Search results for: Caputo derivative

Authors: Hakiki Kheira, Belhamiti Omar

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In this paper, we propose a new compartmental fractional order model for the simulation of epidemic obesity dynamics. Using the Legendre wavelet method combined with the decoupling and quasi-linearization technique, we demonstrate the validity and applicability of our model. We also present some fractional differential illustrative examples to demonstrate the applicability and efficiency of the method. The fractional derivative is described in the Caputo sense.

Keywords: Caputo derivative, epidemiology, Legendre wavelet method, obesity

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Authors: Trilok Mathur, Shivi Agarwal

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This paper deals with the solutions of fuzzy-fractional-order Riccati equations under Caputo-type fuzzy fractional derivatives. The Caputo-type fuzzy fractional derivatives are defined based on Hukuhura difference and strongly generalized fuzzy differentiability. The Laplace-Adomian-Pade method is used for solving fractional Riccati-type initial value differential equations of fractional order. Moreover, we also displayed some examples to illustrate our methods.

Keywords: Caputo-type fuzzy fractional derivative, Fractional Riccati differential equations, Laplace-Adomian-Pade method, Mittag Leffler function

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Authors: Yacine Arioua

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In this paper, we investigate the existence and uniqueness of solutions for a coupled system of nonlinear Caputo-type Katugampola fractional differential equations with integral boundary conditions. Based upon a contraction mapping principle, Schauders fixed point theorems, some new existence and uniqueness results of solutions for the given problems are obtained. For application, some examples are given to illustrate the usefulness of our main results.

Keywords: fractional differential equations, coupled system, Caputo-Katugampola derivative, fixed point theorems, existence, uniqueness

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Authors: Sadia Arshad

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The new coronavirus disease or COVID-19 still poses an alarming situation around the world. Modeling based on the derivative of fractional order is relatively important to capture real-world problems and to analyze the realistic situation of the proposed model. Weproposed a mathematical model for the investigation of COVID-19 dynamics in a generalized fractional framework. The new model is formulated in the Caputo sense and employs a nonlinear time-varying transmission rate. The existence and uniqueness solutions of the fractional order derivative have been studied using the fixed-point theory. The associated dynamical behaviors are discussed in terms of equilibrium, stability, and basic reproduction number. For the purpose of numerical implementation, an effcient approximation scheme is also employed to solve the fractional COVID-19 model. Numerical simulations are reported for various fractional orders, and simulation results are compared with a real case of COVID-19 pandemic. According to the comparative results with real data, we find the best value of fractional orderand justify the use of the fractional concept in the mathematical modelling, for the new fractional modelsimulates the reality more accurately than the other classical frameworks.

Keywords: fractional calculus, modeling, stability, numerical solution

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Authors: Kamel Al-Khaled

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Reaction-diffusion equations are commonly used in population biology to model the spread of biological species. In this paper, we propose a fractional reaction-diffusion equation, where the classical second derivative diffusion term is replaced by a fractional derivative of order less than two. Based on the symbolic computation system Mathematica, Adomian decomposition method, developed for fractional differential equations, is directly extended to derive explicit and numerical solutions of space fractional reaction-diffusion equations. The fractional derivative is described in the Caputo sense. Finally, the recent appearance of fractional reaction-diffusion equations as models in some fields such as cell biology, chemistry, physics, and finance, makes it necessary to apply the results reported here to some numerical examples.

Keywords: fractional partial differential equations, reaction-diffusion equations, adomian decomposition, biological species

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Authors: Golden J. Zhang, Dongbao Zhou

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Darcy’s law is the fundamental theory in fluid dynamics and engineering applications. Although Darcy linearity was found to be valid for slow, viscous flow, non-linear and non-Darcian flow has been well documented under both small and large velocity fluid flow. Various classical models were proposed and used widely to quantify non-Darcian flow, including the well-known Forchheimer, Izbash, and Swartzendruber models. Applications, however, revealed limitations of these models. Here we propose a general model built upon the Caputo fractional derivative to quantify non-Darcian flow for various flows (laminar to turbulence).Real-world applications and model comparisons showed that the new fractional-derivative model, which extends the fractional model proposed recently by Zhou and Yang (2018), can capture the non-Darcian flow in the relatively small velocity in low-permeability deposits and the relatively high velocity in high-permeability sand. A scale effect was also identified for non-Darcian flow in fractured rocks. Therefore, fractional calculus may provide an efficient tool to improve classical models to quantify fluid dynamics in aquatic environments.

