Search results for: hybrid multistep method

Authors: Adamu S. Salawu, Ibrahim O. Isah

Abstract:

Direct solution to several forms of fourth-order ordinary differential equations is not easily obtained without first reducing them to a system of first-order equations. Thus, numerical methods are being developed with the underlying techniques in the literature, which seeks to approximate some classes of fourth-order initial value problems with admissible error bounds. Multistep methods present a great advantage of the ease of implementation but with a setback of several functions evaluation for every stage of implementation. However, hybrid methods conventionally show a slightly higher order of truncation for any k-step linear multistep method, with the possibility of obtaining solutions at off mesh points within the interval of solution. In the light of the foregoing, we propose the continuous form of a hybrid multistep method with Chebyshev polynomial as a basis function for the numerical integration of fourth-order initial value problems of ordinary differential equations. The basis function is interpolated and collocated at some points on the interval [0, 2] to yield a system of equations, which is solved to obtain the unknowns of the approximating polynomial. The continuous form obtained, its first and second derivatives are evaluated at carefully chosen points to obtain the proposed block method needed to directly approximate fourth-order initial value problems. The method is analyzed for convergence. Implementation of the method is done by conducting numerical experiments on some test problems. The outcome of the implementation of the method suggests that the method performs well on problems with oscillatory or trigonometric terms since the approximations at several points on the solution domain did not deviate too far from the theoretical solutions. The method also shows better performance compared with an existing hybrid method when implemented on a larger interval of solution.

Keywords: Chebyshev polynomial, collocation, hybrid multistep method, initial value problems, interpolation

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Authors: A. A. James, A. O. Adesanya, M. R. Odekunle, D. G. Yakubu

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This paper examines the development of one step, five hybrid point method for the solution of first order initial value problems. We adopted the method of collocation and interpolation of power series approximate solution to generate a continuous linear multistep method. The continuous linear multistep method was evaluated at selected grid points to give the discrete linear multistep method. The method was implemented using a constant order predictor of order seven over an overlapping interval. The basic properties of the derived corrector was investigated and found to be zero stable, consistent and convergent. The region of absolute stability was also investigated. The method was tested on some numerical experiments and found to compete favorably with the existing methods.

Keywords: interpolation, approximate solution, collocation, differential system, half step, converges, block method, efficiency

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Authors: James Adewale, Joshua Sunday

Abstract:

In this paper, we developed a linear multistep method, which is implemented in predictor corrector-method. The corrector is developed by method of collocation and interpretation of power series approximate solutions at some selected grid points, to give a continuous linear multistep method, which is evaluated at some selected grid points to give a discrete linear multistep method. The predictors were also developed by method of collocation and interpolation of power series approximate solution, to give a continuous linear multistep method. The continuous linear multistep method is then solved for the independent solution to give a continuous block formula, which is evaluated at some selected grid point to give discrete block method. Basic properties of the corrector were investigated and found to be zero stable, consistent and convergent. The efficiency of the method was tested on some linear, non-learn, oscillatory and stiff problems of first order, initial value problems of ordinary differential equations. The results were found to be better in terms of computer time and error bound when compared with the existing methods.

Keywords: predictor, corrector, collocation, interpolation, approximate solution, independent solution, zero stable, consistent, convergent

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Authors: A. M. Sagir

Abstract:

Discrete linear multistep block method of uniform order for the solution of first order Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs) is presented in this paper. The approach of interpolation and collocation approximation are adopted in the derivation of the method which is then applied to first order ordinary differential equations with associated initial conditions. The continuous hybrid formulations enable us to differentiate and evaluate at some grids and off – grid points to obtain four discrete schemes, which were used in block form for parallel or sequential solutions of the problems. Furthermore, a stability analysis and efficiency of the block method are tested on ordinary differential equations, and the results obtained compared favorably with the exact solution.

Keywords: block method, first order ordinary differential equations, hybrid, self-starting

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Authors: Y. A. Yahaya, Ahmad Tijjani Asabe

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This paper focuses on the formulation of 3-step hybrid Adams type method for the solution of first order differential equation (ODE). The methods which was derived on both grid and off grid points using multistep collocation schemes and also evaluated at some points to produced Block Adams type method and Adams moulton method respectively. The method with the highest order was selected to serve as the corrector. The convergence was valid and efficient. The numerical experiments were carried out and reveal that hybrid Adams type methods performed better than the conventional Adams moulton method.

