Distributed order equations as boundary value problems

Two Alternating Direction Implicit Difference Schemes for Two-Dimensional Distributed-Order Fractional Diffusion Equations

“…So this topic is discussed in the present work. Two effective ADI difference schemes, with the convergence orders O(τ 2 | ln τ | + h 2 1 + h 2 2 + α 2 ) and O(τ 2 | ln τ | + h 4 1 + h 4 2 + α 4 ) in a discrete L 1 (L ∞ ) norm, respectively, are constructed for the two-dimensional time-fractional diffusion equations of distributed order. The energy method is used to proceed with the corresponding theoretical analysis.…”

“…The exact solution is given by u(x, y, t) = 2 21 t 5 x 4 (1 − x) 4 y 4 (1 − y) 4 . The choice of the power 21 is to make that max…”

“…Diethelm and Ford [32,33] presented the numerical methods for solving the distributed-order ordinary differential equations, where the distributed-order integral was firstly approximated using the quadrature formula and then the multi-term fractional differential equations were resulted in, which were finally reduced to a system of single-term equations. The idea was followed by Ford and Morgado [4] still for the distributed-order ordinary differential equations. The matrix approach to the solution of distributed-order differential equations was introduced by Podlubny et al [34].…”

“…In other words, the single-order time-fractional diffusion equation can not characterize this process and it must be generalized, among which the distributed-order time-fractional diffusion equation is a good alternative choice. Recent researches show that many other applications of the distributed-order differential equation have appeared in modelling, for example, the stress behavior of an elastic medium, the torsional phenomenon of anelastic or dielectric spherical shells and infinite planes, the rheological properties of composite materials, dielectric induction and diffusion and so on [4].…”

Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method

“…With the growing interest on this type of equation, numerical methods started being developed, for example in the work by Diethelm and Ford [16] where they present a basic framework for the numerical solution of distributed order differential equations (see also [17,18]). A more recent increase in interest in the use of distributed order differential equations (particularly in the case where the derivatives are given in the Caputo sense) led Ford and Morgado [24] to discuss the existence and uniqueness of solutions for this type of equation, and also to propose a numerical method for their approximation in the case where the initial conditions are not known (with boundary conditions being given away from the origin). Two years later, Katsikadelis [38] devised a numerical method for the solution of the distributed order FDE approximating them with a multiterm FDE (that is solved by adjusting appropriately the numerical method developed for multi-term FDEs).…”