Keywords: fractional derivative, darcy’s law, non-darcian flow, fluid dynamics

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Authors: A. Guezane-Lakoud, S. Bensebaa

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In this paper, we study a boundary value problem of nonlinear fractional differential equation. Existence and positivity results of solutions are obtained.

Keywords: positive solution, fractional caputo derivative, Banach contraction principle, Avery and Peterson fixed point theorem

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Authors: Fernando Maass, Pablo Martin, Jorge Olivares

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Fractional derivatives have become very important in several areas of Engineering, however, the solutions of simple differential equations are not known. Here we are considering the simplest first order nonhomogeneous differential equations with Bessel regular functions of the first kind, in this way the solutions have been found which are hypergeometric solutions for any fractional derivative of order α, where α is rational number α=m/p, between zero and one. The way to find this result is by using Laplace transform and the Caputo definitions of fractional derivatives. This method is for values longer than one. However for α entire number the hypergeometric functions are Kumer type, no integer values of alpha, the hypergeometric function is more complicated is type ₂F₃(a,b,c, t2/2). The argument of the hypergeometric changes sign when we go from the regular Bessel functions to the modified Bessel functions of the first kind, however it integer seems that using precise values of α and considering no integers values of α, a solution can be obtained in terms of two hypergeometric functions. Further research is required for future papers in order to obtain the general solution for any rational value of α.

Keywords: Caputo, fractional calculation, hypergeometric, linear differential equations

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Authors: Asma Hanif, A. I. K. Butt, Shabir Ahmad, Rahim Ud Din, Mustafa Inc

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This paper is devoted to a study of the fuzzy fractional mathematical model reviewing the transmission dynamics of the infectious disease Covid-19. The proposed dynamical model consists of susceptible, exposed, symptomatic, asymptomatic, quarantine, hospitalized and recovered compartments. In this study, we deal with the fuzzy fractional model defined in Caputo’s sense. We show the positivity of state variables that all the state variables that represent different compartments of the model are positive. Using Gronwall inequality, we show that the solution of the model is bounded. Using the notion of the next-generation matrix, we find the basic reproduction number of the model. We demonstrate the local and global stability of the equilibrium point by using the concept of Castillo-Chavez and Lyapunov theory with the Lasalle invariant principle, respectively. We present the results that reveal the existence and uniqueness of the solution of the considered model through the fixed point theorem of Schauder and Banach. Using the fuzzy hybrid Laplace method, we acquire the approximate solution of the proposed model. The results are graphically presented via MATLAB-17.

Keywords: Caputo fractional derivative, existence and uniqueness, gronwall inequality, Lyapunov theory

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Authors: Mohamed Houas, Maamar Benbachir

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In this paper, under weak assumptions, we study the existence and uniqueness of solutions for a nonlinear fractional boundary value problem. New existence and uniqueness results are established using Banach contraction principle. Other existence results are obtained using scheafer and krasnoselskii's fixed point theorem. At the end, some illustrative examples are presented.

Keywords: caputo derivative, boundary value problem, fixed point theorem, local conditions

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Authors: Shatha S. Alhily

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The purpose of this research paper is to show all the possible values of the pth power of the integrable function which make the integral means of the derivative of univalent function existing and finite.

Keywords: derivative, integral means, self conformal maps, univalent function

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Authors: Aysan Esgandanian, Sabalan Daneshvar

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The study is done to determine the comparison between proportional-integral-derivative controller (PID controller) and tilt-integral-derivative (TID controller) for cardiac pacemaker systems, which can automatically control the heart rate to accurately track a desired preset profile. The controller offers good adaption of heart to the physiological needs of the patient. The parameters of the both controllers are tuned by particle swarm optimization (PSO) algorithm which uses the integral of time square error as a fitness function to be minimized. Simulation results are performed on the developed cardiovascular system of humans and results demonstrate that the TID controller produces superior control performance than PID controllers. In this paper, all simulations were performed in Matlab.

