Using the local time-space calculus of Peskir (2005) and the method developed in Mijatovic (2010), we derive a new integral representation for the backward probability associated with the first-passage time (FPT) of a diffusion process through a time-dependent barrier. We present a complete three-step numerical algorithm: first, the problem is reduced to a Volterra-type integral equation; second, its kernel is approximated by a Markov chain; finally, the resulting equation is solved using a quadrature rule. The method is implemented for several representative examples, and its convergence properties are established. An extension to double barrier problems is carried out
In this paper, we study a linear convection–diffusion equation with time-dependent coefficients on a bounded interval, motivated by associated physical problems. We apply and adapt the unified transform method (UTM), also known as the Fokas method, which handles both time-varying coefficients and non-zero boundary da...
Konstantinos Kalimeris, T. Özsarı· Proceedings of the Royal Soc...· 1 citation
This work studies the expansion of the level sets of the solutions of a Fisher-KPP type reaction- diffusion equation, the diffusion being given by a nonnegative, compactly supported, even kernel. Because the latter is not assumed to be spherically symmetric, the level sets advance at an asymptotic speed given by a Frei...
We establish the existence and uniqueness of solutions to a $(1+1)$-dimensional fractional stochastic Burgers-type equation featuring a Caputo fractional time derivative, a fractional Laplacian, and space-time white noise forcing modified by a Riemann-Liouville fractional integral. From a physical perspective, equation...
We develop and analyze a Monte Carlo method for sampling killed anomalous diffusions obtained by time-changing Brownian motion with drift by the inverse of a subordinator. The method targets probabilistic representations of time-nonlocal, including time-fractional, Cauchy--Dirichlet problems on bounded domains. Since i...
I. Biočić, Daniel E. Cedeño-Girón, Aleksandar Mijatovic et al.· 0 citations
This paper introduces an explicit finite-difference scheme (FDS) for time–space fractional-order (FO) advection equations whose temporal and spatial derivatives have different FOs. The discretization is constructed from a FO Taylor expansion and produces a forward-Euler–type approximation of Caputo derivatives. We prov...
Amer Dababneh, H. Al-Zoubi· International Journal of Ope...· 0 citations
For a one-dimensional time-fractional diffusion-wave equation, we investigate joint recovery of its fractional order and the initial velocity from measurements taken at one interior point. Uniqueness is proved through analytic continuation, Laplace transforms, an integral mean value theorem, and Mittag-Leffler function...
Kun Wang, Jun Xian, Lei Zhang· Mathematics· 0 citations
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