A Constructive Differential Algebraic Framework for Nonlinear Real-Order Multivariate Integral Equations:Explicit Solutions Through Exterior Integration and Combinatorial Analysis

This paper establishes a constructive differential algebraic framework for obtaining explicit analytic solutions to broad classes of nonlinear real-order multivariate integral equations. We define the nonlinear real-order integral algebraic closure KRNLIE, a differentially closed field extension constructed through a recursive adjunction process that incorporates solutions to linearized real-order integral equations, multi-index radical extensions, roots of unity, and a predefined set of nonlinear special functions adapted to real-order operators. Within this closure, we prove that solutions to nonlinear multivariate real-order integral equations with analytic kernels admit a unified representation.The framework rigorously addresses the multi-dimensional, infinite-dimensional, and non-local challenges inherent in real-order integral equations. We provide constructive proofs, derive explicit combinatorial expressions for nonlinear correction coefficients in the real-order setting, and establish convergence criteria for the iterative construction of nonlinear basis functions. Building upon the theory of exterior integration with real-order measures, we demonstrate how multiple real-order integration concepts are unified in a coordinate-independent geometric framework.Detailed algorithms with complexity analysis are presented, including stability guarantees and adaptive precision control for real-order operators. A rigorous validation framework with certified error bounds is established, employing interval arithmetic and cross-verification against high-order numerical methods for real-order equations. This work demonstrates that while closed-form solutionsin elementary functions are impossible for many nonlinear real-order integral equations, explicit analytic solutions exist within the appropriately extended and constructively defined nonlinear real-order integral algebraic closure KRNLIE.