Some Fractional Calculus Results Pertaining To Mittag-Leffler Type Functions

In this paper, we study the generalized fractional operators pertaining to the generalized Mittag-Leffler function and multi-index Mittag-Leffler function. Some applications of the established results associated with generalized Wright function are also deduced as corollaries. The results are useful in solving the problems of science, engineering and technology where the Mittag-Leffler function occurs naturally.

Numerical Solutions of the Modified Burgers’ Equation by Finite Difference Methods

In this study, a numerical solution of the modified Burgers’ equation is obtained by the finite difference methods. For the solution process, two linearization techniques have been applied to get over the non-linear term existing in the equation. Then, some comparisons have been made between the obtained results and those available in the literature. Furthermore, the error norms L2 and L∞ are computed and found to be sufficiently small and compatible with others in the literature. The stability analysis of the linearized finite difference equations obtained by two different linearization techniques has been separately conducted via Fourier stability analysis method.

Analytic approximation of Volterra’s population model

In this paper, the Volterra’s population model is studied for population growth of a species within a closed system. Modified Adomian decomposition method (MADM) in conjunction with Pade technique is formally proposed to obtain an analytic approximation for the solution of the model, which is a nonlinear intgro-differential equation. The results of the method are compared with the existing exact results, confirming the accuracy and the efficiency of the proposed approach.

LARGE Estimation of the stress-strength reliability of progressively censored inverted exponentiated Rayleigh distributions

Based on progressively Type-II censored samples, this paper deals with the estimation of R = P(X < Y) when X and Y come from two independent inverted exponentiated rayleigh distributions with different shape parameters, but having the same scale parameter. The maximum likelihood estimator and UMVUE of R is obtained. Different confidence intervals are presented. The Bayes estimator of R and the corresponding credible interval using the Gibbs sampling technique are also proposed. Monte Carlo simulations are performed to compare the performances of the different methods. One illustrative example is provided to demonstrate the application of the proposed method.

Feasible Generalized Stein-Rule Restricted Ridge Regression Estimators

Several versions of the Stein-rule estimators of the coefficient vector in a linear regression model are proposed in the literature. In the present paper, we propose new feasible generalized Stein-rule restricted ridge regression estimators to examine multicollinearity and autocorrelation problems simultaneously for the general linear regression model, when certain additional exact restrictions are placed on these coefficients. Moreover, a Monte Carlo simulation experiment is performed to investigate the performance of the proposed estimator over the others.

Finite Population Mean Estimation through a Two-Parameter Ratio Estimator Using Auxiliary Information in Presence of Non-Response

In surveys covering human populations it is observed that information in most cases are not obtained at the first attempt even after some callbacks. Such problems come under the category of non-response. Surveys suffer with non-response in various ways. It depends on the nature of required information, either surveys is concerned with general or sensitive issues of a society. Hansen and Hurwitz (1946) have considered the problem of non-response while estimating the population mean by taking a subsample from the non-respondent group with the help of extra efforts and an estimator was suggested by combining the information available from the response and nonresponse groups. We also mention that in survey sampling auxiliary information is commonly used to improve the performance of an estimator of a quantity of interest. For estimating the population mean using auxiliary information in presence of non-response has been discussed by various authors. In this paper, we have developed estimators for estimating the population mean of the variable under interest when there is non-response error in the study as well as in the auxiliary variable. We have studied properties of the suggested estimators under large sample approximation. Comparison of the suggested estimators with usual unbiased estimator reported by Hansen and Hurwitz (1946) and the ratio estimator due to Rao (1986) have been made. The results obtained are illustrated with aid of an empirical study.

Approximated Solutions of Linear Quadratic Fractional Optimal Control Problems

In this study we apply the Adomian decomposition method (ADM) to approximate the solution of fractional optimal control problems (FOCPs) where the dynamic of system is a linear control system with constant coefficient and the cost functional is defined in a quadratic form. First we stated the necessary optimality conditions in a form of fractional two point boundary value problem (TPBVP), then the ADM is used to solve the resulting fractional differential equations (FDEs). Some examples are provided to demonstrate the validity and applicability of the proposed method.

A Neural Computational Intelligence Method Based on Legendre Polynomials for Fuzzy Fractional Order Differential Equation

In this article, Legendre simulated annealing, neural network (LSANN) is designed for fuzzy fractional order differential equations, which is employed on fractional fuzzy initial value problem (FFIVP) with triangular condition. Here, Legendre polynomials are used to modify the structure of neural networks with a Taylor series approximation of the tangent hyperbolic as activation function while the network adaptive coefficients are trained in the procedure of simulated annealing to optimize the residual error. The computational results are depicted in terms of numerical values to compare them with previous results.

New Fractional Complex Transform for Conformable Fractional Partial Differential Equations

Conformable fractional complex transform is introduced in this paper for converting fractional partial differential equations to ordinary differential equations. Hence analytical methods in advanced calculus can be used to solve these equations. Conformable fractional complex transform is implemented to fractional partial differential equations such as space fractional advection diffusion equation and space fractional telegraph equation to obtain the exact solutions of these equations.

Some Generalized Inequalities Involving Local Fractional Integrals and their Applications for Random Variables and Numerical Integration

We establish generalized pre-Grüss inequality for local fractional integrals. Then, we obtain some inequalities involving generalized expectation, p−moment, variance and cumulative distribution function of random variable whose probability density function is bounded. Finally, some applications for generalized Ostrowski-Grüss inequality in numerical integration are given.