An Integro-Differential Equation of the Fractional Form: Cauchy Problem and Solution

  • Fernando Olivar-RomeroEmail author
  • Oscar Rosas-Ortiz
Part of the CRM Series in Mathematical Physics book series (CRM)


We solve the Cauchy problem defined by the fractional partial differential equation \([\partial _{tt}-\kappa \mathbb {D}]u=0\), with \(\mathbb {D}\) the pseudo-differential Riesz operator of first order, and the initial conditions \(u(x,0)=\mu (\sqrt {\pi }x_0)^{-1}e^{-(x/x_0)^2}\), ut(x, 0) = 0. The solution of the Cauchy problem resulting from the substitution of the Gaussian pulse u(x, 0) by the Dirac delta distribution φ(x) = μδ(x) is obtained as corollary.


Fractional partial differential equations Fox H-functions Dirac delta distribution Pseudo-differential Riesz operator Complementary equation 



Financial support from Ministerio de Economía y Competitividad (Spain) grant MTM2014-57129-C2-1-P, Consejería de Educación, Junta de Castilla y León (Spain) grants VA057U16, and Consejo Nacional de Ciencia y Tecnología (México) project A1-S-24569 is acknowledged.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Physics DepartmentCinvestavMéxico CityMexico

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