Problem 14 - Applied Partial Differential Equations

Question

Show that the "backwards" heat equation $$ \frac{\partial u}{\partial t}=-k \frac{\partial^{2} u}{\partial x^{2}} $$ suhject to \(u(0, t)=u(L, t)=0\) and \(u(x, 0)=f(x)\), is not well posed. [Hint: Show that if the data are changed an arbitrarily small amount, for example $$ f(x) \longrightarrow f(x)+\frac{1}{n} \sin \frac{n \pi x}{L} $$ for large \(n\), then the solution \(u(x, t)\) changes by a large amount.]

Step-by-Step Solution

Answer

Because the solution does not have continuous dependence on the initial data, the backwards heat equation is not a well-posed problem.

Step 1: Analyzing and Setting up the Equation

We start by considering the backwards heat equation which is given as \[ \frac{\partial u}{\partial t}=-k \frac{\partial^{2} u}{\partial x^{2}} \] Subject to \[ u(0, t)=u(L, t)=0 \] and \[ u(x, 0)=f(x) \] Now, proceed by making an arbitrarily small change to the initial condition \(f(x)\) by introducing a tweak as given: \[ f(x) \rightarrow f(x)+\frac{1}{n} \sin \frac{n \pi x}{L} \] for large n

Step 2: Checking the Unbounded Variation

Points of interest are the solutions to the backwards heat equation with this new initialization. Now solving the equation with \( f(x)=\frac{1}{n} \sin \frac{n \pi x}{L} \), while keeping in mind the boundary conditions, we get the solution: \[ u(x, t) = e^{k(n \pi /L)^2 t} \sin (n \pi x/L) \] We can observe that as \( t \rightarrow 0^+ \), this value increases without bound for fixed \( x \), which clearly contradicts the concept of continuous dependence on the initial data.

Step 3: Drawing Conclusion

Given that a minor modification in function \( f(x) \) at \( t=0 \) led to an unbounded alteration in solution \( u(x, t) \) as \( t \rightarrow 0^+ \), we can conclude that the backwards heat equation is not well-posed, because it does not meet the criteria of the solution depending continuously on the data.

Key Concepts

Well-posedness

Understanding whether a mathematical problem is well-posed is crucial. The concept of well-posedness comes from the French mathematician Jacques Hadamard. This idea is primarily used in the context of differential equations. A problem is deemed well-posed if it satisfies three key criteria:

• A solution exists.
• The solution is unique.
• The solution's behavior changes continuously with the initial conditions.

If any of these criteria fail, we call the problem ill-posed. In the case of the backwards heat equation example, a small change in initial conditions (like adding a small sinusoidal function) caused drastic changes in the solution.
This illustrates a lack of continuous dependency, leading us to conclude that the backwards heat equation is not well-posed. Problems that are ill-posed can be difficult to solve and may exhibit large variations in outcome due to small changes in input.

Partial Differential Equations

Partial differential equations, or PDEs, are fundamental in modeling a variety of phenomena in science and engineering. These equations contain unknown multivariable functions and their partial derivatives. The power of PDEs lies in their ability to describe systems with varying conditions over continuous variables such as space and time.
In general, PDEs can be classified based on their characteristics such as linearity, order, and homogeneity. A common form of PDEs is the heat equation, which models the distribution of heat (or variation in temperature) in a given region over time.

• Order: The order of a PDE is determined by the highest derivative order it contains.
• Linearity: A linear PDE implies all terms are of the first degree in the function and its derivatives.

The backwards heat equation is a PDQ known for its ill-posed nature, highlighting the complexity and variability present in partial differential equations.

Initial Boundary Value Problem

An initial boundary value problem (IBVP) in the context of PDEs combines initial conditions with boundary conditions to specify a unique solution. Initial conditions define the state of a system at the beginning of the problem, while boundary conditions specify the behavior of the system at the given spatial limits.
For example, in a heat equation problem, these conditions can include the initial temperature distribution (initial condition) and the temperatures at the ends of a rod (boundary conditions).

• Initial Condition: Specifies the starting point or state of the system.
• Boundary Condition: Establishes constraints at the spatial boundaries of the problem.

Solving an IBVP requires adhering to both these conditions to ensure the solution is accurate. In the provided exercise, the conditions were set as:

• Initial condition: \( u(x, 0) = f(x) \)
• Boundary conditions: \( u(0, t) = u(L, t) = 0 \)

These specifications help generate solutions, providing a framework necessary to solve PDEs and understanding how they evolve over time and space.