Keywords: adam-moulton type (amt), corrector method, off-grid, block method, convergence analysis

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Authors: A. M. Sagir

Abstract:

The aim of this paper is to investigate the performance of the developed linear multistep block method for solving first order initial value problem of Ordinary Differential Equations (ODEs). The method calculates the numerical solution at three points simultaneously and produces three new equally spaced solution values within a block. The continuous formulations enable us to differentiate and evaluate at some selected points to obtain three discrete schemes, which were used in block form for parallel or sequential solutions of the problems. A stability analysis and efficiency of the block method are tested on ordinary differential equations involving practical applications, and the results obtained compared favorably with the exact solution. Furthermore, comparison of error analysis has been developed with the help of computer software.

Keywords: block method, first order ordinary differential equations, linear multistep, self-starting

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Authors: A. M. Sagir

Abstract:

This paper derived four newly schemes which are combined in order to form an accurate and efficient block method for parallel or sequential solution of third order ordinary differential equations of the form y^'''= f(x,y,y^',y^'' ), y(α)=y_0,〖y〗^' (α)=β,y^('' ) (α)=μ with associated initial or boundary conditions. The implementation strategies of the derived method have shown that the block method is found to be consistent, zero stable and hence convergent. The derived schemes were tested on stiff and non-stiff ordinary differential equations, and the numerical results obtained compared favorably with the exact solution.

Keywords: block method, hybrid, linear multistep, self-starting, third order ordinary differential equations

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Authors: Hudson Akewe

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We prove that the convergence of explicit Mann, explicit Ishikawa, explicit Noor, explicit SP, explicit multistep and explicit multistep-SP fixed point iterative procedures are equivalent for the classes of multi-valued phi-contraction, phi-Zamfirescu and phi-quasi-contractive mappings in the framework of modular function spaces. Our results complement equivalence results on normed and metric spaces in the literature as they elegantly cut out the triangle inequality.

Keywords: multistep iterative procedures, multivalued mappings, equivalence results, fixed point

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Authors: T. A. Biala

Abstract:

This paper deals with the numerical integration of the general second order multipoint boundary value problems. This has been achieved by the development of a continuous linear multistep method (LMM). The continuous LMM is used to construct a main discrete method to be used with some initial and final methods (also obtained from the continuous LMM) so that they form a discrete analogue of the continuous second order boundary value problems. These methods are used as boundary value methods and adapted to cope with the integration of the general second order multipoint boundary value problems. The convergence, the use and the region of absolute stability of the methods are discussed. Several numerical examples are implemented to elucidate our solution process.

Keywords: linear multistep methods, boundary value methods, second order multipoint boundary value problems, convergence

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Authors: Mohamed Suleiman, Zarina Bibi Ibrahim, Nor Ain Azeany, Khairil Iskandar Othman

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In this paper, a class of variable order fully implicit multistep Block Backward Differentiation Formulas (VOBBDF) using uniform step size for the numerical solution of stiff ordinary differential equations (ODEs) is developed. The code will combine three multistep block methods of order four, five and six. The order selection is based on approximation of the local errors with specific tolerance. These methods are constructed to produce two approximate solutions simultaneously at each iteration in order to further increase the efficiency. The proposed VOBBDF is validated through numerical results on some standard problems found in the literature and comparisons are made with single order Block Backward Differentiation Formula (BBDF). Numerical results shows the advantage of using VOBBDF for solving ODEs.

Keywords: block backward differentiation formulas, uniform step size, ordinary differential equations

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Authors: J. P. Chollom, G. M. Kumleng, S. Longwap

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The search for higher order A-stable linear multi-step methods has been the interest of many numerical analysts and has been realized through either higher derivatives of the solution or by inserting additional off step points, supper future points and the likes. These methods are suitable for the solution of stiff differential equations which exhibit characteristics that place a severe restriction on the choice of step size. It becomes necessary that only methods with large regions of absolute stability remain suitable for such equations. In this paper, high order block implicit multi-step methods of the hybrid form up to order twelve have been constructed using the multi-step collocation approach by inserting one or more off step points in the multi-step method. The accuracy and stability properties of the new methods are investigated and are shown to yield A-stable methods, a property desirable of methods suitable for the solution of stiff ODE’s. The new High Order Block Implicit Multistep methods used as block integrators are tested on stiff differential systems and the results reveal that the new methods are efficient and compete favourably with the state of the art Matlab ode23 code.

