A review on integral transforms: Aboodh and finite Mellin transforms

Introduction

Integral transforms are fundamental and basic mathematical tools used to convert differential equations into algebraic equations (Liaqat et al., 2022) [1] . This transformation simplifies the search for analytical or approximate solutions in various domains of science and engineering. Among the wide lists of transforms, the Aboodh transform and the Mellin transform represent critical frameworks for solving complex problems in applied mathematics, and theoretical physics. The Aboodh transform, first introduced by Khalid Aboodh in 2013, has interest due to its simplicity in solving initial value problems (Qureshi et al., 2019) [2] , (Aboodh et al., 2022) [3] . On the other hand, the Mellin transform particularly its finite version, it has wide application in signal sampling theories, and complex ordinary and partial differential equation in higher physics (Butzer et al., 1999) [4] , (Atale, 2024) [5] . This paper review on these transforms, examining their theoretical foundations and expanding practical applications, limitation and future scope.

The Aboodh transform

Mathematical definition

The Aboodh Transform with parameter s of the function 𝑓(𝑡) is denoted by 𝐴 { 𝑓(𝑡) } = 𝐹(𝑠) and is given by

The integral kernel is 𝑒 -𝑠𝑡 (Alshikh et al., 2016) [6] , (Benattia et al., 2020) [7] , (Iqbal et al., 2024) [8] .

Variations and extensions

The Aboodh transform has developed to tackle more complex mathematical structures, some versions of it:

Local Fractional Aboodh Transform (LFAT):

This transform is use to convert linear local fractional differential equations into algebraic equation, using the Mittag-Leffler function in its kernel for nonsmooth functions (Ziane et al., 2022) [9] .

Double Aboodh Transform:

Used to solve two variables' problems in fractional order (Alfaqeih et al., 2019) [10] .

Triple Laplace-Aboodh-Sumudu Transform: A hybrid transform used for solving three variable 3D Helmholtz equation and partial differential equations (Hunaiber, 2025) [11] , (Al-aati et al., 2022) [12] .

Aboodh Transform Series Decomposition Method (ATSDM): the Taylor series combine with transform for solving higher-order nonlinear boundary value problems by decomposition methods (Akinola et al., 2023) [13] .

Applications of the Aboodh transform

Ordinary and partial differential equations

The Aboodh transform is effective for solving linear and nonlinear ordinary differential equations (ODEs) with variable coefficients (Benattia et al., 2019) [14] .

The Variational Iteration Method (VIM) or the Homotopy Perturbation Method (HPM), solved Klein-Gordon equations and partial differential equations (PDEs) (Aboodh et al., 2022) [3] , (Sedeeg et al., 2016) [15] , (Cherif et al., 2018) [16] .

Fractional calculus and nonlinear dynamics

Korteweg-de Vries (KdV) and Kawahara Equations: It play fundamental role for understanding of nonlinear wave dynamics provide and analytical solutions for the modified KdV and fifthorder Kawahara equations (Almusawa et al., 2024) [17] , (Iqbal et al., 2024) [8] , (Shah et al., 2020) [18] .

Zakharov-Kuznetsov Equation: multi-dimensional fractional ZK equations solve using the Adomian Decomposition Method (Rashid et al., 2021) [19] .

Fokker-Planck Equations:

The ZZ transform is related to the Aboodh transform for analysis of fractional-view Fokker-Planck equations (Alshehry et al., 2022) [20] .

Telegraph Equations: In transmission lines, transform solve fractional-order telegraph equations to model signals (Abdlrasol et al., 2022) [21] , (Ali et al., 2020) [22] .

Keller-Segel Equation: KS equation applied to fractional modelling in mathematical biology [23] .

Physics, engineering, and mechanics

Quantum Mechanics: Solving time-fractional quantum mechanics problems (Liaqat et al., 2022) [1] .

MEMS Systems: Application of the homotopy perturbation method (Amir et al., 2024) [24] .

Fluid Dynamics and Plasma: Modeling nonlinear waves in plasma and fluids (Shah et al., 2022) [25] , (Mukhtar, 2023) [26] .

Vibrating Systems: Analyzing vibrating beams and rotating systems (Aggarwal et al., 2019) [27] , (Mackerle, 1999) [28] .

Cryptography and data security

It has been proposed for use in image encryption and as a mathematical framework for online examination data security (Sedeeg et al., 2016) [29] .

Its ability to map functions into a transform domain provides a layer of complexity useful for cryptographic algorithms.

The transformation of functions from the original domain into a suitable transform domain introduces additional mathematical complexity, which can be exploited in cryptographic algorithms. This property may be utilized to develop encryption and decryption procedures, enhance data confidentiality, and provide additional security for sensitive information transmitted or stored in digital systems. In particular, transform-domain representations can be incorporated into image encryption schemes to alter the structure and statistical characteristics of the original image, thereby making unauthorized reconstruction more difficult. Furthermore, the transform-based approach may be investigated for securing online examination systems, where confidentiality and integrity of question papers, student responses, and examination-related data are important. Thus, the proposed transform provides a mathematically structured framework that can potentially contribute to the development of secure information-processing techniques.

