89 lines
3.3 KiB
Markdown
89 lines
3.3 KiB
Markdown
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date: 04.09.2024
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type: math
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---
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## Integral Curve
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- **What is it?**
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An **integral curve** of a vector field is a curve that is tangent to the vector field at every point. In simpler terms, given a vector field (which can be thought of as arrows pointing in various directions), an integral curve is a path that follows the directions of these arrows.
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For a vector field $\mathbf{F}(x, y) = (P(x, y), Q(x, y))$ in 2D, an integral curve $\mathbf{r}(t) = (x(t), y(t))$ is a solution to the system of ordinary differential equations:
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$$
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\frac{dx}{dt} = P(x(t), y(t)), \quad \frac{dy}{dt} = Q(x(t), y(t)).
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$$
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**Example:**
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Consider a simple vector field defined by $\mathbf{F}(x, y) = (y, -x)$. The integral curves of this field are solutions to the differential equations:
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$$
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\frac{dx}{dt} = y, \quad \frac{dy}{dt} = -x.
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$$
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Solving this system, we get solutions of the form:
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$$
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x(t) = A \cos(t) + B \sin(t), \quad y(t) = -A \sin(t) + B \cos(t),
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$$
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which represent circles centered at the origin.
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## Cauchy Problem
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The Cauchy problem is a fundamental concept in the study of partial differential equations (PDEs[^1]}). It refers to the problem of finding a solution to a PDE given initial conditions along a certain hypersurface[^2].
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- **How do we solve it?**
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To solve a Cauchy problem for a PDE, we generally follow these steps:
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1. **Formulate the PDE and Initial Conditions:**
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Define the PDE and the initial conditions. The initial conditions are given on a hypersurface, such as a line (in 2D) or a plane (in 3D). For example, consider the wave equation in one dimension:
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$$
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\frac{\partial^2 u}{\partial t^2} - c^2 \frac{\partial^2 u}{\partial x^2} = 0.
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$$
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The initial conditions could be:
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$$
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u(x, 0) = f(x), \quad \frac{\partial u}{\partial t}(x, 0) = g(x),
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$$
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where$f(x)$and$g(x)$are given functions.
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2. **Find a General Solution:**
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Solve the PDE using a method that applies to the type of PDE (e.g., separation of variables, Fourier transforms, or characteristic methods). For the wave equation, the general solution can be written using d'Alembert's formula:
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$$
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u(x, t) = \frac{1}{2} \left( f(x - ct) + f(x + ct) \right) + \frac{1}{2c} \int_{x - ct}^{x + ct} g(s) \, ds.
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$$
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3. **Apply the Initial Conditions:**
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Substitute the initial conditions into the general solution to find specific forms of the arbitrary functions or constants.
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4. **Verify the Solution:**
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Check that the obtained solution satisfies both the PDE and the initial conditions.
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**Example:**
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For the heat equation in one dimension:
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$$
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\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2},
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$$
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with initial condition$u(x, 0) = f(x)$, the solution is:
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$$
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u(x, t) = \frac{1}{\sqrt{4 \pi \alpha t}} \int_{-\infty}^\infty e^{-\frac{(x - \xi)^2}{4 \alpha t}} f(\xi) \, d\xi,
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$$
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which uses a convolution of the initial condition$f(x)$with a Gaussian kernel.
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[^1]: Partial differential equation
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[^2]: A **hypersurface** is a generalization of the concept of a surface to higher dimensions. Similar to matrices in linear algebra. In 3D, it's a plane, in 2D it's a curve/line.
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