Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution

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Abstract

The paper deals with the solution of some fractional partial differential equations obtained by substituting modified Riemann-Liouville derivatives for the customary derivatives. This derivative is introduced to avoid using the so-called 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 by-passing 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) =Eα (hαDxα)f(x).

AMS Mathematics Subject Classification

26A33 49K20 44A10 

Key words and phrases

Fractional PDE Riemann-Liouville derivative fractional Taylor series Mittag-Leffler function Lagrange characteristics Lagrange constant variation 

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Copyright information

© Korean Society for Computational & Applied Mathematics and Korean SIGCAM 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Quebec at MontrealMontrealCanada

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