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The purpose of the present work is to introduce a wavelet method for the solution of linear and nonlinear psi-Caputo fractional initial and boundary value problem.

The authors have introduced the new generalized operational matrices for the psi-CAS (Cosine and Sine) wavelets, and these matrices are successfully utilized for the solution of linear and nonlinear psi-Caputo fractional initial and boundary value problem. For the nonlinear problems, the authors merge the present method with the quasilinearization technique.

The authors have drived the orthogonality condition for the psi-CAS wavelets. The authors have derived and constructed the psi-CAS wavelets matrix, psi-CAS wavelets operational matrix of psi-fractional order integral and psi-CAS wavelets operational matrix of psi-fractional order integration for psi-fractional boundary value problem. These matrices are successfully utilized for the solutions of psi-Caputo fractional differential equations. The purpose of these operational matrices is to make the calculations faster. Furthermore, the authors have derived the convergence analysis of the method. The procedure of implementation for the proposed method is also given. For the accuracy and applicability of the method, the authors implemented the method on some linear and nonlinear psi-Caputo fractional initial and boundary value problems and compare the obtained results with exact solutions.

Since psi-Caputo fractional differential equation is a new and emerging field, many engineers can utilize the present technique for the numerical simulations of their linear/non-linear psi-Caputo fractional differential models. To the best of the authors’ knowledge, the present work has never been introduced and implemented for psi-Caputo fractional differential equations.

The purpose of this paper is to report on a sequential three-stage analysis of inflation persistence using monthly data from 11 inflation targeting (IT) countries and, for comparison, the USA, a non-IT country with a history of credible monetary policy.

First, the authors estimate inflation persistence in a rolling-window fractional-integration setting using the semiparametric estimator suggested by Phillips (2007). Second, the authors use tests for unknown structural breaks as a means to identify effects of the regime switch and the global financial crisis on inflation persistence. The authors use the sequences of estimated persistence measures from the first stage as dependent variables in the Bai and Perron (2003) structural break tests. Finally, the authors reapply the Phillips (2007) estimator to the subsamples defined by the breaks.

Four countries (Canada, Iceland, Mexico, and South Korea) experience a structural break in inflation persistence that coincide with the implementation of the IT regime, and three IT countries (Sweden, Switzerland, and the UK), as well as the USA experience a structural break in inflation persistence that coincides with the global financial crisis.

The authors find that in most cases the estimates of inflation persistence switch from mean-reversion nonstationarity to mean-reversion stationarity.

Monetary policy implications differ between pre- and post-global financial crisis.

Global financial crisis affected the persistence of inflation rates.

First paper to consider the effect of the global financial crisis on inflation persistence.

This paper's aim is to test for the presence of fractional integration, or long memory, in the daily returns of the Portuguese stock market using autoregressive fractionally integrated moving average (ARFIMA), generalised autoregressive conditional heteroskedasticity (GARCH) and ARFIMA‐FIGARCH models.

The data cover two periods: 4 January 1993‐13 January 2006 (full sample), and 1 February 2002‐13 January 2006 (that is, data are considered after the merger of the Portuguese Stock Exchange with Euronext).

The results from the full sample show strong evidence of long memory in stock returns. When data after the merger are considered, weaker evidence of long memory is found. It is concluded that the Portuguese stock market is more efficient after the merger with Euronext.

The findings of this paper are helpful to financial managers and investors dealing with Portuguese stock indices.

The purpose of this paper is to address a fractional order fuzzy PID (FOFPID) control approach for solving the problem of enhancing high precision tracking performance and robustness against to different reference trajectories of a 6-DOF Stewart Platform (SP) in joint space.

For the optimal design of the proposed control approach, tuning of the controller parameters including membership functions and input-output scaling factors along with the fractional order rate of error and fractional order integral of control signal is tuned with off-line by using particle swarm optimization (PSO) algorithm. For achieving this off-line optimization in the simulation environment, very accurate dynamic model of SP which has more complicated dynamical characteristics is required. Therefore, the coupling dynamic model of multi-rigid-body system is developed by Lagrange-Euler approach. For completeness, the mathematical model of the actuators is established and integrated with the dynamic model of SP mechanical system to state electromechanical coupling dynamic model. To study the validness of the proposed FOFPID controller, using this accurate dynamic model of the SP, other published control approaches such as the PID control, FOPID control and fuzzy PID control are also optimized with PSO in simulation environment. To compare trajectory tracking performance and effectiveness of the tuned controllers, the real time validation trajectory tracking experiments are conducted using the experimental setup of the SP by applying the optimum parameters of the controllers. The credibility of the results obtained with the controllers tuned in simulation environment is examined using statistical analysis.

