Vasily E Tarasov 2009 J. Phys. A: Math. Theor. 42 465102 doi:10.1088/1751-8113/42/46/465102
Vasily E Tarasov1
Show affiliationsDiscrete maps with long-term memory are obtained from nonlinear differential equations with Riemann–Liouville and Caputo fractional derivatives. These maps are generalizations of the well-known universal map. The memory means that their present state is determined by all past states with special forms of weights. To obtain discrete maps from fractional differential equations, we use the equivalence of the Cauchy-type problems and to the nonlinear Volterra integral equations of the second kind. General forms of the universal maps with memory, which take into account general initial conditions for the cases of the Riemann–Liouville and Caputo fractional derivative, are suggested.
45E05 Integral equations with kernels of Cauchy type (See also 35J15)
45J05 Integro-ordinary differential equations (See also 34K05, 34K30, 47G20)
Issue 46 (20 November 2009)
Received 6 August 2009, in final form 8 September 2009
Published 26 October 2009
Vasily E Tarasov 2009 J. Phys. A: Math. Theor. 42 465102
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