Related papers: The Finite-Difference Analysis and Time Flow
A fractional Hamiltonian formalism is introduced for the recent combined fractional calculus of variations. The Hamilton-Jacobi partial differential equation is generalized to be applicable for systems containing combined Caputo fractional…
In this paper we present a rather general phenomenological theory of tick-by-tick dynamics in financial markets. Many well-known aspects, such as the L\'evy scaling form, follow as particular cases of the theory. The theory fully takes into…
Statistics of stochastic processes are crucially influenced by the boundary conditions. In one spatial dimension, for example, the first passage time distribution in semi-infinite space (one absorbing boundary) is markedly different from…
The "theoretical limit of time-frequency resolution in Fourier analysis" is thought to originate in certain mathematical and/or physical limitations. This, however, is not true. The actual origin arises from the numerical (technical) method…
Studies have shown that standard lectures and instructional laboratory experiments are not effective at teaching interference and diffraction. In response, the author created an interactive computer program that simulates interference and…
Scattering amplitudes are often split up into their color (su(N)) and kinematic components. Since the su(N) gauge part can be described using flows of color, one may anticipate that the double su(2) kinematic part can be described in terms…
Mathematical models that accurately predict the mechanical behavior of blood can contribute to the development of biomedical devices and medications which are relevant in clinical applications. The models existing in the literature are…
Fluctuations in the return time statistics of a dynamical system can be described by a new spectrum of dimensions. Comparison with the usual multifractal analysis of measures is presented, and difference between the two corresponding sets…
We define a time-changed fractional Ornstein-Uhlenbeck process by composing a fractional Ornstein-Uhlenbeck process with the inverse of a subordinator. Properties of the moments of such process are investigated and the existence of the…
We introduce the notion of structural derivative on time scales. The new operator of differentiation unifies the concepts of fractal and fractional order derivative and is motivated by lack of classical differentiability of some…
We develop and analyze a numerical method for stochastic time-fractional diffusion driven by additive fractionally integrated Gaussian noise. The model involves two nonlocal terms in time, i.e., a Caputo fractional derivative of order…
The main objective of this addendum to the mentioned article by Park is to provide some remarks on bifurcation theories for nonlinear partial differential equations (PDE) and their applications to fluid dynamics problems. We only wish to…
These notes contains an introduction to the theory of Brownian and diffusion local time, as well as its relations to the Tanaka Formula, the extended Ito-Tanaka formula for convex functions, the running maximum process, and the theory of…
The present paper deals with the numerical solution of time-fractional advection-diffusion equations involving the Caputo derivative with source term by means of an unconditionally stable implicit finite difference method on quasi-uniform…
Consider a time series with missing observations but a known final point. Using control theory ideas we estimate/predict these missing observations. We obtain recurrence equations which minimize sum of squares of a control sequence. An…
In this work, an inverse problem in the fractional diffusion equation with random source is considered. Statistical moments are used of the realizations of single point observation $u(x_0,t,\omega).$ We build the representation of the…
We propose an extension of the cable equation by introducing a Caputo time fractional derivative. The fundamental solutions of the most common boundary problems are derived analitically via Laplace Transform, and result be written in terms…
Inspired by the works of \cite{baz2} and \cite{kian}, this study develops an abstract framework for analyzing differential equations with space-dependent fractional time derivatives and bounded operators. Within this framework, we establish…
In this paper, we consider some aspects of the numerical analysis of the mathematical model of fractional Duffing with a derivative of variable fractional order of the Riemann-Liouville type. Using numerical methods: an explicit…
The first part of this paper introduces sufficient conditions to determine conservation laws of diffusion equations of arbitrary fractional order in time. Numerical methods that satisfy a discrete analogue of these conditions have…