# Questions tagged [boundary-value-problem]

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

53 questions
18k views

### Difference between “essential boundary conditions” and “natural boundary conditions”?

In a boundary value problem, what's the difference between "essential boundary conditions" and "natural boundary conditions"?
575 views

### Minimizing a functional with a free boundary condition

Find the extremals of the functional $$\text{J}(y)= y^2(1) + \int_0^1 y'^2(x)dx , \ \ y(0)=1.$$ Answer: $y(x)=1-\frac{1}{2}x$ My solution: $F (x,y,y')=y'^2(x)$ After solving the Euler ...
682 views

1k views

### Value of $u(0)$ of the Dirichlet problem for the Poisson equation

Pick an integer $n\geq 3$, a constant $r>0$ and write $B_r = \{x \in \mathbb{R}^n : |x| <r\}$. Suppose that $u \in C^2(\overline{B}_r)$ satises \begin{align} -\Delta u(x)=f(x), & \qquad x\...
320 views

178 views

### Solve Boundary Value Problem for $y''+ y' + e^xy = f(x)$

Consider to solve Boundary Value Problem : $y''+ y' + e^xy = f(x)$ with $0 < x < 1$ and $y(0)=y(1)=0$ with exact solution $y(x) = \sin \pi x$ $f(x)=(e^x- \pi^2)\sin \pi x + \pi \cos \pi x$...
707 views

### Laplace Equation on the Corners and Boundary of a Rectangle?

Consider for some rectangle $[a,b] \times [c,d] \in \mathbb{R}^2$, we have a generic boundary value problem: \begin{equation*} \begin{cases} \frac{\partial ^2 u}{\partial x ^2}+\frac{\partial ^2 u}{\...
815 views

### Uniqueness of a holomorphic function with certain boundary values on an arc

Is it true that if a holomorphic function in the unit disk converges uniformly to the $0$ function some connected arc of the unit circle, this function is globally null? If that is true, this would ...
818 views

### 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 ...
68 views

1k views

### Biharmonic Equation on a square (fourier series solution needed)

$\nabla^{4}f=1$ for $f$ defined in a square from $x\in[-1,1]$ and $y\in[-1,1]$ The boundary conditions are: $f=0,f_{xx}=0$ on $x=\pm 1$ $f=0,f_{yy}=\mp 1$ on $y=\pm 1$ I intend to solve this ...
56 views

86 views

### Solution of $y''(x) -k = \delta(x-x_0)y(x)$

I need to solve following differential equation $y''(x) -k = y\delta(x-x_0)$ subject to boundary conditions \begin{eqnarray} y(x=-a) = 0 \\ y(x=b) = p \end{eqnarray} I am not sure if it is possible ...