Keywords: block linear multistep methods, high order, implicit, stiff differential equations

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Authors: Mahmood Otadi, Maryam Mosleh

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The hybrid differential equations have a wide range of applications in science and engineering. In this paper, the homotopy analysis method (HAM) is applied to obtain the series solution of the hybrid differential equations. Using the homotopy analysis method, it is possible to find the exact solution or an approximate solution of the problem. Comparisons are made between improved predictor-corrector method, homotopy analysis method and the exact solution. Finally, we illustrate our approach by some numerical example.

Keywords: fuzzy number, fuzzy ODE, HAM, approximate method

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Authors: Mahdi Kiani, Hassan Salarieh, Aria Alasty, S. Mahdi Darbandi

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The design and implementation of the hybrid control method for a three-pole active magnetic bearing (AMB) is proposed in this paper. The system is inherently nonlinear and conventional nonlinear controllers are a little complicated, while the proposed hybrid controller has a piecewise linear form, i.e. linear in each sub-region. A state-feedback hybrid controller is designed in this study, and the unmeasurable states are estimated by an observer. The gains of the hybrid controller are obtained by the Linear Quadratic Regulator (LQR) method in each sub-region. To evaluate the performance, the designed controller is implemented on an experimental setup in static mode. The experimental results show that the proposed method can efficiently stabilize the three-pole AMB system. The simplicity of design, domain of attraction, uncomplicated control law, and computational time are advantages of this method over other nonlinear control strategies in AMB systems.

Keywords: active magnetic bearing, three pole AMB, hybrid control, Lyapunov function

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Authors: M. Y. Waziri, M. A. Aliyu

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The paper proposes an approach to improve the performance of Broyden’s method for solving systems of nonlinear equations. In this work, we consider the information from two preceding iterates rather than a single preceding iterate to update the Broyden’s matrix that will produce a better approximation of the Jacobian matrix in each iteration. The numerical results verify that the proposed method has clearly enhanced the numerical performance of Broyden’s Method.

Keywords: mulit-step Broyden, nonlinear systems of equations, computational efficiency, iterate

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Authors: Miriam Eduful

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The Electricity Company of Ghana (ECG) is under obligation, demanded by the Public Utility and Regulation Commission to meet set performance indices. However, in certain parts of the country, bamboo related power interruptions have become a challenge. Growth rate of the bamboo is such that the cost of regular vegetation maintenance along route of the overhead power lines has become prohibitive. To address the problem, several methods and techniques of bamboo eradication have being used. Some of these methods involved application of chemical compounds that are considered inimical and dangerous to the environment. In this paper, three methods of bamboo eradication along the route of the ECG overhead power lines have been investigated. A hybrid method has been found to be very effective and ecologically friendly. The method is locally available and comparatively inexpensive to apply.

Keywords: bamboo, eradication, hybrid method, gly gold

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Authors: Jaechun No

Abstract:

This paper focuses on I/O optimizations of N-hybrid (New-Form of hybrid), which provides a hybrid file system space constructed on SSD and HDD. Although the promising potentials of SSD, such as the absence of mechanical moving overhead and high random I/O throughput, have drawn a lot of attentions from IT enterprises, its high ratio of cost/capacity makes it less desirable to build a large-scale data storage subsystem composed of only SSDs. In this paper, we present N-hybrid that attempts to integrate the strengths of SSD and HDD, to offer a single, large hybrid file system space. Several experiments were conducted to verify the performance of N-hybrid.

Keywords: SSD, data section, I/O optimizations, hybrid system

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Authors: F. J. Adeyeye, D. Eni, K. M. Okedoye

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In this paper we present a Hybrid of Adomian decomposition method (ADM). This is the substitution of a One-step method of Taylor’s series approximation of orders I and II, into the nonlinear part of Adomian decomposition method resulting in a convergent series scheme. This scheme is applied to solve some Logistic problems represented as Abelian differential equation and the results are compared with the actual solution and Runge-kutta of order IV in order to ascertain the accuracy and efficiency of the scheme. The findings shows that the scheme is efficient enough to solve logistic problems considered in this paper.

