Constructive Differential Algebraic Framework for Real-Order Nonlinear Partial Differential Equations

This paper establishes a constructive differential algebraic framework for obtaining explicit analytic solutions to broad classes of real-order nonlinear partial differential equations (real-order PDEs). We define the real-order nonlinear partial differential algebraic closure KRNLPDE, a differentially closed field extension constructed through a recursive adjunction process that incorporates solutions to real-order linearized PDEs, multi-index radical extensions, roots of unity, and a predefined set of real-order nonlinear special functions.Within this closure, we prove that solutions to real-order n-th order nonlinear PDEs with analytic coefficients, satisfying appropriate real-order Cauchy–Kovalevskaya type conditions, admit a unified representation. The framework rigorously addresses the multi-dimensional, non-local, and memory-dependent challenges inherent in real-order PDEs. We provide constructive proofs, derive explicit combinatorial expressions for real-order nonlinear correction coefficients, and establish convergence criteria for the iterative construction of real-order nonlinear basis functions.Detailed algorithms with complexity analysis are presented, including stability guarantees and adaptive precision control. A rigorous validation framework with certified error bounds is established, employing interval arithmetic and cross verification against high-order numerical methods. This work demonstrates that while closed-form solutions in elementary functions are impossible for many real-order nonlinear PDEs, explicit analytic solutions exist within the appropriately extended and constructively defined real-order nonlinear partial differential algebraic closure KRNLPDE. The framework is shown to be consistent with classical real-order PDE theory while extending the solution space to include real-order nonlinear special functions and combinatorial correction structures.