Search results for: Boolean equations

Authors: Saurabh Rawat, Anushree Sah

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K Maps are generally and ideally, thought to be simplest form for obtaining solution of Boolean equations. Cubical Representation of Boolean equations is an alternate pick to incur a solution, otherwise to be meted out with Truth Tables, Boolean Laws, and different traits of Karnaugh Maps. Largest possible k- cubes that exist for a given function are equivalent to its prime implicants. A technique of minimization of Logic functions is tried to be achieved through cubical methods. The main purpose is to make aware and utilise the advantages of cubical techniques in minimization of Logic functions. All this is done with an aim to achieve minimal cost solution.r

Keywords: K-maps, don’t care conditions, Boolean equations, cubes

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Authors: U. M. Swamy

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A compact Hausdorff and totally disconnected topological space are known as Boolean space in view of the stone duality between Boolean algebras and such topological spaces. A sheaf over X is a triple (S, p, X) where S and X are topological spaces and p is a local homeomorphism of S onto X (that is, for each element s in S, there exist open sets U and G containing s and p(s) in S and X respectively such that the restriction of p to U is a homeomorphism of U onto G). Here we mainly concern on sheaves over Boolean spaces. From a given sheaf over a Boolean space, we obtain an algebraic structure in such a way that there is a one-to-one correspondence between these algebraic structures and sheaves over Boolean spaces.

Keywords: Boolean algebra, Boolean space, sheaf, stone duality

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Authors: Marcin Michalak

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Biclustering is the way of two-dimensional data analysis. For several years it became possible to express such issue in terms of Boolean reasoning, for processing continuous, discrete and binary data. The mathematical backgrounds of such approach — proved ability of induction of exact and inclusion–maximal biclusters fulfilling assumed criteria — are strong advantages of the method. Unfortunately, the core of the method has quite high computational complexity. In the paper the basics of Boolean reasoning approach for biclustering are presented. In such context the problems of computation parallelization are risen.

Keywords: Boolean reasoning, biclustering, parallelization, prime implicant

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Authors: Sekou Kangoye, Alexis Todoskoff, Mihaela Barreau

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Modified Condition/Decision Coverage (MC/DC) is a structural coverage criterion that aims to prove that all conditions involved in a Boolean expression can influence the result of that expression. In the context of automotive, MC/DC is highly recommended and even required for most security and safety applications testing. However, due to complex Boolean expressions that often embedded in those applications, generating a set of MC/DC compliant test cases for any of these expressions is a nontrivial task and can be time consuming for testers. In this paper we present an approach to automatically generate MC/DC test cases for any Boolean expression. We introduce novel techniques, essentially based on binary trees to quickly and optimally generate MC/DC test cases for the expressions. Thus, the approach can be used to reduce the manual testing effort of testers.

Keywords: binary trees, MC/DC, test case generation, nontrivial task

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Authors: Finley Lawson

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The triune nature of God is one of the most complex doctrines of Christianity, and its complexity is further compounded when one considers the incarnation. However, many of the difficulties and paradoxes associated with our idea of the divine arise from our adherence to reductionist ontology. In order to move our theological discourse forward, in respect to divine and human nature, a holistic interpretation of our profession of faith is necessary. The challenge of a holistic interpretation is that it questions our ability to make any statement about the genuine, ontological individuation of persons (both divine and human), and in doing so raises the issue of whether we are, ontologically, bound to descend in to a form of pan(en)theism. In order to address the ‘inevitable’ slide in to pan(en)theism. The impact of two forms of holistic interpretation, Boolean and Non-Boolean, on our concept of personhood will be examined. Whilst a Boolean interpretation allows for a greater understanding of the relational nature of the Trinity, it is the Non-Boolean interpretation which has greater ontological significance. A Non-Boolean ontology, grounded in our scientific understanding of the nature of the world, shows our quest for individuation rests not in ontological fact but in epistemic need, and that it is our limited epistemology that drives our need to divide that which is ontologically indivisible. This discussion takes place within a ‘methodological’, rather than ‘doctrinal’ approach to science and religion - examining assumptions and methods that have shaped our language and beliefs about key doctrines, rather than seeking to reconcile particular Christian doctrines with particular scientific theories. Concluding that Non-Boolean holism is the more significant for our doctrine is, in itself, not enough. A world without division appears much removed from the distinct place of man and divine as espoused in our creedal affirmation, to this end, several possible interpretations for understanding Non-Boolean human – divine relations are tentatively put forward for consideration.

