A popular reaction-diffusion model fractional Fitzhugh-Nagumo equation: analytical and numerical treatment

The main objective of this article is to obtain the new analytical and numerical solutions of fractional Fitzhugh-Nagumo equation which arises as a model of reaction-diffusion systems, transmission of nerve impulses, circuit theory, biology and the area of population genetics. For this aim conformable derivative with fractional order which is a well behaved,understandable and applicable definition is used as a tool. The analytical solutions were got by utilizing the fact that the conformable fractional derivative provided the chain rule. By the help of this feature which is not provided by other popular fractional derivatives, nonlinear fractional partial differential equation is turned into an integer order differential equation. The numerical solutions which is obtained with the aid of residual power series method are compared with the analytical results that obtained by performing sub equation method. This comparison is made both with the help of three-dimensional graphical representations and tables for different values of the γ.     

An efficient analytical approach for fractional Lakshmanan-Porsezian-Daniel model

In this paper, the q -homotopy analysis transform method (q -HATM) is applied to find the solution for the fractional Lakshmanan-Porsezian-Daniel (LPD) model. The LPD model is the generalization of the non-linear Schrödinger (NLS) equation. The proposed method is graceful fusions of Laplace transform technique with q -homotopy analysis scheme, and the derivative is considered in Caputo sense. In order to validate and illustrate the efficiency of the proposed method, we analysed the projected model in terms of fractional order. Moreover, the physical behaviour of the obtained solution has been captured for the three different cases in terms of 3D and contour plots for diverse values of the fractional order. The obtained results confirm that the future method is easy to implement, highly methodical, and very effective to analyse the behaviour of complex non-linear fractional differential equations exist in the connected areas of science and engineering.     

Novel parameter estimation techniques for a multi-term fractional dynamical epidemic model of dengue fever

Numerical Algorithms - An inverse problem to identify parameters for the single-term (and multi-term) fractional-order system of an outbreak of dengue fever is considered. Firstly, we propose a...     

Novel numerical techniques for the finite moment log stable computational model for European call option

Option pricing models are often used to describe the dynamic characteristics of prices in financial markets. Unlike the classical Black–Scholes (BS) model, the finite moment log stable (FMLS) model can explain large movements of prices during small time steps. In the FMLS, the second-order spatial derivative of the BS model is replaced by a fractional operator of order α which generates an α-stable Lévy process. In this paper, we consider the finite difference method to approximate the FMLS model. We present two numerical schemes for this approximation: the implicit numerical scheme and the Crank–Nicolson scheme. We carry out convergence and stability analyses for the proposed schemes. Since the fractional operator routinely generates dense matrices which often require high computational cost and storage memory, we explore three methods for solving the approximation schemes: the Gaussian elimination method, the bi-conjugate gradient stabilized method (Bi-CGSTAB) and the fast Bi-CGSTAB (FBi-CGSTAB) in order to compare the cost of calculations. Finally, two numerical examples with exact solutions are presented where we also use extrapolation techniques to achieve higher-order convergence. The results suggest that the proposed schemes are unconditionally stable and convergent, and the FMLS model is useful for pricing options.     

What's the common ground of all numerical and analytical techniques for nonlinear problems?

In this paper the author attempts to find out in very general meanings the common ground of all numerical and analytical techniques for nonlinear problems. By means of analyzing the finiteness or infiniteness of the fundamental operations necessary for mathematically solving a problem, the author points out that the key of any nonlinear technique is the way to transform a nonlinear problem to an infinite series of sub-problems which can be solved by finite number of fundamental operations.     

On vaccination controls for the SEIR epidemic model

This paper presents a simple continuous-time linear vaccination-based control strategy for a SEIR (susceptible plus infected plus infectious plus removed populations) disease propagation model. The model takes into account the total population amounts as a refrain for the illness transmission since its increase makes more difficult contacts among susceptible and infected. The control objective is the asymptotically tracking of the removed-by-immunity population to the total population while achieving simultaneously the remaining population (i.e. susceptible plus infected plus infectious) to asymptotically tend to zero.     

Numerical techniques for simulating a fractional mathematical model of epidermal wound healing

A number of mathematical models investigating certain aspects of the complicated process of wound healing are reported in the literature in recent years. However, effective numerical methods and supporting error analysis for the fractional equations which describe the process of wound healing are still limited. In this paper, we consider the numerical simulation of a fractional mathematical model of epidermal wound healing (FMM-EWH), which is based on the coupled advection-diffusion equations for cell and chemical concentration in a polar coordinate system. The space fractional derivatives are defined in the Left and Right Riemann-Liouville sense. Fractional orders in the advection and diffusion terms belong to the intervals (0,1) or (1,2], respectively. Some numerical techniques will be used. Firstly, the coupled advection-diffusion equations are decoupled to a single space fractional advection-diffusion equation in a polar coordinate system. Secondly, we propose a new implicit difference method for simulating this equation by using the equivalent of Riemann-Liouville and Grünwald-Letnikov fractional derivative definitions. Thirdly, its stability and convergence are discussed, respectively. Finally, some numerical results are given to demonstrate the theoretical analysis.     

A numerical approximation for Volterra’s population growth model with fractional order

This paper presents a numerical scheme for approximate solutions of the fractional Volterra’s model for population growth of a species in a closed system. In fact, the Bessel collocation method is extended by using the time-fractional derivative in the Caputo sense to give solutions for the mentioned model problem. In this extended of the method, a generalization of the Bessel functions of the first kind is used and its matrix form is constructed. And then, the matrix form based on the collocation points is formed for the each term of this model problem. Hence, the method converts the model problem into a system of nonlinear algebraic equations. We give some numerical applications to show efficiency and accuracy of the method. In applications, the reliability of the technique is demonstrated by the error function based on accuracy of the approximate solution.     

