Numerical Solution of the Anomalous Diffusion Equation in a Rectangular Domain via Hypermatrix Equations

Matthew Harker, Paul O'Leary

Research output: Contribution to journalArticleResearchpeer-review


This paper presents a fundamentally new approach to the numerical solution of partial fractional differential equations (PFDE) in higher dimensions by means of hypermatrix equations. By generalizing matrices to their higher dimensional form, i.e., hypermatrices, we show that there is a one-to-one correspondence between PFDE and hypermatrix equations. It is shown that the resulting hypermatrix equation can be solved in an expedient manner, namely by an O (n4) algorithm for an l x m x n discretized integral surface with l ~ m ~ n. Given that previous algorithms were of order O (n9) this represents a massive improvement in computational complexity. The proposed algorithm is demonstrated for a problem in two spatial and one time dimension; however, the algorithm can be extended to higher dimensions as well.
Original languageEnglish
Pages (from-to)9730-9735
Number of pages6
Issue number1
Publication statusE-pub ahead of print - 18 Oct 2017


  • Fractional Derivative
  • Fractional Order Systems
  • Hypermatrix
  • Matrix Equations
  • Numerical Approximation
  • Partial Fractional Differential Equation

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