An Integral Transform Solution of the Differential Equation for the Transverse Motion of an Elastic Beam

It is well known* that certain types of partial differential equation may be solved using integral transforms with suitable kernels. In general, these equations may be solved by the classical method of separating variables, but the use of an integral transform yields the solution in a more direct way in the sense that the boundary values are contained in the solution. It is the purpose of this note to apply this technique to obtain the solution of the differential equation associated with the transverse motion of an elastic beam for a wide class of boundary conditions. The inversion theorem for a finite integral transform is in the form of a series expansion of the characteristic functions of the differential system, and the normalisation of these functions may involve laborious integrations. It is shown in §3 how the expansion coefficients may be obtained algebraically, thereby simplifying the process.