Open Access Original Research Article

Probabilities and Probable Solutions of a Modified KdV Type Nonlinear Partial Differential Equation

J. R. Bogning, R. Njikue, J. P. Ngantcha, H. M. Omanda, C.T. Djeumen Tchaho

Asian Research Journal of Mathematics, Page 1-13
DOI: 10.9734/arjom/2022/v18i130350

The goal of this work is not only the search for the solutions of a nonlinear partial differential equation, but how to locate and choose a form of solution verifying the nonlinear partial differential equation. In this work, we use the probabilities of appearance of the pairs (n, m) linked to iB-functions for which certain terms of the range of coefficients equations are grouped together to locate and then determine the solutions of the partial differential equation of the KdV type. The pairs (n, m) when identified, indicate with precision the iB-function which will choose from the start as the solution function which we want to build. The probabilities here are essential data to select the analytical sequences of the solutions to be investigated.

Open Access Original Research Article

Generalized Estimators for Finite Population Variance Using Measurable and Affordable Auxiliary Character

Isah Muhammad, Yahaya Zakari, Ahmed Audu

Asian Research Journal of Mathematics, Page 14-30
DOI: 10.9734/arjom/2022/v18i130351

In this paper, a generalized exponential-type estimator for estimating the population variance using measurable and affordable auxiliary character in single phase sampling has been proposed. Some special cases of the proposed generalized estimator with a = 1 and a = 2 have also been discussed. The expressions for the mean square error and bias of the proposed generalized estimator have been derived. The efficiency of the proposed generalized estimator has been compared theoretically with the existing estimators and the conditions under which the proposed estimators are better than some existing estimators have also been given.  Numerical examples with five real data sets to compute MSEs and PREs were conducted to ascertain the efficiency of the proposed estimators over others and the results showed that both of the proposed estimators were more efficient than the estimators considered in literature. Thus, the estimator with  perform better than the estimator with a = 1.

Open Access Original Research Article

Exact Solutions of Second-order Fractional Fredholm Integro-differential Equations

Abdelhalim Ebaid, Ohoud A. Alshahrani

Asian Research Journal of Mathematics, Page 31-38
DOI: 10.9734/arjom/2022/v18i130352

The present paper analyzes the second-order fractional Fredholm integro-differential equations by means of the Caputo definition in the fractional calculus (FC). The exact solutions are obtained for two examples utilizing a direct solution method. Furthermore, it is shown that the present solutions in fractional forms reduce to the corresponding classical ones in the relevant literature, with integer derivatives, as special cases.

Open Access Original Research Article

Spectral Radius of a Normal Operator

Achiles Nyongesa Simiyu, Philis Alosa, Fanuel Olege

Asian Research Journal of Mathematics, Page 39-52
DOI: 10.9734/arjom/2022/v18i130353

Let X be a Complex Banach space and T be a bounded operator in X. The number sup {|λ| : λ σ(T)} (where σ(T) is the spectrum of T and σ(T) ̸= ϕ) is called the spectral radius of T and denoted by r(T). Since λ ≤ ∥Tfor all λ σ(T), it follows that r(T) ≤ ∥T. The spectral mapping theorem implies that r(Tn) = (r(T))n for every positive integer n. It frequently turns out that it is easy to compute the spectral radius of an operator even if it is hard to _nd the spectrum. This is often made easy by the spectral radius formula. Let H be a Hilbert space and T be a bounded linear operator in H. In this paper we show that if T is normal, then Tn is normal for each n N and Tn= Tn. Consequently, we use the spectral radius formula to show that r(T) = ∥T. Moreover, we show that if X is a Complex Banach space and T is bounded in X then there is a λ belonging to the spectrum of T such that |λ| = r(T). Let H be a Complex Hilbert space and T be a bounded operator in H which is normal; we show that ∥T= sup {|Tx, x| : x H and x= 1} and the residual spectrum of T is void.

Open Access Original Research Article

One-step Hybrid Block Method for Directly Solving Fifth-order Initial Value Problems of Ordinary Differential Equations

M. K. Duromola, A. L. Momoh, J. M. Adeleke

Asian Research Journal of Mathematics, Page 53-64
DOI: 10.9734/arjom/2022/v18i130354

An effective one-step hybrid block for getting the approximate solution of a fifth-order IVP with applications to problems in the sciences and engineering is constructed in this study. The mathematical formulation of the method is based on the principle of interpolation and collocation of the trial solution and its derivatives at the chosen equidistant grid and off-grid points. The basic properties of the derived method are examined, and it has an order greater than one, zero stable, consistent, and hence convergent. The derived method is applied to solve five different linear and nonlinear fifth-order initial value problems. Comparison of the absolute errors obtained using the derived method with a few existing ones in the literature supports its good performance.

Open Access Original Research Article

Modified Classes of Regression-Type Estimators of Population Mean in the Presences of Auxiliary Attribute

A. Audu, S. A. Abdulazeez, A. Danbaba, Y. M. Ahijjo, A. Gidado, M. A. Yunusa

Asian Research Journal of Mathematics, Page 65-89
DOI: 10.9734/arjom/2022/v18i130355

The use of relevant information from auxiliary variable at the estimation stage and design stage to obtain reliable and efficient estimate is a common practice is a sample survey. But situations arise when the available auxiliary information are attribute in nature. There are some existing estimators based on auxiliary attribute in literature, however, they are less efficient when the bi-serial correlation between the study variable and auxiliary attribute is negative. Also, some depend on an unknown parameter of the study variable (Cy) which makes their applicability of the estimators in real life situations not possible unless if the value is estimated using a large sample which requires additional resources. In this work, the concept of regression base estimator was used to obtain estimators that are independent of unknown population parameter of the study variable and applicable for both negative and positive correlations. The properties (Biases and MSEs) of the modified estimators were derived up to the first order of approximation using Taylor series approach. The efficiency conditions of the proposed estimation over the existing estimator considered in the study were established. The empirical studies were conducted using both existing population parameters and stimulation to investigate the efficiency of the proposed estimators over the efficiency of the existing estimators. The results revealed that the proposed estimators have minimum MSEs and higher PREs among all the competing estimators. These imply that the proposed estimators are more efficient and can produce better estimate of the population mean compared to other existing estimators considered in the study.