A note on analytical solutions of nonlinear fractional 2D heat equation with non-local integral terms

In this paper, we consider the (2+1) nonlinear fractional heat equation with non-local integral terms and investigate two different cases of such non-local integral terms. The first has to do with the time-dependent non-local integral term and the second is the space-dependent non-local integral term. Apart from the nonlinear nature of these formulations, the complexity due to the presence of the non-local integral terms impelled us to use a relatively new analytical technique called q-homotopy analysis method to obtain analytical solutions to both cases in the form of convergent series with easily computable components. Our numerical analysis enables us to show the effects of non-local terms and the fractional-order derivative on the solutions obtained by this method.     

New fractional derivative with sigmoid function as the kernel and its models

Based on the idea of the fractional derivative with respect to another function, a new fractional derivative operator with sigmoid function as the kernel in this article, is proposed for the first time. Then, we make use of this new fractional operator to model various nonlinear phenomena from different fields of applications in science, such as the population growth, the shallow water wave phenomena and reaction-diffusion processes, and so on. As a result, we hope that the new fractional operator can be used to discover more evolutionary mechanisms of these phenomena.     

A second-order accurate numerical method for the two-dimensional fractional diffusion equation

Spatially fractional order diffusion equations are generalizations of classical diffusion equations which are used in modeling practical superdiffusive problems in fluid flow, finance and others. In this paper, we present an accurate and efficient numerical method to solve a fractional superdiffusive differential equation. This numerical method combines the alternating directions implicit (ADI) approach with a Crank–Nicolson discretization and a Richardson extrapolation to obtain an unconditionally stable second-order accurate finite difference method. The stability and the consistency of the method are established. Numerical solutions for an example super-diffusion equation with a known analytic solution are obtained and the behavior of the errors are analyzed to demonstrate the order of convergence of the method.     

A new type of equation of motion and numerical method for a harmonic oscillator with left and right fractional derivatives

The aim of this research is to propose a new fractional Euler-Lagrange equation for a harmonic oscillator. The theoretical analysis is given in order to derive the equation of motion in a fractional framework. The new equation has a complicated structure involving the left and right fractional derivatives of Caputo-Fabrizio type, so a new numerical method is developed in order to solve the above-mentioned equation effectively. As a result, we can see different asymptotic behaviors according to the flexibility provided by the use of the fractional calculus approach, a fact which may be invisible when we use the classical Lagrangian technique. This capability helps us to better understand the complex dynamics associated with natural phenomena.     

A simple and efficient numerical scheme to integrate non-local potentials

As nuclear wave functions have to obey the Pauli principle, potentials issued from reaction theory or Hartree-Fock formalism using finite-range interactions contain a non-local part. Written in coordinate space representation, the Schrödinger equation becomes integro-differential, which is difficult to solve, contrary to the case of local potentials, where it is an ordinary differential equation. A simple and powerful method was proposed several years ago, with the trivially equivalent potential method, where the non-local potential is replaced by an equivalent local potential, which is state dependent and has to be determined iteratively. Its main disadvantage, however, is the appearance of divergences in potentials if the wave functions have nodes, which is generally the case. We will show that divergences can be removed by a slight modification of the trivially equivalent potential method, leading to a very simple, stable and precise numerical technique to deal with non-local potentials. Examples will be provided with the calculation of the Hartree-Fock potential and associated wave functions of ¹⁶O using the finite-range N³LO realistic interaction.     

Amorphous track models: A numerical comparison study

We present an open-source code library for amorphous track modelling which is suppose to faciliate the application and numerical comparability as well as serve as a frame-work for the implementation of new models. We show an example of using the library indicating the choice of submodels has a significant impact on the modelling outcome.     

Fokker-Planck equation with fractional coordinate derivatives

Using the generalized Kolmogorov-Feller equation with long-range interaction, we obtain kinetic equations with fractional derivatives with respect to coordinates. The method of successive approximations, with averaging with respect to a fast variable, is used. The main assumption is that the correlation function of probability densities of particles to make a step has a power-law dependence. As a result, we obtain a Fokker-Planck equation with fractional coordinate derivative of order 1<α<2.     

Langevin equation with two fractional orders

A new type of fractional Langevin equation of two different orders is introduced. The solutions for this equation, known as the fractional Ornstein-Uhlenbeck processes, based on Weyl and Riemann-Liouville fractional derivatives are obtained. The basic properties of these processes are studied. An example of the spectral density of ocean wind speed which has similar spectral density as that of Weyl fractional Ornstein-Uhlenbeck process is given.     

Helmholtz equation and non-singular boundary elements applied to multi-disciplinary physical problems

The famous scientist Hermann von Helmholtz was born 200 years ago. Many complex physical wave phenomena in engineering can effectively be described using one or a set of equations named after him: the Helmholtz equation. Although this has been known for a long time, from a theoretical point of view, the actual numerical implementation has often been hindered by divergence-free and/or curl-free constraints. There is further a need for a numerical method that is accurate, reliable and takes into account radiation conditions at infinity. The classical boundary element method satisfies the last condition, yet one has to deal with singularities in the implementation. We review here how a recently developed singularity-free three-dimensional boundary element framework with superior accuracy can be used to tackle such problems only using one or a few Helmholtz equations with higher order (quadratic) elements which can tackle complex curved shapes. Examples are given for acoustics (a Helmholtz resonator among others) and electromagnetic scattering.     

