Shifted genocchi polynomials operational matrix for solving fractional order stiff system

In this paper, we solve the fractional order stiff system using shifted Genocchi poly�nomials operational matrix. Different than the well known Genocchi polynomials, we shift the interval from [0, 1] to [1, 2] and name it as shifted Genocchi polynomials. Using the nice prop�erties of shifted Genocchi polynomials which inherit from classical Genocchi polynomials, the shifted Genocchi polynomials operational matrix of fractional derivative will be derived. Collo�cation scheme are used together with the operational matrix to solve some fractional order stiff system. From the numerical examples, it is obvious that only few terms of shifted Genocchi polynomials is sufficient to obtain result in high accurac

Differential equation and complex network approaches for epidemic modelling

This study consists of three parts. The first part focuses on bifurcation analysis of epidemic models with sub-optimal immunity and saturated treatment/recovery rate as well as nonlinear incidence rate. The second part of the research focuses on estimating the domain of attraction for sub-optimal immunity epidemic models. In the third part of the research, we develop a bond percolation model for community clustered networks with an arbitrarily specified joint degree distribution

New operational matrix of derivative for solving non-linear fractional differential equations via Genocchi polynomials

In this research, new operational method based on Genocchi polynomials for numerical solutions of nonlinear fractional differential equations (NFDEs) is proposed. The Genocchi operational matrix of fractional derivative is first constructed by using some important properties of Genocchi polynomials. These operational matrices together with the collocation method are used to reduce the NFDEs into a system of nonlinear algebraic equations. The error bound for this proposed method is shown. Some examples are given to display the simplicity and accuracy of the proposed technique

Numerical solution of fredholm fractional integro-differential equation with right-sided caputo’s derivative using bernoulli polynomials operational matrix of fractional derivative

In this article, fractional integro-differential equation (FIDE) of Fredholm type involving right-sided Caputo’s fractional derivative with multi-fractional orders is considered. Analytical expressions of the expansion coefficient ck by Bernoulli polynomials approximation have been derived for both approximation of single- and double-variable function. The Bernoulli polynomials operational matrix of right-sided Caputo’s fractional derivative Pα −;B is derived. By approximating each term in the Fredholm FIDE with right-sided Caputo’s fractional derivative in terms of Bernoulli polynomials basis, the equation is reduced to a system of linear algebraic equation of the unknown coefficients ck. Solving for the coefficients produces the approximate solution for this special type of FIDE

Design for sustainability (D4S) in product development using solidworks software: a case study of redesign a sugar cane extractor machine

The Design for Sustainability (D4S) is a recognized method to produce sustainable product that embrace three main elements: environment, economic and social. However, the emphasization on the D4S is immature among industries in Malaysia. The objectives of this study are to propose a user friendly D4S assessment checklist base on SolidWorks software features and to implement on a case study. Implementation of this study done by established a D4S assessment checklist. The checklist consists of mapping between the sustainability main elements and SolidWorks software. The D4S case study is redesign a Sugar Cane Extractor Machine based on the proposed assessment checklist using Solidworks software. The results of study have shown improvement on all three main elements of sustainability. On the environment element, the redesign SCEM has been designed with less environmental impact than original design as following; Carbon Footprint 17.4%, Total Energy Consumed 19.0%, Air Acidification 16.2% and Water Eutrophication 49.4%. On the economic element, the redesign SCEM has been designed less material and production cost than original design for USD28.00 and USD58.50 respectively. On the social element, the redesign SCEM has been equipped with extra safety cover at exposed area and ergonomics handles to ease pushing, pulling and lifting operation. In conclusion, the D4S study has been done successful with consideration on all three main sustainability

Numerical solution of fractional diffusion wave equation and fractional klein–gordon equation via two-dimensional genocchi polynomials with a ritz–galerkin method

