A new Integral Transform was introduced in this paper. Fundamental properties of this transform were derived and presented such as the convolution identity, and step Heaviside function. It is proven and tested to solve some basic linear-differential equations and had succesfully solved the Abel's Generalized equation and derived the Volterra Integral Equation of the second kind by means of Initial Value Problem. The Natural Logarithm (e.g log ln e xx  ) has been established and defined by means of modifying the Euler Definite Integral based on the Rangaig's fomulation. Hence, this transform may solve some different kind of integral and differential equations and it competes with other known transforms like Laplace, Sumudu and Elzaki Transform.

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In this paper, we discussed the existence of a four point boundary value problem for q-fractional differential equation () () () () , , 0 q q D x t f t x t D x t α β = = in a Banach space, particulary, using the Banach contraction principle for certain conditions on f.

In this study, the nuclear fragmentation of the secondary particles produced when water is irradiated with protons and carbons were investigated. Proton beams with varying incident energies of 100 MeV, 130 MeV, 150 MeV and 160 MeV were used with corresponding 12C ion beams of about 187.50 MeV/u, 241.67 MeV/u, 285.42 MeV/u and 308.33 MeV/u respectively. The kinetic energy distribution and energy deposition of primary and secondary particles were studied via Monte Carlo simulation with the aid of GATE v.8.0 via GEANT4 simulation toolkit version 10.3.2 with 1 x 106 incident beams. The physics list used was QGSP_BIC (Quark Gluon String Pre-compound Binary Cascade). When the primary 12C ion and proton beams interact with water, secondary light-charged and heavy-charged particles are produced with atomic number Z > 2 are produced. In general, it was shown that the incident 12C ions are less scattered as they traverse mater compared to the incident protons. Thus, the energy deposition of 12C ions is well-defined and is better in terms of conformation.