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) =E α (h α D x α )f(x).