2010
DOI: 10.1016/j.jmaa.2009.08.014
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On the local fractional derivative

Abstract: We present the necessary conditions for the existence of the Kolwankar-Gangal local fractional derivatives (KG-LFD) and introduce more general but weaker notions of LFDs by using limits of certain integral averages of the difference-quotient. By applying classical results due to Stein and Zygmund (1965) [16] we show that the KG-LFD is almost everywhere zero in any given intervals. We generalize some of our results to higher dimensional cases and use integral approximation formulas obtained to design numerical … Show more

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Cited by 111 publications
(70 citation statements)
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“…[1][2][3][4][5][6][7][8] The theory of the local fractional derivative (LFD) is a mathematical tool for describing fractals, that was used to model the fractal complexity in shallow water surfaces, 14 LC-electric circuit, 15 traveling-wave solution of the Burgerstype equation, 16 PDEs, [17][18][19][20] ODEs, 21 and inequalities. 22,23 The useful models for the LFD were considered [24][25][26][27][28][29] and discussed. 30 However, the nonlinear local fractional Boussinesq equations and their non-differentiable-type traveling-wave solutions have not yet been tackled.…”
Section: Introductionmentioning
confidence: 99%
“…[1][2][3][4][5][6][7][8] The theory of the local fractional derivative (LFD) is a mathematical tool for describing fractals, that was used to model the fractal complexity in shallow water surfaces, 14 LC-electric circuit, 15 traveling-wave solution of the Burgerstype equation, 16 PDEs, [17][18][19][20] ODEs, 21 and inequalities. 22,23 The useful models for the LFD were considered [24][25][26][27][28][29] and discussed. 30 However, the nonlinear local fractional Boussinesq equations and their non-differentiable-type traveling-wave solutions have not yet been tackled.…”
Section: Introductionmentioning
confidence: 99%
“…It can be used to describe the jump discontinuities of a waveform (Zhang, 1996, Chen et al, 2010). An example for BMO is the one-dimensional step function…”
Section: Theorymentioning
confidence: 99%
“…Recently, the concept of local fractional derivatives have gained relevance, namely because they kept some of the properties of ordinary derivatives, although they loss the memory condition inherent to the usual fractional derivatives. For example, in [9], a concept similar to the Caputo fractional derivative is presented, but the firstorder derivative f 0 .t / is replaced by another operator:…”
Section: Introductionmentioning
confidence: 99%