Intereting Posts

How to solve polynomials?
Probability to find the sequence “Rar!” in a random (uniform) bytes stream at a position $\le n$
Proving that $|x|^p,p \geq 1$ is convex
Prove that any two cyclic groups of the same order are isomorphic?
If $(y_{2n}-y_n) \to 0$ then $\lim_{n\to \infty} y_n$ exists
Verifying Carmichael numbers
Proof of equivalence of algebraic and geometric dot product?
For 3 numbers represented with $n$ bits in binary, how many bits is required for their product.
Help Understanding Complex Roots
Interpretation of $\frac{22}{7}-\pi$
Coloring $\mathbb R^n$ with $n$ colors always gives us a color with all distances.
Intuitive bernoulli numbers
Who is buried in Weierstrass' tomb?
$30$ points in a $5\times5$ square
Simplify series involving derivatives

We have the Laplace equation in polar coordinates:

$$u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta \theta}=0, 0 \leq r <a, 0 \leq \theta \leq 2 \pi$$

With the separation of variables, the solution is in the form $u(r \theta)=R(r) \Theta(\theta)$

Then after calculations, we get:

$$u(r, \theta)=\sum_{n=0}^{\infty}{[A_n \cos{(n \theta)}+B_n \sin{(n \theta)}]r^n}, 0 \leq \theta \leq 2 \pi, 0 \leq r <a$$

- What is a general solution to a differential equation?
- Particular solution to a Riccati equation $y' = 1 + 2y + xy^2$
- Green's function in a moving frame for a constant heat source
- self similar solution for porous medium equation 2
- How do you solve the Initial value probelm $dp/dt = 10p(1-p), p(0)=0.1$?
- Help with solving differential Equation using Exact Equation method

The boundary condition is $h(\theta)=u(r=a, \theta)$

$h(\theta)$ is a periodic function with period $ 2 \pi$, so we can write it as a Fourier series.

After calculations we get the following formula (Poisson formula):

$$u(r, \theta)=\frac{a^2-r^2}{2 \pi} \int_0^{2 \pi}{ \frac{h( \phi) d \phi}{a^2+r^2-2 a r \cos{(\theta-\phi)}}}$$

$$$$

$|r’|=a, |\overrightarrow{r}-\overrightarrow{r’}|^2=|\overrightarrow{r}|^2+|\overrightarrow{r’}|^2-2|\overrightarrow{r}||\overrightarrow{r’}| \cos{(\theta-\phi)}=r^2+a^2-2ar \cos{(\theta – \phi)}$

So we can write the equation also:

$$u(r, \theta)=\frac{a^2-r^2}{2 \pi a} \int_{|\overrightarrow{r’}|=a}{\frac{u(\overrightarrow{r’})ds}{|\overrightarrow{r}-\overrightarrow{r’}|^2}}(*)$$

Could you tell me how we get to the relation $(*)$??

$$$$

$\frac{ds}{a}= d \phi$

For $r=0$

$$u(0)=\frac{1}{2 \pi a} \int_{| \overrightarrow{r’}|=a}{u(\overrightarrow{r’})}ds$$

**This is the mean value of the field at the circumference of the circle.**

Could you explain me the sentence above?

- General Solution of a Differential Equation using Green's Function
- What exactly is steady-state solution?
- Solving a simple 2nd order ODE with boundary values
- applications of linear systems of differential equations
- Under what conditions can a function $ y: \mathbb{R} \to \mathbb{R} $ be expressed as $ \dfrac{z'}{z} $?
- Proof Strategy for a Dynamical System of Points on the Plane
- How do we deduce that $c_1 \phi_1(x)+c_2 \phi_2(x)$ is a solution of the specific initial value problem?
- If an IVP does not enjoy uniqueness, then there are infinitely many solutions.
- Show $f''+vf' +\alpha^2 f(1-f)=0$ has solutions satisfying $\lim_{x \to - \infty}f=0$ and $\lim_{x \to \infty}f=1$ given $v\leq -2\alpha < 0$
- Stability of periodic solution

For instance the following curve traces a circle of radius $a$:

$$\vec r(s) = a\cos \frac s {a} \hat \imath + a\sin \frac s {a} \hat \jmath$$

where in this case $s$ is the distance along the curve.

Integration of a function $f(\vec r) = f(r,\theta)$ over a circle with radius $a$ can then be written in the following ways:

$$\int_{|\vec r|=a} f(\vec r) |d\vec r| = \int_{|\vec r|=a} f(\vec r) ds = \int_{r=a} f(r,\theta) ds = \int_0^{2\pi} f(a,\theta)\ a d\theta$$

Substitute the relevant formulas to find (*).

The mean value of a function $f(\vec r)$ on a curve is:

$$\text{Mean value} = \frac 1 {\text{Length of curve}} \int_{\text{curve}} f(\vec r) ds $$

- Galois ring extension
- How calculate the probability density function of $Z = X_1/X_2$
- Finding an explicit formula for $a_n$ defined recursively by
- How prove this $|x_{p}-y_{q}|>0$
- What does it mean to say “a divides b”
- irreducibility of a polynomial over $\mathbb{F}_2$
- How to prove that $\det(A+B) ≥ \det A +\det B$?
- Definition of Sinc function
- Proving connectedness of the $n$-sphere
- Proof for the funky trace derivative : $d (\operatorname{trace} (ABA'C))$?
- Construction of special $\omega_1$-Aronszajn tree
- Show that $a_n<\sqrt{2}$ for every $n\in\mathbb{N}$.
- How many segments are there in the Cantor set?
- The dual of subspace of a normed space is a quotient of dual: $X' / U^\perp \cong U'$
- Is $\mathbb{E}\exp \left( k \int_0^T B_t^2 \, dt \right)<\infty$ for small $k>0$?