Keywords: integral of time square error, pacemaker systems, proportional-integral-derivative controller, PSO algorithm, tilt-integral-derivative controller

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Authors: Kassim A. Jassim

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In the present paper, we introduce and discuss a new class Kp(λ,α) of meromorphic multivalent functions in the punctured unit disk U*={z∈¢:0<|z|<1} defined by Ruscheweyh derivative. We obtain some sufficient conditions for the functions belonging to the class Kp(λ,α).

Keywords: meromorphic multivalent function, Ruscheweyh derivative, hadamard product

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Authors: Mohammad Mehdi Mazarei, Ali Asghar Behroozpoor

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In this paper, we define an optimization problem corresponding to smooth and nonsmooth functions which its optimal solution is the first derivative of these functions in a domain. For this purpose, a linear programming problem corresponding to optimization problem is obtained. The optimal solution of this linear programming problem is the approximate generalized first derivative. In fact, we approximate generalized first derivative of nonsmooth functions as tailor series. We show the efficiency of our approach by some smooth and nonsmooth functions in some examples.

Keywords: general derivative, linear programming, optimization problem, smooth and nonsmooth functions

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Authors: Mohammed Abdulhameed, Sagir M. Abdullahi

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In this paper, based on the applications of nanoparticle, the blood flow along with nanoparticles through stenosed artery is studied. The blood is acted by periodic body acceleration, an oscillating pressure gradient and an external magnetic field. The mathematical formulation is based on Caputo-Fabrizio fractional derivative without singular kernel. The model of ordinary blood, corresponding to time-derivatives of integer order, is obtained as a limiting case. Analytical solutions of the blood velocity and temperature distribution are obtained by means of the Hankel and Laplace transforms. Effects of the order of Caputo-Fabrizio time-fractional derivatives and three different nanoparticles i.e. Fe3O4, TiO4 and Cu are studied. The results highlights that, models with fractional derivatives bring significant differences compared to the ordinary model. It is observed that the addition of Fe3O4 nanoparticle reduced the resistance impedance of the blood flow and temperature distribution through bell shape stenosed arteries as compared to TiO4 and Cu nanoparticles. On entering in the stenosed area, blood temperature increases slightly, but, increases considerably and reaches its maximum value in the stenosis throat. The shears stress has variation from a constant in the area without stenosis and higher in the layers located far to the longitudinal axis of the artery. This fact can be an important for some clinical applications in therapeutic procedures.

Keywords: nanoparticles, blood flow, stenosed artery, mathematical models

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Authors: Yildiray Keskin, Omer Acan, Murat Akkus

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In this paper, the solution of fractional diffusion equations is presented by means of the reduced differential transform method. Fractional partial differential equations have special importance in engineering and sciences. Application of reduced differential transform method to this problem shows the rapid convergence of the sequence constructed by this method to the exact solution. The numerical results show that the approach is easy to implement and accurate when applied to fractional diffusion equations. The method introduces a promising tool for solving many fractional partial differential equations.

Keywords: fractional diffusion equations, Caputo fractional derivative, reduced differential transform method, partial

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Authors: Ramzi B. Albadarneh

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In this paper, we use the new definition of fractional derivative called conformable fractional derivative to derive some finite difference formulas and its error terms which are used to solve fractional differential equations and fractional partial differential equations, also to derive fractional Euler method and its error terms which can be applied to solve fractional differential equations. To provide the contribution of our work some applications on finite difference formulas and Euler Method are given.

Keywords: conformable fractional derivative, finite difference formula, fractional derivative, finite difference formula

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Authors: Pablo Martin, Jorge Olivares, Fernando Maass

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The use of fractional derivatives to different problems in Engineering and Physics has been increasing in the last decade. For this reason, we have here considered partial derivatives when the integral is a spherical Bessel function of the first kind in both regular and modified ones simple initial conditions have been also considered. In this way, the solution has been found as a combination of hypergeometric functions. The case of a general rational value for α of the fractional derivative α has been solved in a general way for alpha between zero and two. The modified spherical Bessel functions of the first kind have been also considered and how to go from the regular case to the modified one will be also shown.

