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# 数学代写|实分析代写Real Analysis代考|Math444 Applications to some partial differential equations

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## 数学代写|实分析代写Real Analysis代考|Applications to some partial differential equations

We mentioned earlier that a crucial property of the Fourier transform is that it interchanges differentiation and multiplication by polynomials. We now use this crucial fact together with the Fourier inversion theorem to solve some specific partial differential equations.

The time-dependent heat equation on the real line
In Chapter 4 we considered the heat equation on the circle. Here we study the analogous problem on the real line.

Consider an infinite rod, which we model by the real line, and suppose that we are given an initial temperature distribution $f(x)$ on the rod at time $t=0$. We wish now to determine the temperature $u(x, t)$ at a point $x$ at time $t>0$. Considerations similar to the ones given in Chapter 1 show that when $u$ is appropriately normalized, it solves the following partial differential equation:
$(8)$
$$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2},$$
called the heat equation. The initial condition we impose is $u(x, 0)=f(x)$.

## 数学代写|实分析代写Real Analysis代考|The steady-state heat equation in the upper half-plane

The equation we are now concerned with is
$$\triangle u=\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$$
in the upper half-plane $\mathbb{R}{+}^2={(x, y): x \in \mathbb{R}, y>0}$. The boundary condition we require is $u(x, 0)=f(x)$. The operator $\triangle$ is the Laplacian and the above partial differential equation describes the steady-state heat distribution in $\mathbb{R}{+}^2$ subject to $u=f$ on the boundary. The kernel that solves this problem is called the Poisson kernel for the upper half-plane, and is given by
$$\mathcal{P}_y(x)=\frac{1}{\pi} \frac{y}{x^2+y^2} \quad \text { where } x \in \mathbb{R} \text { and } y>0 \text {. }$$

## 数学代写|实分析代写Real Analysis代考|Applications to some partial differential equations

$(8)$
$$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2},$$

## 数学代与写|实分析代写Real Analysis代考|The steady-state heat equation in the upper half-plane

$$\triangle u=\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$$

$$\mathcal{P}_y(x)=\frac{1}{\pi} \frac{y}{x^2+y^2} \quad \text { where } x \in \mathbb{R} \text { and } y>0 \text {. }$$

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