Fractional Equations and Models

The analysis of fractional differential equations, carried out by means of fractional calculus and integral transforms (Laplace, Fourier), leads to certain special functions of Mittag-Leffler (M-L) and Wright types. These useful special functions are investigated systematically as relevant cases of the general class of functions which are popularly known as Fox H-functions, named after Charles Fox, who initiated a detailed study of these functions as symmetrical Fourier kernels (Fox, Trans Am Math Soc 98:395, 1961). Definitions, some properties, relations, asymptotic expansions, and Laplace transform formulas for the M-L type functions and Fox H-function are given in this chapter. At the beginning of the twentieth century, Swedish mathematician Gösta Mittag-Leffler introduced a generalization of the exponential function, today known as the M-L function (Mittag-Leffler, C R Acad Sci Paris 137:554, 1903). The properties of the M-L function and its generalizations had been totally ignored by the scientific community for a long time due to their unknown application in the science. In 1930 Hille and Tamarkin solved the Abel-Volterra integral equation in terms of the M-L function (Hille and Tamarkin, Ann Math 31:479, 1930). The basic properties and relations of the M-L function appeared in the third volume of the Bateman project in the Chapter XVIII: Miscellaneous Functions (Erdélyi et al., Higher Transcendental Functions, vol. 3. McGraw-Hill, New York, 1955). More detailed analysis of the M-L function and their generalizations as well as the fractional derivatives and integrals were published later, and it has been shown that they are of great interest for modeling anomalous diffusion and relaxation processes. Similarly, Fox H-function, introduced by Fox (Trans Am Math Soc 98:395, 1961), is of great importance in solving fractional differential equations and for analysis of anomalous diffusion processes. The Fox H-function has been used to express the fundamental solution of the fractional diffusion equation obtained from a continuous time random walk model. Therefore, in this chapter we will give the most important definitions, relations, asymptotic expansions of these functions which represent a basis for investigation of anomalous diffusion and non-exponential relaxation in different complex systems.

From the time of discovery of calculus by Leibniz, he studied the problem of fractional differentiations. 30 September 1695, the day when Leibniz sent a letter to L’Hôpital with a reply of the L’Hôpital’s question related to the differentiation of a function of order n = 1∕2, became a birthday of the fractional calculus. By using the Leibniz product rule and the binomial theorem he obtained some paradoxical results. Euler partially resolved the Leibniz paradox by introducing the gamma function as 1 ⋅ 2 ⋅… ⋅ n = n! = Γ(n + 1). Therefore, the fractional calculus has attracted attention to a range of celebrated mathematicians and physicists, such as Leibniz, Euler, Laplace, Lacroix, Fourier, Abel, Liouville, Riemann, Grünwald, Letnikov, to name but a few.

We now analyze Cauchy type problems of differential equations of fractional order with Hilfer and Hilfer-Prabhakar derivative operators. The existence and uniqueness theorems for n-term nonlinear fractional differential equations with Hilfer fractional derivatives of arbitrary orders and types will be proved. Cauchy type problems for integro-differential equations of Volterra type with generalized Mittag-Leffler function in the kernel will be considered as well. Using the operational method of Mikusinski, the solution of a Cauchy type problem for a linear n-term fractional differential equations with Hilfer fractional derivatives will be obtained. We will show utility of operational method to solve Cauchy type problems of a wide class of integro-differential equations with variable coefficients, involving Prabhakar integral operator and Laguerre derivatives. For this purpose, following some recent works, we choose the examples which, by means of fractional derivatives, generalize the well-known ordinary differential equations and partial differential equations, related to time fractional heat equations, free electronic laser equation, some evolution and boundary value problems, and finally some Cauchy type problems for the generalized fractional Poisson process.

In this chapter we pay our attention to the CTRW theory and the related fractional diffusion and Fokker-Planck equations. In the literature the mostly used fractional diffusion equations are equivalent.

The Langevin equation is connected to the Brownian motion formulated by Einstein and Smoluchowski. The Langevin equation for a free particle with mass m is given by Langevin (CR Acad Sci Paris 146:530, 1908)

where x(t) is the particle displacement, v(t) is its velocity, γ is the friction coefficient and ξ(t) is a Gaussian random noise with zero mean 〈ξ(t)〉 = 0 (so-called white noise).

FGLEs are generalizations of the GLE where the integer order derivatives is substituted by fractional derivatives. Recently, some GLE models for a particle driven by single or multiple fractional Gaussian noise have been investigated in order to describe generalized diffusion processes, such as accelerating and retarding diffusion.