Numerical Solution of System of Fractional Differential Equations in Imprecise Environment

Fractional calculus and fuzzy calculus theory, mutually, are highly applicable for showing different aspects of dynamics appearing in science. This chapter provides comprehensive discussion of system of fractional differential models in imprecise environment. In addition, presenting a new vast area to investigate numerical solutions of fuzzy fractional differential equations, numerical results of proposed system are carried out by the Grünwald‐Letnikov\u27s fractional derivative. The stability along with truncation error of the Grünwald‐Letnikov’s fractional approach is also proved. Moreover, some numerical experiments are performed and effective remarks are concluded on the basis of efficient convergence of the approximated results towards the exact solutions and on the depictions of error bar plots

Tracking uncertainty in a spatially explicit susceptible-infected epidemic model

In this paper we conceive an interval-valued continuous cellular automaton for describing the spatio-temporal dynamics of an epidemic, in which the magnitude of the initial outbreak and/or the epidemic properties are only imprecisely known. In contrast to well-established approaches that rely on probability distributions for keeping track of the uncertainty in spatio-temporal models, we resort to an interval representation of uncertainty. Such an approach lowers the amount of computing power that is needed to run model simulations, and reduces the need for data that are indispensable for constructing the probability distributions upon which other paradigms are based

Quantum canonical tensor model and an exact wave function

Tensor models in various forms are being studied as models of quantum gravity. Among them the canonical tensor model has a canonical pair of rank-three tensors as dynamical variables, and is a pure constraint system with first-class constraints. The Poisson algebra of the first-class constraints has structure functions, and provides an algebraically consistent way of discretizing the Dirac first-class constraint algebra for general relativity. This paper successfully formulates the Wheeler-DeWitt scheme of quantization of the canonical tensor model; the ordering of operators in the constraints is determined without ambiguity by imposing Hermiticity and covariance on the constraints, and the commutation algebra of constraints takes essentially the same from as the classical Poisson algebra, i.e. is first-class. Thus one could consistently obtain, at least locally in the configuration space, wave functions of "universe" by solving the partial differential equations representing the constraints, i.e. the Wheeler-DeWitt equations for the quantum canonical tensor model. The unique wave function for the simplest non-trivial case is exactly and globally obtained. Although this case is far from being realistic, the wave function has a few physically interesting features; it shows that locality is favored, and that there exists a locus of configurations with features of beginning of universe.Comment: 17 pages. Section 2 expanded to include fuzzy-space interpretation, and other minor change

The utilization of coffee waste into fired clay brick

The rapid growth of coffee industry is accompanied by a staggering increase in the amount of agriculture waste produced. In coffee producing countries, coffee wastes constitute a source of severe contamination and a serious environmental problem. In this study, the investigation on the possibility to utilize the coffee waste (CW) incorporated into the fired clay brick was carried out. The main purpose of this study is to determine the physical, mechanical properties and leach ability test of bricks incorporated with different percentages of CW. In this methodology, control brick (CB) and three different percentages of coffee waste brick (CWB) (1%, 3% and 5%) were manufactured and fired at 1050 °C. Physical and mechanical properties including shrinkage, density and compressive strength were reported and discussed. Additionally, leaching of heavy metals from manufactured clay brick was tested by using Toxicity Characteristics Leaching Procedure (TCLP). The results reported that with the incorporation of CW, the shrinkage increased linearly but still comply with minimum standard below 8% and good quality of brick was manufactured. Meanwhile, the results showed that density value decreased up to 30% from the normal brick with increased percentages of CW. The decreased compressive strength value of all the manufactured brick is still complies with minimum standard. On the other hand, heavy metals concentration leach out from different percentages of coffee waste brick is not exceeding the limit of 5 mg/L allowed by United States Environmental Protection Agency (USEPA). As a conclusion, the incorporation of CW into fired clay brick gives some advantages to the brick properties and also provides alternative solution on disposing the CW. In addition, the CW could also be a potential of low cost waste additive for the production of a brick

Status of the differential transformation method

Further to a recent controversy on whether the differential transformation method (DTM) for solving a differential equation is purely and solely the traditional Taylor series method, it is emphasized that the DTM is currently used, often only, as a technique for (analytically) calculating the power series of the solution (in terms of the initial value parameters). Sometimes, a piecewise analytic continuation process is implemented either in a numerical routine (e.g., within a shooting method) or in a semi-analytical procedure (e.g., to solve a boundary value problem). Emphasized also is the fact that, at the time of its invention, the currently-used basic ingredients of the DTM (that transform a differential equation into a difference equation of same order that is iteratively solvable) were already known for a long time by the "traditional"-Taylor-method users (notably in the elaboration of software packages --numerical routines-- for automatically solving ordinary differential equations). At now, the defenders of the DTM still ignore the, though much better developed, studies of the "traditional"-Taylor-method users who, in turn, seem to ignore similarly the existence of the DTM. The DTM has been given an apparent strong formalization (set on the same footing as the Fourier, Laplace or Mellin transformations). Though often used trivially, it is easily attainable and easily adaptable to different kinds of differentiation procedures. That has made it very attractive. Hence applications to various problems of the Taylor method, and more generally of the power series method (including noninteger powers) has been sketched. It seems that its potential has not been exploited as it could be. After a discussion on the reasons of the "misunderstandings" which have caused the controversy, the preceding topics are concretely illustrated.Comment: To appear in Applied Mathematics and Computation, 29 pages, references and further considerations adde