Keywords: caputo fractional derivatives, hypergeometric functions, linear differential equations, spherical Bessel functions

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Authors: Chin-Yun Chen

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Numerical integration is an essential tool for deriving different physical quantities in engineering and science. The effectiveness of a numerical integrator depends on different factors, where the crucial one is the error estimation. This work presents an error estimator that combines a derivative free method to improve the performance of verified numerical quadrature.

Keywords: numerical quadrature, error estimation, derivative free method, interval computation

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Authors: Chuang-Chang Chang, Keng-Yu Ho, Yu-Jen Hsiao, Hsin-Ni Yang

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Using a sample of detailed ownership data of 1,032 listed commercial bank observations in 30 European countries from 2004 to 2010, we explore what categories of shareholder are more likely to use derivatives and how different types of owners affect the bank value. We find that a shift in equity from bank investors to either non-financial companies or institutional investors have increase incentives to use derivatives. Moreover, we have significant evidence that a shift in equity from bank investors to either family or manager shareholders who attend derivative activities will decrease bank value. However, a shift in equity from bank investors to non-financial companies who use derivative instrument will increase the bank value. Our results are also robustness to address for the potential endogeneity problems.

Keywords: derivative usage, ownership structure, bank value

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Authors: Sachin Kumar

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Fuzzy fractional diffusion equation is widely useful to depict different physical processes arising in physics, biology, and hydrology. The motive of this article is to deal with the fuzzy fractional diffusion equation. We study a mathematical model of fuzzy space-time fractional diffusion equation in which unknown function, coefficients, and initial-boundary conditions are fuzzy numbers. First, we find out a fuzzy operational matrix of Legendre polynomial of Caputo type fuzzy fractional derivative having a non-singular Mittag-Leffler kernel. The main advantages of this method are that it reduces the fuzzy fractional partial differential equation (FFPDE) to a system of fuzzy algebraic equations from which we can find the solution of the problem. The feasibility of our approach is shown by some numerical examples. Hence, our method is suitable to deal with FFPDE and has good accuracy.

Keywords: fractional PDE, fuzzy valued function, diffusion equation, Legendre polynomial, spectral method

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Authors: Jorge Olivares, Fernando Maass, Pablo Martin

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Fractional derivatives have been considered recently as a way to solve different problems in Engineering. In this way, second kind modified Bessel functions are considered here. The order α fractional differential equations of second kind Bessel functions, Kᵥ(x), are studied with simple initial conditions. The Laplace transform and Caputo definition of fractional derivatives are considered. Solutions have been found for ν=1/3, 1/2, 2/3, -1/3, -1/2 and (-2/3). In these cases, the solutions are the sum of two hypergeometric functions. The α fractional derivatives have been for α=1/3, 1/2 and 2/3, and the above values of ν. No convergence has been found for the integer values of ν Furthermore when α has been considered as a rational found m/p, no general solution has been found. Clearly, this case is more difficult to treat than those of first kind Bessel Function.

Keywords: Caputo, modified Bessel functions, hypergeometric, linear fractional differential equations, transform Laplace

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Authors: Dong Xie, Leah Howard, Yiming Weng

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The objective of this study was to synthesize and characterize 5-acryloyloxy-3,4-dichlorocrotonolactone (a furanone derivative), use this derivative to modify a dental restorative, and study the effect of the derivative on the antibacterial activity and compressive strength of the formed restorative. In this study, a furanone derivative was synthesized, characterized, and used to formulate a dental restorative. Compressive strength (CS) and S. mutans viability were used to evaluate the mechanical strength and antibacterial activity of the formed restorative. The fabricated restorative specimens were photocured and conditioned in distilled water at 37oC for 24 h, followed by direct testing for CS or/and incubating with S. mutans for 48 h for antibacterial testing. The results show that the modified dental restorative showed a significant antibacterial activity without substantially decreasing the mechanical strengths. With addition of the antibacterial derivative up to 30%, the restorative kept its original CS nearly unchanged but showed a significant antibacterial activity with 68% reduction in the S. mutans viability. Furthermore, the antibacterial function of the modified restorative was not affected by human saliva. The aging study also indicates that the modified restorative may have a long-lasting antibacterial function. It is concluded that this experimental antibacterial restorative may potentially be developed into a clinically attractive dental filling restorative due to its high mechanical strength and antibacterial function.