Keywords: holism, individuation, ontology, Trinitarian relations

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Authors: Riddhi Jangid, Gajendra Pratap Singh

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Petri nets, a formalized modeling language that was introduced back around 50-60 years, have been widely used for modeling discrete event dynamic systems and simulating their behavior. Reachability analysis of Petri nets gives many insights into a modeled system. This idea leads us to study the reachability technique and use it in the reachability problem in the state space of reachable markings. With the same concept, Crisp Boolean Petri nets were defined in which the marking vectors that are boolean are distinct in the reachability analysis of the nets. We generalize the concept and define ‘Crisp’ Petri nets that generate the marking vectors exactly once in their reachability-based analysis, not necessarily Boolean.

Keywords: marking vector, n-vector, Petri nets, reachability

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Authors: Riddhi Jangid, Gajendra Pratap Singh

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Petri nets, a mathematical tool, is used for modeling in different areas of computer sciences, biological networks, chemical systems and many other disciplines. A Petri net model of a given system is created by the graphical representation that describes the properties and behavior of the system. While looking for the behavior of any system, 1-safe Petri nets are of particular interest to many in the application part. Boolean Petri nets correspond to those class in 1- safe Petri nets that generate all the binary n-vectors in their reachability analysis. We study the class by changing different parameters like the token counts in the places and how the structure of the tree changes in the reachability analysis. We discuss here an extended class of Boolean Petri nets that generates n-ary trees in their reachability-based analysis.

Keywords: marking vector, n-vector, petri nets, reachability

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Authors: Fikre Belay Tekulu

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The Adigrat town lies in the Tigray region of Ethiopia. This region is mountainous and experiences a semiarid type of climate. Most of the rainfall occurs in four months of the year, which are June to September. During this season, flood is a common natural disaster, especially in urban areas. In this paper, an attempt is made to identify flood-prone areas in Adigrat town using Boolean logic with GIS and remote sensing techniques. Three parameters were incorporated as land use type, elevation, and slope. Boolean logic was used as land use equal to buildup land, elevation less than 2430 m, and slope less than 5 degrees. As a result, 0.575 km² was identified severely affected by floods during the rainy season.

Keywords: flood, GIS, hydrology, Adigrat

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Authors: Zhangquan Zhou, Guilin Qi

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Materialization is an important reasoning service for applications built on the Web Ontology Language (OWL). To make materialization efficient in practice, current research focuses on deciding tractability of an ontology language and designing parallel reasoning algorithms. However, some well-known large-scale ontologies, such as YAGO, have been shown to have good performance for parallel reasoning, but they are expressed in ontology languages that are not parallelly tractable, i.e., the reasoning is inherently sequential in the worst case. This motivates us to study the problem of parallel tractability of ontology materialization from a theoretical perspective. That is we aim to identify the ontologies for which materialization is parallelly tractable, i.e., in the NC complexity. Since the NC complexity is defined based on Boolean circuit that is widely used to investigate parallel computing problems, we first transform the problem of materialization to evaluation of Boolean circuits, and then study the problem of parallel tractability based on circuits. In this work, we focus on datalog rewritable ontology languages. We use Boolean circuits to identify two classes of datalog rewritable ontologies (called parallelly tractable classes) such that materialization over them is parallelly tractable. We further investigate the parallel tractability of materialization of a datalog rewritable OWL fragment DHL (Description Horn Logic). Based on the above results, we analyze real-world datasets and show that many ontologies expressed in DHL belong to the parallelly tractable classes.

Keywords: ontology materialization, parallel reasoning, datalog, Boolean circuit

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Authors: Ali Khwaldeh, Nimer Adwan

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In this paper, a method for constructing a controlled balanced Boolean function satisfying the criterion of a Strictly Avalanche Criteria (SAC) effect is proposed. The proposed method is based on the use of three orthogonal nonlinear components which is unlike the high-order SAC functions. So, the generator synthesized by the proposed method has separate sets of control and information inputs. The proposed method proves its simplicity and the implementation ability. The proposed method allows synthesizing a SAC function generator with fixed control and information inputs. This ensures greater efficiency of the built-in oscillator compared to high-order SAC functions that can be used as a generator. Accordingly, the method is completely formalized and implemented as a software product.

