The Modified Galerkin Method for Solving the Helmholtz Equation for Low Frequencies on Planet Mars

The objective of this paper is to investigate numerical solutions of several boundary value problems for the Helmholtz equation for two smooth surfaces. The superellipsoid is a shapethat is controlled by two parameters. There are some numerical issues in this type of an analysis;any integration method is affected by the wave number k, because of the oscillatory behavior ofthe fundamental solution. The Biconcave Disk is a closed, simply connected bounded shapemodified from a sphere where the two sides concave toward the center, mapped by a sine curve. This project was funded by NASA RI Space Grant and the NASA EPSCoR Grant for testingof boundary conditions for these shapes. One practical value of all these computations can be a shape for the part of the space shuttle that might one day land on planet Mars. Theatmospheric condition on Mars is conducive for small atmospheric wave numbers or lowfrequencies. We significantly reduced the number of terms in the infinite series needed tomodify the original integral equation and used the Green's theorem to solve the integralequation on the boundary of the surface