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Cauchy problems with fractal-fractional operators and applications to groundwater dynamics

Fractals, ISSN: 1793-6543, Vol: 28, Issue: 8
2020
  • 28
    Citations
  • 0
    Usage
  • 5
    Captures
  • 0
    Mentions
  • 0
    Social Media
Metric Options:   Counts1 Year3 Year

Metrics Details

  • Citations
    28
    • Citation Indexes
      28
  • Captures
    5

Article Description

As the Riemann-Liouville derivative is a derivative of a convolution of a function and the power law, the fractal-fractional derivative of a function is the fractal derivative of a convolution of that function with the power law or exponential decay. In order to further open new doors on ongoing investigations with field of partial differential equations with non-conventional differential operators, we introduce in this paper new Cauchy problems with fractal-fractional differential operators. We consider two cases, when the operator is constructed with power law and when it is constructed with exponential decay law with Delta-Dirac property. For each case, we present the conditions under which the exact solution exists and is unique. We suggest a suitable and accurate numerical scheme that can be used to solve such differential equation numerically. We present illustrative examples where an application to a partial differential equation and to a model of groundwater flow within the confined aquifer are done with numerical simulations provided. The clear variation of water level shows the impact of the fractal-fractional derivative on the dynamics.

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