A stochastic method for solving time-fractional differential equations Articles uri icon

authors

  • GUIDOTTI, NICOLAS L.
  • ACEBRON DE TORRES, JUAN ANTONIO
  • MONTEIRO, JOSE

publication date

  • April 2024

start page

  • 240

end page

  • 253

volume

  • 159

abstract

  • We present a stochastic method for efficiently computing the solution of time-fractional partial differential equations (fPDEs) that model anomalous diffusion problems of the subdiffusive type. After discretizing the fPDE in space, the ensuing system of fractional linear equations is solved resorting to a Monte Carlo evaluation of the corresponding Mittag-Leffler matrix function. This is accomplished through the approximation of the expected value of a suitable multiplicative functional of a stochastic process, which consists of a Markov chain whose sojourn times in every state are Mittag-Leffler distributed. The resulting algorithm is able to calculate the solution at conveniently chosen points in the domain with high efficiency. In addition, we present how to generalize this algorithm in order to compute the complete solution. For several large-scale numerical problems, our method showed remarkable performance in both shared-memory and distributed-memory systems, achieving nearly perfect scalability up to

subjects

  • Mathematics

keywords

  • monte carlo method; time-fractional differential equations; anomalous diffusion; mittag-leffler function; matrix functions; parallel algorithms; high performance computing