Keywords: antibacterial, dental restorative, compressive strength, S. mutans viability

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Authors: Aasheesh Raturi, Vimal Kothiyal, P. D. Semalty

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We studied mechanical properties of Eucalyptus tereticornis using FT-NIR spectroscopy. Firstly, spectra were pre-processed to eliminate useless information. Then, prediction model was constructed by partial least squares regression. To study the influence of pre-processing on prediction of mechanical properties for NIR analysis of wood samples, we applied various pretreatment methods like straight line subtraction, constant offset elimination, vector-normalization, min-max normalization, multiple scattering. Correction, first derivative, second derivatives and their combination with other treatment such as First derivative + straight line subtraction, First derivative+ vector normalization and First derivative+ multiplicative scattering correction. The data processing methods in combination of preprocessing with different NIR regions, RMSECV, RMSEP and optimum factors/rank were obtained by optimization process of model development. More than 350 combinations were obtained during optimization process. More than one pre-processing method gave good calibration/cross-validation and prediction/test models, but only the best calibration/cross-validation and prediction/test models are reported here. The results show that one can safely use NIR region between 4000 to 7500 cm-1 with straight line subtraction, constant offset elimination, first derivative and second derivative preprocessing method which were found to be most appropriate for models development.

Keywords: FT-NIR, mechanical properties, pre-processing, PLS

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Authors: Cletus Abhulimen, L. A. Ukpebor

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In this paper, we construct a class of four-step third derivative exponential fitting integrator of order six for the numerical integration of stiff initial-value problems of the type: y’= f(x,y); y(x₀) =y₀. The implicit method has free parameters which allow it to be fitted automatically to exponential functions. For the purpose of effective implementation of the proposed method, we adopted the techniques of splitting the method into predictor and corrector schemes. The numerical analysis of the stability of the new method was discussed; the results show that the method is A-stable. Finally, numerical examples are presented, to show the efficiency and accuracy of the new method.

Keywords: third derivative four-step, exponentially fitted, a-stable, stiff differential equations

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Authors: Atia Alam

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Growing liberalisation and intense market competition increase firm’s risk exposure and induce corporations to use derivatives extensively as a risk management instrument, which results in decrease in firm’s risk, and increase in value. Present study contributes towards existing literature by providing an in-depth analysis regarding the effect of extent of derivative usage on firm’s risk and value by using panel data models and seemingly unrelated regression technique. New evidence is established in current literature by dividing the sample data based on firm’s Exchange Rate (ER) and Interest Rate (IR) exposure. Analysis is performed for the effect of extent of derivative usage on firm’s risk and value and its variation with respect to the ER and IR exposure. Sample data consists of 166 Pakistani firms listed on Pakistan stock exchange for the period of 2004-2010. Results show that extensive usage of derivative instruments significantly increases firm value and reduces firm’s risk. Furthermore, comprehensive analysis depicts that Pakistani corporations having higher exchange rate exposure, with respect to foreign sales, and higher interest rate exposure, on the basis of industry adjusted leverage, have higher firm value and lower risk. Findings from seemingly unrelated regression also provide robustness to results obtained through panel data analysis. Study also highlights the role of derivative usage as a risk management instrument in high and low ER and IR risk and helps practitioners in understanding how value increasing effect of extent of derivative usage varies with the intensity of firm’s risk exposure.

Keywords: extent of derivative usage, firm value, risk, Pakistan, non-financial firms

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Authors: Luke Ukpebor, C. E. Abhulimen

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Stiff initial value problems in ordinary differential equations are problems for which a typical solution is rapidly decaying exponentially, and their numerical investigations are very tedious. Conventional numerical integration solvers cannot cope effectively with stiff problems as they lack adequate stability characteristics. In this article, we developed a new family of four-step second derivative exponentially fitted method of order six for the numerical integration of stiff initial value problem of general first order differential equations. In deriving our method, we employed the idea of breaking down the general multi-derivative multistep method into predator and corrector schemes which possess free parameters that allow for automatic fitting into exponential functions. The stability analysis of the method was discussed and the method was implemented with numerical examples. The result shows that the method is A-stable and competes favorably with existing methods in terms of efficiency and accuracy.

