A FINITE DIFFERENCES SOLUTION TO THE VIBRATING MEMBRANE PROBLEM

Abstract A realistic approach to the solution of mechanical systems containing multiple parameters must take into account the fact that dependent variables depend not only on, but on more space variables. The modelling of such problems leads to partial differential equations (P.D.Es), rather than Ordinary Differential Equations (O.D.Es). Here, the wave equation, a P.D E that governs the vibrating membrane problem is considered. A finite difference method (F.D.M) is provided as an alternative to the analytic methods. F.D.Ms  basically involve three steps; dividing the solution into grids of nodes, approximating the given differential equation by finite difference equivalences that relate the solutions to grid points and solving the difference equations subject to the prescribed boundary and/or initial conditions. It is shown here that the error in the result is relatively negligible, and the conclusion made that the method developed can further be used to solve certain non-linear P.D.Es. Key words; Wave Equation, Vibrating membrane, Fouling, Analytic solution, Numerical solution, Local truncation Error, Stabilit

Elastic Scattering Reaction of on Partial Wave Scattering Matrix, Differential Cross-Section and Reaction Cross-Section at Laboratory Energies of 5-15 Mev: An Optical Model Analysis.

The nuclear optical model has been used in the analysis of elastic scattering for the reaction . This model has six optical parameters; the depth, Coulomb radius and the diffuseness on both the real part and imaginary part potentials. Out of the six, five parameters were chosen and for this case diffuseness parameter on the imaginary part was kept constant. The five parameters were used to calculate the partial wave S-matrix, the differential cross section and the reaction cross section as a ratio to Rutherford cross section. The partial wave scattering data was obtained basing on the quantum mechanical optical code for all the stated Laboratory energies. The angular distribution for the reaction  for both reaction cross section to the Rutherford cross-section and differential cross section ranging from centre of mass angles  of  were also obtained for all the energies, (  and  MeV) and whose data and graphs are presented. Key words: Differential cross-section, Reaction cross-section and Partial scattering matrix