Keywords: Adomian decomposition method, nonlinear part, one-step method, Taylor series approximation, hybrid of Adomian polynomial, logistic problem, Malthusian parameter, Verhulst Model

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Authors: Melody Yin

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Concussions impact a large amount of adolescents; they make up as much as half of the diagnosed concussions in America. This research proposes a hybrid machine learning model based on the combination of human/knowledge-based domains and computer-generated feature rankings to improve the accuracy of diagnosing sports related concussion (SRC). Using a data set of symptoms collected on the sideline post-SRC events, the symptom selection criteria method has been developed by using Google AutoML's important score function to identify the top 10 symptom features. In addition, symptom domains have been introduced as another parameter, categorizing the symptoms into physical, cognitive, sleep, and emotional domains. The hybrid machine learning model has been trained with a combination of the top 10 symptoms and 4 domains. From the results, the hybrid model was the best performer for symptom resolution time prediction in 2 and 4-week thresholds. This research is a proof of concept study in the use of domains along with machine learning in order to improve concussion prediction accuracy. It is also possible that the use of domains can make the model more efficient due to reduced training time. This research examines the use of a hybrid method in predicting sports-related concussion. This achievement is based on data preprocessing, using a hybrid method to select criteria to achieve high performance.

Keywords: hybrid model, machine learning, sports related concussion, symptom resolution time

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Authors: M. N. Ervina Efzan, H. J. Kong, C. K. Kok

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The present work deals with a study on the influences of hybrid modifier on LM 6 added through ladle metallurgy. In this study, LM 6 served as the reference alloy while Na2CO3 and Ti powders were used as the hybrid modifier. The effects of hybrid modifier on the micro structural enhancement of LM 6 were investigated using optical microscope (OM) and Scanning Electron Microscope (SEM). The results showed fragmented Si-rich needles and strength enhanced petal/ globular-like structures without obvious formation of soft primary α-Al and β-Fe-rich inter metallic compound (IMC) after the hybrid modification. Hardness test was conducted to examine the mechanical improvement of hybrid modified LM 6. 10% of hardness improvement was recorded in the hybrid modified LM 6 through ladle metallurgy.

Keywords: Al-Si, hybrid modifier, ladle metallurgy, hardness

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Authors: Atilla Bayram

Abstract:

This paper presents the trajectory tracking control of a spatial redundant hybrid manipulator. This manipulator consists of two parallel manipulators which are a variable geometry truss (VGT) module. In fact, each VGT module with 3-degress of freedom (DOF) is a planar parallel manipulator and their operational planes of these VGT modules are arranged to be orthogonal to each other. Also, the manipulator contains a twist motion part attached to the top of the second VGT module to supply the missing orientation of the endeffector. These three modules constitute totally 7-DOF hybrid (parallel-parallel) redundant spatial manipulator. The forward kinematics equations of this manipulator are obtained, then, according to these equations, the inverse kinematics is solved based on an optimization with the joint limit avoidance. The dynamic equations are formed by using virtual work method. In order to test the performance of the redundant manipulator and the controllers presented, two different desired trajectories are followed by using the computed force control method and a switching control method. The switching control method is combined with the computed force control method and genetic algorithm. In the switching control method, the genetic algorithm is only used for fine tuning in the compensation of the trajectory tracking errors.

Keywords: computed force method, genetic algorithm, hybrid manipulator, inverse kinematics of redundant manipulators, variable geometry truss

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Authors: Atul K. Desai, Hemal J. Shah

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The tall windmill towers are designed as monopole tower or lattice tower. In the present research, a 125-meter high hybrid tower which is a combination of lattice and monopole type is proposed. The response of hybrid tower is compared with conventional monopole tower. The towers were analyzed in finite element method software considering nonlinear seismic time history load. The synthetic seismic time history for different soil is derived using the SeismoARTIF software. From the present research, it is concluded that, in the hybrid tower, we are not getting resonance condition. The base shear is less in hybrid tower compared to monopole tower for different soil conditions.

Keywords: dynamic analysis, hybrid wind mill tower, resonance condition, synthetic time history

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Authors: P.Kiruthika, K. Ramani

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This paper presented the design and advancement of a single-phase 27-level Hybrid Multilevel DC-AC Converter interfacing with Photo Voltaic. In this context, the Multicarrier Pulse Width Modulation method can be implemented in 27-level Hybrid Multilevel Inverter for generating a switching pulse. Perturb & Observer algorithm can be used in the Maximum Power Point Tracking method for the Photo Voltaic system. By implementing Maximum Power Point Tracking with three separate solar panels as an input source to the 27-level Hybrid Multilevel Inverter. This proposed method can be simulated by using MATLAB/simulink. The result shown that the proposed method can achieve silky output wave forms, more flexibility in voltage range, and to reduce Total Harmonic Distortion in medium-voltage drives.

