Viscoelastic flows with fractional derivative models: Computational approach by convolutional calculus of Dimovski

Abstract

An initial-boundary value problem for the velocity distribution of a viscoelastic flow with generalized fractional Oldroyd-B constitutive model is studied. The model contains two Riemann-Liouville fractional derivatives in time. The eigenfunction expansion of the solution is constructed. The behavior of the time-dependent components of the solution is studied and the results are used to establish convergence of the series under some conditions. Further, applying the convolutional calculus approach proposed by Dimovski (I.H. Dimovski, Convolutional Calculus, Kluwer, Dordrecht (1990)), a Duhamel-type representation of the solution is found, containing two convolution products of particular solutions and the given initial and source functions. A non-classical convolution with respect to spatial variable is used. The obtained representation is applied for numerical computation of the solution in the case of a generalized second grade fluid. Numerical results for several one-dimensional examples are given and the present technique is compared to a finite difference method in terms of efficiency, accuracy, and CPU time.

Completely monotone functions and some classes of fractional evolution equations

“…The developed technique is also applicable to other related problems, for example the Rayleigh-Stokes problem for the generalized second grade fluid with fractional derivative model, see e.g. [3,4], or to more general abstract Volterra integral equations with kernel k(t), which Laplace transform k(s) is well-defined for s > 0 and is such that ( k(s)) −1 is a Bernstein function.…”

Subordination Principle for a Class of Fractional Order Differential Equations

“…Alternatively, this expansion can be deduced, inserting the series expansion of the function g(s) s e −τ g(s) in (31) and using the Laplace transform pair (12).…”

“…Numerical analysis of Problem (1) with A being the one-or two-dimensional Laplacian with Dirichlet boundary conditions is carried out in [6][7][8][9][10][11]. In [12], a compact Duhamel-type representation of the solution is obtained and used for its numerical computation.…”

Tau method for the numerical solution of a fuzzy fractional kinetic model and its application to the oil palm frond as a promising source of xylose

“…We mention that the fractional derivatives and integrals provide more precise models of the systems under consideration. The researchers have proved the existence of fractional calculus in anomalous diffusion [11], medicine [12], viscoelastic [13], random and disordered media [14], signal processing [15], and so on.…”