# Boundary Value Problems

Boundary value problem solvers for ordinary differential equations

Boundary value problems (BVPs) are ordinary differential equations that are subject to boundary conditions. Unlike initial value problems, a BVP can have a finite solution, no solution, or infinitely many solutions. The initial guess of the solution is an integral part of solving a BVP, and the quality of the guess can be critical for the solver performance or even for a successful computation. The `bvp4c` and `bvp5c` solvers work on boundary value problems that have two-point boundary conditions, multipoint conditions, singularities in the solutions, or unknown parameters. For more information, see Solving Boundary Value Problems.

## Functions

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 `bvp4c` Solve boundary value problem — fourth-order method `bvp5c` Solve boundary value problem — fifth-order method `bvpinit` Form initial guess for boundary value problem solver
 `bvpget` Extract properties from options structure created with bvpset `bvpset` Create or alter options structure of boundary value problem
 `deval` Evaluate differential equation solution structure `bvpxtend` Form guess structure for extending boundary value solutions

## Topics

Solving Boundary Value Problems

Background information, solver capabilities and algorithms, and example summary.

Solve BVP with Two Solutions

This example uses `bvp4c` with two different initial guesses to find both solutions to a BVP problem.

Solve BVP with Unknown Parameter

This example shows how to use `bvp4c` to solve a boundary value problem with an unknown parameter.

Solve BVP with Multiple Boundary Conditions

This example shows how to solve a multipoint boundary value problem, where the solution of interest satisfies conditions inside the interval of integration.

Solve BVP with Singular Term

This example shows how to solve Emden's equation, which is a boundary value problem with a singular term that arises in modeling a spherical body of gas.

Solve BVP Using Continuation

This example shows how to solve a numerically difficult boundary value problem using continuation, which effectively breaks the problem up into a sequence of simpler problems.

Verify BVP Consistency Using Continuation

This example shows how to use continuation to gradually extend a BVP solution to larger intervals.