On Caputo-Type Cable Equation: Analysis and Computation

In this paper,a special case of nonlinear time fractional cable equation is studied.For the equation defined on a bounded domain,the existence,uniqueness,and regularity of the solution are firstly studied.Furthermore,it is numerically studied via the weighted and shifted Grünwald difference(WSGD)methods/the local discontinuous Galerkin(LDG)finite element methods.The derived numerical scheme has been proved to be stable and convergent with order O(
t2+hk+1),where
t,h,k are the time stepsize,the spatial stepsize,and the degree of piecewise polynomials,respectively.Finally,a numerical experiment is presented to verify the theoretical analysis.

NUMERICAL SIMULATIONS FOR A VARIABLE ORDER FRACTIONAL CABLE EQUATION

In this article, Crank-Nicolson method is used to study the variable order fractional cable equation. The variable order fractional derivatives are described in the Riemann- Liouville and the Griinwald-Letnikov sense. The stability analysis of the proposed technique is discussed. Numerical results are provided and compared with exact solutions to show the accuracy of the proposed technique.

A NEW MODEL AND IMPROVED CABLE FUNCTION FOR REPRESENTING THE ACTIVATING PERIPHERAL NERVES BY A TRANSVERSE ELECTRIC FIELD DURING MAGNETIC STIMULATION

Objective Previous studies of peripheral nerves activation during magnetic stimulation have focused almost exclusively on the cause of high external parallel electric field along the nerves, whereas the effect of the transverse component has been ignored. In the present paper, the classical cable function is modified to represent the excitation of peripheral nerves stimulated by a transverse electric field during magnetic stimulation. Methods Responses of the Ranvier nodes to a transverse-field are thoroughly investigated by mathematic simulation. Results The simulation demonstrates that the excitation results from the net inward current driven by an external field. Based on a two-stage process, a novel model is introduced to describe peripheral nerves stimulated by a transverse-field. Based on the new model, the classical cable function is modified. Conclusion Using this modified cable equation, the excitation threshold of peripheral nerves in a transverse field during MS is obtained. The modified cable equation can be used to represent the response of peripheral nerves by an arbitrary electric field.

Steady-State Configuration and Tension Calculations of Marine Cables Under Complex Currents via Separated Particle Swarm Optimization?

Under complex currents, the motion governing equations of marine cables are complex and nonlinear, and the calculations of cable configuration and tension become difficult compared with those under the uniform or simple currents. To obtain the numerical results, the usual Newton-Raphson iteration is often adopted, but its stability depends on the initial guessed solution to the governing equations. To improve the stability of numerical calculation, this paper proposed separated the particle swarm optimization, in which the variables are separated into several groups, and the dimension of search space is reduced to facilitate the particle swarm optimization. Via the separated particle swarm optimization, these governing nonlinear equations can be solved successfully with any initial solution, and the process of numerical calculation is very stable. For the calculations of cable configuration and tension of marine cables under complex currents, the proposed separated swarm particle optimization is more effective than the other particle swarm optimizations.