### Modified beta generalized linear failure rate distribution: theory and applications

JPRM-Vol. 16 (2020), Issue 2, pp. 123 – 148 Open Access Full-Text PDF
Farrukh Jamal, Ibrahim Elbatal, Christophe Chesneau, Mohammed Elgarhy, Amal S. Hassan
Abstract: In this paper we introduce a new comprehensive six-parameter distribution called the modified beta generalized linear failure rate distribution. One of the interest of this distribution is to generalize some well-known flexible distributions discussed in the literature, such as (i) the beta linear failure rate distribution, (ii) the generalized linear failure rate distribution, (iii) the beta exponential distribution, (iv) the beta Rayleigh distribution and (v) the generalized exponential distribution, among others. We derive some of its statistical properties, including the moments, the moment generating function, the quantile function, the order statistics and the mean deviations. We propose the method of maximum likelihood for estimating the model parameters. A simulation study is performed in order to investigate the performance of the maximum likelihood estimators. A real data set is used to illustrate the importance and the flexibility of the new distribution.

### Existence of a solution for integral Urysohn type equations system via fixed points technique in complex valued extended $$b$$-metric spaces

JPRM-Vol. 16 (2020), Issue 2, pp. 109 – 122 Open Access Full-Text PDF
Abdelkader Belhenniche, Sfya Benahmed, Liliana Guran
Abstract: In this article, we obtain fixed point results and we give a common fixed point theorem for ‘Ciri’c type operators on complex valued extended b-metric spaces which may satisfy very general assumptions. Our results extend and generalize the results of Kiran et al. [1], as well as some known results in the literature. Then an illustrative application to Urysohn type integral equations system is given.

### Improvement of the Hardy inequality involving $$k$$-fractional calculus

JPRM-Vol. 16 (2020), Issue 2, pp. 89 – 108 Open Access Full-Text PDF
Abstract: The major idea of this paper is to establish some new improvements of the Hardy inequality by using $$k$$-fractional integral of Riemann-type, Caputo $$k$$-fractional derivative, Hilfer $$k$$-fractional derivative and Riemann-Liouville $$(k,r)$$-fractional integral. We discuss the $$\log$$-convexity of the related linear functionals. We also deduce some known results from our general results.

### Multiplicative Shingali and Kanabour indices for Bismuth Tri-Iodide

JPRM-Vol. 16 (2020), Issue 2, pp. 80 – 88 Open Access Full-Text PDF
Abaid ur Rehman Virk
Abstract: Topological indices helps us to collect information about algebraic graphs and gives us mathematical approach to understand the properties of algebraic structures. Here, we will introduce Multiplicative version of Shingali and Kanabour indices and compute these indices for Bismuth Tri-Iodide chain $$m-Bil_{3}$$ and sheet $$Bil_{3}(m\times n)$$.

### Novel fractional differential operator and its application in fluid dynamics

JPRM-Vol. 16 (2020), Issue 2, pp. 69 – 79 Open Access Full-Text PDF
Abstract: Theoretical analysis of unsteady incompressible viscous fluid has been carried with constant proportional Caputo fractional derivative namely constant proportional Caputo type with singular kernel. The modeled considered in this paper is the fundament problem of fluid dynamics. The resulting governing equations are modeled with hybrid fractional operator of singular kernel and its solution obtained by using Laplace transform method and expressed in terms of series. Some graphs are captured for fractional parameter $$\alpha$$ for large and small time and found that velocity shows dual trend for small and large values of time for different values of fractional parameter $\alpha$. Further, compared the present results with the results obtained with new fractional operators and found that constant proportional Caputo type operator portrait better velocity decay. Moreover, for increasing time, momentum boundary layer thickness increases while for grater values of fractional parameter it reduces.

### A new approach for the enumeration of components of Digraphs over quadratic maps

JPRM-Vol. 16 (2020), Issue 2, pp. 56 – 66 Open Access Full-Text PDF
M. Haris Mateen, M. Khalid Mahmood
Abstract: Various partial attempts to count cycles and components of digraphs from congruences have been made earlier. While the problem is still open till date. In this work, we introduce a new approach to solve the problem over quadratic congruence equations. Define a mapping $$g:Z_{m}\mapsto Z_{m}$$ by $$g(t)=t^{2}$$, where $$Z_{m}$$ is the ring of residue classes modulo $$m$$. The digraph $$G(2,m)$$ over the set of residue classes assumes an edge between the residue classes $$\overline{x}$$ and $$\overline{y}$$ if and only if $$g(\overline{x})\equiv \overline{y}~(\text{mod}~m)$$ for $$m\in Z^{+}$$. Classifications of cyclic and non-cyclic vertices are proposed and proved using basic modular arithmetic. Finally, explicit formulas for the enumeration of non-isomorphic components are proposed followed by simple proofs from number theory.

### Numerical solutions of the fractional SIS epidemic model via a novel technique

JPRM-Vol. 16 (2020), Issue 2, pp. 44 – 55 Open Access Full-Text PDF
Abstract: This article introduces a novel technique called modified fractional Taylor series method (MFTSM) to find numerical solutions for the fractional SIS epidemic model. The fractional derivative is considered in the sense of Caputo. The most important feature of the MFTSM is that it is very effective, accurate, simple, and more computational than the methods found in literature. The validity and effectiveness of the proposed technique are investigated and verified through numerical example.

### A new efficient method for time-fractional Sine-Gordon equation with the Caputo and Caputo-Fabrizio operators

JPRM-Vol. 16 (2020), Issue 2, pp. 27 – 43 Open Access Full-Text PDF
Abstract: In this work, a new efficient method called, Elzaki’s fractional decomposition method (EFDM) has been used to give an approximate series solutions to time-fractional Sine-Gordon equation. The time-fractional derivatives are described in the Caputo and Caputo-Fabrizio sense. The EFDM is based on the combination of two different methods which are: the Elzaki transform method and the Adomian decomposition method. To demonstrate the accuracy and efficiency of the proposed method, a numerical example is provided. The obtained results indicate that the EFDM is simple and practical for solving the fractional partial differential equations which appear in various fields of applied sciences.

### The new auxiliary method in the solution of the generalized Burgers-Huxley equation

JPRM-Vol. 16 (2020), Issue 2, pp. 16 – 26 Open Access Full-Text PDF
Muhammad Asim Khan, M. Ali Akbar, Norhashidah Hj. Mohd. Ali, Muhammad Abbas
Abstract: A recently developed direct method known as the new auxiliary method shown significant improvement for solving non-leaner partial differential equations (PDEs) and gives more exact solutions compared to the traditional direct method. In this paper, we used the new auxiliary method for solitary wave solutions of the generalized Burgers Huxley equation (B-HE). The new auxiliary method is a very powerful, felicitous, effective method to get solitary wave solutions of PDEs.