Questions tagged [boundary-value-problem]

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

1,013 questions
269 views

146 views

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 ...
184 views

209 views

Integrating a sum of delta functions?

I know that the "hand-wavy" definition of the $\delta (x)$ function is $$\delta(x) = \begin{cases} \infty &\quad\ x=0 \\ 0 &\quad\text{otherwise} \end{cases}$$ ...
355 views

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 ...
226 views

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 ...
232 views

335 views

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 $\phi$ ...
49 views

180 views

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 ...
112 views

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{...
151 views

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 ...
380 views

128 views

379 views

Is Helmholtz equation in arbitrary regular polygon solvable in closed form?

A Helmholtz equation $\Delta f=-\lambda^2 f$ with Dirichlet boundary conditions can easily be solved in a square and also not too hard to solve in equilaterial triangle. In both cases the solution is ...
93 views

955 views

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 ...
124 views

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 ...
1k views

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 ...
266 views

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 ...
134 views

How solve inhomogeneous ODE with boundary conditions?

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 ...
86 views

338 views

Finding the frequencies of vibration of a drum; PDE

I want to find the frequencies of vibration of a circular and square drum. To do this, I need to solve a 2-dimensional wave equation (PDE) with boundary conditions. Every method that I have ...
274 views

Derive $u(x,t)$ as a solution to the initial/boundary-value problem.

Given $g : [0,\infty) \to \mathbb{R}$, with $g(0)=0$, derive the formula $$u(x,t)=\frac{x}{\sqrt{4\pi}}\int_0^t \frac 1{(t-s)^{3/2}}e^{-\frac{x^2}{4(t-s)}}g(s)\,ds$$ for a solution of the initial/...