In this report, a new simple meshless method is presented for the solution of incompressible
inviscid fluid flow problems with moving boundaries. A Lagrangian formulation established on
pressure, as a potential equation, is employed. In this method, the approximate solution is
expressed by a linear combination of exponential basis functions (EBFs), with complex-valued
exponents, satisfying the governing equation. Constant coefficients of the solution series are
evaluated through point collocation on the domain boundaries via a complex discrete transformation
technique. The numerical solution is performed in a time marching approach using an implicit
algorithm. In each time step, the governing equation is solved at the beginning and the end of the
step, with the aid of an intermediate geometry. The use of EBFs helps to find boundary velocities
with high accuracy leading to a precise geometry updating. The developed Lagrangian meshless
algorithm is applied to variety of linear and nonlinear benchmark problems. Non-linear sloshing
fluids in rigid rectangular two-dimensional basins are particularly addressed.