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Abstract
This paper extends the differential algebraic closure framework from ordinary differential equations to a broad class of linear partial differential equations (PDEs). We construct a rigorously defined partial differential algebraic closure (KPDE) and demonstrate that solutions to linear PDEs
with analytic coefficients and non-characteristic initial/boundary conditions can be analytically expressed within this closure.We provide a detailed constructive framework, derive corrected combinatorial expressions with complete proofs, present a complete algorithmic description, and situate our results within classical PDE theory and differential Galois theory. Numerical experiments confirm spectral convergence and demonstrate the necessity of combinatorial corrections for higher-order equations. This work establishes that explicit analytic solutions exist in the appropriately constructed partial differential algebraic closure KPDE for a significant class of linear PDEs, providing a new algebraic perspective on PDE solvability that complements existing numerical methods such as finite element and spectral techniques.
Supplementary materials
Title
Extension of the Differential Algebraic Framework to Nonlinear Partial Differential Equations: A Constructive Approach to Unified Analytic Solutions
Description
This paper establishes a constructive differential algebraic framework for obtaining explicit analytic solutions to broad classes of nonlinear partial differential equations (PDEs). We define the nonlinear partial differential algebraic closure KNLPDE, a differentially closed field extension constructed
through a recursive adjunction process that incorporates solutions to linearized PDEs, multi-index radical extensions, roots of unity, and a predefined set of nonlinear special functions. Within this closure, we prove that solutions to n-th order nonlinear PDEs with analytic coefficients, satisfying the conditions of the Cauchy-Kovalevskaya theorem, admit a unified representation. The framework rigorously addresses the multi-dimensional and infinite-dimensional challenges inherent in PDEs. We provide constructive proofs, derive explicit combinatorial expressions for nonlinear correction coefficients, and establish convergence criteria for the iterative construction of nonlinear basis functions. Detailed algorithms with complexity analysis are presented, including stability guarantees and adaptive precision control.
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