3.3 KiB
| date | type |
|---|---|
| 04.09.2024 | math |
Integral Curve
- What is it?
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.
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:
\frac{dx}{dt} = P(x(t), y(t)), \quad \frac{dy}{dt} = Q(x(t), y(t)).
Example:
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:
\frac{dx}{dt} = y, \quad \frac{dy}{dt} = -x.
Solving this system, we get solutions of the form:
x(t) = A \cos(t) + B \sin(t), \quad y(t) = -A \sin(t) + B \cos(t),
which represent circles centered at the origin.
Cauchy Problem
The Cauchy problem is a fundamental concept in the study of partial differential equations (PDEs1}). It refers to the problem of finding a solution to a PDE given initial conditions along a certain hypersurface2.
- How do we solve it?
To solve a Cauchy problem for a PDE, we generally follow these steps:
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Formulate the PDE and Initial Conditions: 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:
\frac{\partial^2 u}{\partial t^2} - c^2 \frac{\partial^2 u}{\partial x^2} = 0.The initial conditions could be:
u(x, 0) = f(x), \quad \frac{\partial u}{\partial t}(x, 0) = g(x),where$f(x)$and$g(x)$are given functions.
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Find a General Solution: 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:
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. -
Apply the Initial Conditions: Substitute the initial conditions into the general solution to find specific forms of the arbitrary functions or constants.
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Verify the Solution: Check that the obtained solution satisfies both the PDE and the initial conditions.
Example: For the heat equation in one dimension:
\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2},
with initial condition$u(x, 0) = f(x)$, the solution is:
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,
which uses a convolution of the initial condition$f(x)$with a Gaussian kernel.