On an Ecological Model of Mutualisim Between Two Species With a Mortal Predator

In this paper, we study an ecological model of a three-space food chain consists of two logically growing mutual species and third species acts as a predator to second mutual species with Holling type II functional response. This model is constituted by a system of nonlinear decoupled ordinary differential equations. By using perturbed method, we identify the nature of the system at each equilibrium point and also global stability is investigated for this model using Lypanov function at the possible equilibrium points

A NEW ROOT–FINDING ALGORITHM USING EXPONENTIAL SERIES

In this paper, we present a new root-finding algorithm to compute a non-zero real root of the transcendental equations using exponential series. Indeed, the new proposed algorithm is based on the exponential series and in which Secant method is special case. The proposed algorithm produces better approximate root than bisection method, regula-falsi method, Newton-Raphson method and secant method. The implementation of the proposed algorithm in Matlab and Maple also presented. Certain numerical examples are presented to validate the efficiency of the proposed algorithm. This algorithm will help to implement in the commercial package for finding a real root of a given transcendental equation

A NEW ROOT–FINDING ALGORITHM USING EXPONENTIAL SERIES

In this paper, we present a new root-finding algorithm to compute a non-zero real root of the transcendental equations using exponential series. Indeed, the new proposed algorithm is based on the exponential series and in which Secant method is special case. The proposed algorithm produces better approximate root than bisection method, regula-falsi method, Newton-Raphson method and secant method. The implementation of the proposed algorithm in Matlab and Maple also presented. Certain numerical examples are presented to validate the efficiency of the proposed algorithm. This algorithm will help to implement in the commercial package for finding a real root of a given transcendental equation

ON NEW HYBRID ROOT-FINDING ALGORITHMS FOR SOLVING TRANSCENDENTAL EQUATIONS USING EXPONENTIAL AND HALLEY'S METHODS

The objective of this paper is to propose two new hybrid root finding algorithms for solving transcendental equations. The proposed algorithms are based on the well-known root finding methods namely the Halley's method, regula-falsi method and exponential method. We show using numerical examples that the proposed algorithms converge faster than other related methods. The first hybrid algorithm consists of regula-falsi method and exponential method (RF-EXP). In the second hybrid algorithm, we use regula falsi method and Halley's method (RF-Halley). Several numerical examples are presented to illustrate the proposed algorithms, and comparison of these algorithms with other existing methods are presented to show the efficiency and accuracy. The implementation of the proposed algorithms is presented in Microsoft Excel (MS Excel) and the mathematical software tool Maple

A New Hybrid Root-Finding Algorithm to Solve Transcendental Equations Using Arcsine Function

The objective of this paper is to propose a new hybrid root finding algorithms for solving non-linear equations (NLEs) or transcendental equations (TEs). The proposed algorithm is based on the trigonometrical algorithm using arcsine function to find a root. Several numerical examples are presented to illustrate the proposed algorithms, and comparisons are presented with other existing methods to show efficiency and accuracy. Implementation of the proposed algorithms is presented in a mathematical software tool Maple

Numerical Analysis of Friction Stir Welding on an Alumunium Butt Joint

In this paper, we present a three-dimensional numerical analysis of friction stir welding on an alumunium butt joint. A thin sheet of aluminum marking material was embedded into the 6061-aluminum alloy panel and its rear weld path. The positions after friction stir welding were investigated by metallographic techniques. Looking at the visualized material flow pattern, a three-dimensional model was developed to numerically simulate the temperature profile and plastic effects. The calculated velocity profile for plastic flow in the immediate vicinity of the tool generally agrees with the visualized results. Increasing the tool speed while maintaining a constant tool feed rate increases the material flow near the pin. The shape and size of the predicted weld zone match the experimentally measured ones

A Three Species Ecological Model with Holling Type-II Functional Response

This paper elucidates an ecological model of a three-space food chain consists of two competing species and a third species, which is supporting the first competing species in non-linear manner of Holling type-II functional response and ingests the second competitive species in linear style, within the limited resources. This model is constituted by a system of non-linear decoupled first order ordinary differential equations. By using perturbed method, we investigate the local stability nature of the system at each possible equilibrium point. Further, the numerical illustrations at suitable parametric values to the model are presented by observing the species survivalness in nature for long time

Implementation of a Reducing Algorithm for Differential-Algebraic Systems in Maple

This paper discusses the implementation of a reduction algorithm for differential-algebraic systems in Maple. Using the proposed Maple package, the given system of differential-algebraic equations can be transformed into another simpler system having the same properties. Sample computations are presented to illustrate the proposed algorithm. This algorithm will be helpful to implement this in commercial software packages such as Mathematica, Matlab, Singular, SCIlab etc

A Symbolic Algorithm for Polynomial Interpolation with Integral Conditions

This paper presents a new symbolic algorithm for polynomial interpolation with integral conditions at arbitrary points. For expressing the integral conditions in the present algorithm, we employ the algebra of integro-differential operators. We also present another algorithm for computing the polynomial interpolation with Stieltjes conditions (combination of general, differential and integral conditions) as a quotient of two determinants. Error due to the formulation of a given function by the proposed interpolation is discussed and its symbolic formulation is presented. This algorithm helps OR would help to implement the manual calculations in commercial packages such as Maple, Mathematica, Matlab, Singular, etc. Certain numerical examples are presented to verify the proposed algorithms

