Closure techniques integral equations

The preferred numerical method is then a combination of Newton-Raphson and Picard schemes first proposed by Gillan, together with the Ng method for handling long-range Coulomb potentials. It is a completely general technique and can be used with any closure or potential model to solve any integral equation for molecules of arbitrary symmetry. We will demonstrate the application of the method to the SSOZ equation. For this purpose, it is convenient to write the SSOZ equation in the form... 

Much more flexible is to describe the solute-solvent system within the supercell technique employing a set of 3D plane waves. The convolution in the OZ equation can be then approximated by means of the 3D fast Fourier transform (3D-FFT) procedure. For the first time, this approach was elaborated by Beglov and Roux [16] for numerical solution of the 3D-OZ integral equation with the hypernetted chain (HNC) closure for the 3D distribution of a simple LJ solvent around non-polar solutes... 

For bubbly flows most of the early papers either adopted a macroscopic population balance approach with an inherent discrete discretization scheme as described earlier, or rather semi-empirical transport equations for the contact area and/or the particle diameter. Actually, very few consistent source term closures exist for the microscopic population balance formulation. The existing models are usually solved using discrete semi-integral techniques, as will be outlined in the next sub-section. 

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