BASHA G W, WOYA A A, TEKILE A K

**037037** BASHA G W, WOYA A A, TEKILE A K (Statistics Dep, Bahir Dar Univ, Amhara, Ethiopia, Email: garewa2010@gmail.com) : **Prevalence and risk factors of active trachoma among primary school children of Amhara Region, Northwest Ethiopia.** Indian J Ophthalmol 2020, 68(5), 750-4.

Trachoma is the leading infectious cause of blindness in the world. It is caused by conjunctival infection with the bacterium Chlamydia trachomatis. The objective of this study was to determine the prevalence and risk factors of active trachoma among primary school children in Amhara region, Ethiopia. A cross‑sectional study was conducted from April to May 2018. Data on socio demographic and health characteristics of a child were collected using a structured questionnaire and eye examination for this study. Bivariate and multiple logistic regression statistical analyses were used to determine the prevalence and risk factors of active trachoma among primary school children living in Amhara region,Northwest Ethiopia. The prevalence of active trachoma among primary school children living in Amhara region was 10.3% in this study. The results of multiple logistic regression analysis revealed that children who wash their face at least two times per day (AOR = 0.37, 95% CI: 0.14–0.97), children with noflies on their face (AOR = 0.12, 95% CI: 0.10–0.30) had significantly lower risk of being infected by trachoma and children of households who spend 30 min to 1 h to fetch water (AOR = 10.02, 95% CI: 1.10–93.53) had significantly higher risk of being infected by trachoma in the study area. The result of this study showed that risk factors: frequency of face washing per day, presence of flies on child’s face during interview, and time required to fetch water for the household were found to be significantly associated with trachoma. Therefore, the study recommends that any concerned bodies directed at prevention and control of trachoma among primary school children living in Amhara region should give special attention to these factors. Implementing the World Health Organization (WHO) endorsed SAFE (Surgery, Antibiotics, Facial cleanliness, Environmental improvement) strategy was recommended for the effective prevention and control of trachoma in the study area since the prevalence of active trachoma, 10.3% was higher than the WHO recommended thresholds (>10% prevalence).

2 tables, 18 ref

BHUYAN M, CHUTIA C

**037134** BHUYAN M, CHUTIA C (Mathematics Dep, DCB Girls’ Coll, Jorhat, Assam, Email: moloyabhuyan@yahoo.com) : **Existence of smooth epimorphism from fuchsian group to the group of automorphisms of compact riemann surface to the point group of carbon tetrachloride.** Ultra Scient phys Sci-Sect A 2020, 32(3), 13-21.

A finite group G acts as a group of automorphisms on a compact Riemann surface S of genus g if and only if there exist a Fuchsian group G and an epimorphism ɸ: G ® G such that kerɸ = K is a surface group of genus g. And then ɸ is named as smooth homomorphism. The objective of this paper is to establish aset of necessary and sufficient conditions for the existence of smooth epimorphism from a Fuchsian group G to the finite group of symmetries of Carbon Tetra chloride molecule, whose abstract group representation is {a,b|a^{4}= b^{3}=(ab)^{2}}.

1 tables, 18 ref

HOODA E, VERMA U, HOODA B K

**035935** HOODA E, VERMA U, HOODA B K (Mathematics & Statistics Dep, CCS Haryana Agricultural Univ, Hisar- 125 004, Email: ektahooda@gmail.com) : **ARIMA and State-Space models for sugarcane (Saccharum officinarum) yield forecasting in Northern agro-climatic zone of Haryana.** J Appl Nat Sci 2020, 12(1), 53-8.

Advance estimates of significant cereal and commercial crops are given by the Directorate of Economics and Statistics and the central Ministry of Agriculture, Cooperation & Farmers’ Welfare. However, the final estimates are released a few months after the actual harvest of the crops. In this study, ARIMA and State-Space models have been developed for sugarcane yield forecasting in Ambala and Karnal districts of Haryana. The above-mentioned models have been developed using yield data of sugarcane crop for the time period 1966-67 to 2009-10 of Ambala and Karnal districts. The validity of fitted models has been tested over the years 2010-11 to 2016-17. The forecasting performance of the developed models has been studied using percent deviations of sugarcane yield forecasts in relation to the actual yield, and root means squared errors. It has been observed that state-space models outperform the popular ARIMA models for forecasting of sugarcane yield in Northern Agro-climatic Zone of Haryana.

3 illus, 8 tables, 15 ref

LOGANATHAN P, DEEPA K

**035816** LOGANATHAN P, DEEPA K (Mathematics Dep, Anna Univ, Chennai- 600 025, Email: logu@annauniv.edu) : **Heat and mass transfer analysis of casson fluid flow on a permeable riga-plate.** Indian J Pure Appl Phys 2020, 58(02), 79-86.

Numerical analysis has been carried out for a casson fluid flow on a riga-plate with temperature dependent thermal conductivity. The physical model which governs the transport properties is solved numerically. This investigation emphasizes the consequence of variable thermal conductivity and electrically conducting magnetic field on the fluid flow. Rate of heat transfer is elevated, while the flow exposed to constructive case of variable thermal conductivity. The flow speed is enhanced for the improved values of modified Hartmann number. Correlation of the results with the similarity solutions declares the accuracy.

11 illus, 33 ref

CHAUDHARY S, CHAUDHARY S, CHAUDHARY M K

**035815** CHAUDHARY S, CHAUDHARY S, CHAUDHARY M K (Mathematics Dep, Malaviya National Institute of Technology, Jaipur- 302 017, Email: d11.santosh@yahoo.com) : **Numerical investigation of unsteady MHD flow and radiation heat transfer past a stretching surface in porous media with viscous dissipation and heat generation/absorption.** Indian J Pure Appl Phys 2020, 58(02), 71-8.

Viscous dissipation and radiation effects on unsteady flow of laminar incompressible viscous electrically conducting fluid through stretching surface in a porous media with magnetic field and heat generation/absorption have been investigated. Taking suitable similarity variables, the governing boundary layer equations are converted into a set of ordinary differential equations and solved numerically by Runge-Kutta fourth order method along with shooting technique. The effects of the various physical dimensionless parameters such as unsteadiness parameter, permeability parameter, magnetic parameter, radiation parameter, Prandtl number, heat generation/absorption parameters and Eckert number for velocity and temperature distributions have been analyzed in detail through graphical representations. Further skin friction coefficient and Nusselt number at the surface are numerated and compared with previous researchers.

11 illus, 2 tables, 23 ref

MANDAVA R K, VUNDAVILLI P R

**025039** MANDAVA R K, VUNDAVILLI P R (School of Mechanical Sciences, Bhubaneswar- 751013, Email: rm19@iitbbs.ac.in) : **Implementation of modified chaotic invasive weed optimization algorithm for optimizing the PID controller of the biped robot.** Sadhana 2018, 43(5), 66.

