Formal Power Series Solutions in Differential Equations

Formal power series solutions underpin much of modern analysis of differential and difference equations, offering algebraic expansions that satisfy equations term by term even when actual convergence is not guaranteed. Originating in the work of Poincaré and Borel, this approach has matured into a systematic theory encompassing singular ordinary and partial differential equations, q-difference equations and moment differential systems. Central themes include the classification of formal series by their coefficient growth (Gevrey classes), the identification of summability properties that permit reconstruction of genuine analytic solutions via Borel–Laplace transforms and multisummation methods, and the use of Newton polygon techniques to detect irregular singularities and bifurcate solutions into inner and outer expansions. Recent advances have broadened applications to nonlinear problems exhibiting vanishing or blow-up behaviour near singular points, to parametric families of equations in perturbation theory and to functional equations with accelerating operators. The global significance of this body of work is reflected in its impact on fluid mechanics, quantum field theory, control systems and special-function theory, where precise asymptotic descriptions of solutions guide both numerical computation and qualitative understanding.

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Formal Power Series Solutions in Differential Equations publication trend

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Technical terms

Formal power series solution: an infinite series that satisfies a differential equation when substituted term by term, without initial concern for convergence.

Gevrey class: a hierarchy of formal series classified by factorial growth rates of coefficients, indicating levels of analytic regularity.

Summability: a technique to assign a genuine analytic function to a divergent formal series through Borel transformation and Laplace integration.

q-difference equation: a functional equation in which the independent variable is scaled by a constant factor q, generalising discrete shifts.

Newton polygon: a geometric construction used to analyse the balance of terms in a differential operator and predict asymptotic scales of solutions.

References

1. Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point. Advances in Nonlinear Analysis (2024).
2. Gevrey regularity of the solutions of inhomogeneous nonlinear partial differential equations. Electronic Journal of Differential Equations (2023).
3. On the summability and convergence of formal solutions of linear q-difference-differential equations with constant coefficients. Mathematische Annalen (2023).