The Weighted Time-Domain Laplace-Fourier Transform: A Comprehensive Study

Abstract

Traditional integral transforms like Fourier and Laplace have been foundational tools for signal and system analysis in physics, engineering, and mathematics. However, these classical methods are often inadequate for analyzing nonlinear, non-stationary, and time-varying signals that are increasingly prevalent in real-world systems. In response to these limitations, this study introduces the Weighted Time-Domain Laplace-Fourier Transform (WTLFT), which integrates the benefits of Fourier and Laplace transforms with a time-domain weighting function to enhance analytical flexibility. Prior methods lack robustness in capturing transient behaviors and localized features in dynamic systems. There is limited development of transform techniques that address both fractional dynamics and time-varying structures within a single framework. This research aims to define the WTLFT, prove its core properties, and demonstrate its effectiveness in solving complex differential equations involving non-integer derivatives and dynamic systems. The WTLFT is analytically validated through properties such as linearity, time-shifting, convolution, and differentiation. It is successfully applied to exponential and Mittag-Leffler functions, and solves Caputo-type fractional models, diffusion equations, and differential-algebraic systems. Unlike conventional transforms, the WTLFT allows for flexible weighting strategies that adapt to specific application domains, enabling enhanced signal analysis and accurate reconstruction of solutions. The WTLFT establishes a powerful new direction for signal processing, control theory, and fractional calculus applications, especially in engineering, biomedical, and physical science fields.