The present research work involves the implementation of Modiﬁed Chaotic Invasive Weed Optimization (MCIWO) algorithm for optimizing the gains of torque based proportional integral and derivative (PID) controller used to control the motors of the biped robot while walking on ﬂat surface. While designing the controller, the dynamics of the biped robot has been derived using the well-known Lagrange-Euler (L-E) formulation. Subsequently, manual tuning procedure is employed to ﬁnd the ranges of the gains of PID controller used in the developed algorithm. Once it is optimized, the effectiveness of the proposed algorithm is then compared with the Differential Evolution (DE) algorithm, in terms of variation of error, torque required, zero moment point (ZMP) and dynamic balance margin (DBM) of the biped robot. It has been observed that the MCIWO algorithm tuned PID controller is found to perform better than DE tuned controller. Further, the optimal gait obtained through the developed algorithm is validated by executing it on the real robot. It has been observed that the robot has successfully negotiated the ﬂat terrain with the gaits obtained by the optimal PID controller.

14 illus, 5 tables, 45 ref

GUPTA A K, KUMAR D, SAMUAL P

**025038** GUPTA A K, KUMAR D, SAMUAL P (Motilal Nehru National Institute of Technology, Allahabad - 211004, Email: akhileshgupta08@gmail.com) : **A meta-heuristic cuckoo search and eigen permutation approach for model order reduction.** Sadhana 2018, 43(5), 65.

In this paper, a new model order reduction technique is presented by combining the beneﬁts of the meta-heuristic cuckoo search optimization and Eigen permutation methods for order reduction of higher order continuous-time systems. In the proposed approach, the numerator and the denominator polynomials of reduced order model are determined by Cuckoo search and Eigen permutation approaches, respectively. The proposed approach preserves the stability of the original system into the lower order model as the Eigen permutation retains the dominant pole with simultaneous cluster formation of the remaining real and complex poles. The effectiveness of the proposed method is validated by single-input single-output and multiple-inputs multipleoutputs numerical examples.

8 illus, 8 tables, 52 ref

GHOSH U, BANERJEE J, SARKAR S, DAS S

**025037** GHOSH U, BANERJEE J, SARKAR S, DAS S (Mathematics Dep, Calcutta Univ, Kolkata- 700 009, Email: uttam_math@yahoo.co.in) : **Fractional Klein-Gordon equation composed of Jumarie fractional derivative and its interpretation by a smoothness parameter.** Pramana J Phys 2018, 90(6), 0074.

Klein–Gordon equation is one of the basic steps towards relativistic quantum mechanics. In this paper, we have formulated fractional Klein–Gordon equation via Jumarie fractional derivative and found two types of solutions. Zero-mass solution satisﬁes photon criteria and non-zero mass satisﬁes general theory of relativity. Further, we have developed rest mass condition which leads us to the concept of hidden wave. Classical Klein– Gordon equation fails to explain a chargeless system as well as a single-particle system. Using the fractional Klein–Gordon equation, we can overcome the problem. The fractional Klein–Gordon equation also leads to the smoothness parameter which is the measurement of the bumpiness of space. Here, by using this smoothness parameter, we have deﬁned and interpreted the various cases.

4 illus, 19 ref

BHAT M A, KHAN A

**025036** BHAT M A, KHAN A (Mathematics Dep, Jamia Millia Islamia, New Delhi- 110 025, Email: mzfar012@gmail.com) : **Multiswitching combination synchronisation of non-identical fractional-order chaotic systems.** Pramana J Phys 2018, 90(6), 0073.

In this paper, multiswitching combination synchronisation (MSCS) scheme has been investigated in a class of three non-identical fractional-order chaotic systems. The fractional-order Lorenz and Chen systems are taken as the drive systems. The combination of multidrive systems is then synchronised with the fractional-order Lü chaotic system. In MSCS, the state variables of the two drive systems synchronise with different state variables of the response system, simultaneously. Based on the stability of fractional-order chaotic systems, the MSCS of three fractional-order non-identical systems has been investigated. For the synchronisation of three non-identical fractional-orderchaoticsystems,suitablecontrollershavebeendesigned.Theoreticalanalysisandnumericalresults are presented to demonstrate the validity and feasibility of the applied method.

4 illus, 39 ref

BONKILE M P, AWASTHI A, LAKSHMI C, MUKUNDAN V, V S ASWIN

**025035** BONKILE M P, AWASTHI A, LAKSHMI C, MUKUNDAN V, V S ASWIN (Mathematics Dep, National Institute of Technology, Kozhikode - 673 601, Email: aawasthi@nitc.ac.in) : **A systematic literature review of Burgers` equation with recent advances.** Pramana J Phys 2018, 90(6), 0069.

Even if numerical simulation of the Burgers’ equation is well documented in the literature, a detailed literature survey indicates that gaps still exist for comparative discussion regarding the physical and mathematical significance of the Burgers’ equation. Recently, an increasing interest has been developed within the scientific community, for studying non-linear convective–diffusive partial differential equations partly due to the tremendous improvement in computational capacity. Burgers’ equation whose exact solution is well known, is one of the famous non-linear partial differential equations which is suitable for the analysis of various important areas. A brief historical review of not only the mathematical, but also the physical significance of the solution of Burgers’ equation is presented, emphasising current research strategies, and the challenges that remain regarding the accuracy, stability and convergence of various schemes are discussed. One of the objectives of this paper is to discuss the recent developments in mathematical modelling of Burgers’ equation and thus open doors for improvement. No claim is made that the content of the paper is new. However, it is a sincere effort to outline the physical and mathematical importance of Burgers’ equation in the most simplified ways. We throw some light on the plethora of challenges which need to be overcome in the research areas and give motivation for the next breakthrough to take place in a numerical simulation of ordinary/partial differential equations.

6 illus, 210 ref

KHAN A, TYAGI A

**025034** KHAN A, TYAGI A (Mathematics Dep, Jamia Millia Islamia, New Delhi - 110 025, Email: artityagi28@gmail.com) : **Disturbance observer-based adaptive sliding mode hybrid projective synchronisation of identical fractional-order financial systems.** Pramana J Phys 2018, 90(5), 0067.

