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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.

The purpose of this paper is to develop a block method of order five for the general solution of the first-order initial value problems for Volterra integro-differential equations (VIDEs).

A collocation approximation method is adopted using the shifted Legendre polynomial as the basis function, and the developed method is applied as simultaneous integrators on the first-order VIDEs.

The new block method possessed the desirable feature of the Runge–Kutta method of being self-starting, hence eliminating the use of predictors.

In this paper, some information about solving VIDEs is provided. The authors have presented and illustrated the collocation approximation method using the shifted Legendre polynomial as the basis function to investigate solving an initial value problem in the class of VIDEs, which are very difficult, if not impossible, to solve analytically. With the block approach, the non-self-starting nature associated with the predictor corrector method has been eliminated. Unlike the approach in the predictor corrector method where additional equations are supplied from a different formulation, all the additional equations are from the same continuous formulation which shows the beauty of the method. However, the absolute stability region showed that the method is A-stable, and the application of this method to practical problems revealed that the method is more accurate than earlier methods.

The purpose of this paper is to apply the fractional sub-equation method to research on coupled fractional variant Boussinesq equation and fractional approximate long water wave equation.

The algorithm is implemented with the aid of fractional Ricatti equation and the symbol computational system Mathematica.

New travelling wave solutions, which include generalized hyperbolic function solutions, generalized trigonometric function solutions and rational solutions, for these two equations are obtained.

The algorithm is demonstrated to be direct and precise, and can be used for many other nonlinear fractional partial differential equations. The fractional derivatives described in this paper are in the Jumarie's modified Riemann-Liouville sense.

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.

The purpose of this paper is to propose a Tau method for solving nonlinear Blasius equation which is a partial differential equation on a flat plate.

The operational matrices of derivative and product of modified generalized Laguerre functions are presented. These matrices together with the Tau method are then utilized to reduce the solution of the Blasius equation to the solution of a system of nonlinear equations.

The paper presents the comparison of this work with some well‐known results and shows that the present solution is highly accurate.

This paper demonstrates solving of the nonlinear Blasius equation with an efficient method.

To provide a progress report into research conducted to establish guidelines for the development of guidance vision aids.

The first stage of the research is to establish a coherent engineering basis for the methods of (visual) motion perception and control to inform the design of pilot aids that will support flight in degraded visual conditions, particularly when close to the ground. The next stage will then be to construct and evaluate synthetic displays that recover the visual cues necessary to allow flight in degraded visual conditions for a range of manoeuvres using the flight simulation facilities at the University of Liverpool (UoL). The research is guided by tau (time to contact) theory from the field of ecological psychology.

The closure of spatial gaps for a number of aircraft manoeuvres are presented in the tau domain. Analysis of the landing flare manoeuvre suggest that both a constant rate of change of tau strategy and an intrinsic tau‐guidance strategy will yield benefits in terms of touchdown descent rate if presented as display symbology.

Results are presented from trials where only one professional pilot was used. Results from a wider population of pilots need to be analysed to ensure that the observed trends are generic.

The reported results are being used in the next phase of the research project to inform the design of a guidance vision‐aid for the flare manoeuvre. These displays will be tested in flight simulation trials.

The research takes a theory of motion perception and applies it to aircraft guidance display technology.

This paper aims to investigate analytical solutions of natural frequencies and mode shapes of Euler-Bernoulli beams with step changes in the stiffness.

In this work, analytical solutions for a beam with a single discontinuity was performed. Subsequently, based on an effective matrix formulation, the closed-form expressions of the single discontinuity beam could be conveniently extended to stepped beams with multiple stiffness discontinuities.

The results of the study show that the natural frequency of the beam can be adjusted by the local stiffness variation, and step location plays a significant role in free vibration responses.

The effects of the stiffness of the segment and step location on the natural frequencies of the stepped beams under different boundary conditions were examined using the proposed analytical scheme. This study provides insights into the design of variable-stiffness beam structures with the capability to adjust natural frequencies.

Laminarization of commercial aircraft surfaces is the most promising technology to reduce fuel consumption and ecological impact. As laminar flow highly depends on cross-flow effects, there is the question in which way simple estimations and simplifications for application in conceptual aircraft design can be used to capture these cross-flow influences. This paper aims to show the accuracy of 2D methods for estimating laminar flow regions on 3D wing objects.

Several methods, relating 3D and 2D flow conditions, are analyzed with regard to capture cross-flow influences. The 3D pressure distributions depending on utilized transformation method are compared to Reynolds-averaged Navier–Stokes (RANS) solutions. With the most precise transformation method, the laminar flow area on a conventional wing of a short range aircraft is determined and compared to the laminar area obtained with the RANS pressure distributions as input. Further, hybrid laminar flow control component sizing is carried out to obtain the net benefit in fuel reduction of simplified method compared to RANS method for a conventional short range aircraft.

In this particular case, the solutions calculated with the simplified methods show high deviations from those obtained with RANS.

This investigation underlines the need of proper methods for fast and accurate estimation of cross-flow effects to be able to assess the full potential of laminar flow control within conceptual aircraft design.

The purpose of this paper is to present a hybridized technique for analyzing the behavior of an industrial system stochastically utilizing vague, imprecise, and uncertain data. The press unit of a paper mill situated in a northern part of India, producing 200 tons of paper per day, has been considered to demonstrate the proposed approach. Sensitivity analysis of system's behavior has also been done.

In the proposed approach, two important tools namely traditional Lambda‐Tau technique and genetic algorithm have been hybridized to build genetic algorithms‐based Lambda‐Tau (GABLT) technique to analyze the behavior of complex repairable industrial systems stochastically up to a desired degree of accuracy. This technique has been demonstrated by computing six well‐known reliability indices used for behavior analysis of the considered system in more promising way.

The behavior analysis results computed by GABLT technique have reduced region of prediction in comparison of existing Lambda‐Tau technique region, i.e. uncertainties involved in the analysis are reduced. Thus, it may be a more useful analysis tool to assess the current system conditions and involved uncertainties. The paper suggested an approach to improve the system's performance.

The paper suggests a hybridized technique for analyzing the stochastic behavior of an industrial subsystem by computing six well‐known reliability indices in the form of fuzzy membership function.

The proposed hybridized framework provides a new performance optimization-based paradigm for analysing the failure behaviour of paneer unit (PU) in the dairy industry.

A novel fuzzy Jaya-based Lambda–Tau Optimization (JBLTO) approach-based mathematical modelling was developed for calculating various reliability indices of the considered unit. Failure mode and effect analysis (FMEA) was carried using qualitative information gathered from system's expert opinions. Fuzzy-complex proportional assessment (FCOPRAS) approach was integrated within FMEA to recognize the most critical failure causes associated with various subsystem/components.

The availability of the unit falls by 0.053% as the uncertainty level increases from ±15 to ±25% and further decreases to 0.323% as the uncertainty level increases from ±25 to ±60%. Failure causes, namely wearing in gears of gearbox (MST4), an impeller's cavitation and/or corrosion (CFP4), winding failure of electric motor (WS9), were recognized as the most critical failure causes with FCOPRAS final performance scores of 100, 100 and 100 and fuzzy combinative distance-based assessment (FCODAS) resultant assessment score of 0.5997, 1.1898 and 1.6135.

JBLTO approach-based reliability results were compared with traditional particle swarm optimization-based Lambda–Tau (PSOBLT) and traditional fuzzy Lambda–Tau (FLT) approaches for confirming the downward trend in the system's availability. The ranking results of qualitative analysis are compared with the implementation of FCODAS technique. Sensitivity analysis was executed to evaluate the robustness of the proposed hybridized framework.