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We outline a statistical theory of turbulence based on the Lagrangian formulation of fluid motion. We derive a hierarchy of evolution equations for Lagrangian N-point probability distributions as well as a functional equation for a suitably…
We propose and analyze an algorithm to approximate distribution functions and densities of perpetuities. Our algorithm refines an earlier approach based on iterating discretized versions of the fixed point equation that defines the…
In this paper we investigate the solution of generalized distributed order diffusion equations with composite time fractional derivative by using the Fourier-Laplace transform method. We represent solutions in terms of infinite series in…
A deterministic pathogen transmission model based on high-fidelity physics has been developed. The model combines computational fluid dynamics and computational crowd dynamics in order to be able to provide accurate tracing of viral matter…
A distributed parameter model of pipeline transmission line behaviour is presented, based on a Galerkin method incorporating frequency-dependent friction. This is readily interfaced to an existing model of the pumping dynamics of a plunger…
Cerebral blood flow and perfusion can be estimated using tracer dilution experiments. Accurate estimation of blood flow parameters is a crucial part of medical imaging for effective diagnosis and treatment. This study explores two themes:…
Sun and Farooq [2] showed that random samples can be efficiently drawn from an arbitrary n-dimensional hyperellipsoid by transforming samples drawn randomly from the unit n-ball. They stated that it was a straightforward to show that, given…
We consider Taylor dispersion for tracer particles in micro-fluidic planar channels with strong confinement. In this context, the channel walls modify the local diffusivity tensor and also interactions between the tracer particles and the…
A fractional diffusion equation with advection term is rigorously derived from a kinetic transport model with a linear turning operator, featuring a fat-tailed equilibrium distribution and a small directional bias due to a given vector…
We study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers, of order $s \in (0,1)$, of symmetric, coercive, linear, elliptic, second-order operators in bounded domains…
From a suitable integral representation of the Laplace transform of a positive semi-definite quadratic form of independent real random variables with not necessarily identical densities a univariate integral representation is derived for…
A new definition of a fractional derivative has recently been developed, making use of a fractional Dirac delta function as its integral kernel. This derivative allows for the definition of a distributional fractional derivative, and as…
In this paper we explain how to use the Fast Fourier Transform (FFT) to solve partial differential equations (PDEs). We start by defining appropriate discrete domains in coordinate and frequency domains. Then describe the main limitation of…
In view of the role of reaction equations in physical problems, the authors derive the explicit solution of a fractional reaction equation of general character, that unifies and extends earlier results. Further, an alternative shorter…
Curse of Dimensionality is an unavoidable challenge in statistical probability models, yet diffusion models seem to overcome this limitation, achieving impressive results in high-dimensional data generation. Diffusion models assume that…
The goal of this paper is to study the Sawi transform and its relationship to Hilfer-Prabhakar and regularized Hilfer-Prabhakar fractional derivatives, as well as to present some lemmas related to the Sawi transform. Additionally, the paper…
This paper is concerned with the inverse random source problem for a stochastic time fractional diffusion equation, where the source is assumed to be driven by a Gaussian random field. The direct problem is shown to be well-posed by…
A physical-mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. The fundamental solution (for the…
Diffusion theory is a central tool of modern population genetics, yielding simple expressions for fixation probabilities and other quantities that are not easily derived from the underlying Wright-Fisher model. Unfortunately, the textbook…
A theoretical framework is developed to describe the transformation that distributes probability density functions uniformly over space. In one dimension, the cumulative distribution can be used, but does not generalize to higher…