Matrix Science Mathematic (MSMK)

This study establishes mathematically a comprehensive approach for evaluating the solutions of Bessel’s function of the first kind for different values of the order n. The study revealed that the knowledge of gamma function, Maclaurin’s series and basic laws of indices plays a significant role in evaluating the Bessel’s function of order n. The results showed that the solutions, J-1(x) = J-2(x) = J-3(x) = 0 for x = 0, 1, 2, 3, 4…….12; the pattern of variation for the various graphs of J0(x), J1/2(x), J-1/2(x), J1(x), J3/2(x), J-3/2(x), J2(x), J5/2(x), J-5/2(x) and J3(x) were investigated and the results showed that there is a slight decrease in the values of J0(x) from x = 0 to 3 which then increase steadily from x = 4 to 12, for J1/2(x), J3/2(x), J2(x) and J3(x), the values increases from x = 0 to 12. The values of J-1/2(x) increases steadily from x = 0 to 12 in the form of a parabola. For J1(x), the graph increase from x = 0 to 2 then decreases at x = 3 and increases continuously from x = 4 to 12. The figure depicting J-3/2(x) showed that when x = 0, J-3/2(0) = ∞ , negative values were obtained at x = 1 and 2, the values then increases steadily from x = 4 to 12. The figure for J5/2(x) shows that the values increases from x = 0 to 2 and decreases negatively from x = 3 to 8 and increases steadily from x = 9 to 12. The figure for J-5/2(x) depicts that when x = 0, J-5/2(0) = ∞, the values decrease from x = 1 to 2 and decreases further but negatively from x = 3 to 12.