Search results

This paper aims to propose an efficient and convenient numerical algorithm for two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations (of Hammerstein and mixed types).

The main idea of the presented algorithm is to combine Bernoulli polynomials approximation with Caputo fractional derivative and numerical integral transformation to reduce the studied two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations to easily solved algebraic equations.

Without considering the integral operational matrix, this algorithm will adopt straightforward discrete data integral transformation, which can do good work to less computation and high precision. Besides, combining the convenient fractional differential operator of Bernoulli basis polynomials with the least-squares method, numerical solutions of the studied equations can be obtained quickly. Illustrative examples are given to show that the proposed technique has better precision than other numerical methods.

The proposed algorithm is efficient for the considered two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations. As its convenience, the computation of numerical solutions is time-saving and more accurate.

This paper aims to study fractional Brownian motion and its applications to nonlinear stochastic integral equations. Bernstein polynomials have been applied to obtain the numerical results of the nonlinear fractional stochastic integral equations.

Bernstein polynomials have been used to obtain the numerical solutions of nonlinear fractional stochastic integral equations. The fractional stochastic operational matrix based on Bernstein polynomial has been used to discretize the nonlinear fractional stochastic integral equation. Convergence and error analysis of the proposed method have been discussed.

Two illustrated examples have been presented to justify the efficiency and applicability of the proposed method. The corresponding obtained numerical results have been compared with the exact solutions to establish the accuracy and efficiency of the proposed method.

To the best of the authors’ knowledge, nonlinear stochastic Itô–Volterra integral equation driven by fractional Brownian motion has been for the first time solved by using Bernstein polynomials. The obtained numerical results well establish the accuracy and efficiency of the proposed method.

In this chapter we demonstrate the construction of inverse test confidence intervals for the turning-points in estimated nonlinear relationships by the use of the marginal or first derivative function. First, we outline the inverse test confidence interval approach. Then we examine the relationship between the traditional confidence intervals based on the Wald test for the turning-points for a cubic, a quartic, and fractional polynomials estimated via regression analysis and the inverse test intervals. We show that the confidence interval plots of the marginal function can be used to estimate confidence intervals for the turning-points that are equivalent to the inverse test. We also provide a method for the interpretation of the confidence intervals for the second derivative function to draw inferences for the characteristics of the turning-point.

This method is applied to the examination of the turning-points found when estimating a quartic and a fractional polynomial from data used for the estimation of an Environmental Kuznets Curve. The Stata do files used to generate these examples are listed in Appendix A along with the data.

The collection of chapters in this 30th volume of Advances in Econometrics provides a well-deserved tribute to Thomas B. Fomby and R. Carter Hill, who have served as editors of the Advances in Econometrics series for 25 and 21 years, respectively. Volume 30 contains a more varied collection of chapters than previous volumes, in essence mirroring the wide variety of econometric topics covered by the series over 30 years. Volume 30 starts with a chapter discussing the history of this series over the last 30 years. The next five chapters can be broadly categorized as focusing on model specification and testing. Following this section are three contributions that examine instrumental variables models in quite different settings. The next four chapters focus on applied macroeconomics topics. The final chapter offers a practical guide to conducting Monte Carlo simulations.

The space fractional PDEs (SFPDEs) play an important role in the fractional calculus field. Proposing a high-order, stable and flexible numerical procedure for solving SFPDEs is the main aim of most researchers. This paper devotes to developing a novel spectral algorithm to solve the FitzHugh–Nagumo models with space fractional derivatives.

The fractional derivative is defined based upon the Riesz derivative. First, a second-order finite difference formulation is used to approximate the time derivative. Then, the Jacobi spectral collocation method is employed to discrete the spatial variables. On the other hand, authors assume that the approximate solution is a linear combination of special polynomials which are obtained from the Jacobi polynomials, and also there exists Riesz fractional derivative based on the Jacobi polynomials. Also, a reduced order plan, such as proper orthogonal decomposition (POD) method, has been utilized.

A fast high-order numerical method to decrease the elapsed CPU time has been constructed for solving systems of space fractional PDEs.

