On fundamental solutions of higher-order space-fractional Dirac equations

Starting from the pseudo-differential decomposition D=(-Delta)H-1/2 of the Dirac operator D= Sigma(n)(j=1)e(j)partial derivative(xj) in terms of the fractional operator (-Delta)(1/2) of order 1 and of the Riesz-Hilbert type operator H we will investigate the fundamental solutions of the space-fractional Dirac equation of Levy-Feller type partial derivative(t)Phi(alpha)(x, t; theta) = -(-Delta)alpha/2exp(i pi theta/2H)Phi(alpha)(x, t; theta) involving the fractional Laplacian -(-Delta)(alpha/2) of order alpha, with 2m <= alpha < 2m + 2 (m is an element of N), and the exponentiation operator exp(i pi theta/2H) as the hypercomplex counterpart of the fractional Riesz-Hilbert transform carrying the skewness parameter theta, with values in the range vertical bar theta vertical bar <= min{alpha - 2m, 2m + 2 - alpha}. Such model problem permits us to obtain hypercomplex counterparts for the fundamental solutions of higher-order heat-type equations partial derivative F-t(M)(x, t) = kappa(M)(partial derivative(x))F-M(M)(x, t) (M = 2,3, ... ) in case where the even powers resp. odd powers D-2m = (-Delta)(m) (M = alpha = 2m) resp. D2m+1 = (-Delta)Hm+1/2 (M = alpha = 2m + 1) of D are being considered.