Keywords: boolean function, controlled balanced boolean function, strictly avalanche criteria, orthogonal nonlinear

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Authors: Amritpal Singh Nafria, Rohit Sharma, Md. Shami Ansari

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Up to now only five different equations of motion can be derived from velocity time graph without needing to know the normal and frictional forces acting at the point of contact. In this paper we obtained all possible requisite conditions to be considering an equation as an equation of motion. After that we classified equations of motion by considering two equations as fundamental kinematical equations of motion and other three as additional kinematical equations of motion. After deriving these five equations of motion, we examine the easiest way of solving a wide variety of useful numerical problems. At the end of the paper, we discussed the importance and educational benefits of classification of equations of motion.

Keywords: velocity-time graph, fundamental equations, additional equations, requisite conditions, importance and educational benefits

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Authors: Lev Idels, Arcady Ponosov

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Stochastic fractional differential equations have recently attracted considerable attention, as they have been used to model real-world processes, which are subject to natural memory effects and measurement uncertainties. Compared to conventional hereditary differential equations, one of the advantages of fractional differential equations is related to more realistic geometric properties of their trajectories that do not intersect in the phase space. In this report, a Peano-like existence theorem for nonlinear stochastic fractional differential equations is proven under very general hypotheses. Several specific classes of equations are checked to satisfy these hypotheses, including delay equations driven by the fractional Brownian motion, stochastic fractional neutral equations and many others.

Keywords: delay equations, operator methods, stochastic noise, weak solutions

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Authors: Gulgassyl Nugmanova, Zhanat Zhunussova, Kuralay Yesmakhanova, Galya Mamyrbekova, Ratbay Myrzakulov

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In this paper, we consider some integrable Heisenberg Ferromagnet Equations with self-consistent potentials. We study their Lax representations. In particular we derive their equivalent counterparts in the form of nonlinear Schr\"odinger type equations. We present the integrable reductions of the Heisenberg Ferromagnet Equations with self-consistent potentials. These integrable Heisenberg Ferromagnet Equations with self-consistent potentials describe nonlinear waves in ferromagnets with some additional physical fields.

Keywords: Heisenberg Ferromagnet equations, soliton equations, equivalence, Lax representation

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Authors: Ming-Yang Guo, Cheng-Xian Wu, Wei-Xiang Chen, Chun-Yuan Lin, Yen-Jen Lin, Ann-Shyn Chiang

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With more and more Drosophila Driver and Neuron images, it is an important work to find the similarity relationships among them as the functional inference. There is a general problem that how to find a Drosophila Driver image, which can cover a set of Drosophila Driver/Neuron images. In order to solve this problem, the intersection/union region for a set of images should be computed at first, then a comparison work is used to calculate the similarities between the region and other images. In this paper, three encoding schemes, namely Integer, Boolean, Decimal, are proposed to encode each image as a one-dimensional structure. Then, the intersection/union region from these images can be computed by using the compare operations, Boolean operators and lookup table method. Finally, the comparison work is done as the union region computation, and the similarity score can be calculated by the definition of Tanimoto coefficient. The above methods for the region computation are also implemented in the multi-core CPUs environment with the OpenMP. From the experimental results, in the encoding phase, the performance by the Boolean scheme is the best than that by others; in the region computation phase, the performance by Decimal is the best when the number of images is large. The speedup ratio can achieve 12 based on 16 CPUs. This work was supported by the Ministry of Science and Technology under the grant MOST 106-2221-E-182-070.

Keywords: Drosophila driver image, Drosophila neuron images, intersection/union computation, parallel processing, OpenMP

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Authors: A. W. Gbolagade, M. O. Olayiwola, K. O. Kareem

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In this paper, further to our result on recent paper on the solution of nonlinear advection equations, we present further results on the nonlinear nonhomogeneous advection equations using a modified variational iteration method.

Keywords: lagrange multiplier, non-homogeneous equations, advection equations, mathematics

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Authors: Chi Zhang, Jun Jiang

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Constitutive equations are very important in finite element (FE) modeling, and the accuracy of the material constants in the equations have significant effects on the accuracy of the FE models. A wide range of constitutive equations are available; however, fitting the material constants in the constitutive equations could be complex and time-consuming due to the strong non-linearity and relationship between the constants. This work will focus on the development of a set of unified MATLAB programs for fitting the material constants in the constitutive equations efficiently. Users will only need to supply experimental data in the required format and run the program without modifying functions or precisely guessing the initial values, or finding the parameters in previous works and will be able to fit the material constants efficiently.

