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October 7, 2025Open Access

Application of Natural Generalized-Laplace Transform and Its Properties

Key Points

The natural generalized-laplace transform enables the solution of complex equations, including linear telegraph equations.
Key properties such as the existence condition and convolution theorem are discussed in detail, enhancing its applicability.
This methodology provides a novel approach for solving the singular one-dimensional Boussinesq equation.
The research illustrates the NGLT's potential in solving various differential equations, indicating its broader implications.

Abstract

In this work, we combine the Natural Transform and generalized-Laplace Transform into a new transform called, the Natural Generalized-Laplace Transform, (NGLT) and some of its properties are provided. Moreover, the existence condition, convolution theorem, periodic theorem, and non-constant coefficient partial derivatives are proved with some details. The (NGLT) is applied to gain the solutions of linear telegraph and partial integro-differential equations. Also, we obtained the solution of the singular one-dimensional Boussinesq equation by employing the Natural Generalized-Laplace Transform Decomposition Method, (NGLTDM).

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