Fuzzy Solutions to Second Order Three Point Boundary Value Problem

In this manuscript, the proposed work is to study the existence of second-order differential equations with three point boundary conditions. Existence is proved using fuzzy set valued mappings of a real variable whose values are normal, convex, upper semi continuous and compactly supported fuzzy sets. The sufficient conditions are also provided to establish the existence results of fuzzy solutions of second order differential equations for three point boundary value problem. By using Banach fixed point principle, a new existence theorem of solutions for these equations in the metric space of normal fuzzy convex sets with distance given by the maximum of the Hausdorff distance between level sets is obtained. Then to further establish the existence, fixed point theorem for absolute retracts is used by taking consideration that space of fuzzy sets can be embedded isometrically as a cone in Banach space. Finally, an example is provided to illustrate the result

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

The Phase Diagram of Scalar Field Theory on the Fuzzy Disc

Using a recently developed bootstrapping method, we compute the phase diagram of scalar field theory on the fuzzy disc with quartic even potential. We find three distinct phases with second and third order phase transitions between them. In particular, we find that the second order phase transition happens approximately at a fixed ratio of the two coupling constants defining the potential. We compute this ratio analytically in the limit of large coupling constants. Our results qualitatively agree with previously obtained numerical results.Comment: 1+17 pages, v2: typos fixed, published versio

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