Review on Triple Integral Transforms and Their Inverses

Shatha Haider Theyb; Zainab A.  Khudhai; Nour K.  Salman; Emad A.  Kuffi

Authors

Shatha Haider Theyb Department of Mathematics, College of Basic Education, Mustansiriyah University
Zainab A. Khudhai Department of Mathematics, College of Basic Education, Mustansiriyah University
Nour K. Salman Department of Mathematics, College of Basic Education, Mustansiriyah University
Emad A. Kuffi Department of Mathematics, College of Basic Education, Mustansiriyah University

Keywords:

Integral transforms, Triple integral transforms, Inverse of Triple Integral Transform

Abstract

Transformations involving triple-integrals are the basis for the solutions of a number of three-dimensional partial differential equations that arise in physics, engineering, and applied sciences. Although many triple transforms are available (e.g., Laplace, Fourier, Aboodh, Shehu, Ezaki, and various hybrid triple transforms), to the best of our knowledge, a detailed systematic comparative review is not yet available that compares different triple transforms with respect to their definitions and kernels, inverse formulations, and applications. This paper fills this gap by giving an overview of most important, widespread triple integral transforms, whereby introducing their formal definitions, inverse transforms as well as structural properties. Analytical behaviors of benign configurations are presented in solving different problems such as heat, wave, diffusion, and boundary value problems, which differ due to differences in kernel structures, convergence conditions, and their parameter configuration. We show that all the triple transforms impose preservation of linearity and algebraic simplification of differential operators, but each possess respective benefits specific to final transformed domain characteristics and boundary conditions. The results highlight the significance of careful selection of an adequate transform based on the structural characteristic of the problem under investigation and that future research should include generalized and hybrid triple transforms coupled with numerical methods to provide a more time-efficient analytical–numerical model

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