Keywords: Multi Carrier Pulse Width Modulation Technique (MCPWM), Multi Level Inverter (MLI), Maximum Power Point Tracking (MPPT), Perturb and Observer (P&O)

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Authors: Maryam Mosleh, Mahmood Otadi

Abstract:

The hybrid differential equations have a wide range of applications in science and engineering. In this paper, the homotopy analysis method (HAM) is applied to obtain the series solution of the hybrid differential equations. Using the homotopy analysis method, it is possible to find the exact solution or an approximate solution of the problem. Comparisons are made between improved predictor-corrector method, homotopy analysis method and the exact solution. Finally, we illustrate our approach by some numerical example.

Keywords: Fuzzy number, parametric form of a fuzzy number, fuzzy integrodifferential equation, homotopy analysis method

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Authors: Heri Suryoatmojo, Adi Kurniawan, Feby A. Pamuji, Nursalim, Syaffaruddin, Herbert Innah

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Photovoltaic (PV) system hybrid with diesel system is utilized widely for electrification in remote area. PV output power fluctuates due to uncertainty condition of temperature and sun irradiance. When the penetration of PV power is large, the reliability of the power utility will be disturbed and seriously impact the unstable frequency of system. Therefore, designing a robust frequency controller in PV-diesel hybrid power system is very important. This paper proposes new method of frequency control application in hybrid PV-diesel system based on artificial neural network (ANN). This method can minimize the frequency deviation without smoothing PV output power that controlled by maximum power point tracking (MPPT) method. The neural network algorithm controller considers average irradiance, change of irradiance and frequency deviation. In order the show the effectiveness of proposed algorithm, the addition of battery as energy storage system is also presented. To validate the proposed method, the results of proposed system are compared with the results of similar system using MPPT only. The simulation results show that the proposed method able to suppress frequency deviation smaller compared to the results of system using MPPT only.

Keywords: energy storage system, frequency deviation, hybrid power generation, neural network algorithm

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Authors: Manoj Reghu, Prabhu Rajagopal, C. V. Krishnamurthy, Krishnan Balasubramaniam

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A thorough understanding of guided ultrasonic wave behavior in structures is essential for the application of existing Non Destructive Evaluation (NDE) technologies, as well as for the development of new methods. However, the analysis of guided wave phenomena is challenging because of their complex dispersive and multimodal nature. Although numerical solution procedures have proven to be very useful in this regard, the increasing complexity of features and defects to be considered, as well as the desire to improve the accuracy of inspection often imposes a large computational cost. Hybrid models that combine numerical solutions for wave scattering with faster alternative methods for wave propagation have long been considered as a solution to this problem. However usually such models require modification of the base code of the solution procedure. Here we aim to develop Generic Hybrid models that can be directly applied to any two different solution procedures. With this goal in mind, a Numerical Hybrid model and an Analytical-Numerical Hybrid model has been developed. The concept and implementation of these Hybrid models are discussed in this paper.

Keywords: guided ultrasonic waves, Finite Element Method (FEM), Hybrid model

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Authors: Ahcene Habbi, Yassine Boudouaoui

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This paper deals with the problem of automatic rule generation for fuzzy systems design. The proposed approach is based on hybrid artificial bee colony (ABC) optimization and weighted least squares (LS) method and aims to find the structure and parameters of fuzzy systems simultaneously. More precisely, two ABC based fuzzy modeling strategies are presented and compared. The first strategy uses global optimization to learn fuzzy models, the second one hybridizes ABC and weighted least squares estimate method. The performances of the proposed ABC and ABC-LS fuzzy modeling strategies are evaluated on complex modeling problems and compared to other advanced modeling methods.

