- Approximate solutions of the advection diffusion equation for spatially variable flows PHYSICS OF FLUIDS 2022; 34 (3)
- Lie group solutions of advection-diffusion equations PHYSICS OF FLUIDS 2021; 33 (4)
Black-Scholes Theory and Diffusion Processes on the Cotangent Bundle of the Affine Group
2020; 22 (4)
The Black-Scholes partial differential equation (PDE) from mathematical finance has been analysed extensively and it is well known that the equation can be reduced to a heat equation on Euclidean space by a logarithmic transformation of variables. However, an alternative interpretation is proposed in this paper by reframing the PDE as evolving on a Lie group. This equation can be transformed into a diffusion process and solved using mean and covariance propagation techniques developed previously in the context of solving Fokker-Planck equations on Lie groups. An extension of the Black-Scholes theory with coupled asset dynamics produces a diffusion equation on the affine group, which is not a unimodular group. In this paper, we show that the cotangent bundle of a Lie group endowed with a semidirect product group operation, constructed in this paper for the case of groups with trivial centers, is always unimodular and considering PDEs as diffusion processes on the unimodular cotangent bundle group allows a direct application of previously developed mean and covariance propagation techniques, thereby offering an alternative means of solution of the PDEs. Ultimately these results, provided here in the context of PDEs in mathematical finance may be applied to PDEs arising in a variety of different fields and inform new methods of solution.
View details for DOI 10.3390/e22040455
View details for Web of Science ID 000537222600079
View details for PubMedID 33286229
View details for PubMedCentralID PMC7516939
The unusual fluid dynamics of particle electrophoresis
JOURNAL OF COLLOID AND INTERFACE SCIENCE
2019; 553: 845–63
The classical problem of the electrophoretic motion of a spherical particle has been treated theoretically by Overbeek in his 1941 PhD thesis and almost 40 years later by O'Brien & White. Although both approaches used identical assumptions, the details are quite different. Overbeek solved for the pressure, velocity fields as well as the electrostatic potential, whereas O'Brien & White obtained the electrophoretic mobility without the need to consider the pressure and velocity explicitly. In this paper, we establish the equivalence of these two approaches that allow us to show that the tangential component of the fluid velocity has a maximum near the surface of the particle and outside the double layer, the velocity decays as 1/r3, where r is the distance from the sphere, instead of 1/r in normal Stokes flow. Associated with this behavior is that of an irrotational outer flow field. This is consistent with the fact that a sphere moving with a constant electrophoretic velocity experiences zero net force. A study of the forces on the particle also provides a physical explanation of the independence of the electrophoretic mobility on the electrostatic boundary conditions or dielectric permittivity of the particle. These results are important in situations where inter-particle interaction is considered, for instance, in electrokinetic deposition.
View details for DOI 10.1016/j.jcis.2019.06.029
View details for Web of Science ID 000483454400088
View details for PubMedID 31306934