This article introduces the Generalized Fourier Series (GFS), a novel spectral method that extends the clas- sical Fourier series to non-periodic functions. GFS addresses key challenges such as the Gibbs phenomenon and poor convergence in non-periodic settings by decomposing functions into periodic and aperiodic com- ponents. The periodic part is represented using standard Fourier modes and efficiently computed via the Fast Fourier Transform (FFT). The aperiodic component employs adaptive, low-rank sinusoidal functions with non-harmonic modes, dynamically tuned to capture discontinuities and derivative jumps across domain boundaries. Unlike conventional Fourier extension methods, GFS achieves high accuracy without requiring compu- tational domain extensions, offering a compact and efficient representation of non-periodic functions. The adaptive low-rank approach ensures accuracy while minimizing computational overhead, typically involving additional complex modes for the aperiodic part. Furthermore, GFS demonstrates a high-resolution power, with degrees of freedom comparable to FFT in periodic domains, and maintains N log2(N) computational complexity. The effectiveness of GFS is validated through numerical experiments, showcasing its ability to approximate functions and their derivatives in non-periodic domains accurately. With its robust framework and minimal computational cost, GFS holds significant potential for advancing applications in numerical PDEs, signal processing, machine learning, and computational physics by providing a robust and efficient tool for high-accuracy function approximations.

This paper introduces the conceptual design of a Photon Driven Reactor (PDR), an innovative subcritical reactor designed for energy generation driven by a synchrotron radiation. The PDR concept overcomes key technological challenges of conventional accelerator-driven systems, particularly the target's structural durability and its thermal management, by employing synchrotron photons directly interacting with fissile material to induce photonuclear reactions. Computational analyses involved criticality and fixed-source simulation using MCNPx and SERPENT Monte Carlo codes, providing robust evaluation of the neutron production, moderation, and multiplication mechanisms. The main focus of this study was to evaluate the system's capability to achieve a positive net energy gain, specifically assessing the thermal power output agains the electrical power absorbed from the grid. Furthermore, the adoption of spent nuclear fuel for the subcritical reactor core loading has been investigated, highlighting the sustainability and environmental benefits of the proposed PDR design. The proposed system is able to exploit a modularity feature. For each large synchrotron, up to fifty beam lines may be operated simultaneously, each delivering photons to an independent subcritical reactor core. With a photon flux on the order of $8.8 \times 10^{17}$ photons per second in each beamline, the results indicate that each individual reactor can achieve a thermal output up to 8 MW, while requiring about 435-660 kW of electrical input from the grid, thereby demonstrating the feasibility of energy amplification in the PDR.