Qualitative and numerical study of bianchi IX models

The continuous evolution of the Mixmaster universe toward the cosmological singularity contains features that differ substantially from its discrete counterpart. We examine here the determination and interpretation of the Liapunov exponent of the continuous orbit. It is briefly mentioned that this is not the only aspect of the Mixmaster dynamics to be affected when we switch from continuous to discrete mode of evolution.     

A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions

Combining order reduction approach and L1 discretization, a box-type scheme is presented for solving a class of fractional sub-diffusion equation with Neumann boundary conditions. A new inner product and corresponding norm with a Sobolev embedding inequality are introduced. A novel technique is applied in the proof of both stability and convergence. The global convergence order in maximum norm is O(τ^2−α + h²). The accuracy and efficiency of the scheme are checked by two numerical tests.     

A fractional order diffusion-wave equation for time-dispersion media

Electromagnetic fields in time-dispersion media with a power-law dependence on time are analyzed. It is shown that these media are fractal and their fractal dimension is determined. Equations for scalar and vector potentials are derived using analogues of Maxwell??s equations for these types of media with the use of Caputo fractional derivatives. Electromagnetic fields in a bounded domain are numerically calculated for arbitrary functions of charge and current.     

Complete reduction of fermion-antifermion bethe-salpeter equation with static kernel

The Bethe-Salpeter (BS) equation for a $\text{spin}\text{-}\text{1}\text{2}$ fermion-antifermion bound system is considered for the case in which the kernel is static and is the fourth component (i.e., three-scalar part) of a vector potential. Relative time (or relative energy) dependence can be eliminated easily. The 16 BS bispinor amplitudes are reexpressed in the usual way in terms of corresponding tensor amplitudes which satisfy 16 coupled integrodifferential equations. If Lorentz-, parity-, and charge-conjugation invariance are used, these equations can be reduced through a sequence of transformations to single eigenvalue equations, involving scalar and three-vector wavefunctions for singlet and triplet states, respectively. The effective Hamiltonians obtained in these equations are correct to all orders in the coupling constant and have a simple structure, consisting in general of a scalar, a spin-orbit, and a tensor part, which are explicitly exhibited.Although the equations could well be used for consideration of a general particleantiparticle system (e.g., quark-antiquark), for the present only positronium with a Coulomb interaction kernel is treated as an illustrative example. There exists a singlettriplet splitting in leading order mα⁶ ln α even though no spin-spin forces are directly introduced in the kernel. The splitting is calculated in detail in perturbation theory to order mα⁶ ln α and mα⁶.     

Similarity solution of coagulation equation with an inverse kernel

We analyse a model for the aggregation of polystyrene particles which arises in an electrorheology system in which linear clusters grow upon the application of an alternating electromagnetic field. We consider a coagulation kernel involving negative powers of cluster sizes and investigate the reduction of the governing equations to a similarity solution in the large-time limit. Comparison between the experimental results and the theory presented here shows a good collapse of the data onto a single curve, which matches the theoretical results particularly well at the larger cluster sizes.     

Exact solutions of discrete complex cubic-quintic Ginzburg-Landau equation with non-local quintic term

In this paper, via the extended tanh-function approach, the abundant exact solutions for discrete complex cubic-quintic Ginzburg-Landau equation, including chirpless bright soliton, chirpless dark soliton, constant magnitude solution (plane wave solution), triangular function solutions and some solutions with alternating phases, etc. are obtained. Meanwhile, the range of parameters where some exact solution exist are given. Among these solutions, solutions with alternating phases do not have continuous analogs. Moreover, in the lattice, the points of singularity of tan-type and sec-type solutions can be ‘between sites’ and thus the singularities can be avoided.     

Snaking of radial solutions of the multi-dimensional Swift-Hohenberg equation: A numerical study

The bifurcation structure of localized stationary radial patterns of the Swift-Hohenberg equation is explored when a continuous parameter n is varied that corresponds to the underlying space dimension whenever n is an integer. In particular, we investigate how 1D pulses and 2-pulses are connected to planar spots and rings when n is increased from 1 to 2. We also elucidate changes in the snaking diagrams of spots when the dimension is switched from 2 to 3.     

A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems

How to characterize the memory property of systems is a challenging issue in the modeling and analysis of complex systems. This study makes a comparative investigation of integer-order derivative, constant-order fractional derivative and two types of variable-order fractional derivatives in characterizing the memory property of systems. The advantages and potential applications of two variable-order derivative definitions are highlighted through a comparative analysis of anomalous relaxation process.     

A causal and fractional all-frequency wave equation for lossy media

This work presents a lossy partial differential acoustic wave equation including fractional derivative terms. It is derived from first principles of physics (mass and momentum conservation) and an equation of state given by the fractional Zener stress-strain constitutive relation. For a derivative order α in the fractional Zener relation, the resulting absorption α(k) obeys frequency power-laws as α(k) ∝ ω(1+α) in a low-frequency regime, α(k) ∝ ω(1-α/2) in an intermediate-frequency regime, and α(k) ∝ ω(1-α) in a high-frequency regime. The value α=1 corresponds to the case of a single relaxation process. The wave equation is causal for all frequencies. In addition the sound speed does not diverge as the frequency approaches infinity. This is an improvement over a previously published wave equation building on the fractional Kelvin-Voigt constitutive relation.