BvP

Boundary value problem (BVP) is a problem in mathematics that asks for a function to satisfy a differential equation together with specified boundary conditions. BVPs are a central object of study in analysis and its applications. They are distinguished from initial value problems (IVPs) by the presence of conditions at two or more points or on the boundary of a domain.

There are two major categories: ordinary differential equation BVPs (ODE-BVPs) and partial differential equation BVPs (PDE-BVPs). An ODE-BVP typically involves an equation on an interval [a, b] with conditions at a and/or b, such as y(a) = α and y(b) = β. A PDE-BVP involves a differential equation on a spatial domain with boundary values specified on the domain boundary. Boundary conditions can be Dirichlet (the solution’s value is specified), Neumann (the value of a derivative is specified), or Robin (a linear combination of the solution and its derivative is specified).

Existence and uniqueness results depend on the problem. For linear BVPs, well-developed theories (including Sturm–Liouville theory for second-order self-adjoint problems) provide conditions under which solutions exist and are unique. Nonlinear BVPs may have multiple, unique, or no solutions, and often require numerical methods or fixed-point theorems to study.

Solution methods include analytic techniques such as separation of variables, Green’s functions, and eigenfunction expansions for linear problems, as well as numerical approaches like finite difference methods, shooting methods, and finite element methods.

Applications of BVPs span physics and engineering to chemistry and biology, modeling steady-state and dynamic phenomena such as heat conduction, vibrations, electrostatics, quantum mechanics, and fluid flow.