In this paper, we have studied the hybrid projective synchronisation for incommensurate, integer and commensurate fractional-order financial systems with unknown disturbance. To tackle the problem of unknown bounded disturbance, fractional-order disturbance observer is designed to approximate the unknown disturbance. Further, we have introduced simple sliding mode surface and designed adaptive sliding mode controllers incorporating with the designed fractional-order disturbance observer to achieve a bounded hybrid projective synchronisation between two identical fractional-order financial model with different initial conditions. It is shown that the slave system with disturbance can be synchronised with the projection of the master system generated through state transformation. Simulation results are presented to ensure the validity and effectiveness of the proposed sliding mode control scheme in the presence of external bounded unknown disturbance. Also, synchronisation error for commensurate, integer and incommensurate fractional-order financial systems is studied in numerical simulation.

15 illus, 63 ref

Z IQBAL, MEHMOOD Z, AHMAD B

**025033** Z IQBAL, MEHMOOD Z, AHMAD B (Mathematics Dep, HITEC Univ, Pakistan, Email: bilalahmadgondal@yahoo.com) : **Numerical study of entropy generation and melting heat transfer on MHD generalised non-Newtonian fluid (GNF): Application to optimal energy.** Pramana J Phys 2018, 90(5), 0064.

This paper concerns an application to optimal energy by incorporating thermal equilibrium on MHD- generalised non-Newtonian fluid model with melting heat effect. Highly nonlinear system of partial differential equations is simplified to a nonlinear system using boundary layer approach and similarity transformations. Numerical solutions of velocity and temperature profile are obtained by using shooting method. The contribution of entropy generation is appraised on thermal and fluid velocities. Physical features of relevant parameters have been discussed by plotting graphs and tables. Some noteworthy findings are: Prandtl number, power law index and Weissenberg number contribute in lowering mass boundary layer thickness and entropy effect and enlarging thermal boundary layer thickness. However, an increasing mass boundary layer effect is only due to melting heat parameter. Moreover, thermal boundary layers have same trend for all parameters, i.e., temperature enhances with increase in values of significant parameters. Similarly, Hartman and Weissenberg numbers enhance Bejan number.

19 illus, 2 tables, 22 ref

KHATER M M A, SEADAWY A Y R, LU D

**025032** KHATER M M A, SEADAWY A Y R, LU D (Mathematics Dep, Taibah Univ, Saudi Arabia, Email: Aly742001@yahoo.com) : **New optical soliton solutions for nonlinear complex fractional schrödinger equation via new auxiliary equation method and novel (GJ/ G)-expansion method.** Pramana J Phys 2018, 90(5), 0059.

In this research, we apply two different techniques on nonlinear complex fractional nonlinear Schrödinger equation which is a very important model in fractional quantum mechanics. Nonlinear Schrödinger equation is one of the basic models in fibre optics and many other branches of science. We use the conformable fractional derivative to transfer the nonlinear real integer-order nonlinear Schrödinger equation to nonlinear complex fractional nonlinear Schrödinger equation. We apply new auxiliary equation method and novel GJ/G -expansion method on nonlinear complex fractional Schrödinger equation to obtain new optical forms of solitary travelling wave solutions. We find many new optical solitary travelling wave solutions for this model. These solutions are obtained precisely and efficiency of the method can be demonstrated.

49 ref

RAJOTIA D

**025031** RAJOTIA D (Mathematics Dep, Rajasthan Univ, Jaipur - 302 004, Email: rajotia.dinesh@gmail.com) : **Effects of radiation on unsteady three-dimensional boundary layer flow and heat transfer over a permeable axisymmetric shrinking sheet with suction.** Indian J Pure Appl Phys 2018, 56(6), 475-82.

In the present analysis, we have investigated the effects of thermal radiation on an unsteady three-dimensional boundary layer flow of an incompressible viscous fluid and heat transfer due to a permeable axisymmetric shrinking sheet with suction and power-law variation in wall temperature. The similarity transformations have been used to convert the governing partial differential equations into non-linear ordinary differential equations and then solved numerically by shooting technique. The conditions of the existence, non-existence and duality of the similarity solution have been explored by the investigation where the solution depends not only on suction parameter but on unsteadiness parameter also. Further, it has been emerged that the range of the suction parameter S, where the similarity solution exists, is increased with the increase of unsteadiness parameter A. Also, the graphs of velocity profile, temperature profile, skin-friction coefficient and rate of heat transfer at the sheet have been drawn with various values of the parameters and then discussed these characteristic features in detail.

13 illus, 1 table, 30 ref

CHEN L, WANG Y, ZHOU Y

**025040** CHEN L, WANG Y, ZHOU Y (Mathematics Dep, Harbin Institute of Technology, Harbin-150 001, People’s Republic of China, Email: chenlili_02_06@163.com) : **Association analysis of multiple traits by an approach of combining P values.** J Genet 2018, 97(1), 79-85.

Increasing evidence shows that one variant can affect multiple traits, which is a widespread phenomenon in complex diseases. Joint analysis of multiple traits can increase statistical power of association analysis and uncover the underlying genetic mechanism. Although there are many statistical methods to analyse multiple traits, most of these methods are usually suitable for detecting common variants associated with multiple traits. However, because of low minor allele frequency of rare variant, these methods are not optimal for rare variant association analysis. In this paper, we extend an adaptive combination of P values method (termed ADA) for single trait to test association between multiple traits and rare variants in the given region. For a given region, we use reverse regression model to test each rare variant associated with multiple traits and obtain the P value of single-variant test. Further, we take the weighted combination of these P values as the test statistic. Extensive simulation studies show that our approach is more powerful than several other comparison methods in most cases and is robust to the inclusion of a high proportion of neutral variants and the different directions of effects of causal variants.

5 illus, 1 table, 37 ref

KAMOH NM, ABADA A, COMFORT S M

**025029** KAMOH NM, ABADA A, COMFORT S M (Mathematics/Statistics Dep, Bingham Univ, Nigeria) : **A block procedure with continuous coefficients for the direct solution of general second order initial value problems of (ODEs) using shifted Legendre polynomials as basis function.** Int J Multidiscip Res Dev 2018, 5(4), 236-41.

This paper presents a self starting block method for the direct solution of general second order initial value problems of ordinary differential equations. The method was developed via interpolation and collocation of the shifted Legendre polynomial as basis function. A continuous linear multistep method was generated and was evaluated at some desired points to give the discrete block method. The block method was investigated and was found to be consistent, zero stable and convergent. The method was applied on some nonlinear as well as linear ordinary differential equations problems and the performance was relatively better than those constructed by ^{[1]}, ^{[12]} and ^{[15]} respectively.

3 tables, 20 ref

CHOUDHARY R

**025028** CHOUDHARY R (Pt. Ravishankar Shukla Univ, Chhattisgarh) : **Application of "dependent t-test" by using SPSS for conducting physical education researches.** Int J Multidiscip Res Dev 2018, 5(3), 149-52.

