This paper presents the development of fractional order PI^{λ}D^{μ} and fuzzy- controllers for control of dynamic plant. The fractional order derivative and integral are described. The designs of fractional order PI^{λ}D^{μ} - and fuzzy controllers have been done. The parameters of fractional order controllers are turned using a real coded Genetic Algorithm (GA). The performances of the proposed control systems are illustrated through application examples. Fractional calculus has shown improvement in time response characteristics of feedback control system through the use of non-integer order derivatives and integrals.

In this paper, we implemented a very important method to solve nonlinear partial differential equations known as the exp(-Φ(ξ))-expansion method. Recently, there are several methods being constructed for finding analytical solutions of nonlinear partial differential equations. However, the exp(-Φ(ξ))-expansion method is more effective and useful for solving the nonlinear evolution equations. With the help of this method, we are investigated the exact traveling wave solutions of the Zhiber-Shabat equation. The obtaining exact solutions of this equation are describe many physical phenomena in mathematical physics such as solid state physics, plasma physics, nonlinear optics, chemical kinetics and quantum field theory. Further, three-dimensional plots of the solutions such as solitons, cuspon, periodic, singular kink and bell type are also given to visualize the dynamics of the equation.

In this paper we estimate the growth rate parameter of Gompertz tumour growth model and prove the uniqueness theorem and discuss the sensitivity analysis for same.

In reliability analysis for improving the system performance, the scale parameter of the life time model has mainly considered to obtain equivalence factors for the system designs. In this paper, we propose a new approach through modifying the shape parameter of the Burr type X distribution. The proposed approach is applied to the general series parallel systems. Three different methods are used to improve the system reliability: (i) the reduction method, (ii) the hot duplication method and (iii) the cold duplication method. Numerical example is presented to compare performance of the applied methods, to find limitations for the equivalence factors and to illustrate the overall theoretical analysis.

Due to the need and the necessity to express a physical phenomenon in terms of an effective and comprehensive analytical form, this paper is devoted to study of Airy functions, which arise from the Airy differential equations, by means of integral transforms. Illustrative examples are also provided. The result reveals that the integral transforms are very useful tools to solve differential equations.

The distribution of ratio of two random variables are of interest in many areas of the sciences. The study of the ratio of same family and finding their exact density expression was examined by many authors for decades. Some are solved and others gave approximations, but many still unsolved. In this paper, we consider the ratio of two independent Hypoexponential distributions. We find the exact expressions for the probability density function, the cumulative distribution function, moment generating function, the reliability function and hazard function, which was proved to be a linear combination of the Generalized-F distribution.

The objective of this work was to find the numerical solution of the Impendence problem for the Helmholtz equation for a smooth superellipsoid. The superellipsoid is a shape that is controlled by two parameters. There are some numerical issues in this type of an analysis; any integration method is affected by the wave number k, because of the oscillatory behavior of the fundamental solution. The Helmholtz equation, which is the modified wave equation, is used in many scattering problems. This project was funded by NASA RI Space Grant for testing of the Robin boundary condition for the shape of the superellipsoid. One practical value of all these computations can be getting a shape for the engine nacelles in a ray tracing the space shuttle. We significantly reduced the number of terms in the infinite series needed to modify the original integral equation and used the Green's theorem to solve the integral equation for the boundary of the surface.

Background: In vaccination studies with complex sample survey, survival functions have been used since 2002. Recent publications have proposed several methods for evaluating the adjusted survival functions in non-population-based studies. However, alternative methods for calculating adjusted survival functions for complex sample survey have not been described. Objectives: Propose two methods for calculating adjusted survival functions in the complex sample survey setting; apply the two methods to 2011 National Immunization Survey (NIS) child data with SUDAAN software package. Methods: The inverse probabilities of being in a certain group are defined as the new weights and applied to obtain the inverse probability weighting (IPW) adjusted Kaplan-Meier (KM) survival function. Survival functions are evaluated for each of the unique combination of all levels of predictors in complex sample survey obtained from Cox proportional hazards (PH) model, and the weighted average of these individual functions is defined as the Cox corrected group (CCG) adjusted survival function. Results: The IPW and CCG methods were applied to generate adjusted cumulative vaccination coverage curves across children’s age in days receiving the first dose of varicella by family mobility status. The IPW adjusted cumulative varicella vaccination coverage curves could be consistent estimates of the true coverage curves, the IPW adjustment made the curve for moved family closer to the curve for not-moved family, and the IPW method significantly reduced the standard errors of the cumulative vaccination coverage across children age in days receiving the first dose of varicella comparing to the unadjusted KM method. The Cox PH assumption is not valid for 2011 NIS data. Conclusions: If the Cox PH assumption is not met, then the IPW adjusted KM method is the only good choice, if adjusted survival estimates are desired. If the Cox PH assumption is valid, either the IPW or CCG methods can be used.

For a connected graph G, a subset S = {s_{1}, s_{2},...., s_{k}} of vertices of G and each vertex x of G we associate a pair of k-dimensional vectors (u,v), where u = (d(x, s_{1}), d(x, s_{2}),...., d(x, s_{k})) and v = (δ(x, s_{1}), δ(x, s_{2}),...., (x, s_{k})), where d(x, s_{i}) and δ(x, s_{i}) respectively denote the lengths of a shortest and longest paths between x and s_{i}. The subset S is said to bi-resolve G if no two distinct vertices receive the same pair. The minimum cardinality of a bi-resolving set is called bi-metric dimension of G. In this paper we show bi-metric dimension is lesser than or equal to the metric dimension and determine bi-metric dimensions of some standard graphs.