laplace transform analysis

This work addresses the absence of a rigorous formalization of the Laplace transform and its inversion in existing interactive theorem provers. It presents the first complete formalization in Lean 4 of the Laplace transform for complex-valued functions, along with fundamental operational rules and a Bromwich-type inversion theorem grounded in real integrals and Dirichlet integrals. By integrating classical analysis with formal verification techniques, the framework is successfully applied to the harmonic oscillator problem. The development not only verifies core aspects of Laplace transform theory but also formally derives the solution and proves that its transform coincides with the standard transform of $\sin(\omega t)$. This demonstrates the feasibility and rigor of formalized mathematics in engineering analysis.

This study systematically compares MATLAB, Mathematica, and Maple in solving ordinary differential equations (ODEs), partial differential equations (PDEs), and systems of differential equations. A unified benchmark suite—grounded in analytically tractable reference solutions—is employed to empirically evaluate the tools across five dimensions: syntactic usability, numerical accuracy, computational efficiency, visualization capability, and specialized solver functionality. Crucially, the work introduces a novel, problem-driven software selection framework that classifies tasks by intrinsic characteristics—including equation type, stiffness, and boundary condition complexity. Results indicate that Mathematica excels in symbolic solution derivation and medium-scale ODE accuracy; MATLAB demonstrates superior performance in large-scale numerical simulation and engineering-oriented PDE applications; and Maple offers distinctive advantages in special-function handling and analytic derivation. This is the first systematic, multidimensional comparative study of these major mathematical software platforms, thereby bridging a critical gap in computational tool evaluation and providing actionable, evidence-based guidance for scientific and engineering practice.

High computational cost and poor parallel scalability hinder time-domain simulations of low-frequency electromagnetic eddy current problems. To address these challenges, this paper proposes a novel domain decomposition method integrating tree-cotree edge handling with isogeometric tearing and interconnecting dual-primal (IETI-DP). For the first time, tree-cotree regularization is embedded within the IETI-DP framework, synergistically combining isogeometric analysis, implicit time discretization, and non-overlapping domain decomposition to enable physics-driven variable reduction and interface continuity enforcement. Numerical experiments demonstrate that the method significantly improves convergence rates and strong/weak scalability; on multiple complex geometries, it reduces solution time by over 70% compared to conventional approaches. The proposed framework establishes a new paradigm for large-scale transient eddy current simulation—achieving high accuracy, numerical robustness, and superior parallel efficiency.

This work addresses the injectivity testing problem for univariate polynomial maps over finite fields, motivated by algebraic decomposition and controllable design in discrete dynamical systems. Methodologically, it integrates algebraic dynamics theory, finite-field polynomial analysis, and synchronous/alternating execution modeling. The main contribution is the first coefficient-wise complete algebraic characterization of injective polynomials—explicit structural conditions on coefficients are derived. Based on this characterization, the paper proposes the first polynomial-time injectivity test, with time complexity $O(n^2)$, markedly improving upon exponential brute-force enumeration. This resolves a long-standing fundamental decision problem and provides a computationally tractable tool for algebraically structured modeling and control of dynamical systems.

Solving high-dimensional partial integro-differential equations (PIDEs) numerically remains challenging due to computational intractability and poor interpretability. To address this, we propose FEX-PG, a finite-expression method featuring a novel parameter grouping (PG) strategy that drastically reduces the number of trainable coefficients required for high-dimensional function approximation. FEX-PG explicitly approximates nonlocal integral terms via truncated Taylor series, balancing computational efficiency with enhanced accuracy. The method yields compact, physically meaningful explicit analytical solutions. Evaluated on multiple high-dimensional benchmark PIDEs, FEX-PG achieves relative errors on the order of single-precision machine epsilon (~1×10⁻⁷), substantially outperforming conventional finite element and finite difference methods as well as state-of-the-art deep learning approaches. Thus, FEX-PG simultaneously delivers high accuracy, strong interpretability, and computational feasibility.

This work addresses the efficient algebraic representation and factorization of linear ordinary differential operators over compatible derivation modules. By implementing differential operators as first-class objects in Scratchpad II, the approach supports standard notation and provides a unified treatment of left and right module structures. For operators with coefficients in a field or polynomial ring, it integrates Ore localization, pseudo-division, and construction of right fraction fields to enable left and right division, computation of greatest common divisors, least common multiples, and extended Euclidean algorithms. Furthermore, by combining Riccati equations with Newton polygon analysis, the method effectively characterizes the singularities of factors. This framework facilitates constructive factorization and algebraic manipulation of operators with constant, elementary, rational, and even matrix-valued coefficients.

This work addresses the computational inefficiency associated with solving the dense matrix resulting from the discretization of the three-dimensional electric field integral equation (EFIE). To this end, the authors propose a spectral truncation filtering method based on the spherical Hankel transform. By constructing a spectral representation of the Green’s function, they apply an analytical spectral filter to the integral operator, marking the first application of this technique to operator compression and regularization for the three-dimensional EFIE. The proposed approach significantly improves the spectral distribution of both continuous and discrete operators in static and dynamic regimes, thereby substantially enhancing the convergence rate and computational efficiency of both iterative and direct solvers.

This work proposes a novel data-driven approach to directly learn soliton dynamics from scattering data without requiring prior knowledge of the underlying scattering equations. By integrating the inverse scattering transform framework with weak-form system identification, the method constructs low-dimensional, interpretable dynamical models in the scattering domain—eliminating the need to assume explicit analytical equations or fit soliton trajectories. It is applicable to both perturbed and near-integrable systems and has been successfully validated on synthetic and experimental KdV-type shallow water wave data, accurately recovering effective dynamics consistent with classical inverse scattering theory. This demonstrates the method’s innovation and practical utility in data-driven modeling of nonlinear wave phenomena.

This work addresses the well-known ill-conditioning of the electric field integral equation (EFIE) at low frequencies, high frequencies, and fine discretizations, which hinders efficient numerical solution. The authors propose a unified preconditioning strategy based on shifted Helmholtz operator regularization, integrating a novel preconditioner design with a fast matrix-vector product algorithm of quasi-linear complexity. This approach effectively overcomes the limitations of conventional pseudo-inverse methods in handling the shift operator. The resulting solver exhibits remarkably stable iteration counts across a wide range of frequencies and mesh resolutions, achieving—for the first time—a unified, efficient treatment of all three canonical ill-conditioned regimes while attaining quasi-linear computational complexity for EFIE solutions.

This study addresses the challenge of efficiently and stably simulating nonlinear Föppl–von Kármán plate vibrations, which is hindered by the high computational cost of modal coupling. The authors propose an explicit energy-stable method based on modal synthesis: nonlinear terms are evaluated in physical space using a pseudospectral approach, while derivatives are handled exactly in the modal domain. Discrete sine and cosine transforms naturally enforce simply supported boundary conditions, and a scalar auxiliary variable enables explicit time integration. This approach is the first within a modal framework to simultaneously achieve explicit stability, spectral accuracy in frequency representation, and computational efficiency, substantially reducing the overhead associated with nonlinear modal coupling. The method’s efficacy is demonstrated through high-fidelity acoustic simulations.