# Questions tagged [boundary-value-problem]

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

504 questions
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 ...
151 views

### Monotonic convergence of Newton's method for boundary value problems

I’m interested in solving nonlinear elliptic boundary value problems of the type $$-a\Delta u + f\left(u\right) = 0, \\ u\big\vert_\Gamma = u_0$$ by Newton’s method when its convergence is global ...
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

63 views

955 views

### Weak form of steady Navier-Stokes equations with special boundary condition

Suppose we want to solve the steady low-Mach-number Navier-Stokes equations coupled with a passive scalar $\xi$, which read: \begin{align} \nabla \cdot \rho \mathbf{v} & = 0,\\ \rho \mathbf{v} \...
106 views

### Sufficient Boundary Condition to a General PDE on a General Domain

We know that for an ODE of $n^{th}$ order we need $n$ different boundary conditions. In PDEs, for example, for Laplace equation $\nabla^2 U=0$ (which is a second order PDE) we need only one B.C. (e.g ...
694 views

### Conditions for solvability of Poisson's equation with Neumann boundary condition

Suppose I have: $$\begin{cases}-\Delta u= f, &\text{ on } \Omega\\ \nabla u \cdot n = g &\text{ on } \partial \Omega\\ \int_\Omega u = \operatorname{const}. \end{cases}$$ I'm supposed to ...
27 views

I am given the PDE, $\ u_t=u_{xx},$ with boundary conditions $u(0,t)=A, \ u(1,t)=B$ and $u(x,0)=f(x)$. I have found the solution of this PDE is u(x,t)=A+(B-A)x+\sum_{n=1}^{\infty}B_ne^{n^2\pi^2 t}\... 0answers 36 views ### How to solve wave equations with boundary condition u_x(0,t)=h(t)? \begin{align*} u_{tt}-c^2u_{xx}=0, x>0\\ u(x,0)=u_t(x,0)=0\\ u_x(0,t)=\frac{t}{1+t^2},t>0 \end{align*} According to the textbook, I should look for solutions in the form u(x,t)=F(x-ct) and ... 0answers 73 views ### Uniqueness of the potential flow past a cylinder I have a question regarding the uniqueness of the potential flow past a cylinder. Consider a two dimensional uniform potential flow in x_1-direction past the cylinder B_R = \{ x = (x_1, x_2) \in \... 0answers 90 views ### Converting between Solution forms using Green's Functions in Linear Differential Equation EDIT: Bounty is over tomorrow so I tried to clean up the question a bit, and put the additional work below as optional to read. I summarized the current results and the solution form I am trying to ... 0answers 61 views ### PDE Similarity Solutions Boundary Value Problem and Solution: Explanation of Steps Requested I have the following similarity solutions problem and solution: Problem u_t = ku_{xx} for all x > 0, with u_x (0, t) = 1, u(x, t) \to 0 as x \to \infty, and u(x, 0) = 0 for x &... 0answers 33 views ### Help Solving Textbook Heat Conduction Laplace Transforms PDE Problem 2 This problem is related to this question. If you can answer this, then you might be able to also answer the other question, so please have a look at it. I am trying to solve the following problem:...
$\bullet$ I have the following heat equation with a time-periodic transport term: $$\kappa u_{xx} - a \sin(\omega t)u_x = cu_t$$ I'm considering a 1D domain over $-l<x<l$. I'd like to be able ...