In the field of physical education, quantitative and qualitative researches are conducted. To analyze data parametric and non-parametric statistics are used. Parametric statistics are used in case of interval and ratio scale. Dependent t-test is parametric statistics used to compare two different observations of the same group. The assumption of normality is required to apply dependent t-test. If assumption of normality is not fulfilled, option of parallel non-parametric statistics is available.

5 illus, 1 table, 7 ref

IKRAM Z J W, IRAWAN E B, DWIYANA, QOHAR A, AZIS N

**025027** IKRAM Z J W, IRAWAN E B, DWIYANA, QOHAR A, AZIS N (Mathematics Education Dep, State Univ of Malang, Malang, Indonesia) : **Development of pythagoras puzzle media to enhance understanding pythagoras theorem concept.** Int J Multidiscip Res Dev 2018, 5(3), 106-9.

The purpose of this study is to produce a valid and practical learning media in learning mathematics, especially in the material of Pythagoras theorem, to evaluate the product of the development result to be revised and test the product of the development result to know its practicality in the learning. The type of this research is Research and Development with the model used consist of several stages: initial research, development stage, and assessment phase. The result obtained is the value of validity or media feasibility obtained by analyzing the media validation sheet of 2 validators. The value of the feasibility of the Pythagoras puzzle media is 3 which means Pythagorean puzzle media is valid and feasible to use. The value of media practicability is obtained by analyzing student questionnaire. The value of practical obtained is 2.6 which means Pythagorean puzzle media is quite practical and need some revisions.

2 illus, 2 tables, 7 ref

RASYID A A, ANSYARI K, SINAR T S

**025026** RASYID A A, ANSYARI K, SINAR T S (Linguistics Dep, Sumatera Utara Univ, Indonesia) : **Error analysis of arabic numeral usage by Indonesian learners: Contrastive studies.** Int J Multidiscip Res Dev 2018, 5(3), 19-23.

Arabic numeralia (or al-'adad) is a numerical construction that has an impact on the depth of understanding to Arabic numeral contexts adjacent to the Indonesian system. Therefore, the depth is used as the target of research in order to find and analyze the errors Indonesian learners make in forming the constructions. From the counter-analysis and error analysis, there are implications of numerals to Indonesian learners. Qualitative methods are used and focused on the use of data in the form of words, phrases, sentences, drawings, or charts and quantitative methods is focused on data expressed in terms of numbers, both nominal and interval data. From the operation of the theories and methods it is found that there are errors in the numerical control of Arabic by Indonesian learners. Quantitatively, the errors are closely related to the correlation of the linguistic domain of the learners and the depth of numerical knowledge of Indonesian and Arabic languages to numerical learning in Indonesia.

12 ref

ASHMA , GOEL R, KHAN M

**025023** ASHMA , GOEL R, KHAN M (Mathematics and Statistics Dep, Chaudhary Charan Singh Haryana Agricultural Univ, Hisar, Email: mkhanstat@gmail.com) : **Newspapers reading frequency and readership pattern towards Swachh Bharat mission.** Clim Chang Environ Sustain 2018, 6(1), 72-8.

Print media is the oldest communicating tool still being used in all the life stages. The importance of print media especially newspaper can be accounted from the fact that millions of people across the country depend on the information that appears in the newspapers. Newspapers are a source of information from last many years. However, with the dissemination of television news channels, readers’ choice of getting news and information has now changed. This creation of new media has expended the horizons for news and information gatherers; therefore, the focus of this study is on newspapers reading frequency and readership pattern of respondents towards Swachh Bharat Mission among Hisar city of Haryana state. The study also identifies language preferences and time spent on reading newspapers. A sample of 100 urban respondents was interviewed randomly from two wards (50 from each ward) of the Hisar city. The results of readership behaviour signifies that majority of the respondents were having the print as well as electronic media possession, listen/watched programmes related to Swachh Bharat Mission, got new information from friends, aware about national newspapers in both Hindi and English languages, read newspapers regularly and spent 15–30 min per day in reading newspapers. Most of the respondents read newspapers for improving their knowledge and preferred Dainik Bhaskar newspaper. About one-fourth of the respondents preferred political section in the newspapers. Majority of the respondents read news related to Swachh Bharat Mission followed by campaigns.

9 tables, 22 ref

MOHANTY D J, KUMAR R S, GOSWAMI A

**025030** MOHANTY D J, KUMAR R S, GOSWAMI A (Mathematics Dep, Indian Institute of Technology, Kharagpur- 721 302, Email: dipanajyoti80@gmail.com) : **Trade-credit modeling for deteriorating item inventory system with preservation technology under random planning horizon.** Sadhana 2018, 43(3), 45.

This paper deals with a stochastic deteriorating item inventory model with preservation technology and trade-credit finance. The planning horizon of many seasonal or fashionable items stochastically varies to some extent. On the other side, the demand of such items increases in initial phase followed by a constant. This type of demand pattern can be modeled as ramp-type demand. Deterioration of such products is a characteristic, which can be reduced by making the investment on latest equipment and technology. Consumption of such items within shelf life prevents to deterioration, which can be achieved by bulk sale. In order to stimulate the selling, trade-credit policy is also considered here. In these regards, this study examines the joint effect of preservation technology investment and trade-credit policy, wherein shortage is allowed and mixture of partial backlog and lost sales. Six cases may arise depending upon three parameters of time µ (demand increases up to µ), T (when on-hand inventory reaches to zero) and M (trade-credit period). The mathematical models are mainly categorized in two cases: (i) µ ≤ M and (ii) µ > M. The model is illustrated through numerical experiments, sensitivity analysis, and graphical representation.

7 illus, 2 tables, 36 ref

WEI Z, RAJAGOPAL K, ZHANG W, KINGNI S T, AKGUL A

**025025** WEI Z, RAJAGOPAL K, ZHANG W, KINGNI S T, AKGUL A (School of Mathematics and Physics, China University of Geosciences, Wuhan- 430074, China, Email: weizhouchao@163.com) : **Synchronisation, electronic circuit implementation, and fractional-order analysis of 5D ordinary differential equations with hidden hyperchaotic attractors.** Pramana – J Phys 2018, 90(4), 50.

Hidden hyperchaotic attractors can be generated with three positive Lyapunov exponents in the proposed 5D hyperchaotic Burke–Shaw system with only one stable equilibrium. To the best of our knowledge, this feature has rarely been previously reported in any other higher-dimensional systems. Unidirectional linear error feedback coupling scheme is used to achieve hyperchaos synchronisation, which will be estimated by using two indicators: the normalised average root-mean squared synchronisation error and the maximum cross-correlation coefficient. The 5D hyperchaotic system has been simulated using a specially designed electronic circuit and viewed on an oscilloscope, thereby confirming the results of the numerical integration. In addition, fractional-order hidden hyperchaotic system will be considered from the following three aspects: stability, bifurcation analysis and FPGA implementation. Such implementations in real time represent hidden hyperchaotic attractors with important consequences for engineering applications.

