Cable equations with fractional order temporal operators are introduced to model electrotonic properties of spiny neuronal dendrites. These equations are derived from Nernst-Planck equations with fractional order operators to model the anomalous subdiffusion that arises from trapping properties of dendritic spines. The fractional cable models predict that postsynaptic potentials propagating alo...

In this paper, we consider a variable-order fractional advection-diffusion equation with a nonlinear source term on a finite domain. Explicit and implicit Euler approximations for the equation are proposed. Stability and convergence of the methods are discussed. Moveover, we also present a fractional method of lines, a matrix transfer technique, and an extrapolation method for the equation. Som...

A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation. The suggested method is the development of Podlubny’s matrix approach (Fractional Calculus and Applied Analysis, ...

in this paper we introduce a type of fractional-order polynomials basedon the classical chebyshev polynomials of the second kind (fcss). also we construct the operationalmatrix of fractional derivative of order $ gamma $ in the caputo for fcss and show that this matrix with the tau method are utilized to reduce the solution of some fractional-order differential equations.

The variable order wave equation plays a major role in acoustics, electromagnetics, and fluid dynamics. In this paper, we consider the space-time variable order fractional wave equation with variable coefficients. We propose an effective numerical method for solving the aforementioned problem in a bounded domain. The shifted Jacobi polynomials are used as basis functions, and the variable-order...

The cable equation describes the voltage in a straight cylindrical cable, and this model has been employed to model electrical potential in dendrites and axons. However, sometimes this equation might give incorrect predictions for some realistic geometries, in particular when the radius of the cable changes significantly. Cables with a nonconstant radius are important for some phenomena, for ex...

The paper is concerned with the finite-time stabilization of a coupled PDE–ODE system describing motion an overhead crane flexible cable. dynamics cable described by wave equation variable coefficient which affine function curvilinear abscissa along Using several changes variables, backstepping transformation, and stable second-order ODE for conveniently chosen variable, we prove that global oc...

In this paper, the modified Korteweg-de Vries (mKdV) equation with variable coefficients (vc-mKdV equation) is investigated via two kinds of approaches and symbolic computation. On the one hand, we firstly reduce the vc-mKdV equation to a second-order nonlinear nonhomogeneous ODE using travelling wave-like similarity transformation. And then we obtain its many types of exact fractional solution...

Abstract. We consider Hadamard fractional derivatives and integrals of variable fractional order. A new type of fractional operator, which we call the Hadamard–Marchaud fractional derivative, is also considered. The objective is to represent these operators as series of terms involving integer-order derivatives only, and then approximate the fractional operators by a finite sum. An upper bound ...