A Educação Funcional como Elo entre as Revoluções Agrícola e Industrial: Uma Abordagem de Modelagem Epidemiológica

Education has been regarded as a major route to economic prosperity. Its potentials, if properly exploited, have the tendency to revolutionize a nation’s productivity, income, and development. [Objetive] In this work, a mathematical model was formulated, and an epidemiological modeling approach was explored to derive the necessary and sufficient conditions for agricultural and industrial transformation in Nigeria in terms of education. The use of mathematics to quantify such a phenomenon is relatively new. [Methodology] Ample mathematics theorems were employed to test for the existence, boundedness, and positivity of the model’s solutions which are basic features of a valid epidemiological model. The model was solved to derive the equilibrium points, and the analytic threshold that governed revolution in industry and agriculture in terms of education was derived. Stability and sensitivity analyses were conducted via the stability theory of nonlinear differential equations and normalized forward sensitivity index, respectively. Simulations were later conducted using the Maple software to validate the analytical results.  [Results] A government expenditure of 25% on education with or without corruption did not make education functional. Also, an increase in expenditure on education to 65% but with corruption of more than 30% retarded functional education. In general, education became functional when government investment in education attained 60%, with corruption levels lower than 20%. [Conclusions] Both the analytical and numerical results linked agricultural transformation and industrial revolution through functional education to massive investment in education and victory over corruption.La educación ha sido considerada como una ruta importante hacia la prosperidad económica. Sus potencialidades, si se explotan adecuadamente, tienden a revolucionar la productividad, los ingresos y el desarrollo de una nación. [Objetivo] En este trabajo se formuló un modelo matemático y se exploró un enfoque de modelado epidemiológico para derivar las condiciones necesarias y suficientes para la transformación agrícola e industrial en Nigeria en términos de educación. El uso de las matemáticas para cuantificar tal fenómeno es bastante nuevo. [Metodología] Se emplearon amplios teoremas matemáticos para comprobar la existencia, la delimitación y la positividad de las soluciones del modelo, características básicas de un modelo epidemiológico válido. Se resolvió el modelo para derivar los puntos de equilibrio y se derivó el umbral analítico que rigió la revolución en la industria y la agricultura en términos de educación. Los análisis de estabilidad y sensibilidad se realizaron mediante la teoría de estabilidad de ecuaciones diferenciales no lineales y el índice de sensibilidad directo normalizado, respectivamente. Posteriormente se realizaron simulaciones a través de un software de arce para validar los resultados analíticos. [Resultados] Un gasto público del 25 % en educación con corrupción o sin ella no hizo que la educación fuera funcional. Asimismo, un aumento del gasto en educación al 65 %, pero con una corrupción de más del 3 0% retrasó la educación funcional. En general, la educación se volvió funcional cuando la inversión del gobierno en educación alcanzó 60 % con un nivel de corrupción inferior al 20 %. [Conclusiones] Tanto los resultados analíticos como los numéricos vincularon la transformación agrícola y la revolución industrial a través de la educación funcional con la inversión masiva en educación y la victoria sobre la corrupción.A educação tem sido considerada como um importante caminho para a prosperidade econômica. Seu potencial, se devidamente explorado, tende a revolucionar a produtividade, a renda e o desenvolvimento de uma nação. [Objetivo] Este artigo formulou um modelo matemático e explorou uma abordagem de modelagem epidemiológica para derivar as condições necessárias e suficientes para a transformação agrícola e industrial na Nigéria em termos de educação. O uso da matemática para quantificar tal fenômeno é relativamente novo. [Metodologia] Extensos teoremas matemáticos foram usados para testar a existência, a delimitação e a positividade das soluções do modelo, que são características básicas de um modelo epidemiológico válido. O modelo para derivar pontos de equilíbrio foi resolvido e o limiar analítico que orientou a revolução na indústria e na agricultura em termos de educação foi derivado. As análises de estabilidade e sensibilidade foram realizadas utilizando-se a teoria da estabilidade das equações diferenciais não lineares e o índice de sensibilidade direta normalizada, respectivamente. Posteriormente, foram realizadas simulações por meio de um software Maple para validação dos resultados analíticos. [Resultados] Um gasto público de 25% em educação com ou sem corrupção não tornou a educação funcional. Da mesma forma, um aumento nos gastos com educação para 65%, mas com uma corrupção de mais de 30%, atrasou a educação funcional. No geral, a educação tornou-se funcional quando o investimento do governo em educação atingiu 60%, com um nível de corrupção abaixo de 20%. [Conclusões] Tanto os resultados analíticos quanto numéricos ataram a transformação agrícola e a revolução industrial através da educação funcional com o investimento maciço na educação e a vitória sobre a corrupção