ODE Initial-, Boundary-, and Eigenvalue Problems

An ordinary differential equation becomes a well-defined problem only after its interval, data, endpoint behavior, free parameters, and success test are specified. This card distinguishes the principal problem classes and maps each method to the evidence that can expose failure. For prerequisite treatment, see Differential Equations for Relativity. Numerical applications are organized under ODE, Boundary-Value, and Spectral Computation.

Classify the ODE problem

Write an $n$-component system as

$$y'=f(x,y,\lambda), \qquad a\le x\le b.$$

Problem	Defining data	Decisive question
Initial-value problem	All $n$ components of $y(a)$	Does the solution exist uniquely across the claimed interval, and does integration preserve the required accuracy or invariant?
Boundary-value problem	A total of $n$ independent conditions distributed between $a$ and $b$	Are the conditions compatible, and how sensitive is the solution to their data?
Eigenvalue problem	Homogeneous boundary conditions and a nonzero solution	Which values of $\lambda$ permit a solution? Its overall normalization remains free.
Resonance problem	Frequency-dependent ingoing, outgoing, regular, or decaying endpoint branches	Which continued spectral problem and analytic sheet define the complex root?

The same differential equation can realize different rows when its domain or boundary conditions change. NIST separates numerical initial-, boundary-, and Sturm–Liouville problems in NIST DLMF 2026, §§ 3.7(ii)–(iv). For a boundary problem, a small endpoint mismatch is not sufficient: the conditioning of the continuum problem and the interior defect also matter Ascher, Mattheij, and Russell 1995, ch. 3 § 3.4, pp. 110–126.

Ordinary, regular-singular, and irregular endpoints

For

$$y''+P(x)y'+Q(x)y=0,$$

$x_0$ is an ordinary point when $P$ and $Q$ are analytic there. It is regular singular when $(x-x_0)P(x)$ and $(x-x_0)^2Q(x)$ are analytic. Then

$$y=(x-x_0)^r\sum_{k=0}^{\infty}a_k(x-x_0)^k,$$

and the lowest-power coefficient gives the indicial equation. If the two indicial roots differ by an integer, a logarithm in the second solution is permitted but not forced. On a chosen branch, the Frobenius series converge at least up to the nearest other singularity of the differential equation in the complex plane; the actual radius may be larger, and analytic continuation is needed only beyond that radius. These qualifications follow from NIST DLMF 2026, § 2.7(i), Eqs. (2.7.1)–(2.7.6) and the convergence statement following Eq. (2.7.6). At an irregular point, exponential factors and asymptotic series can replace a convergent power series; branch and sector data then form part of the endpoint condition NIST DLMF 2026, § 2.7(ii), Eqs. (2.7.7)–(2.7.8).

Match the method to the failure mode

Method	Appropriate use	Minimum discriminating check
Adaptive integration	Regular initial-value propagation	Vary tolerance and precision separately; check a residual, invariant, or exact limit.
Shooting	Moderately conditioned two-point problems	Evolve sensitivity equations; compare single with multiple shooting or collocation.
Matching or transfer matrices	Independent local bases with an overlap	Verify basis rank, Wronskians, matching-point independence, and scaling.
Collocation or relaxation	Global boundary problems and stiff spatial directions	Test the equation away from collocation nodes and inspect boundary residuals and mesh or coefficient convergence.
Continued fractions	Frobenius recurrences whose minimal solution enforces the second endpoint	Sweep fraction depth, precision, inversion index, tail model, and recurrence residual.
Spectral pencils	Regularized eigenvalue and resonance problems	Vary resolution, basis, map, precision, endpoint treatment, and an independent method.

Stiffness is not an intrinsic label attached to the equation alone. It depends on scaling, interval, data, tolerance, and the chosen stability function. For $y'=Ay$, an explicit step must keep the relevant $h\lambda_j(A)$ inside its stability region even after a fast mode has decayed. Adaptive local-error control does not prove stability, and an implicit method is not automatically more accurate; the standard distinctions are developed in Hairer and Wanner 1996, ch. IV §§ 1–3, pp. 2–55.

Shooting replaces missing initial data by a root of the far-end boundary residual. A growing fundamental solution can make that root exponentially sensitive, so endpoint agreement may conceal a poor interior solution. Single and multiple shooting are treated in Ascher, Mattheij, and Russell 1995, ch. 4 §§ 4.1–4.3, pp. 132–154, and collocation checks in Ascher, Mattheij, and Russell 1995, ch. 5 § 5.4, pp. 218–225, and §§ 5.6.2–5.6.4, pp. 247–265.

For radiative black-hole ODEs, factor the future-horizon and outer-outgoing branches before expanding or discretizing. A continued fraction selects the minimal recurrence solution only after that branch choice is fixed; its derivation and coupled angular–radial conditions are in Leaver 1985, Eqs. (12)–(14), pp. 287–289, and Eqs. (18)–(27), pp. 289–291. With $e^{-i\omega t}$, stable resonances satisfy $\operatorname{Im}\omega\lt0$. A finite-depth zero is not sufficient: follow the same root under deeper fractions, greater precision, an adjacent inversion, and an independent spectral calculation.

Spectral convergence can be rapid only when the solution is regular in each domain and every interface condition is supplied. Basis, mapping, compactification, and multidomain checks are given in Grandclément and Novak 2009, §§ 2.1–2.6, § 3.1.2, especially Eq. (89), and § 4. A small matrix backward residual does not establish a small continuum error; test the original ODE off grid and retain only digits stable under independent changes.

References

• Ascher, U. M., Mattheij, R. M. M., and Russell, R. D. (1995). Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Philadelphia: Society for Industrial and Applied Mathematics. DOI.
• Grandclément, P., and Novak, J. (2009). “Spectral Methods for Numerical Relativity.” Living Reviews in Relativity 12, 1. DOI.
• Hairer, E., and Wanner, G. (1996). Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd revised edition. Springer, Berlin. DOI.
• Leaver, E. W. (1985). “An Analytic Representation for the Quasi-Normal Modes of Kerr Black Holes.” Proceedings of the Royal Society A 402, 285–298. DOI.
• National Institute of Standards and Technology. Digital Library of Mathematical Functions. Version 1.2.7, 15 June 2026. § 2.7 and § 3.7.