In this paper, two-dimensional Genocchi polynomials and the Ritz–Galerkin method were developed to investigate the Fractional Diffusion Wave Equation (FDWE) and the Fractional Klein–Gordon Equation (FKGE). A satisfier function that satisfies all the initial and boundary conditions was used. A linear system of algebraic equations was obtained for the considered equation with the help of two-dimensional Genocchi polynomials along with the Ritz–Galerkin method. The FDWE and FKGE, including the nonlinear case, were reduced to solve the linear system of the algebraic equation. Hence, the proposed method was able to greatly reduce the complexity of the problems and provide an accurate solution. The effectiveness of the proposed technique is demonstrated through several examples

Investigation of cutting temperature and cutting force from mist flow pattern in MQL technique

Minimum Quantity Lubrication (MQL) is an alternative method to supply the cutting fluid in the formation of mist. MQL has proven to reduce machining cost and increase machining performance. Previous research have stated that machining performance is affected by the lubricant type, flow rate, the distance between nozzle and tool tip, and the workpiece material. These important parameters are not reported in many research documents. MQL is known for its many benefits but no one was able to prove that the statement is true or ever suggested a systematic procedure to prove MQL’s efficiency. The effectiveness and the working principle of MQL are still questionable with very few explanations provided. The present study is about investigation of cutting temperature and cutting force from mist flow pattern in MQL technique The MQL nozzle distance and cutting fluid flow pattern are among the factors that can provide optimum machining performance in term of cutting force and cutting temperature. The objective of this study is to conduct machining process using MQL technique with different combination of spray parameters and to optimize spray parameters for minimum machining temperature and cutting forces. The four nozzle distances of 3, 6, 7 and 9 mm were selected based on the results obtained from Phase Doppler Anemometry (PDA). The machining performance was evaluated under three levels of cutting speed and two levels of feed rate at constant depth of cut. The cutting force was measured using a set of dynamometer and cutting temperature using thermal imager. The most suitable mist flow pattern during machining was the largest spray cone angle supplied under 0.4 MPa input air pressure. The results obtained from the machining process shows a significant reduction of cutting force and cutting temperature at the nozzle distance in the range of 6 to 9 mm under 0.4 MPa input air pressure for larger diameter OD30 nozzle

On the new properties of Caputo–Fabrizio operator and its application in deriving shifted Legendre operational matrix

In this paper, we study the recently introduced Caputo and Fabrizio operator, which this new operator was derived by replacing the singular kernel in the classical Caputo derivative with the regular kernel. We introduce some useful properties based on the definition by Caputo and Fabrizio for a general order n < α < n + 1, n ∈ N. Here, we extend the associated integral of Caputo–Fabrizio sense to n < α < n + 1, n ∈ N. We also find the general formula for the Caputo–Fabrizio operator of (t − a)β . Then, we derive Legendre operational matrix based on this new operator and together with Tau method, we use it to solve the differential equations defined in the Caputo–Fabrizio sense. As far as we know, the operational matrix method has yet been derived or attempted for solving the differential equations in Caputo–Fabrizio sense, while it has been successfully used to solve fractional calculus problems involving the classical Caputo sense. Some numerical examples are given to display the simplicity and accuracy of the proposed technique

Urban rail transit PPPs: Lessons from East Asian cities

Private sector participation in urban rail transit has proliferated in the past two decades. The large metropolises of East Asia have had decades of experience with private sector participation in the provision of heavy metro services. The design of these public–private partnerships (PPP) are varied. The diverse experiences of Tokyo, Hong Kong, Singapore and Beijing contain valuable lessons for other cities. Using a case study approach, this paper discusses three features of urban rail transit developments in the context of East Asian cities, viz., farebox recovery, land value capture mechanisms, and vertical structure of the industry. Super vertical integration between rail transit and real estate development as land value capture strategy to finance urban rail transit has proven to be successful in Japanese cities and Hong Kong. Singapore’s experience illustrates that vertically unbundled PPPs could cut off avenues for cross-subsidisation, reduce information flows as well as economies of scale and scope, introduce principal agent problems, and result in underinvestment in capital stock and maintenance. We conclude that (i) a combination of high farebox recovery ratios and successful land value capture contributed significantly to the development of urban rail transit in East Asia cities; (ii) given the complexities and high costs of heavy metros, the optimal structure is a vertically integrated public-owned and driven system, with the public sector entering into selective partnerships with the private sector where risk sharing is clearly defined and allocated