IN two previous notes1, Prof. Max Born and I have shown that one can obtain a theory of superconductivity by taking account of the fact that the interaction of the electrons with the ionic lattice is appreciable only near the boundaries of Brillouin zones, and particularly strong near the corners of these. This leads to the criterion that the metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.

On the theory of superconductivity

A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations

A review of definitions of fractional derivatives and other operators

Abstract Variable-order (VO) fractional differential equations (FDEs) with a time (t), space (x) or other variables dependent order have been successfully applied to investigate time and/or space dependent dynamics. This study aims to provide a survey of the recent relevant literature and findings in primary definitions, models, numerical methods and their applications. This review first offers an overview over the existing definitions proposed from different physical and application backgrounds, and then reviews several widely used numerical schemes in simulation. Moreover, as a powerful mathematical tool, the VO-FDE models have been remarkably acknowledged as an alternative and precise approach in effectively describing real-world phenomena. Hereby, we also make a brief summary on different physical models and typical applications. This review is expected to help the readers for the selection of appropriate definition, model and numerical method to solve specific physical and engineering problems.

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A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications

This paper obtains soliton solutions to optical couplers by two methods. These are sine–cosine function method and Bernoulli’s equation approach. There are four laws that are touched upon in this paper. These are Kerr law, power law, parabolic law and dual-power law. The first integration scheme is applicable to Kerr and power laws only where bright soliton solutions are retrievable. The second tool is applicable to parabolic and dual-power laws only that leads to dark and singular solitons for these two nonlinear media.

Optical solitons in nonlinear directional couplers by sine–cosine function method and Bernoulli’s equation approach

The cable equation plays a central role in many areas of electrophysiology and in modeling neuronal dynamics. This paper reports an accurate spectral collocation method for solving one- and two-dimensional variable-order fractional nonlinear cable equations. The proposed method is based on shifted Jacobi collocation procedure in conjunction with the shifted Jacobi operational matrix for variable-order fractional derivatives, described in the sense of Caputo. The main advantage of the proposed method is to investigate a global approximation for spatial and temporal discretizations. In addition, the method reduces the variable-order fractional nonlinear cable equation to a simpler problem that consists of solving a system of algebraic equations. The validity and effectiveness of the method are demonstrated by solving three numerical examples. The convergence of the method is graphically analyzed. The results demonstrate that the proposed method is a powerful algorithm with high accuracy for solving the variable-order nonlinear partial differential equations.

Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation

On the formulation and numerical simulation of distributed-order fractional optimal control problems

While several high-order methods have been extensively developed for fixed-order fractional differential equations (FDEs), there are no such methods for variable-order FDEs. In this paper, we propose an accurate and robust approach to approximate the solution of functional Dirichlet boundary value problem with a type of variable-order Caputo fractional derivative. The proposed method is principally based on the shifted Chebyshev polynomials as basis functions and the matrix representation of variable-order fractional derivative of such polynomials. The underline variable-order FDE is then reduced to a system of algebraic equations, which greatly simplifies the solution process. Through numerical results, we confirm that the proposed scheme is very efficient and accurate for handling such problem.

Numerical algorithm for the variable-order Caputo fractional functional differential equation

In many dynamic processes, the fractional differential operators not only appear as discrete fractional, but they also possess a continuous nature in a sense that their order is distributed over a given range. This paper is concerned with the optimization of systems whose governing equations contain a fractional derivative of distributed order, in the Caputo sense. By using the Lagrange multiplier within the calculus of variations and by applying the fractional integration-by-parts formula, the necessary optimality conditions are derived in terms of a nonlinear two-point distributed-order fractional boundary value problem. A Legendre spectral collocation method is developed for solving such problem. The solution method involves the use of three-term recurrence relations for both the left- and right-sided fractional integrals of the shifted Legendre polynomials. The convergence of the proposed method is discussed. The optimal profiles show the performance of the numerical solution and the effect of the fractional derivatives in the optimal results.

Mahmoud A. Zaky

Papers

A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations

Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation

On the formulation and numerical simulation of distributed-order fractional optimal control problems

Numerical algorithm for the variable-order Caputo fractional functional differential equation

A Legendre collocation method for distributed-order fractional optimal control problems