Consider the Cauchy problem
U, = kUxx, XER, t > 0; u(x, 0) =Ф(x), X E R.
( Solve the above problem for (x) = 6, tIl S1
(il) For @(x) = (e*, *≥0
x<0*

The given Cauchy problem is:

\[
U_t = kU_{xx}, \quad x \in \mathbb{R}, \ t > 0; \quad U(x, 0) = \Phi(x), \quad x \in \mathbb{R}.
\]

This is a **heat equation** with initial condition \( U(x, 0) = \Phi(x) \). We will solve this problem for two different initial conditions:

1. \( \Phi(x) = \delta(x) \) (Dirac delta function)
2. \( \Phi(x) = e^x \) for \( x \geq 0 \) and \( \Phi(x) = 0 \) for \( x < 0 \).

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### (i) Solution for \( \Phi(x) = \delta(x) \)

The **fundamental solution** (also called the heat kernel) of the heat equation is the solution when the initial condition is a Dirac delta function \( \delta(x) \). This is given by:

\[
U(x,t) = \frac{1}{\sqrt{4\pi kt}} \exp\left( -\frac{x^2}{4kt} \right).
\]

This is the Green's function for the heat equation and represents how heat diffuses from a point source at \( x = 0 \) over time.

Thus, for \( \Phi(x) = \delta(x) \), the solution is:

\[
U(x,t) = \frac{1}{\sqrt{4\pi kt}} \exp\left( -\frac{x^2}{4kt} \right).
\]

### (ii) Solution for \( \Phi(x) = \begin{cases} e^x, & x \geq 0, \\ 0, & x < 0 \end{cases} \)

In this case, the initial condition is a piecewise function, and we will use **Duhamel's principle** or convolution with the heat kernel to solve this.

The solution of the heat equation for a general initial condition \( \Phi(x) \) is given by the convolution of the initial condition with the heat kernel:

\[
U(x,t) = \int_{-\infty}^{\infty} \Phi(\xi) \frac{1}{\sqrt{4\pi kt}} \exp\left( -\frac{(x-\xi)^2}{4kt} \right) d\xi.
\]

Substitute the given \( \Phi(x) \):

\[
U(x,t) = \int_0^{\infty} e^{\xi} \frac{1}{\sqrt{4\pi kt}} \exp\left( -\frac{(x-\xi)^2}{4kt} \right) d\xi.
\]

This integral is more complex but can be evaluated using techniques from convolution integrals or error functions. The exact solution involves careful handling of the boundary between \( x \geq 0 \) and \( x < 0 \). The result is typically expressed in terms of complementary error functions or an integral representation.

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Do you want more details or have any questions?

Here are 5 related questions to expand on this:
1. How is the heat equation derived from physical principles?
2. What are some properties of the heat kernel?
3. How does the solution change if \( k \) is variable?
4. Can you explain the method of separation of variables for PDEs?
5. How is the error function involved in the solution of the heat equation?

**Tip:** When solving PDEs, look for symmetry in the problem. Symmetry often leads to simpler solutions.

Consider the Cauchy problem: U_t = kU_xx, x ∈ ℝ, t > 0; U(x, 0) = Φ(x), x ∈ ℝ. Solve the above problem for (i) Φ(x) = δ(x), (ii) Φ(x) = e^x, x ≥ 0, Φ(x) = 0, x < 0.

Explore the solution to the Cauchy problem for the heat equation U_t = kU_xx with two different initial conditions: the Dirac delta function and a piecewise exponential function. This detailed explanation covers key concepts like Green's function, Duhamel's principle, and convolution integrals.

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