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The main objective of the paper is to study the three-dimensional fractional Fourier Mellin transforms (3DFRFMT), their basic properties and applicability due to mainly use in the radar system, reconstruction of grayscale images, in the detection of the human face, etc. Only the fractional Fourier transform is based on time-frequency distribution, whereas only the fractional Mellin transform is on scale covariant transformation. Both transforms can discover action in the definite assortment. The fractional Fourier transform is applicable for controlling the range of shift, whereas the fractional Mellin transform is accustomed to managing the range of rotation and scaling of the function. So, combining both transformations, we get an elegant expression for 3DFRFMT, which can be used in several fields. The paper introduces the concept of three-dimensional fractional Fourier Mellin transforms and their applications. Modulation property is the most useful concept in the signal system, radar technology, pattern reorganization, and many more in the integral transform. Parseval's identity applies to the conservation of energy in the universe. Thus we establish the modulation theorem, Parseval's theorem, scaling theorem, analytic theorem for three-dimensional fractional Fourier Mellin transform. We also give some examples of three-dimensional fractional Fourier-Mellin transform on some functions. Finally, we provide three-dimensional fractional Fourier-Mellin transform applications for solving homogeneous and non-homogeneous Mboctara partial differential equations that we can apply with advantages to solve the different types of problems in signal processing systems. The transform is beneficial in a maritime strategy as a co-realtor to control moments in any specific three-dimensional space. The concept is the most powerful tool to deal with any information system problems. After obtaining the generalization, we can explore many more ideas in applying three-dimensional fractional Fourier-Mellin transformations in many real word problems.

Выведены и записаны в максимально редуцированном виде замкнутые представления одно- и двукратного преобразований Лапласа хюльтеновской функции Грина уходящей волны, умноженной на потенциал Ямагучи. Выражения для двукратного преобразования используются при вычислении низкоэнергетического фазового сдвига при упругом рассеянии в системах $\alpha$-нуклон, $\alpha$-$He^3$ и $\alpha$-$H^3$. Результаты расчета хорошо согласуются с экспериментальными данными.

We have discussed the concept of three dimensional natural transform and its applications in material science. We have solved the particular type of third order partial differential equations (known as Mboctara equation) using the three dimensional natural transform operator.

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