17 illus, 1 table, 40 ref

CHEN Z, NING L

**025024** CHEN Z, NING L (School of Mathematics and Information Science, Shaanxi Normal Univ, Xian, China, Email: ninglijuan@snnu.edu.cn) : **Impact of depth and location of the wells on vibrational resonance in a triple-well system.** Pramana – J Phys 2018, 90(4), 49.

The effect of depth and location of a triple-well potential on vibrational resonance is investigated in a quintic oscillator driven by a low-frequency force and a high-frequency force. The values of low-frequency ω and amplitude g of the high-frequency force at which vibrational resonance occurs are derived both numerically and theoretically. It is found that: as ω varies, at most one resonance takes place and the response amplitude at resonance depends on the depth and the location of the potential wells. When g is altered, the depth and location of wells can control the number of resonances, resulting in two, three and four resonances. The system parameters can be adjusted by controlling the depth and position of the wells to achieve optimum vibrational resonance. Furthermore, the changes induced by these two quantities in the tristable system are found to be richer than those induced in bistable systems.

10 illus, 35 ref

ABRAR M N, HAQ R U, AWAIS M, RASHID I

**024166** ABRAR M N, HAQ R U, AWAIS M, RASHID I (Mathematics Dep, Capital Univ of Science and Technology, Islamabad- 44000, Email: nasirabrar10@gmail.com) : **Entropy analysis in a cilia transport of nanofluid under the influence of magnetic field.** Nucl Eng Tech 2017, 49(8), 1680- 88.

In this study, analysis is performed on entropy generation during cilia transport of water based titanium dioxide nanoparticles in the presence of viscous dissipation. Moreover, thermal heat flux is considered at the surface of a channel with ciliated walls. Mathematical formulation is constructed in the form of nonlinear partial differential equations. Making use of suitable variables, the set of partial differential equations is reduced to coupled nonlinear ordinary differential equations. Closed form exact solutions are obtained for velocity, temperature, and pressure gradient. Graphical illustrations for emerging flow parameters, such as Hartmann number (Ha), Brinkmann number (Br), radiation parameter (Rn), and flow rate, have been prepared in order to capture the physical behavior of these parameters. The main goal(i.e., the minimizing of entropy generation) of the second law of thermodynamics can be achieved by decreasing the magnitude of Br, Ha and A parameters.

9 illus, 2 tables, 36 ref

KRISHNA P M, SHARMA R P, SANDEEP N

**024165** KRISHNA P M, SHARMA R P, SANDEEP N (Mathematics Dep, VIT Univ, Vellore- 632 014, Email: dr.nsrh@gmail.com) : **Boundary layer analysis of persistent moving horizontal needle in Blasius and Sakiadis magnetohydrodynamic radiative nanofluid flows.** Nucl Eng Tech 2017, 49(8), 1654- 9.

The boundary layer of a two-dimensional forced convective flow along a persistent moving horizontal needle in an electrically conducting magnetohydrodynamic dissipative nanofluid was numerically investigated. The energy equation was constructed with Joule heating, viscous dissipation, uneven heat source/sink, and thermal radiation effects. We analyzed the boundary layer behavior of a continuously moving needle in Blasius (movingfluid) and Sakiadis (quiescentfluid) flows. We considered Cu nano-particles embedded in methanol. The reduced system of governing Partial differential equations (PDEs) was solved by employing the Rungee-Kutta-based shooting process. Computational outcomes of the rate of heat transfer and friction factors were tabulated and discussed. Velocity and temperature descriptions were examined with the assistance of graphical illustrations. Increasing the needle size did not have asignificant influence on the Blasius flow. The heat transfer rate in the Sakiadis flow was high compared with that in the Blasiusflow.

10 illus, 4 tables, 31 ref

HAYAT T, KHAN M I, WAQAS M, ALSAEDI A

**024164** HAYAT T, KHAN M I, WAQAS M, ALSAEDI A (Mathematics Dep, Quaid-I-Azam Univ, Islamabad- 44000, Pakistan, Email: jazfmg_khan@yahoo.com) : **On the performance of heat absorption/generation and thermal stratification in mixed convective flow of an Oldroyd-B fluid.** Nucl Eng Tech 2017, 49(8), 1645- 53.

This investigation explores the thermally stratified stretchable flow of an Oldroyd-B material bounded by a linear stretched surface. Heat transfer characteristics are addressed through thermal stratification and heat generation/absorption. Formulation is arranged for mixed convection. Application of suitable transformations provides ordinary differential systems through partial differential systems. The homotopy concept is adopted for the solution of nonlinear differential systems. The influence of several arising variables on velocity and temperature is addressed. Besides this, the rate of heat transfer is calculated and presented in tabular form. It is noticed that velocity and Nusselt number increase when the thermal buoyancy parameter is enhanced. Moreover, temperature is found to decrease for larger values of Prandtl number and heat absorption parameter. Comparative analysis for limiting study is performed and excellent agreement is found.

9 illus, 4 tables, 48 ref

QAYYUM S, HAYAT T, SHEHZAD S A, ALSAEDI A

**024163** QAYYUM S, HAYAT T, SHEHZAD S A, ALSAEDI A (Mathematics Dep, Quaid-I-Azam Univ, Islamabad- 44000, Email: Sajidqayyum94@gmail.com) : **Effect of a chemical reaction on magnetohydrodynamic (MHD) stagnation point flow of Walters-B nanofluid with newtonian heat and mass conditions.** Nucl Eng Tech 2017, 49(8), 1636- 44.

The main purpose of this article is to describe the magnetohydrodynamic stagnation pointflow of Walter-B nanofluid over a stretching sheet. The phenomena of heat and mass transfer are based on the involvement of thermal radiation and chemical reaction. Characteristics of Newtonian heating are given special attention. The Brownian motion and thermophoresis models are introduced in the temperature and concentration expressions. Appropriate variables are implemented for the transformation of partial differential frameworks into sets of ordinary differential equations. Plots for velocity, temperature, and nanoparticle concentration are displayed and analyzed for governing parameters. The skin friction co-efficient and local Nusselt and Sherwood numbers are studied using numerical values. The temperature and heat transfer rate are enhanced within the frame of the thermal conjugate parameter.