A numerical technique for solving fractional variational problems

This paper presents an accurate numerical method for solving a class of fractional variational problems (FVPs). The fractional derivative in these problems is in the Caputo sense. The proposed method is called fractional Chebyshev finite difference method. In this technique, we approximate FVPs and end up with a finite‐dimensional problem. The method is based on the combination of the useful properties of Chebyshev polynomials approximation and finite difference method. The Caputo fractional derivative is replaced by a difference quotient and the integral by a finite sum. The fractional derivative approximation using Clenshaw and Curtis formula introduced here, along with Clenshaw and Curtis procedure for the numerical integration of a non‐singular functions and the Rayleigh–Ritz method for the constrained extremum, is considered. By this method, the given problem is reduced to the problem for solving a system of algebraic equations, and by solving this system, we obtain the solution of FVPs. Special attention is given to study the convergence analysis and evaluate an error upper bound of the obtained approximate formula. Illustrative examples are included to demonstrate the validity and applicability of the proposed technique. A comparison with another method is given. Copyright © 2012 John Wiley & Sons, Ltd.     

Implicit numerical approximation scheme for the fractional Fokker-Planck equation

In this paper, we consider an anomalous subdiffusion process, governed by fractional Fokker-Planck equation. An effective numerical method for approximating Fokker-Planck equation in a bounded domain is presented. The stability and convergence of the numerical method are analyzed. Some numerical examples are presented to show the application of the present technique. The numerical results exhibit the good performance of our theoretical analysis.     

On the numerical solutions for the fractional diffusion equation

Fractional differential equations have recently been applied in various area of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional diffusion equation (FDE) is considered. The fractional derivative is described in the Caputo sense. The method is based upon Chebyshev approximations. The properties of Chebyshev polynomials are utilized to reduce FDE to a system of ordinary differential equations, which solved by the finite difference method. Numerical simulation of FDE is presented and the results are compared with the exact solution and other methods.     

Higher order numerical methods for solving fractional differential equations

In this paper we introduce higher order numerical methods for solving fractional differential equations. We use two approaches to this problem. The first approach is based on a direct discretisation of the fractional differential operator: we obtain a numerical method for solving a linear fractional differential equation with order 0<α<1. The order of convergence of the numerical method is O(h ^3?α ). Our second approach is based on discretisation of the integral form of the fractional differential equation and we obtain a fractional Adams-type method for a nonlinear fractional differential equation of any order α>0. The order of convergence of the numerical method is O(h ³) for α≥1 and O(h ^1+2α ) for 0<α≤1 for sufficiently smooth solutions. Numerical examples are given to show that the numerical results are consistent with the theoretical results.     

New numerical simulation for fractional Benney–Lin equation arising in falling film problems using two novel techniques

The pivotal aim of the present work is to find the numerical solution for fractional Benney–Lin equation by using two efficient methods, called q ‐homotopy analysis transform method and fractional natural decomposition method. The considered equation exemplifies the long waves on the liquid films. Projected methods are distinct with solution procedure and they are modified with different transform algorithms. To illustrate the reliability and applicability of the considered solution procedures we consider eight special cases with different initial conditions. The fractional operator is considered in Caputo sense. The achieved results are drowned through two and three‐dimensional plots for different Brownian motions and classical order. The numerical simulations are presented to ensure the efficiency of considered techniques. The behavior of the obtained results for distinct fractional order is captured in the present framework. The outcomes of the present investigation show that, the considered schemes are efficient and powerful to solve nonlinear differential equations arise in science and technology.     

Traveling waves for a diffusive SEIR epidemic model with standard incidences

This paper is devoted to the existence of the traveling waves of the equations describing a diffusive susceptible-exposed-infected-recovered(SEIR) model. The existence of traveling waves depends on the basic reproduction rate and the minimal wave speed. We obtain a more precise estimation of the minimal wave speed of the epidemic model, which is of great practical value in the control of serious epidemics. The approach in this paper is to use the Schauder fixed point theorem and the Laplace transform. We also give some numerical results on the minimal wave speed.     

Novel numerical techniques based on Fokas transforms,for the solution of initial boundary value problems

The unified transform method of A. S. Fokas has led to important new developments, regarding the analysis and solution of various types of linear and nonlinear PDE problems. In this work we use these developments and obtain the solution of time-dependent problems in a straightforward manner and with such high accuracy that cannot be reached within reasonable time by use of the existing numerical methods. More specifically, an integral representation of the solution is obtained by use of the A. S. Fokas approach, which provides the value of the solution at any point, without requiring the solution of linear systems or any other calculation at intermediate time levels and without raising any stability problems. For instance, the solution of the initial boundary value problem with the non-homogeneous heat equation is obtained with accuracy 10⁻¹⁵, while the well-established Crank–Nicholson scheme requires 2048 time steps in order to reach a 10⁻⁸ accuracy.     

Simplified analytical expressions for numerical differentiation via cycle index

Based on the Lagrange interpolation to the function f[_x0,⋅]$f[x_0,\cdot ]$ for arbitrarily chosen _x0$x_0$ and logarithmic differentiation, we give a simple approach to analytical expressions for numerical differentiation using cycle index. A detailed analysis for the remainder is also included.     

Generalized fractional programming: Algorithms and numerical experimentation

Several algorithms to solve the generalized fractional program are summarized and compared numerically in the linear case. These algorithms are iterative procedures requiring the solution of a linear programming problem at each iteration in the linear case. The most efficient algorithm is obtained by marrying the Newton approach within the Dinkelbach approach for fractional programming.