Keywords: automatic design, learning, fuzzy rules, hybrid, swarm optimization

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Authors: Donggun Lee, Q-Han Park

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Due to its coarse grid, the Pseudo-Spectral Time Domain (PSTD) method has advantages against the Finite Difference Time Domain (FDTD) method in terms of memory requirement and operation time. However, since the efficiency of parallelization is much lower than that of FDTD, PSTD is not a useful method for a large-scale electromagnetic simulation in a parallel platform. In this paper, we propose the parallelization technique of the hybrid PSTD-FDTD (HPF) method which simultaneously possesses the efficient parallelizability of FDTD and the quick speed and low memory requirement of PSTD. Parallelization cost of the HPF method is exactly the same as the parallel FDTD, but still, it occupies much less memory space and has faster operation speed than the parallel FDTD. Experiments in distributed memory systems have shown that the parallel HPF method saves up to 96% of the operation time and reduces 84% of the memory requirement. Also, by combining the OpenMP library to the MPI library, we further reduced the operation time of the parallel HPF method by 50%.

Keywords: FDTD, hybrid, MPI, OpenMP, PSTD, parallelization

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Authors: Nur Suraya Anis Ahmad Bakhtiar, Hazizan Md Akil

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In the present work, the dielectric properties of Epoxy/MWCNTs-muscovite HYBRID and MIXED composites based on ratio 30:70 were studies. The multi-wall carbon nanotubes (MWCNTs) were prepared by two method; (a) muscovite-MWCNTs hybrids were synthesized by chemical vapor deposition (CVD) and (b) physically mixing of muscovite with MWCNTs. The effect of different preparations of the composites and filler loading was evaluated. It is revealed that the dielectric constants of HYBRID epoxy composites are slightly higher compared to MIXED epoxy composites. It is also indicated that the dielectric constant increased by increases the MWCNTs filler loading.

Keywords: muscovite, epoxy, dielectric properties, hybrid composite

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Authors: Vishnu Goyal, Basant Agarwal

Abstract:

Sentiment analysis research provides methods for identifying the people’s opinion written in blogs, reviews, social networking websites etc. Sentiment analysis is to understand what opinion people have about any given entity, object or thing. Sentiment analysis research can be broadly categorised into three types of approaches i.e. semantic orientation, machine learning and lexicon based approaches. Feature selection methods improve the performance of the machine learning algorithms by eliminating the irrelevant features. Information gain feature selection method has been considered best method for sentiment analysis; however, it has the drawback of selection of threshold. Therefore, in this paper, we propose a hybrid feature selection methods comprising of information gain and proposed feature selection method. Initially, features are selected using Information Gain (IG) and further more noisy features are eliminated using the proposed feature selection method. Experimental results show the efficiency of the proposed feature selection methods.

Keywords: feature selection, sentiment analysis, hybrid feature selection

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Authors: Tinomuvonga Manenji Zhou, Eubert Mahofa, Tatenda Crispen Madzokere

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The nanostructured formulation can increase fertilizer efficacy and uptake ratio of the soil nutrients in agriculture production and save fertilizer resources. Controlled release modes have properties of both release rate and release pattern of nutrients, for fertilizers that are soluble in water might be correctly controlled. Nanoparticles can reduce the rate at which fertilizer nutrients are in the soil by leaching. A slow release NPK-hydroxyapatite nano hybrid fertilizer was synthesized using exfoliated bentonite as filler material. A simple, scalable method was used to synthesize the nitrogen-phosphorus hydroxyapatite nano fertilizer, where calcium hydroxide, phosphoric acid, and urea were used as precursor material, followed by the incorporation of potassium through a liquid grinding method. The product obtained was an NPK-hydroxyapatite nano hybrid fertilizer. A quantitative analysis was done to determine the percentage of nitrogen, phosphorus, and potassium in the hybrid fertilizer. AAS was used to determine the percentage of potassium in the fertilizer. An accelerated water test was conducted to compare the nutrient release behavior of nutrients between the synthesized NPK-hydroxyapatite nano hybrid fertilizer and commercial NPK fertilizer. The rate of release of Nitrogen, phosphorus, and potassium was significantly lower in the synthesized NPK hydroxyapatite nano hybrid fertilizer than in the convectional NPK fertilizer. The synthesized fertilizer was characterized using XRD. NPK hydroxyapatite nano hybrid fertilizer encapsulated in exfoliated bentonite thus prepared can be used as an environmentally friendly fertilizer formulation which could be extended to solve one of the major problems faced in the global fertilization of low nitrogen, phosphorus, and potassium use efficiency in agriculture.

Keywords: NPK hydroxyapatite nano hybrid fertilizer, bentonite, encapsulation, low release

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