Keywords: constitutive equations, FE modelling, MATLAB program, non-linear curve fitting

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Authors: Tarul Garg, P. N. Agrawal

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We propose to define the q-analogue of the α-Bernstein Kantorovich operators and then introduce the q-bivariate generalization of these operators to study the approximation of functions of two variables. We obtain the rate of convergence of these bivariate operators by means of the total modulus of continuity, partial modulus of continuity and the Peetre’s K-functional for continuous functions. Further, in order to study the approximation of functions of two variables in a space bigger than the space of continuous functions, i.e. Bögel space; the GBS (Generalized Boolean Sum) of the q-bivariate operators is considered and degree of approximation is discussed for the Bögel continuous and Bögel differentiable functions with the aid of the Lipschitz class and the mixed modulus of smoothness.

Keywords: Bögel continuous, Bögel differentiable, generalized Boolean sum, K-functional, mixed modulus of smoothness

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Authors: Yu-Kai Ting, Jia-Ying Tu, Chung-Chun Hsiao

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New physical insights into the nonlinear Lorenz equations related to flow resistance is discussed in this work. The chaotic dynamics related to Lorenz equations has been studied in many papers, which is due to the sensitivity of Lorenz equations to initial conditions and parameter uncertainties. However, the physical implication arising from Lorenz equations about convectional motion attracts little attention in the relevant literature. Therefore, as a first step to understand the related fluid mechanics of convectional motion, this paper derives the Lorenz equations again with different forced conditions in the model. Simulation work of the modified Lorenz equations without the viscosity or buoyancy force is discussed. The time-domain simulation results may imply that the states of the Lorenz equations are related to certain flow speed and flow resistance. The flow speed of the underlying fluid system increases as the flow resistance reduces. This observation would be helpful to analyze the coupling effects of different fluid parameters in a convectional model in future work.

Keywords: Galerkin method, Lorenz equations, Navier-Stokes equations, convectional motion

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Authors: Teoman Ozer, Ozlem Orhan

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This study deals with symmetry group properties and conservation laws of partial differential equations. We give a geometrical interpretation of notion of μ-prolongations of vector fields and of the related concept of μ-symmetry for partial differential equations. We show that these are in providing symmetry reduction of partial differential equations and systems and invariant solutions.

Keywords: λ-symmetry, μ-symmetry, classification, invariant solution

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Authors: Leila Motamed-Jahromi, Mohsen Hatami, Alireza Keshavarz

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This research aims at obtaining the equations of pulse propagation in nonlinear plasmonic waveguides created with As₂S₃ chalcogenide materials. Via utilizing Helmholtz equation and first-order perturbation theory, two components of electric field are determined within frequency domain. Afterwards, the equations are formulated in time domain. The obtained equations include two coupled differential equations that considers nonlinear dispersion.

Keywords: nonlinear optics, plasmonic waveguide, chalcogenide, propagation equation

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Authors: Fuziyah Ishak, Siti Norazura Ahmad

Abstract:

Volterra integro-differential equations appear in many models for real life phenomena. Since analytical solutions for this type of differential equations are hard and at times impossible to attain, engineers and scientists resort to numerical solutions that can be made as accurately as possible. Conventionally, numerical methods for ordinary differential equations are adapted to solve Volterra integro-differential equations. In this paper, numerical solution for solving Volterra integro-differential equation using extended trapezoidal method is described. Formulae for the integral and differential parts of the equation are presented. Numerical results show that the extended method is suitable for solving first order Volterra integro-differential equations.

Keywords: accuracy, extended trapezoidal method, numerical solution, Volterra integro-differential equations

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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: Ruchi, A. M. Acu, Purshottam N. Agrawal

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The purpose of this paper to introduce the Durrmeyer type modification of q-generalized-Bernstein operators which include the Bernstein polynomials in the particular α = 0. We investigate the rate of convergence by means of the Lipschitz class and the Peetre’s K-functional. Also, we define the bivariate case of Durrmeyer type modification of q-generalized-Bernstein operators and study the degree of approximation with the aid of the partial modulus of continuity and the Peetre’s K-functional. Finally, we introduce the GBS (Generalized Boolean Sum) of the Durrmeyer type modification of q- generalized-Bernstein operators and investigate the approximation of the Bögel continuous and Bögel differentiable functions with the aid of the Lipschitz class and the mixed modulus of smoothness.

