# Questions tagged [boundary-value-problem]

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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### What is the physical interpretation of $u_x(0,t)$, $u_x(1,t)$ and a negative diffusivity constant in 1D Heat Equation on $(0,1)$?

I'm looking for help with the following problem. We have the heat equation $u_t=Du_{xx}$ modelling the temperature across a conducting material, defined for $0<x<1, t>0$ with boundary ...
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### Boundary points of the set D={(x,y): |y |>= x}

What are the boundary points of the set in xy-plane $$D=\{(x,y): x^2+y^2<1, |y| \geqslant x\}$$ Should it be $bdry(D)=\{(x,y):x^2+y^2=1, |y|=x\}$ or $bdry(D)=\{(x,y):x^2+y^2=1, |y|\geqslant x\}$ ...
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### Solve a Laplace Boundary Value Problem

This is the easiest example from the book, but in the book everything is showed. So how would you solve this problem if you got it on a exam. I dont understand how to solve for $y$ after getting $x$. ...
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### Uniqueness of PDE via energy functional

Assume the pde: $$u_{tt}(t,x) = c^2u_{xx}(t,x) + \sigma u_{txx}(t,x) -\mu u_{t}(t,x), \quad x \in [0,L], t>0$$ $$u_x(t,0) = u(t,L) = 0$$ $$u(0,x) = \phi(x), u_t(0,x) = \theta(x), x\in[0,L]$$ ...
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### Shooting Method

I tried to solve this differential equation: $$y''+\frac{1}{x}y'-\frac{9}{x^2}y=0\text{, where }x \in (1,2), y'(1)=\alpha, y(2)=\beta$$ I should use shooting method but I have a problem with ...
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### Under what condition does this boundary value second order ODE have unique solution?

This is a problem from the textbook The Way of Analysis by Robert Strichartz. So all solutions to $x''(t)=-x(t)$ will have the general form $A\cos t+B\sin t$. Then the problem asks for which values of ...
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### Homogeous Boundary Value Problem (Wave Equation)

I am trying to solve the following boundary value problem, where $0<x<\pi$: $$\frac{\partial^2 u}{\partial t^2}=9\frac{\partial^2u}{\partial t^2}$$ \frac{\partial u}{\partial x}(0,t)=\frac{\...
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### Real analysis. Continuity of a product [duplicate]

Let $X$ $\subset$ $\mathbb{R}$ and $f,g:X\rightarrow\mathbb{R}$ two continuous bounded functions. Is the product $fg$ a uniformly continuous function? Prove or give a counterexample. I would say it ...
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### Truncation error of Crank-Nicolson Scheme with derivative boundary condition

I am handling a heat equation defined on $(t,x) \in [0,T]\times[0,1]$ with Robin boundary condition: \begin{equation} \begin{cases} -u_t + \frac{1}{2}u_{xx} = 0 \\ u(t,0) = u_1(t) \\ ...
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### Help to proof that squared integrable functions form a hilbert space.

Marcus here. I am currently trying to understand the concept Hilbert space, specifically for squared integrable functions. The definition of a hilbert space is "A inner product space \$( \langle \cdot ...
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### Book for Boundary Value ODEs

Can someone suggest me a book on Boundary Value Problems in ODEs, which start from the general theory, and then go on to specialize for self-adjoint values? All the books I have found discuss the self-...