17 illus 4 tables, 40 ref

MAHADEVAN G, IRAVITHUL BASIRA A, SIVAGNANAM C

**024162** MAHADEVAN G, IRAVITHUL BASIRA A, SIVAGNANAM C (Mathematics Dep, Deemed Univ, Gandhigram, Dindigul, Email: drgmaha2014@gmail.com) : **Some characterization of complementary triple connected domination number of 3 ? regular graphs.** J Ind Pollut Control 2017, 33(2), 1639- 44.

A graph is said to be triple connected if any three vertices lie on a path in G. A subset S of V of a nontrivial graph G is said to be complementary triple connected dominating set, if S is a dominating set and the induced sub graph

6 ref

BABU S R R C, VENKATESWARLU S, LAKSHMI K J

**024161** BABU S R R C, VENKATESWARLU S, LAKSHMI K J (Mathematics Dep, JNTUA, Anantapur, Email: ravichandrababusr@gmail.com) : **MHD chemically reacting and radiating nanofluid flow over a vertical cone embedded in a porous medium with variable properties.** J Ultra Scient Phys Sci 2018, 30(2), 110- 20.

In this study, we examine the combined effects of thermal radiation, chemical reaction on MHD hydromagnetic boundary layer flow over a vertical cone filled with nanofluid saturated porous medium under variable properties. The governing flow, heat and mass transfer equations are transformed into ordinary differential equations using similarity variables and are solved numerically by a Galerkin Finite element method. Numerical results are obtained for dimensionless velocity, temperature, nanoparticle volume fraction. The effects of various controlling parameters on these quantities are investigated. Pertinent results are presented graphically and discussed quantitatively. The present results are compared with existing results and found to be good agreement. It is found that the temperature of the fluid remarkably enhances with the rising values of Brownian motion parameter (Nb).

19 illus, 17 ref

REDDY M G, KRISHNA M V

**024160** REDDY M G, KRISHNA M V (Mathematics Dep, Rayalaseema Univ, Andhra Pradesh- 518 007, Email: veerakrishna_maths@yahoo.com) : **Heat and mass transfer on unsteady MHD free convective flow through a porous medium over an infinite vertical plate with time dependent permeability and oscillatory suction.** J Ultra Scient Phys Sci 2017, 29(12), 515- 29.

We considered the free convective flow over an infinite vertical porous plate under the influence of uniform transverse magnate field with time dependent permeability and oscillatory suction. The equations of the flow are solved by perturbation method for small amplitude of the permeability. The closed form solutions for the velocity, temperature and concentration have been derived analytically and also its behavior is computationally discussed with reference to different flow parameters with the help of profiles. The skin fiction on the Nusselt number and Sherwood number are also attained and their behavior computationally discussed.

5 illus, 4 tables, 37 ref

JAIN S, CHOUDHARY R

**024159** JAIN S, CHOUDHARY R (Mathematics & Statistics Dep, Manipal Univ, Jaipur- 303 007, Email: shalini.jain@jaipur.manipal.edu) : **Second law analysis in MHD flow and heat transfer of a rotating Casson fluid over a stretching surface.** Indian J Pure Appl Phy 2018, 56(1), 53- 62.

Second law analysis in three-dimensional MHD boundary layer flow and heat transfer of a rotating Casson fluid over a stretching surface has been investigated. Such flow problems have relevance in extraction and manufacturing of rubber and polymer sheets. Solution of these problems is of great interest as they serve a practical purpose. Partial differential equations governing flow and heat transfer have been transformed into non-linear ordinary differential equations by using suitable similarity transformations. These non-linear differential equations have been solved numerically by using shooting techniques with fourth order Runge-Kutta method. Effects of Casson fluid parameter β, rotation parameter λ, Prandtl number Pr and magnetic field parameter M on velocity profile, temperature profile and skin friction number have been analyzed and depicted through graphs and tables. Entropy generation number and Bejan number have also been obtained and discussed.

18 illus, 5 tables, 37 ref

CHAUDHARY S, CHAUDHARY M K

**024158** CHAUDHARY S, CHAUDHARY M K (Mathematic Dep, Malaviya National Institute of Technology, Jaipur- 302 017, Email: d11.santosh@yahoo.com) : **Viscous dissipation and joule heating effects on an unsteady magnetohydrodynamic flow over a linearly stretching permeable surface with uniform wall temperature.** Indian J Pure Appl Phy 2017, 55(12), 864- 72.

An analysis has been presented to describe the effects of viscous dissipation and Joule heating on an unsteady laminar two-dimensional flow of a viscous incompressible electrically conducting fluid over a stretching permeable surface in the presence of a uniform transverse magnetic field. Similarity solutions for the problem have been formulated and reduced nonlinear ordinary differential equations have been solved numerically using fourth order Runge-Kutta method with shooting technique. Influences of various parameters, namely, mass transfer parameter, unsteadiness parameter, magnetic parameter, Prandtl number and Eckert number on velocity and temperature distributions have been plotted graphically while skin-friction coefficient and Nusselt number have been shown numerically. A comparison of the obtained numerical results has been made with previously published results for non-magnetic case.

9 illus, 3 tables, 47 ref

JAIN S, BOHRA S

**024157** JAIN S, BOHRA S (Mathematics & Statistics Dep, Manipal Univ, Jaipur- 303 007, Email: shalini.jain@jaipur.manipal.edu) : **Heat and mass transfer over a three-dimensional inclined non-linear stretching sheet with convective boundary conditions.** Indian J Pure Appl Phy 2017, 55(12), 847- 56.

This investigation has been made to study of heat and mass on three-dimensional boundary layer flow of an incompressible fluid due to inclined non-linear stretching sheet with convective boundary conditions. The effect of thermophoresis and chemical reaction has been taken into account. Numerical solution has been obtained for the set of non-linear coupled ordinary equations which are reduced by using similarity transformations from non-linear partial differential equations. Transforms system of equations has been solved using Runge–Kutta–Fehlberg fourth–fifth order method along with shooting technique. Effects of various parameters on velocity, temperature and concentration profiles have been obtained. These results have been presented through graphs and also our analysis explores the physical quantities of interest through the tables.

9 illus, 3 tables, 39 ref

LIANG Q, WANG D

**024156** LIANG Q, WANG D (Northwestern Polytechnical Univ, Xi’an Shaanxi- 710 072, Email: liangqingwei@nwpu.edu.cn) : **Influence study of Weibull distribution parameter on double stress down-step accelerated life test.** Indian J Geo-Mar Sci 2018, 47(01), 46-52.