Keywords: Bögel continuous, Bögel differentiable, generalized Boolean sum, Peetre’s K-functional, Lipschitz class, mixed modulus of smoothness

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Authors: Liliana O. Martínez, Juan E González, Manuel Ramírez-Aranda, Ana Cervantes-Herrera

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A serious game category mobile application called Math Dominoes is presented. The main objective of this applications is to strengthen the teaching-learning process of solving algebraic equations and is based on the board game "Double 6" dominoes. Math Dominoes allows the practice of solving first, second-, and third-degree algebraic equations. This application is aimed to students who seek to strengthen their skills in solving algebraic equations in a dynamic, interactive, and fun way, to reduce the risk of failure in subsequent courses that require mastery of this algebraic tool.

Keywords: algebra, equations, dominoes, serious games

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Authors: Arcady Ponosov., Ramazan Kadiev

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The work studies the global moment stability of solutions of systems of nonlinear differential Itô equations with delays. A modified regularization method (W-method) for the analysis of various types of stability of such systems, based on the choice of the auxiliaryequations and applications of the theory of positive invertible matrices, is proposed and justified. Development of this method for deterministic functional differential equations is due to N.V. Azbelev and his students. Sufficient conditions for the moment stability of solutions in terms of the coefficients for sufficiently general as well as specific classes of Itô equations are given.

Keywords: asymptotic stability, delay equations, operator methods, stochastic noise

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Authors: Kamel Al-Khaled

Abstract:

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: Khosrow Maleknejad, Yaser Rostami

Abstract:

In this paper, semi-orthogonal B-spline scaling functions and wavelets and their dual functions are presented to approximate the solutions of integro-differential equations.The B-spline scaling functions and wavelets, their properties and the operational matrices of derivative for this function are presented to reduce the solution of integro-differential equations to the solution of algebraic equations. Here we compute B-spline scaling functions of degree 4 and their dual, then we will show that by using them we have better approximation results for the solution of integro-differential equations in comparison with less degrees of scaling functions.

Keywords: ıntegro-differential equations, quartic B-spline wavelet, operational matrices, dual functions

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Authors: Kamel Al-Khaled

Abstract:

In this paper, numerical solutions for the nonlinear coupled Korteweg-de Vries, (abbreviated as KdV) equations are calculated by Sinc-collocation method. This approach is based on a global collocation method using Sinc basis functions. First, discretizing time derivative of the KdV equations by a classic finite difference formula, while the space derivatives are approximated by a $\theta-$weighted scheme. Sinc functions are used to solve these two equations. Soliton solutions are constructed to show the nature of the solution. The numerical results are shown to demonstrate the efficiency of the newly proposed method.

Keywords: Nonlinear coupled KdV equations, Soliton solutions, Sinc-collocation method, Sinc functions

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Authors: A. I. Ma’ali, R. B. Adeniyi, A. Y. Badeggi, U. Mohammed

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An error estimation of the integrated formulation of the Lanczos tau method for some class of ordinary differential equations was reported. This paper is concern with the generalization of tau approximants and their corresponding error estimates for some class of ordinary differential equations (ODEs) characterized by m + s =3 (i.e for m =1, s=2; m=2, s=1; and m=3, s=0) where m and s are the order of differential equations and number of overdetermination, respectively. The general result obtained were validated with some numerical examples.

Keywords: approximant, error estimate, tau method, overdetermination

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Authors: Divij Gupta

Abstract:

Pendular systems have a range of both mathematical and engineering applications, ranging from modelling the behaviour of a continuous mass-density rope to utilisation as Tuned Mass Dampers (TMD). Thus, it is of interest to study the differential equations governing the motion of such systems. Here we attempt to generalise these equations of motion for the plane compound pendulum with a finite number of N point masses. A Lagrangian approach is taken, and we attempt to find the generalised form for the Euler-Lagrange equations of motion for the i-th bob of the N -bob pendulum. The co-ordinates are parameterized as angular quantities to reduce the number of degrees of freedom from 2N to N to simplify the form of the equations. We analyse the form of these equations up to N = 4 to determine the general form of the equation. We also develop a MATLAB program to compute a solution to the system for a given input value of N and a given set of initial conditions.

Keywords: classical mechanics, differential equation, lagrangian analysis, pendulum

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