Keywords: A-stable, exponentially fitted, four step, predator-corrector, second derivative, stiff initial value problems

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Authors: Fouzi Aboura

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The fractional order proportional-integral-derivative (FOPID) controller or fractional order (PIλDµ) is a proportional-integral-derivative (PID) controller where integral order (λ) and derivative order (µ) are fractional, one of the important application of classical PID is the Automatic Voltage Regulator (AVR).The FOPID controller needs five parameters optimization while the design of conventional PID controller needs only three parameters to be optimized. In our paper we have proposed a comparison between algorithms Differential Evolution (DE) and Hybrid Genetic Algorithm Particle Swarm Optimization (HGAPSO) ,we have studied theirs characteristics and performance analysis to find an optimum parameters of the FOPID controller, a new objective function is also proposed to take into account the relation between the performance criteria’s.

Keywords: FOPID controller, fractional order, AVR system, objective function, optimization, GA, PSO, HGAPSO

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Authors: Chandra Mukherjee

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The present paper is a representation of the work done in the field of ECG signal analysis using MATLAB 7.1 Platform. An accurate and simple ECG feature extraction algorithm is presented in this paper and developed algorithm is validated using BIOPAC software. To detect the QRS peak, ECG signal is processed by following mentioned stages- First Derivative, Second Derivative and then squaring of that second derivative. Efficiency of developed algorithm is tested on ECG samples from different database and real time ECG signals acquired using BIOPAC system. Firstly we have lead wise specified threshold value the samples above that value is marked and in the original signal, where these marked samples face change of slope are spotted as R-peak. On the left and right side of the R-peak, faces change of slope identified as Q and S peak, respectively. Now the inbuilt Detection algorithm of BIOPAC software is performed on same output sample and both outputs are compared. ECG baseline modulation correction is done after detecting characteristics points. The efficiency of the algorithm is tested using some validation parameters like Sensitivity, Positive Predictivity and we got satisfied value of these parameters.

Keywords: first derivative, variable threshold, slope reversal, baseline modulation correction

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Authors: Reyhan Tekin Sitrava

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Teachers should have robust and comprehensive content knowledge for effective mathematics teaching. It was explained that content knowledge includes knowing the facts, truths, and concepts; explaining the reasons behind these facts, truths and concepts, and making relationship between the concepts and other disciplines. By virtue of its importance, it will be significant to explore teachers and prospective teachers’ content knowledge related to variety of topics in mathematics. From this point of view, the purpose of this study was to investigate prospective mathematics teachers’ content knowledge. Particularly, it was aimed to reveal the prospective teachers’ knowledge regarding the definition of limit and derivate. To achieve the purpose and to get in-depth understanding, a qualitative case study method was used. The data was collected from 34 prospective mathematics teachers through a questionnaire containing 2 questions. The first question required the prospective teachers to define the limit and the second one required to define the derivative. The data was analyzed using content analysis method. Based on the analysis of the data, although half of the prospective teachers (50%) could write the definition of the limit, nine prospective teachers (26.5%) could not define limit. However, eight prospective teachers’ definition was regarded as partially correct. On the other hand, twenty-seven prospective teachers (79.5%) could define derivative, but seven of them (20.5%) defined it partially. According to the findings, most of the prospective teachers have robust content knowledge on limit and derivative. This result is important because definitions have a virtual role in learning and teaching of mathematics. More specifically, definition is starting point to understand the meaning of a concept. From this point of view, prospective teachers should know the definitions of the concepts to be able to teach them correctly to the students. In addition, they should have knowledge about the relationship between limit and derivative so that they can explain these concepts conceptually. Otherwise, students may memorize the rules of calculating the derivative and the limit. In conclusion, the present study showed that most of the prospective mathematics teachers had enough knowledge about the definition of derivative and limit. However, the rest of them should learn their definition conceptually. The examples of correct, partially correct, and incorrect definition of both concepts will be presented and discussed based on participants’ statements. This study has some implications for instructors. Instructors should be careful about whether students learn the definition of these concepts or not. In order to this, the instructors may give prospective teachers opportunities to discuss the definition of these concepts and the relationship between the concepts.

Keywords: content knowledge, derivative, limit, prospective mathematics teachers

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