Fractional relaxation equations of distributed order

Abstract

The method of approximation of the tempered convolution based on Laguerre polynomials we are developing here applies to solving nonlinear fractional coupled systems appearing in mechanical (see Stojanović, 2011) [15]) and other fractional convolution equations from life and science (see Stojanović, 2011 [27]). In this paper, we use it as a tool in solving linear and nonlinear relaxation equations of distributed order with constant relaxation parameter, special weight functions, and with a lack of distributional solutions. We expand some special functions such as the Mittag-Leffler function into Laguerre series. A further perspective of a development of this method is generalization to the n -dimensional case with applications to fractional convolution equations in the space S ′ ( R ̄ + n ) = S + ′ ( R ̄ + ) × S + ′ ( R ̄ + ) × ⋯ S + ′ ( R ̄ + ) .