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.

Let $~I = (a, b) ⊆ \mathbb{R}$ be an interval. Let $p, q, r : (a, b) → \mathbb{R}~$ be continuous functions. Consider the linear second order equation given by $$y′′ + p(x)y′ + q(x)y = r(x), \qquad a < x < b.$$Corresponding to this ODE, there are four important kinds of (linear) boundary conditions. They are given by

$1.\quad$Dirichlet or First kind :$$y(a) = η_1,\quad y(b) = η_2,$$ $2.\quad$Neumann or Second kind : $$y′(a) = η_1,\quad y′(b) = η_2,$$ $3.\quad$ Robin or Third or Mixed kind : $$α_1y(a) + α_2y′(a) = η_1, \quad β_1y(b) + β_2y′(b) = η_2,$$ $4.\quad$ Periodic : $$y(a) = y(b),\quad y′(a) = y′(b).$$


There are three types of boundary conditions commonly encountered in the solution of partial differential equations:

$1.\quad$Dirichlet boundary conditions specify the value of the function on a surface} $$T=f(r,t),$$

$2.\quad$ Neumann boundary conditions specify the normal derivative of the function on a surface, $$\frac{\partial T}{\partial n}=\hat{n}\cdot \delta T=f(\vec{r},t), $$ $3.\quad$Robin boundary conditions for an elliptic partial differential equation in a region $\Omega$, Robin boundary conditions specify the sum of $~\alpha u~$ and the normal derivative of $u=f$ at all points of the boundary of $\Omega$, with $\alpha$ and $f$ being prescribed.

References:

"https://en.wikipedia.org/wiki/Boundary_value_problem"

"http://www.math.iitb.ac.in/~siva/ma41707/ode6.pdf"

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Robin BC in the 1D wave equation

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