This thesis is an attempt to solve certain vibrational problems rising methods for the solution of Bessel function dual integral equations by a multiplying factor method developed by Ben Noble (1963). A more general theory of dual integral equations, yielding Noble's result as a special case has been given by Williams (1964) (50). By elegant analysis involving fractional integration, Erdelyi and Sneddon (33) have avoided the heavy manipulation of Noble's analysis, but have reached the same conclusion. The crux of the method considered in this thesis lies in the integration of the kernel of the Fredholm integral equation derived from the dual integral equation. If this kernel can be integrated the problem can be solved for low frequencies. Love (1963) (42) has shown that if the kernel satisfies certain conditions the Fredholm equation possesses a unique solution given by a Liouville-Neumann series, but Love's condition does not apply in this ease. However, in this thesis research has been directed to developing a general system for integrating the kernel and thereby solving problems, where solution for the vibrational case, was unknown previously. To illustrate what has been attempted, the system developed in this thesis is applied to two problems whose solution is already known by other methods. The system is then applied to two further problems whose solution, as far as the writer knows has not been given before. Finally, consideration is given to extending the kernel Integration by removing sone of the restricting conditions.