Nuclear magnetic resonance (NMR) is a physical phenomenon widely used to study complex materials. NMR is governed by the Bloch equation, a first order non-linear differential equation. Fractional order generalization of the Bloch equation provides an opportunity to extend its use to describe a wider range of experimental situations. Here we present a fractional generalization of the Bloch equat...

In this paper we analyze the convergence properties of a new fractional time-stepping technique for the solution of the variable density incompressible Navier–Stokes equations. The main feature of this method is that, contrary to other existing algorithms, the pressure is determined by just solving one Poisson equation per time step. First-order error estimates are proved, and stability of a fo...

Abstract This article adopts a novel technique to numerical solution for fractional time-delay diffusion equation with variable-order derivative (VFDDEs). As matter of fact, the (VFD) that has been used is in Caputo sense. The first step this constructive on construction using shifted Legendre–Laguerre polynomials unknown coefficients. second involves combination collocation method and operatio...

In this paper, a high-order and fast numerical method is investigated for the time-fractional Black-Scholes equation. order to deal with typical weak initial singularity of solution, we construct finite difference scheme variable time steps, where fractional derivative approximated by nonuniform Alikhanov formula sum-of-exponentials (SOE) technique. spatial direction, an average approximation f...

Abstract This paper applies the Heydari–Hosseininia nonsingular fractional derivative for defining a variable-order version of Sobolev equation. The orthonormal shifted discrete Legendre polynomials, as an appropriate family basis functions, are employed to generate operational matrix method this A new related these polynomials is extracted and construct presented method. Using approach, algebr...

BACKGROUND The behavior of the dendritic or axonal membrane voltage due to transcranial magnetic stimulation (TMS) is often modeled with the one-dimensional cable equation. For the cable equation, a length constant λ0 is defined; λ0 describes the axial decay of the membrane voltage in the case of constant applied electric field. In TMS, however, the induced electric field waveform is typically ...

An efficient numerical method to integrate the constitutive response of fractional order viscoelasticity is developed. The method can handle variable time steps. To overcome the problem of the growing amount of data that has to be stored and used in each time step we introduce sparse quadrature. We use an internal variable formulation of the viscoelastic equations where the internal variable is...