Applications of nonstandard finite difference schemes

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Application of a Mickens finite‐difference scheme to the cylindrical Bratu‐Gelfand problem (2004, methodology): “Exactly which method to use to approximate differential equations using difference equations on a discrete grid is a topic that has been extensively addressed by Ronald Mickens of Clark Atlanta University ([15], [16]).”
On the use of nonstandard finite difference methods† (2005, methodology): “However, several important rules have been discovered by Mickens [91,107], which we state shortly.”
Dynamic consistency: a fundamental principle for constructing nonstandard finite difference schemes for differential equations (2005, background): “, [5,7], 1⁄2uðx; tÞ 2 ! um21u m ; ð2:5Þ where x! xm 1⁄4 ðDxÞm and t! tk 1⁄4 ðDtÞk: It should be understood that the NSFD procedures do not require anyaprioriknowledge of the exact solutions to the differential equations.”
Nonstandard discretizations of the generalized Nagumo reaction‐diffusion equation (2003, methodology): “To achieve this aim, the nonstandard framework of Mickens [4, 7, 8] will be used.”
Nonstandard finite difference schemes for differential equations (2014, methodology): “Table 2 shows the error Emax = max 0≤x≤1 |u(tn) − un| obtained with our NSFD method uk+1 = uk − h(2κukuk+1), when the integration is performed on the interval [0, 1].”