Questions tagged [boundary-value-problem]

Question 1

For the system $$ \begin{cases} x'(t)=-a(t) \cdot x(t)\cdot y(t)+y(t)-\varepsilon \cdot (z(t)+y(t))\cdot y(t)\\ y'(t)=k\cdot z(t)-y(t)+\varepsilon \cdot(y(t))^2=k\cdot (1-x(t))-(k+1)y(t)+\varepsilon \...

Question 2

I am trying to proof this by induction Proposition (Boundary values under clamped knots): Let $\{t_i\}_{i=1}^{m=n+k}$ be a clamped knot sequence of order $k$ on the interval $[a,b]$, that is, $$ t_1 = ...

Question 3

I am reviewing the method of Green's functions for solving boundary value problems of the form: $$ \mathcal{L}u = f(x), \quad \text{with} \quad u(0) = A, \quad u(L) = B, $$ where $\mathcal{L}$ is a ...

Question 4

Let $M$ be a compact Riemannian manifold with connected smooth boundary $\Sigma$. Let $\Sigma_D$ and $\Sigma_N$ be two disjoint smooth domains of $\Sigma$ with $\Sigma = \overline{\Sigma_D\cup\Sigma_N}...

Question 5

Consider the Complex variable Partial Differential Equation: $$ \Delta^2 w =f $$ with boundary conditions $$ w=\varphi_0 ~~\text{and}~~ \partial_{\bar{z}}w =\varphi_1. $$ A unique solution to this PDE ...

Question 6

I've been reading THIS PAPER to handle a particular boundary value problem in my field, and the paper characterizes the finalized form of this problem as a "special problem of mathematical ...

Question 7

Suppose I start a Brownian motion $X_t$ at position $(0,0,\varepsilon)$ in $\mathbb R^3$. Suppose also I have a cone with its apex at the origin, given by $U=\{(x,y,z):\sqrt{x^2+y^2}< z\tan\alpha\}$...

Question 8

I'm trying to solve the following PDE using the Fourier series method: \begin{align} &\partial_t u(t, x) - t\partial^2 u(t, x) = 0 && x \in [0, \pi], \; t\in\mathbb{R}^+ \\ &u(t, 0) = ...

Question 9

Let $\Omega$ be a domain, and consider the Dirichlet problem $\Delta u=0$ in $\Omega$, $u=f$ in $\partial\Omega$. If $\Omega$ is bounded, then the maximum principle says that solutions, if they exist, ...

Question 10

Let $D\subset\mathbb{R}^3$ be a bounded $C^2$ domain. On its boundary $\partial D$, place two disjoint, tiny patches $S_\varepsilon$ and $T_\delta$ with diameters $\varepsilon,\delta\ll1$. Consider ...

Question 11

I apologize beforehand for the long question. Here is the case: A vertical pipe is in (assumed perfect) contact with the surrounding ground. We consider the initial temperature to be homogeneous for ...

Question 12

Consider the frequency domain problem $$\begin{align}v_{xx} -\lambda v &= \delta(x),\\-v_x(0)&=i\omega \alpha v(0),\\ v_x(1)=0,\end{align}$$ where $\lambda := -\omega^2$. I want to find the ...

Question 13

My question comes from Example 3.11 (p.184) of Applied Mathematics 4th ed by J. David Logan. The example is on finding a uniform approximation to a boundary value problem. Here's the relevant snippet ...

Question 14

I am learning about the Dirichlet-Problem, especially the probabilistic Solution by Kakutani. I am following chapter 3.1 from the book 'Brownian Motion' written by Peter Mörters and Yuval Peres , ...

Question 15

I am currently reading this paper covering a numerical scheme for solving the guiding-center Vlasov equation on a 2d square grid. The system is: $$\frac{\partial \rho}{\partial t} + \textbf{v}_D\cdot \...

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Questions tagged [boundary-value-problem]

Proving bounds for differential equation system?

Boundary values for b splines under clamped knots

Why must the Green's function for a linear ODE satisfy homogeneous boundary conditions?

$H^{\frac{3}{2}}$-regularity of a specific elliptic mixed boundary valued problem.

Complex variable Biharmonic BVP

Does anyone know the identification of this boundary value problem? The coupled boundary conditions are what make it special

Boundary value problem for Laplace's equation / probability a BMP leaves cone via the ice cream rather than the biscuit

Problem on choosing the sign of the solution of a PDE using Fourier series

How unique are unbounded solutions to the Dirichlet Problem?

Capacity–duty flux split for gated vs fixed window

Transient cylindrical heat equation in 1D, boundary conditions and Bessel functions

Analytical solution to the following BVP

Dominant balance in finding a uniform approximation to a BVP

Example for a domain which does not satisfy the Poincaré-Cone-Condition but has a unique solution for the Dirichlet-Problem

Definition of 'natural boundary conditions' for Vlasov problem

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