Generalized distributed order diffusion equations with composite time fractional derivative

Abstract

In this paper we investigate the solution of generalized distributed order diffusion equations with composite time fractional derivative by using the Fourier-Laplace transform method. We represent solutions in terms of infinite series in Fox H-functions. The fractional and second moments are derived by using Mittag-Leffler functions. We observe decelerating anomalous subdiffusion in case of two composite time fractional derivatives. Generalized uniformly distributed order diffusion equation, as a model for strong anomalous behavior, is analyzed by using Tauberian theorem. Some previously obtained results are special cases of those presented in this paper.MSC2010. Primary: 26A33, 33E12; Secondary: 34A08, 35R11.PACS numbers:The initial condition is usually taken as W (x, 0+) = δ(x), and the boundary conditions are given by W (±∞, t) = ∂ ∂x W (±∞, t) = 0 in unbounded domain in space of Lebesgue integrable functions. These are natural boundary conditions which are used in order to have an unique solution of the equation, as well as the solution to be bounded and differentiable. Note that these conditions are also used in order the Fourier transform (F [f (x)] =

Applications of Distributed-Order Fractional Operators: A Review

“…Asymptotic solutions to initial and boundary value problems based on the DO time-fractional diffusion equations can be found in [ 293 , 294 ]. Some additional and important mathematical aspects, such as the existence of the solution to different types of DO diffusion equations, the solvability of DO diffusion equations, subordination properties, and positivity of the solution were addressed in [ 59 , 63 , 263 , 287 , 295 , 296 , 297 , 298 , 299 , 300 ]. In a series of papers [ 71 , 72 , 301 ], Luchko analyzed the well-posedness of the DO formulation via maximal principles, and obtained a priori norm estimates for solutions to both linear and nonlinear DO diffusion equations.…”

Time-fractional telegraph equation of distributed order in higher dimensions

“…Applying the inverse Laplace transform to (19) and taking into account Theorem 2.1, and expressions ( 25), ( 26), ( 27), (30), (31), and (32), we obtain…”

“…Boundary value problems for the generalized time-fractional diffusion equation of distributed order were studied in [20] and maximum principles for such equation were presented in [2]. More recently, we can find the work of Sandeva et al [30] where it was investigated the solution of generalized distributed order diffusion equations with composite time-fractional derivative by using the Fourier-Laplace transform method. There are also works dealing with numerical methods for solving these equations but our focus is on the analytical analysis for FPDE of distributed order.…”

On a jump-telegraph process driven by an alternating fractional Poisson process

“…[28] (vii) Linear death process n 0 E ν,1 (−μt ν ) [30] (viii) Sublinear death process [30] (ix) Telegraph process 0 [26] and for k = 3 and λ 1 = λ 2 = λ 3 = 1 in case (iv). The asymptotic behaviour of the means given in Table 1 is finally provided in Table 2, obtained thanks to Equation (4.4.16) of [16] and Equation (A.3) of [37]. In case (viii) of Table 2, γ 0.577 216 is the Euler constant and ψ(z) = (z)/ (z) is the digamma function, i.e.…”