Ordinary differential equations Definition

Robin boundary condition

Builds on Boundary value problem Dirichlet boundary condition Neumann boundary condition Normal vector

Definition

Let $\Omega \subseteq \R^d$ be a bounded domain, that is, a connected open set, with boundary $\partial \Omega$ of class $C^1$ and outward unit normal vector $\nu$ . Let $c_1, c_2$ be nonzero real constants and let $g \colon \partial \Omega \to \R$ be a given function. For an equation for $u$ on $\Omega$ , the Robin boundary condition is $c_1\, u(x) + c_2\, \frac{\partial u}{\partial \nu}(x) = g(x) \quad \text{for all } x \in \partial \Omega.$ For a second-order ordinary differential equation on $[a, b]$ , where the outward normal derivatives are $-y'(a)$ and $y'(b)$ , it reads $c_1\, y(a) - c_2\, y'(a) = g_a, \qquad c_1\, y(b) + c_2\, y'(b) = g_b.$

Remarks

If $c_2 = 0$ , the condition becomes a Dirichlet condition; if $c_1 = 0$ , it becomes a Neumann condition. Some authors allow these cases and also let $c_1, c_2$ be functions on the boundary.

Example: $y'' = 0$ on $[0, 1]$ with $y(0) - y'(0) = 0$ and $y(1) + y'(1) = 3$ , so $c_1 = c_2 = 1$ . The function $y = 1 + x$ gives $y(0) - y'(0) = 1 - 1 = 0$ and $y(1) + y'(1) = 2 + 1 = 3$ , so it solves this boundary value problem.

Non-example: the conditions $y(0) = 0$ and $y(1) = 0$ are not of Robin type, because no derivative appears; they are Dirichlet conditions.

A Sturm–Liouville problem usually has separated conditions $\alpha_1 y(a) + \alpha_2 y'(a) = 0$ and $\beta_1 y(b) + \beta_2 y'(b) = 0$ , which include the Robin case.

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Tag	`Y19WM1EMS7`
URI	`m2://Y19WM1EMS7`
Kind	definition
Writing	W0 stub
MSC	34B05, 35J25, 34B24
Defines	Robin boundary condition
Uses	Boundary value problem, Dirichlet boundary condition, Neumann boundary condition, Normal vector
Changed	2 November 2026

Robin boundary condition

Definition

Remarks

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