Boundary value

Boundary Value is a fundamental concept in the field of mathematics, specifically within the area of differential equations. It refers to the values a solution must take on the boundary of the domain. Boundary value problems are distinct from initial value problems, which specify values of the solution at the origin rather than the boundary.

Definition

In mathematics, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.

Types of Boundary Value Problems

There are several types of boundary value problems, including:

• Dirichlet problem: Also known as the first boundary value problem, it involves finding a solution that equals a given function on the boundary of the domain.
• Neumann problem: Also known as the second boundary value problem, it involves finding a solution whose derivative equals a given function on the boundary of the domain.
• Robin problem: Also known as the third boundary value problem, it involves finding a solution for which a linear combination of the solution and its derivative equals a given function on the boundary of the domain.

Applications

Boundary value problems have wide applications in various fields such as physics, engineering, and economics. They are used to model a number of physical phenomena including heat conduction, fluid flow, and electromagnetic fields.

See Also

• Differential equation
• Partial differential equation
• Dirichlet problem
• Neumann problem
• Robin problem