The experimental results clearly demonstrate that the proposed optimal FOFPID controller can improve the control performance and reduce reference trajectory tracking errors of the SP. Also, the proposed PSO optimized FOFPID control strategy outperforms other control schemes in terms of the different difficulty levels of the given trajectories.

To the best of the authors’ knowledge, such a motion controller incorporating the fractional order approach to the fuzzy is first time applied in trajectory tracking control of SP.

Recently, a new formulation has been introduced for non-local mechanics in terms of fractional calculus. Fractional calculus is a branch of mathematical analysis that studies the differential operators of an arbitrary (real or complex) order and is used successfully in various fields such as mathematics, science and engineering. The purpose of this paper is to introduce a new fractional non-local theory which may be applicable in various simple or complex mechanical problems.

In this paper (by using fractional calculus), a fractional non-local theory based on the conformable fractional derivative (CFD) definition is presented, which is a generalized form of the Eringen non-local theory (ENT). The theory contains two free parameters: the fractional parameter which controls the stress gradient order in the constitutive relation and could be an integer and a non-integer and the non-local parameter to consider the small-scale effect in the micron and the sub-micron scales. The non-linear governing equation is solved by the Galerkin and the parameter expansion methods. The non-linearity of the governing equation is due to the presence of von-Kármán non-linearity and CFD definition.

The theory has been used to study linear and non-linear free vibration of the simply-supported (S-S) and the clamped-free (C-F) nano beams and then the influence of the fractional and the non-local parameters has been shown on the linear and non-linear frequency ratio.

A new parameter of the theory (the fractional parameter) makes the modeling more fixable – this model can conclude all of integer and non-integer operators and is not limited to special operators such as ENT. In other words, it allows us to use more sophisticated mathematics to model physical phenomena. On the other hand, in the comparison of classic fractional non-local theory, the theory applicable in various simple or complex mechanical problems may be used because of simpler forms of the governing equation owing to the use of CFD definition.

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.

This paper aims to apply fractional variational iteration method using He's polynomials (FVIMHP) to obtain exact solutions for variable-coefficient fractional heat-like and wave-like equations with fractional order initial and boundary conditions.

The approach is based on FVIMHP. The authors choose as some examples to illustrate the validity and the advantages of the method.

The results reveal that the FVIMHP method provides a very effective, convenient and powerful mathematical tool for solving fractional differential equations.

The variable-coefficient fractional heat-like and wave-like equations with fractional order initial and boundary conditions are solved first. Illustrative examples are included to demonstrate the validity and applicability of the method.

The purpose of this paper is to introduce a new class of positive two‐dimensional (2D) fractional linear systems.

A notion (concept) of order 2D difference is proposed and the solution to the state equations is given.

The classical Cayley‐Hamilton theorem is extended to the positive 2D fractional linear systems. Necessary and sufficient conditions for the positivity of 2D fractional linear systems, reachability and controllability to zero are established.

A method for analysis of positive 2D fractional linear systems is proposed.

In this article, the authors aim to present the homotopy analysis method (HAM) for obtaining the approximate solutions of space‐time fractional differential equations with initial conditions.

The series solution is developed and the recurrence relations are given explicitly. The initial approximation can be determined by imposing the initial conditions.

The comparison of the HAM results with the exact solutions is made; the results reveal that the HAM is very effective and simple. The HAM contains the auxiliary parameter h, which provides a simple way to adjust and control the convergence region of series solution. Numerical examples demonstrate the effect of changing homotopy auxiliary parameter h on the convergence of the approximate solution. Also, they illustrate the effect of the fractional derivative orders a and b on the solution behavior.

The idea can be used to find the numerical solutions of other fractional differential equations.

The paper deals with the existence of positive solutions for a coupled system of nonlinear fractional differential equations with p-Laplacian operator and involving both right Riemann–Liouville and left Caputo-type fractional derivatives. The existence results are obtained by the help of Guo–Krasnosel'skii fixed-point theorem on a cone in the sublinear case. In addition, an example is included to illustrate the main results.

Fixed-point theorems.

No finding.

The obtained results are original.