a PDE also provide insight into the behavior and properties of the solution.

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