Based on the study of time and efficiency of double step-down-stress accelerated life test, the influence of the change of the shape parameters of Weibull distribution on the time and efficiency of the double step-down-stress accelerated life test is studied. Combined with Matlab software, Monte Carlo method is used. The simulation of different shape parameter of Weibull distribution is carred out. Results show that with the increase of sample size, the failure time curve of Weibull distribution stably intersects in 60 %-70 % of the sample size. When shape parameter m is relatively small, small failure sample number of double step-down-stress accelerated life test can be choosen to carry out truncation test, which will help to shorten test time. When shape parameter m is relatively large, the large failure sample number of double step-down-stress accelerated life test can be choosen to carry out truncation test. This conclusion can also be applied to all double stress accelerated life test.

4 illus, 2 tables, 21 ref

ABOUELREGAL A E, ABO-DAHAB S M

**024155** ABOUELREGAL A E, ABO-DAHAB S M (Mathematics Dep, Mansoura Univ, Mansoura- 35516, Email: ahabogal@mans.edu.eg) : **A two-dimensional problem of a mode-I crack in a rotating fibre-reinforced isotropic thermoelastic medium under dual-phase-lag model.** Sadhana 2018, 43(1), 13.

In the present work, a general model of the equations of generalized thermoelasticity for a homogeneous isotropic elastic half-space solid whose surface is subjected to a mode-I crack problem under the effect of rotation is investigated. The normal mode analyses are used to obtain the expressions for the temperature distribution, the displacement component and thermal stresses in the context of the dual-phase-lag theory of thermoelasticity proposed by Tzou. The boundary of the crack is subjected to a prescribed stress distribution and temperature. Some particular cases are also discussed in the context of the problem. The numerical values of the temperature distribution, the displacement components and thermal stresses are also computed for a suitable material and the results are presented graphically. The effects of rotation, reinforcement and the phase lags parameters are discussed in detail in the light of earlier works.

13 illus, 46 ref

DAS P, AKHTER A, SINGH M K

**024154** DAS P, AKHTER A, SINGH M K (Applied Mathematics Dep, Indian Institute of Technology, Dhanbad-826 004, Email: pintudas62@gmail.com) : **Solute transport modelling with the variable temporally dependent boundary.** Sadhana 2018, 43(1), 12.

In this present study, analytical and numerical solutions are obtained for solute transport modelling in homogeneous semi-infinite porous medium. The dispersion coefficient is assumed to be initial dispersion and velocity is assumed to be temporally dependent with initial seepage velocity. Also, the concept of dispersion is directly proportional to the square of the seepage velocity used for finding the solution. Initially, the domain is not solute free. At one end of the domain, source concentration with the effect of different temporally dependent functions taken into account. The concentration gradient assumed to be zero due to no mass flux at other end of the domain. Laplace Transform Technique is used to obtain the exact solution, whereas Explicit Finite Difference method is used for approximate solution. The different types of temporally dependent velocity are used for the graphical representation of the solution. The accuracy of the solution explored by the Relative error analysis.

16 illus, 6 tables, 21 ref

VIJAYASEKHAR J, RAO M V S, SREERAM V

**024153** VIJAYASEKHAR J, RAO M V S, SREERAM V (Mathematics Dep, GITAM Univ, Hyderabad, Email: vijayjaliparthi@gmail.com) : **Vibrational spectra of copper tetramesityl porphyrin using Vibron model.** Oriental J Chem 2018, 34(1), 559-61.

In this paper we calculated the fundamental level vibrational spectra of Metalloporphyrin (bio molecule) copper tetramesityl porphyrin (Cu(TMP)) using Vibron model.

1 illus, 2 tables, 11 ref

SHARMA V, SINGHAL A

**024152** SHARMA V, SINGHAL A (Mathematics Dep, N.A.S. (P.G.) Coll, Uttar Pradesh- 250 002., Email: vireshsharma1@yahoo.com) : **Evaluation of M.T.T.F. and variance of a k-out of N:G redundant complex system under waiting for repair.** Bull Pure Appl Sci- Sect E 2018, 37E(1), 254-9.

This paper presents the investigations for the evaluation of reliability behavior of a k-out of N:G redundant complex system with r repair facilities incorporating the concept of waiting.

3 illus, 5 ref

SUVARNA T, REDDY A M

**024151** SUVARNA T, REDDY A M (Mathematics Dep, SK Univ, Andhra Pradesh- 515 003., Email: suvarna8886@gmail.com) : **Reliability analysis of six unit bridge and parallel series networks with critical and non-critical human errors.** Bull Pure Appl Sci- Sect E 2018, 37E(1), 245-53.

Humans play a pivotal role in the design, development and operational phases of engineering systems. Reliability evaluation of systems without taking into consideration the human element does not provide a realistic picture. Hence, there is a definite need for incorporating the occurrence of human errors in system reliability evaluation. This paper presents reliability analysis of bridge and parallel-series networks with critical and non-critical Human errors. A newly developed approach is used to perform the system reliability analysis. The approach demonstrated in this paper, which consists of six unit bridge network, is a modified version of the block diagram approach.

6 illus, 1 table, 4 ref

RAMESHKUMAR A, KAVITHA G

**024150** RAMESHKUMAR A, KAVITHA G (Mathematics Dep, Srimad Andavan Arts and Science Coll, Tamilnadu-620 005, Email: andavanmathsramesh@gmail.com) : **Two finite simple directed graphs.** Bull Pure Appl Sci- Sect E 2018, 37E(1), 223-9.

The main result of the paper is that in practically interesting situations determining the optimal number of colors in the proposed coloring is an NP-hard problem. A possible conclusion to draw from this result is that for practical transitive tournament search algorithms we have to develop approximate greedy coloring algorithms. It is an empirical fact that coloring the nodes of a graph can be used to speed up clique search algorithms. In directed graphs transitive sub tournaments can play the role of cliques. In order to speed up algorithms to locate large transitive tournaments we propose a scheme for coloring the nodes of a directed graph.

3 illus, 2 illus, 10 ref

SHARMA V, SHARMA R K, SINGHAL A

**024149** SHARMA V, SHARMA R K, SINGHAL A (Mathematics Dep, N.A.S. (P.G.) Coll, Uttar Pradesh-250 003., Email: vireshsharma1@yahoo.com) : **Reliability behaviour of a two-state stand by redundant power plant under arbitrary failure time distribution.** Bull Pure Appl Sci- Sect E 2018, 37E(1), 200-7.

This paper presents the investigations for the evaluation of reliability of the power supply having been carried out with the help of Boolean function orthogonalization algorithm. The motive of the complex system is to supply from MPH to several critical consumers.

