The solution to the Cauchy heat problem for the given partial differential equation

$u_t - k u_{xx} + b u = 0, \quad -\infty < x < \infty, \quad t > 0,$

with the initial condition

$u(x, 0) = g(x), \quad -\infty < x < \infty,$

can be found using the Fourier transform approach. Here's the step-by-step derivation of the formula:

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1. Fourier Transform the PDE

Apply the Fourier transform with respect to $x$ to convert the PDE into an ordinary differential equation in $t$. Denoting the Fourier transform of $u(x, t)$ by $\hat{u}(\xi, t)$:

$\mathcal{F}\{u_t - k u_{xx} + b u \} = 0.$

Using the properties of Fourier transforms:

• $\mathcal{F}\{u_t\} = \frac{\partial \hat{u}}{\partial t}$,
• $\mathcal{F}\{u_{xx}\} = -(2\pi i \xi)^2 \hat{u} = -4\pi^2 \xi^2 \hat{u}$,
• $\mathcal{F}\{u\} = \hat{u}$,

the PDE becomes:

$\frac{\partial \hat{u}}{\partial t} + k(2\pi \xi)^2 \hat{u} - b\hat{u} = 0.$

Simplify:

$\frac{\partial \hat{u}}{\partial t} = -(4\pi^2 k \xi^2 + b)\hat{u}.$

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2. Solve the ODE in Fourier Space

This is a first-order linear ODE for $\hat{u}(\xi, t)$:

$\frac{\partial \hat{u}}{\partial t} + \lambda \hat{u} = 0, \quad \lambda = 4\pi^2 k \xi^2 + b.$

The solution is:

$\hat{u}(\xi, t) = \hat{u}(\xi, 0)e^{-\lambda t} = \hat{g}(\xi)e^{-(4\pi^2 k \xi^2 + b)t}.$

Here, $\hat{g}(\xi)$ is the Fourier transform of the initial condition $g(x)$.

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3. Inverse Fourier Transform

To find $u(x, t)$, apply the inverse Fourier transform:

$u(x, t) = \mathcal{F}^{-1}\{\hat{u}(\xi, t)\}.$

Substitute $\hat{u}(\xi, t)$:

$u(x, t) = \mathcal{F}^{-1}\{\hat{g}(\xi)e^{-(4\pi^2 k \xi^2 + b)t}\}.$

By the convolution theorem, the inverse Fourier transform results in a convolution of $g(x)$ with the kernel corresponding to $e^{-(4\pi^2 k \xi^2)t}$:

$u(x, t) = e^{-bt} \int_{-\infty}^\infty g(y) \frac{1}{\sqrt{4\pi kt}} \exp\left(-\frac{(x-y)^2}{4kt}\right) dy.$

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4. Final Formula

The solution to the Cauchy heat problem is:

$u(x, t) = \frac{e^{-bt}}{\sqrt{4\pi kt}} \int_{-\infty}^\infty g(y) \exp\left(-\frac{(x-y)^2}{4kt}\right) dy.$

This formula represents the evolution of the initial condition $g(x)$ under the influence of heat diffusion (with diffusivity $k$) and the exponential decay due to the constant $b > 0$.

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Key Observations:

1. The term $e^{-bt}$ accounts for the decay due to the positive constant $b$.
2. The Gaussian kernel $\frac{1}{\sqrt{4\pi kt}} \exp\left(-\frac{(x-y)^2}{4kt}\right)$ describes the heat diffusion process.

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Would you like additional details, such as:

1. How Fourier transforms work in solving PDEs?
2. Physical interpretations of $b$ and $k$?
3. How the initial condition $g(x)$ affects the solution?
4. Numerical examples or simulations of this solution?
5. Extensions to other types of boundary conditions?

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Tip: In problems involving heat or wave equations, Fourier transform methods are especially powerful for unbounded domains.