The spectral collocation method is combined with the POD idea to solve the system of space-fractional PDEs. The numerical results are acceptable and efficient for the main mathematical model.

Electricity consumption prediction has been an important topic for its significant impact on electric policies. Due to various uncertain factors, the growth trends of electricity consumption in different cases are variable. However, the traditional grey model is based on a fixed structure which sometimes cannot match the trend of raw data. Consequently, the predictive accuracy is variable as cases change. To improve the model's adaptability and forecasting ability, a novel fractional discrete grey model with variable structure is proposed in this paper.

The novel model can be regarded as a homogenous or non-homogenous exponent predicting model by changing the structure. And it selects the appropriate structure depending on the characteristics of raw data. The introduction of fractional accumulation enhances the predicting ability of the novel model. And the relative fractional order r is calculated by the numerical iterative algorithm which is simple but effective.

Two cases of power load and electricity consumption in Jiangsu and Fujian are applied to assess the predicting accuracy of the novel grey model. Four widely-used grey models, three classical statistical models and the multi-layer artificial neural network model are taken into comparison. The results demonstrate that the novel grey model performs well in all cases, and is superior to the comparative eight models.

A fractional-order discrete grey model with an adaptable structure is proposed to solve the conflict between traditional grey models' fixed structures and variable development trends of raw data. In applications, the novel model has satisfied adaptability and predicting accuracy.

The purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear fractional differential equations involving ψ-Caputo derivative.

An operational matrix to find numerical approximation of ψ-fractional differential equations (FDEs) is derived. This study extends the method to nonlinear FDEs by using quasi linearization technique to linearize the nonlinear problems.

The error analysis of the proposed method is discussed in-depth. Accuracy and efficiency of the method are verified through numerical examples.

The method is simple and a good mathematical tool for finding solutions of nonlinear ψ-FDEs. The operational matrix approach offers less computational complexity.

Engineers and applied scientists may use the present method for solving fractional models appearing in applications.

The purpose of this paper is to introduce a new class of fractional positive continuous‐time and discrete‐time linear systems.

Solutions to the state equations of the fractional systems are given.

Necessary and sufficient conditions are established for the internal and external positivity and of the reachability and controllability to zero of the fractional systems.

A method for analysis of the fractional positive linear systems is proposed.

In particular, in this article, the authors investigate the degree of persistence in the credit-to-gross domestic product (GDP) ratio in 44 Organisation for Economic Co-operation and Development (OECD) economies in the context of nonlinear deterministic trends.

The authors use Chebyshev's polynomials in time, which allow us to model changes in the data in a smoother way than by structural breaks.

This study’s results indicate that approximately one-quarter of the series display non-linear structures, and only Argentina displays a mean reverting pattern.

Policy implications of the results obtained are discussed at the end of the manuscript.

The authors use an approach developed that allows for non-linear trends based on Chebyshev polynomials in time, with the residuals being fractionally integrated or integrated of order d, where d can be any real value.

The classical integer derivative diffusionmodels for fluid flow within a channel of parallel walls, for heat transfer within a rectangular fin and for impulsive acceleration of a quiescent Newtonian fluid within a circular pipe are initially generalized by introducing fractional derivatives. The purpose of this paper is to represent solutions as steady and transient parts. Afterward, making use of separation of variables, a fractional Sturm–Liouville eigenvalue task is posed whose eigenvalues and eigenfunctions enable us to write down the transient solution in the Fourier series involving also Mittag–Leffler function. An alternative solution based on the Laplace transform method is also provided.

In this work, an analytical formulation is presented concerning the transient and passage to steady state in fluid flow and heat transfer within the diffusion fractional models.

From the closed-form solutions, it is clear to visualize the start-up process of physical diffusion phenomena in fractional order models. In particular, impacts of fractional derivative in different time regimes are clarified, namely, the early time zone of acceleration, the transition zone and the late time regime of deceleration.

With the newly developing field of fractional calculus, the classical heat and mass transfer analysis has been modified to account for the fractional order derivative concept.