3 illus, 2 tables, 4 ref

RANI M, HOODA N

**024148** RANI M, HOODA N (Mathematics Dep, Deenbandhu Chhotu Ram Univ of Science and Technology, Haryana-131 039, Email: manjuantil7@gmail.com) : **Some fixed point theorems for mappings on complete Sb?Metric space.** Bull Pure Appl Sci- Sect E 2018, 37E(1), 178-83.

In this paper, we prove some common fixed point theorems in S_{b}-metric space using an increasing function ϕ : [0, ∞) → [0, ∞) with lim_{n→∞} ϕ^{n}(t) = 0 and ϕ(t) < t for each fixed > 0. Our results extend the result of Savitri & Hooda [8] of S-metric space.

10 ref

SATYANARAYANA B, KUMAR Y P, PURNIMA B V

**024147** SATYANARAYANA B, KUMAR Y P, PURNIMA B V (Mathematics Dep, Acharya Nagarjuna Univ, Nagarjuna Nagar- 522 510, Email: drbsn63@yahoo.co.in) : **On Euler-Beta transform of I-function of two variables.** Bull Pure Appl Sci- Sect E 2018, 37E(1), 171-7.

In this paper, the authors establish the Euler-Beta transform of various products involving a General Class of Polynomials, Struve’s function and I-function of two variables. Some similar formulae are also derived as special cases.

11 ref

MADAN D K, GABA A, RAGHAV J S

**024146** MADAN D K, GABA A, RAGHAV J S (Mathematics Dep, Ch. Bansilal Univ, Haryana-127 021, Email: dk_madaan@rediffmail.com) : **Strike-Slip fault model with linear and parabolic dislocations with width-depth ratio.** Bull Pure Appl Sci- Sect E 2018, 37E(1), 157-70.

This paper tackles the problem of static deformation of an orthotropic elastic layer of uniform thickness overline an orthotropic elastic half-space. This is caused by a very long vertical strikeslip fault in the layer with rigid surface and this problem is solved analytically. The effect of ratio of width and depth of the faults has been studied for parabolic and linear slip profiles by varying the horizontal dimensionless displacements and by depicting contour maps for the stresses

2 illus, 12 ref

MAHTO G

**024145** MAHTO G (S N Sinha Coll, Bihar- 805 120, Email: ganeshmahto0201@gmail.com) : **Effect of radiation and oblateness on the equilibrium points in the elliptic restricted three body problem.** Bull Pure Appl Sci- Sect E 2018, 37E(1), 152-6.

In this paper we have obtained the location of equilibrium points in the elliptic restricted three body problem. We suppose that the bigger primary is oblate and the smaller primary as radiating. We find the positions of triangular equilibrium points in our problem and note that the positions of triangular points are shifted away from the line joining the primaries than in the classical case. Thus the positions are affected due to introduction of eccentricity, semi-major axis, radiation and oblateness factors of both primaries.

25 ref

RATH P, DASH R B, GHOSH S K

**024144** RATH P, DASH R B, GHOSH S K (Mathematics Dep, Ravenshaw Univ, Cuttack- 753 003, Email: priyadarsinirath0@gmail.com) : **Solution of fuzzy multi-objective fractional linear programming problem using fuzzy programming technique based on exponential membership function.** Bull Pure Appl Sci- Sect E 2018, 37E(1), 109-16.

A method of solving multi-objective fuzzy fractional linear programming problem (MOFFLPP) is presented. All the coefficients of the fractional multi-objective functions and the constraints are taken to be fuzzy numbers. In this paper a MOFFLPP is first reduced to crisp multi-objective linear programming problem and then the crisp MOLPP is solved by Zimmerman’s technique using exponential membership functions.

17 ref

JAIN D

**024143** JAIN D (Mathematics Dep, Deenbandhu Chhoturam Univ of Science and Technology, Haryana- 131 039., Email: deepakjain.jain6@gmail.com) : **Expansive mappings in digital metric spaces.** Bull Pure Appl Sci- Sect E 2018, 37E(1), 101-8.

In this paper, we prove some common fixed point theorems for expansive mappings in digital metric spaces.

4 illus, 16 ref

DHUMAL M L, KIWNE S B

**024142** DHUMAL M L, KIWNE S B (Mathematics Dep, Deogiri Coll, Maharashtra- 431 005, Email: mldhumal19@gmail.com) : **Numerical treatment of fisher?s equation using finite difference method.** Bull Pure Appl Sci- Sect E 2018, 37E(1), 94-100.

A numerical method is proposed to approximate the solutions of nonlinear Fisher reaction diffusion equation using finite difference method. Fisher’s equation describes the process of interaction between diffusion and reaction. The method is based on replacing each term in the Fisher’s equation using finite difference method. Fisher equation has many applications in science and engineering fields. In this paper we study Du-Fort Frankel finite difference method for Fisher’s equation. This method is derived from Leapfrog explicit scheme.

1 table, 15 ref

RAMESHKUMAR A, STANLY L A J

**024141** RAMESHKUMAR A, STANLY L A J (Mathematics Dep, Srimad Andavan Arts & Science Coll, Tamil Nadu- 620 005., Email: andavanmathsramesh@gmail.com) : **Application of magneto-Williamson model in nanofluid flow.** Bull Pure Appl Sci- Sect E 2018, 37E(1), 82-93.

This paper investigates the effect of the nano particle effect on magneto hydrodynamic boundary layer over a stretching in the current of heat generation or absorption with heat and mass fluxes. The governing partial differential equations of the system are transformed into a system of ordinary differential equations and solved numerically with the effect of the non-Newtonian Williamson parameter, heat capacities ratio parameter, heat generation or absorption parameter, Schmidt number on the fluid properties as well as on the skin friction, heat and mass transfer rates are described and explained in detailed through graph.

8 illus, 16 ref

AHMED N

**024140** AHMED N (Statistics Dep, D.H.S.K. Coll, Assam- 786 001, Email: mdnahmed@yahoo.com) : **Restricted constraints in a max-flow and min-cost problem.** Bull Pure Appl Sci- Sect E 2018, 37E(1), 63-81.

The present paper defines a combination of two most important and old flow problems such as maximum flow and minimum cost flow. In the first one a flow with the maximum value from source node to sink node is sought. The second one seeks a flow with the minimum total transferring costs from source nodes to demand nodes. Here an attempt is made to obtain an optimal route of a more realistic situation as to scheduling some restricted constraint, in which its transferring cost is minimized and its value is maximized. The proposed algorithm is formulated and solved by the lexicographic search approach. It is seen that the time required for the search of the optimal solution is fairly less. Also the algorithm is applied to different order matrices with number of stations = 5, 8, 10, … , 50 and in the dimensions of 6 to exhibit its effectiveness.

9 illus, 17 tables, 26 ref