# Questions tagged [boundary-value-problem]

For questions concerning the properties and solutions to the boundary-value problem for differential equations.

1,035 questions
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### How are boundary conditions formally captured by the jet bundle approach to differential equations?

In the jet bundle approach to differential equations https://en.wikipedia.org/wiki/Jet_bundle#Partial_differential_equations one identifies the equation with the set of a solution of the ...
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### Integrating a sum of delta functions?

I know that the "hand-wavy" definition of the $\delta (x)$ function is $$\delta(x) = \begin{cases} \infty &\quad\ x=0 \\ 0 &\quad\text{otherwise} \end{cases}$$ ...
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### Green Solution to Laplace Equation with Robin Boundary Conditions

Let's say that I know a solution for the Laplace equation in the whole plane: $$\nabla^2u(\mathbf{x})=0\quad \mathbf{x}\in\mathbb{R}^2$$ And I need a solution for the laplace equation in the ...
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### Examples on conceptual problems for eigenvalues in differential equations

I am currently holding a discussion class on diff eqs for engineers and I am looking for an interesting conceptual problem on eigenvalues in diff eqs. Most of the problems in 5 different books that I ...
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### Eigenvalues of Differential Equation with Boundary Condition

Here is a problem from my homework assignment that I am struggling with: Consider the differential equation $\frac{d^2\phi}{dx^2}+\lambda\phi=0$. Determine the eigenvalues $\lambda$ if $\phi$ ...
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### How is “continuous dependence on initial conditions” defined? And how to prove it?

EDIT: I tried to prove the continuous dependence of the problem by somehow use a weak Maximum principle and posted a modified Version of this post on mathoverflow: https://mathoverflow.net/questions/...
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### R.H.S. of Poisson equation localized $\Rightarrow$ Solution localized

Let $\Omega\subset\mathbb{R}^d$ a connected open set (which is not necessarily bounded). Assume that $f\in C_0^\infty(\Omega)$ with $\operatorname{supp}(f)\subset K,$ with $K$ compactand let $u$ be ...
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### Solution to Singular Free Boundary PDE

As part of my research, I have come across the following problem and I am trying to tackle it. Let $(X_t)_{0 \leq t \leq T}$ be a mean controlled Brownian Motion with the following dynamics \begin{...
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### Are there existence results for the heat equation on unbounded Lipschitz domain?

I am looking for a reference/ ideas on the following problem. Let $\Omega\subset\Bbb R^2$ be a Lipschitz domain (if it helps, the domain can be piecewise smooth with only one "kink", for example the ...
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### Green's function using method of images

I'm working on the exercises 1.42 in the textbook "Green's functions and boundary value problems" by Stakgold and Holst, used in a graduate class "Applied Mathematics". Show that the electrostatic ...
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### Due to numerical inaccuracy, the solution of a boundary value problems becomes negative

I treat a toy example to get my point across. In reality I have to deal with a much more complex model. Let us consider a one dimensional boundary value problem using the bvp5c solver in Matlab. Two ...
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### Choosing a boundary for integration

I have the following differential equation $$\frac{d}{dx}\left(\mu e^{cx}f(x)\right) = -\mu\left(\frac{a xe^{-cx}}{a x+x-1}\right)$$ that I am trying to integrate to find $f(x)$ with the boundary ...
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### How to solve Robin problem with general initial data on the half line?

Solve: $$u_t=ku_{xx}, \text{ for } t>0,x>0$$ $$u(x,0)=\phi(x)\text{ for }x>0$$ $$u_x(0,t)-hu(0,t)=0 \text{ for }x=0$$, where $h$ is constant. When $\phi(x)$ is $x$, and $h$ is $2$, we ...
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### Why is the function $\operatorname{Log}(G(t))$ Holder continuous?

I was reading the theory about the Riemann-Hilbert problem $\Phi^+(t)=G(t)\Phi^-(t)$ where $G(t)$ is a Holder continuous function on a closed curve $c$ with index $\operatorname{Ind}_cG(t)=0$. To ...
I'm trying to solve this ODE: $$y''-2xy'+3y=x^3$$ With the conditions: $$\lim_{x\to\pm\infty}e^{-x^2/2}y(x)=\lim_{x\to \pm\infty}e^{-x^2/2}y'(x)=0$$ The homogeneous part is Hermite's equation for ...