Numerical Solution of the Anomalous Diffusion Equation in a Rectangular Domain via Hypermatrix Equations
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In: IFAC-PapersOnLine, Vol. 50.2017, No. 1, 18.10.2017, p. 9730-9735.
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TY - JOUR
T1 - Numerical Solution of the Anomalous Diffusion Equation in a Rectangular Domain via Hypermatrix Equations
AU - Harker, Matthew
AU - O'Leary, Paul
PY - 2017/10/18
Y1 - 2017/10/18
N2 - 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.
AB - 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.
KW - Fractional Derivative
KW - Fractional Order Systems
KW - Hypermatrix
KW - Matrix Equations
KW - Numerical Approximation
KW - Partial Fractional Differential Equation
UR - http://www.scopus.com/inward/record.url?scp=85031811220&partnerID=8YFLogxK
U2 - 10.1016/j.ifacol.2017.08.2176
DO - 10.1016/j.ifacol.2017.08.2176
M3 - Article
AN - SCOPUS:85031811220
VL - 50.2017
SP - 9730
EP - 9735
JO - IFAC-PapersOnLine
JF - IFAC-PapersOnLine
SN - 2405-8963
IS - 1
ER -
