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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.

The purpose of this paper is to develop a new method based on operational matrices of Bernoulli wavelet for solving linear stochastic Itô-Volterra integral equations, numerically.

For this aim, Bernoulli polynomials and Bernoulli wavelet are introduced, and their properties are expressed. Then, the operational matrix and the stochastic operational matrix of integration based on Bernoulli wavelet are calculated for the first time.

By applying these matrices, the main problem would be transformed into a linear system of algebraic equations which can be solved by using a suitable numerical method. Also, a few results related to error estimate and convergence analysis of the proposed scheme are investigated.

Two numerical examples are included to demonstrate the accuracy and efficiency of the proposed method. All of the numerical calculation is performed on a personal computer by running some codes written in MATLAB software.

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.

Reviews previous research on the nature of beta and investigates the stochastic structure of time‐varying beta in Hong Kong, Malaysia and Singapore using the bi‐variate GARCH‐in‐mean model and fractional tests. Develops mathematical models and applies them to 1989‐1998 daily data from all three stock markets. Presents the results, which suggest, in contrast to other findings, that all three time‐varying betas are slowly mean‐reverting (long memory).

The purpose of this paper is to develop a new method based on operational matrices of two-dimensional delta functions for solving two-dimensional nonlinear quadratic integral equations (2D-QIEs) of fractional order, numerically.

For this aim, two-dimensional delta functions are introduced, and their properties are expressed. Then, the fractional operational matrix of integration based on two-dimensional delta functions is calculated for the first time.

By applying the operational matrices, the main problem would be transformed into a nonlinear system of algebraic equations which can be solved by using Newton's iterative method. Also, a few results related to error estimate and convergence analysis of the proposed method are investigated.

Two numerical examples are presented to show the validity and applicability of the suggested approach. All of the numerical calculation is performed on a personal computer by running some codes written in MATLAB software.

The purpose of this paper is to present an algorithm based on operational Tau method (OTM) for solving fractional Fokker‐Planck equation (FFPE) with space‐ and time‐fractional derivatives. Fokker‐Planck equation with positive integer order is also considered.

The proposed algorithm converts the desired FFPE to a set of algebraic equations using orthogonal polynomials as basis functions. The paper states some concepts, properties and advantages of proposed algorithm and its applications for solving FFPE.

Some illustrative numerical experiments including linear and nonlinear FFPE are given and some comparisons are made between OTM and variational iteration method, Adomian decomposition method and homotpy perturbation method.

Results demonstrate some capabilities of the proposed algorithm such as the simplicity, the accuracy and the convergency. Also, this is the first presentation of this algorithm for FFPE.

This paper aims to give a brief review on behavioral economics and behavioral finance and discusses some of the previous research on agents' utility functions, applicable risk measures, diversification strategies and portfolio optimization.

The authors also cover related disciplines such as trading rules, contagion and various econometric aspects.

While scholars could first develop theoretical models in behavioral economics and behavioral finance, they subsequently may develop corresponding statistical and econometric models, this finally includes simulation studies to examine whether the estimators or statistics have good power and size. This all helps us to better understand financial and economic decision-making from a descriptive standpoint.

The research paper is original.

This paper aims to propose a novel approach based on uniform scale-3 Haar wavelets for unsteady state space fractional advection-dispersion partial differential equation which arises in complex network, fluid dynamics in porous media, biology, chemistry and biochemistry, electrode – electrolyte polarization, finance, system control, etc.

Scale-3 Haar wavelets are used to approximate the space and time variables. Scale-3 Haar wavelets converts the problems into linear system. After that Gauss elimination is used to find the wavelet coefficients.

A novel algorithm based on Haar wavelet for two-dimensional fractional partial differential equations is established. Error estimation has been derived by use of property of compactly supported orthonormality. The correctness and effectiveness of the theoretical arguments by numerical tests are confirmed.

Scale-3 Haar wavelets are used first time for these types of problems. Second, error analysis in new work in this direction.

The objective of this work is to provide a novel aircraft allocation model for fractional business aviation. This model may provide decision-makers with alternative routing solutions that take into consideration preventive maintenance and failure prognostics information. The expected results are more efficient routing solutions when compared to conventional planning models, to help decision-makers improve operations and maintenance planning.

The model is a mixed integer linear problem formulation addressing and considering preventive maintenance and failure prognostics for optimal operations. Numerical experiments were performed using both field and synthetic data to validate the proposed method. All instances are solved using branch, price and cut algorithms from open-source software.

The results obtained in this study show that the use of failure prognostics information in aircraft routing can provide improvements in overall planning. By choosing slightly longer flight legs, the flight cost will increase, but putting an aircraft with a higher risk of failure on a leg inbound to a maintenance base can reduce maintenance and overall operating cost.

The model and method provide decision-makers with routing solutions that consider new aspects of planning, not used in previous works, such as failure. Most of the literature focuses on solving routing problems for large commercial airlines. Considering that, few solutions are found in literature for fractional business operators, which have their own operational particularities, such as a company managing a fleet of aircraft belonging to multiple shareowners. In such operation, clients may not always fly in the aircraft that they are shareowners, but an aircraft from the fractional fleet of the same category. Here, the company managing the aircraft guarantees that an aircraft will be ready to attend client demands in minimum time. One of the major differences from other models of operation is the dynamic nature of its flight demands, thus requiring flexible and agile planning limiting the available time to find a routing solution.

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.