Boundary value problems constrain a solution at more than one point or along a boundary, making them structurally different from initial value problems.
Boundary value problem
Definition. A boundary value problem seeks a function satisfying a differential equation on a domain together with boundary conditions. For example, \(-u''(x)=f(x)\) on \([0,1]\), with \(u(0)=u(1)=0\).
Shooting method
Definition. Shooting converts a boundary value problem into an initial value problem with unknown initial slope or parameters. The unknowns are adjusted so the terminal boundary condition is met.
Finite difference discretization
Proposition. Replacing derivatives by finite differences turns a linear boundary value problem into a linear algebra problem. For second-order elliptic problems the resulting matrix often has sparse, structured, positive definite form.
Collocation
Definition. Collocation chooses an approximate function in a finite-dimensional space and enforces the differential equation at selected points. It connects interpolation, residual minimization, and finite element ideas.
Well-posedness first
Remark. A discretization can only be expected to converge to the correct object when the continuous boundary value problem is well posed. Boundary layers, nonunique solutions, and incompatible boundary data require special care.
Two-point boundary conditions
A second-order problem \[ -y''+p(x)y'+q(x)y=f(x),\qquad y(a)=\alpha,\quad y(b)=\beta \] is not an initial-value problem: boundary data constrain both ends, and existence can fail at eigenvalues of the associated homogeneous operator.
Shooting
Introduce unknown initial slope \(s=y'(a)\), integrate the IVP, and solve \[ \Phi(s)=y(b;s)-\beta=0. \] Newton shooting also integrates the sensitivity \[ z'=\partial_yF(x,y)z,\qquad z(a)=\partial_s y(a;s), \] to obtain \(\Phi'(s)\). Shooting is effective for nonstiff, well-conditioned problems but unstable when growing modes make the endpoint map sensitive.
Finite differences
On a uniform grid, \[ -y''(x_i) \approx-\frac{y_{i-1}-2y_i+y_{i+1}}{h^2}. \] For \(-y''+q(x)y=f\) with \(q\ge0\), the resulting tridiagonal matrix is symmetric positive definite.
Theorem. If \(y\in C^4[a,b]\), the centered finite-difference solution satisfies \[ \max_i|y(x_i)-y_i| \le Ch^2\|y^{(4)}\|_\infty \] for the coercive Dirichlet problem.
Proof. Taylor expansion gives local truncation error \(O(h^2)\). The discrete operator is an M-matrix and obeys a maximum-principle inverse bound independent of \(h\). Applying that stability bound to the error equation yields the result.
Collocation
Choose a finite-dimensional trial \(y_h\) and enforce the differential equation at collocation points. Polynomial collocation at Gaussian or Lobatto nodes yields high-order implicit Runge--Kutta and spectral BVP methods. Convergence requires approximation quality plus stability of the collocation operator; dense polynomial bases should use barycentric or orthogonal representations.
Multiple shooting and continuation
Multiple shooting breaks a long interval into subintervals, introduces separate initial data, and enforces continuity as algebraic constraints. It reduces exponential sensitivity of single shooting. Parameter continuation solves a sequence of nearby BVPs and can follow nonlinear branches through folds using pseudo-arclength constraints.