Fractional partial differential equations and modified RiemannLiouville derivative new methods for solution
 Guy Jumarie
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Abstract
The paper deals with the solution of some fractional partial differential equations obtained by substituting modified RiemannLiouville derivatives for the customary derivatives. This derivative is introduced to avoid using the socalled Caputo fractional derivative which, at the extreme, says that, if you want to get the first derivative of a function you must before have at hand its second derivative. Firstly, one gives a brief background on the fractional Taylor series of nondifferentiable functions and its consequence on the derivative chain rule. Then one considers linear fractional partial differential equations with constant coefficients, and one shows how, in some instances, one can obtain their solutions on bypassing the use of Fourier transform and/or Laplace transform. Later one develops a Lagrange method via characteristics for some linear fractional differential equations with nonconstant coefficients, and involving fractional derivatives of only one order. The key is the fractional Taylor series of non differentiable functionf(x + h) =E _{ α } (h ^{ α }D _{ x } ^{ α } )f(x).
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 Title
 Fractional partial differential equations and modified RiemannLiouville derivative new methods for solution
 Journal

Journal of Applied Mathematics and Computing
Volume 24, Issue 12 , pp 3148
 Cover Date
 20070501
 DOI
 10.1007/BF02832299
 Print ISSN
 15985865
 Online ISSN
 18652085
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 26A33
 49K20
 44A10
 Fractional PDE
 RiemannLiouville derivative
 fractional Taylor series
 MittagLeffler function
 Lagrange characteristics
 Lagrange constant variation
 Authors

 Guy Jumarie ^{(1)}
 Author Affiliations

 1. Department of Mathematics, University of Quebec at Montreal, Downtown St, P.O. Box 8888, H3C 3P8, Montreal, Qc, Canada