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Questions tagged [boundary-value-problem]

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For questions concerning the properties and solutions to the boundary-value problem for differential equations. By a Boundary value problem, we mean a system of differential equations with solution and derivative values specified at more than one point. Most commonly, the solution and derivatives are specified at just two points (the boundaries) defining a two-point boundary value problem.

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For the system $ 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 \cdot(y(t))^2 $ ...
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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, $$ t_1 = ...
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Let $M$ be a compact Riemannian manifold with connected smooth boundary $\Sigma$. Let $\Sigma_D$ and $\Sigma_N$ be two disjoint smooth domains of $\Sigma$ with $\Sigma = \overline{\Sigma_D\cup\Sigma_N}...
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I've been reading THIS PAPER to handle a particular boundary value problem in my field, and the paper characterizes the finalized form of this problem as a "special problem of mathematical ...
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I'm trying to solve the following PDE using the Fourier series method: \begin{align} &\partial_t u(t, x) - t\partial^2 u(t, x) = 0 && x \in [0, \pi], \; t\in\mathbb{R}^+ \\ &u(t, 0) = ...
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Let $D\subset\mathbb{R}^3$ be a bounded $C^2$ domain. On its boundary $\partial D$, place two disjoint, tiny patches $S_\varepsilon$ and $T_\delta$ with diameters $\varepsilon,\delta\ll1$. Consider ...
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Consider the frequency domain problem $$\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 ...
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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 , ...
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