Related papers: Old Problems, Classical Methods, New Solutions
Recent progress in the field of artificial intelligence, machine learning and also in computer industry resulted in the ongoing boom of using these techniques as applied to solving complex tasks in both science and industry. Same is, of…
A general class of probability density functions \[u(x,t)=Ct^{-\alpha d}\left (1-\left (\frac{\|x\|}{ct^{\alpha}}\right )^{\beta}\right )_+^{\gamma},\quad x\in \mathbb{R}^d,t>0,\] is considered, containing as particular case the Barenblatt…
A fractional Stefan problem with a boundary convective condition is solved, where the fractional derivative of order $ \alpha \in (0,1) $ is taken in the Caputo sense. Then an equivalence with other two fractional Stefan problems (the first…
We propose a new method for solving optimal stopping problems (such as American option pricing in finance) under minimal assumptions on the underlying stochastic process $X$. We consider classic and randomized stopping times represented by…
This paper investigates the application of the classical Chernoff's theorem to construct explicit solutions for the heat and Schr\"odinger equations on the Heisenberg group $\mathbb{H}^d$. Using semigroup approximation techniques, we obtain…
The understanding of the underlying dynamical mechanisms which determine the macroscopic laws of heat conduction is a long standing task of non-equilibrium statistical mechanics. A better understanding of the mechanism of heat conduction…
A mathematical model for a one-phase change problem (particularly a Stefan problem) with a memory flux, is obtained. The hypothesis that the weighted sum of fluxes back in time is proportional to the gradient of temperature is considered.…
The primary challenge of market making in spot precious metals is navigating the liquidity that is mainly provided by futures contracts. The Exchange for Physical (EFP) spread, which is the price difference between futures and spot, plays a…
We develop estimates for the solutions and derive existence and uniqueness results of various local boundary value problems for Dirac equations that improve all relevant results known in the literature. With these estimates at hand, we…
When prices reflect all available information, they oscillate around an equilibrium level. This oscillation is the result of the temporary market impact caused by waves of buyers and sellers. This price behavior can be approximated through…
Default risk calculus plays a crucial role in portfolio optimization when the risky asset is under threat of bankruptcy. However, traditional stochastic control techniques are not applicable in this scenario, and additional assumptions are…
A new mathematical model for the Black-Scholes equation is proposed to forecast option prices. This model includes new interval for the price of the underlying stock as well as new initial and boundary conditions. Conventional notions of…
We study the supercooled Stefan problem in arbitrary dimensions. First, we study general solutions and their irregularities, showing generic fractal freezing and nucleation, based on a novel Markovian gluing principle. In contrast, we then…
The Stefan problem is a classical free-boundary problem that models phase-change processes and poses computational challenges due to its moving interface and nonlinear temperature-phase coupling. In this work, we develop a physics-informed…
We consider the semilinear heat equation $$ u_t-\Delta u=|u|^{p-1}u,\ \ (t,x)\in\mathbb{R}^+\times\mathbb{R}^n. $$ The well-known difficulty with this problem is that the potential well method cannot be applied directly, due to the scaling…
This book considers posing and the methods of solving simple linear boundary-value problems in classical mathematical physics. The questions encompassed include: the fundamentals of calculus of variations; one-dimensional boundary-value…
Stefan problems relevant to burning oil-water systems are formulated. Two moving boundary sub-problems are defined: burning liquid surface and formation of a distillation ("hot zone") layer beneath it. The basic model considers a heat…
We use machine learning methods to approximate a classical density functional. As a study case, we choose the model problem of a Lennard Jones fluid in one dimension where there is no exact solution available and training data sets must be…
The inflationary flow equations are a frequently used method of surveying the space of inflationary models. In these applications the infinite hierarchy of differential equations is truncated in a way which has been shown to be equivalent…
We present a new explicit and stable numerical algorithm to solve the homogeneous heat equation. We illustrate the performance of the new method in the cases of two 2D systems with highly inhomogeneous random parameters. Spatial…