The finite Mellin transform

Mathematical definition

The Finite-Mellin Transform with parameter p of the function 𝑓(𝑥) is denoted by 𝑀 𝑓 { 𝑓(𝑥) } = 𝐹(𝑝) and is given by

It is classically used for functions on an infinite region (Khairnar et al., 2012) [30] .

The transform is also used in heat transfer and diffusion problem.

The finite Mellin transform is a most recent adaptation for functions (Atale, 2024) [5] , (Butzer et al., 2000) [31] .

The finite Mellin transform for discrete signal analysis and finite-field applications (Katz, 2012) [32] , (Katz, 2012) [33] .

Connections to Ramanujan's master theorem

The finite Mellin transform provides an integral representation for meromorphic functions based on their power series coefficients with connected to Ramanujan's Master Theorem (Atale, 2024) [5] , (Bradshaw et al., 2024) [34] .

This allows for the evaluation of integrals that appear in number theory and mathematical physics.

Applications of the Mellin transform

Signal analysis and sampling theory

Mellin-bandlimited Signals: Analysis of signals with specific integral transform (Butzer et al., 1999) [4] .

Signal Processing: Implementing digital algorithms for signal processing in automated control systems (Makarov et al., 2019) [35] .

Physics and statistical mechanics

Statistical Mechanics: Used to analyze the ideal quantum gases and other problems in thermodynamics with statistical mechanics (Grandy et al., 1981) [36] , (Iwata, 1960) [37] .

Particle Physics: This transform is evaluating loop amplitudes in Feynman diagrams and simplify the calculation of complex scattering processes (Atale, 2024) [5] .

Yang-Mills Theory: Applied to perturbative aspects of deformed Yang-Mills theory (Devika, 2023) [38] .

Electromagnetics: Used in electromagnetic field modeling (Mystilidis et al., 2022) [39] .

Financial mathematics and option pricing

American Options: transformation of the Black-Scholes equation for pricing American options (Frontczak et al., 2008) [40] , (Alobaidi et al., 2014) [41] .

Vanilla Power Options: Valuation of power options with nondividend yields (Nwozo et al., 2014) [42] .

Comparative analysis

Current research focuses on the relationships between different integral transforms.

Aboodh verses Laplace and Elzaki: The Aboodh transform

Comparative studies show performance in solving first and second-order ODEs (Alshikh et al., 2016) [6] .

Aboodh and Mohand:

Comparative studies show both are effective for engineering problems like heat conduction (Aggarwal et al., 2019) [27] .

ZZ Transform: The Aboodh and Laplace transforms is related to the ZZ transform used for solving fractional-order Fokker-Planck equations (Mahgoub et al., 2016) [43] , (Alshehry et al., 2022) [20] , (Jena et al., 2020) [44] .

Ouideen Transform: This new transform offering high accuracy for linear differential equations by introduced as a generalization of the Aboodh and Sumudu transforms [45] .

Limitations of the evidence

The following limitations are identified in the current literature:

Function Class Restrictions: The finite Mellin transform is limited to functions with series expansions (Atale, 2024) [5] .

Inversion Stability: the Mellin transform inversion can be unstable and requiring functional equivalence analysis (Hawkins, 2016) [46] .

Convergence Speed: they may still face challenges with highly complex nonlinear boundary conditions (Aboodh et al., 2022) [

, (Akinola et al., 2023) [13] .

Limited Long-term Projections: For broad theoretical generalization across all possible operators required more time so, the research is focused on demonstrating efficacy for specific equations (Benattia et al., 2020) [7] , (Benattia et al., 2019) [14] .

Future directions and research gaps

The literature points toward several promising research directions:

Semi-Derivative Interpretation: The Aboodh transform using semi-derivatives interpretation expected to future breakthroughs. (Qureshi et al., 2019) [2] .

Loop Amplitudes in Physics: The finite Mellin transform apply to evaluate higher-order loop amplitudes in Feynman diagrams remains a critical gap (Atale, 2024) [5] .

Extension of Ramanujan's Theorem: aims to explore finitelimit analogs of Frullani's theorem (Atale, 2024) [5] and extend research for the class of functions applicable to Ramanujan's Master Theorem.

Enhanced Cryptographic Protocols:

The Aboodh transform developing more robust encryption methods based on the unique mapping properties (Sedeeg et al., 2016) [29] .

Cell Growth Modeling: application of Mellin transforms to functional differential equations arising in cell growth models (Hall et al., 1989) [47] .

Conclusion

In modern mathematics the Aboodh and finite Mellin transforms are essential components for analysis. The Aboodh transform's ability to handle fractional and nonlinear dynamics Meanwhile, the finite

Mellin transform provides application in signal processing, financial mathematics, and theoretical physics. As researchers continue to explore properties, generalizations of these transforms, inversion theorem, uniqueness, hybrid model, the dualities and their impact on scientific and engineering problem-solving. It is expected to grow, offering new insights into the fundamental dynamics of complex systems.

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