arXiv subjects

Fereidoun Sabetghadam

Publications and source records attributed to Fereidoun Sabetghadam.

6 recordsLinked to original sources

arXiv2026

A Homothetic Gauge Theory and the Regularization of the Point Charge

We introduce a Homothetic Hodge de Rham (HHDR) theory that extends the de Rham complex and Hodge decomposition to homothetically dressed differential forms. The dressing, governed by a dilaton field and a Weyl weight $w$, defines the homothetic Hodge machinery. Imposing homothetic symmetry on physical laws yields scale covariant interaction terms that arise canonically from the geometry and can be interpreted as penalty-type couplings in the language of differential equations. On this geometric foundation, we construct a Homothetic Gauge Theory (HGT) for a general weight $w$ and then specialize to $w=1$ to formulate homothetic electromagnetism, obtaining homothetic Maxwell equations for a coupled system of the physical gauge field and a homothetic offset field. As a central application, we revisit the divergence of the self-energy of a point charge: modeling the charge as a boundary condition and choosing an appropriate dilaton profile, we show that both the electric field and its total self-energy remain finite at the origin. The HHDR/HGT framework thus provides a mathematically controlled extension of gauge theory with potential implications for field theory, classical electrodynamics, and numerical penalty methods.

math-ph↗

arXiv2026

Homothetic Hodge$-$de Rham Theory and a Geometric Regularization of Elliptic Boundary Value Problems

We introduce a homothetic extension of classical Weyl integrable geometry by generalizing the usual linear gauge transformations to affine homothetic transformations centered at a distinguished harmonic, scale-invariant form $α_d$. After relinearizing these affine gauge transformations via a suitable shift of variables, we obtain a twisted exterior calculus that is structurally equivalent to the Witten deformation of the de Rham complex. On this basis, we develop a corresponding homothetic Hodge theory: we define a twisted adjoint and homothetic Laplacian, and prove a homothetic Hodge decomposition theorem on compact Riemannian manifolds. In the context of partial differential equations, we show that the scalar homothetic Laplacian provides a rigorous diffuse interface (volume penalization) representation of elliptic boundary value problems. Modeling the Weyl scale field as a fixed distribution localized near a hypersurface, the resulting lower order geometric terms form a penalty layer that enforces Dirichlet, Neumann, or Cauchy data within a single geometric equation. This formulation yields consistent weak solutions even in the presence of classically incompatible Cauchy data. As an application, we construct a nonsingular model for point sources in elliptic field equations, which preserves the correct Coulombian far field while removing the core singularity and yielding finite field energy.

math-ph↗

arXiv2020

Dark Fields do Exist in Weyl Geometry

A generalized Weyl integrable geometry (GWIG) is obtained from simultaneous affine transformations of the tangent and cotangent bundles of a (pseudo)-Riemannian manifold. In comparison with the classical Weyl integrable geometry (CWIG), there are two generalizations here: interactions with an arbitrary dark field, and, anisotropic dilation. It means that CWIG already has interactions with a {\it null} dark field. Some classical mathematics and physics problems may be addressed in GWIG. For example, by derivation of Maxwell's equations and its sub-sets, the conservation, hyperbolic, and elliptic equations on GWIG; we imposed interactions with arbitrary dark fields. Moreover, by using a notion analogous to Penrose conformal infinity, one can impose boundary conditions canonically on these equations. As a prime example, we did it for the elliptic equation, where we obtained a singularity-free potential theory. Then we used this potential theory in the construction of a non-singular model for a point charged particle. It solves the difficulty of infinite energy of the classical vacuum state.

math-ph↗

arXiv2014

An Analytical Framework for Imposition of a Rigid Immersed Surface on the Incompressible Navier--Stokes Equations

The incompressible Navier-Stokes equations are re-formulated to involve an arbitrary time dilation; and in this manner, the modified Navier-Stokes equations are obtained which have some penalization terms in the right hand side. Then, the solid rigid bodies are modeled as the regions where time is dilated infinitely. The physical and mathematical properties of the modified equations and the penalization terms are investigated, and it is shown that the modified equations satisfy the no-slip, no-diffusion, no-advection, and no-pressure coupling conditions. The modified equations can be used in exact imposition of the solid rigid bodies on the incompressible Navier-Stokes equations. To show the capability of the modified equations, three classical exact solutions of the Navier-Stokes equations, that is, the Stokes first problem, the plane stagnation point flow, and the stokes flow over a sphere are re-solved exactly, this time in the presence of a solid region.

physics.flu-dyn↗

arXiv2012

Construction of Solenoidal Immersed Velocity Vectors Using the Kinematic Velocity--Vorticity Relation

The present paper suggests a method for obtaining incompressible solenoidal velocity vectors that satisfy approximately the desired immersed velocity boundary conditions. The method employs merely the mutual kinematic relations between the velocity and vorticity fields (i.e, the curl and Laplacian operators). An initial non-solenoidal velocity field is extended to a regular domain via a zero-velocity margin, where an extended vorticity is found. Re-calculation of the velocities (subjected to appropriate boundary conditions), yields the desired solenoidal velocity vector. The method is applied to the two- and three-dimensional problems for the homogeneous Dirichlet, as well as periodic boundary conditions. The results show that the solenoidality is satisfied up to the machine accuracy for the periodic boundary conditions (employing the Fourier--spectral solution method), while an improvement in the solenoidality is achievable for the homogenous boundary conditions.

math-ph↗

arXiv2011

An Immersed Boundary Fourier Pseudo-spectral Method for Simulation of Confined Two-dimensional Incompressible Flows

The present paper is devoted to implementation of the immersed boundary technique into the Fourier pseudo-spectral solution of the vorticity-velocity formulation of the two-dimensional incompressible Navier--Stokes equations. The immersed boundary conditions are implemented via direct modification of the convection and diffusion terms, and therefore, in contrast to many other similar methods, there is not an explicit external forcing function in the present formulation. The desired immersed boundary conditions are approximated on some regular grid points, using different orders (up to second-order) polynomial extrapolations. At the beginning of each timestep, the solenoidal velocities (also satisfying the desired immersed boundary conditions), are obtained and fed into a conventional pseudo-spectral solver, together with a modified vorticity. The zero-mean pseudo-spectral solution is employed, and therefore, the method is applicable to the confined flows with zero mean velocity and vorticity, and without mean vorticity dynamics. In comparison to the classical Fourier pseudo-spectral solution, the method needs ${\cal O}(4(1+\log N)N)$ more operations for boundary condition settings. Therefore, the computational cost of the method, as a whole, is scaled by ${(N \log N)}$. The classical explicit fourth-order Runge--Kutta method is used for time integration, and the boundary conditions are set at the beginning of each sub-step, in order to increasing the time accuracy. The method is applied to some fixed and moving boundary problems, with different orders of boundary conditions; and in this way, the accuracy and performance of the method are investigated and compared with the classical Fourier pseudo-spectral solutions.

math-ph↗