Processes and materials involving relaxation, creep and fading memory are better understood recently by using fractional LDE representations of the models for the advantage of reducing the number of parameters improving curve fitting schemes and avoiding nonlinearities. A mathematical stringent method to define and solve such equations without an a-priori definition of fractional derivatives is presented. By establishing a "functional calculus" we avoid the well-known difficulties such as fractional initial or boundary conditions and the loss of globality. In particular, using the popular Fractional Calculus (see e.g. [3]) it is necessary to feed in causality to get it out. In the case of constant coefficients we present criteria for existence, continuity and causality of global solutions. Moreover we get a surprisingly simple algorithm for obtaining the solutions.
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