Search results for: iteration solutions

Authors: M. O. Olayiwola

Abstract:

Numerical solutions of the generalized Burger-Fisher are obtained using a Modified Variational Iteration Method (MVIM) with minimal computational efforts. The computed results with this technique have been compared with other results. The present method is seen to be a very reliable alternative method to some existing techniques for such nonlinear problems.

Keywords: burger-fisher, modified variational iteration method, lagrange multiplier, Taylor’s series, partial differential equation

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Authors: Kholod Mohammad Abualnaja

Abstract:

A new algorithm is presented to find some new iterative methods for solving nonlinear equations F(x)=0 by using the variational iteration method. The efficiency of the considered method is illustrated by example. The results show that the proposed iteration technique, without linearization or small perturbation, is very effective and convenient.

Keywords: variational iteration method, nonlinear equations, Lagrange multiplier, algorithms

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Authors: O. M. Bamigbola, A. A. Ibrahim

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Analysis of real life problems often results in linear systems of equations for which solutions are sought. The method to employ depends, to some extent, on the properties of the coefficient matrix. It is not always feasible to solve linear systems of equations by direct methods, as such the need to use an iterative method becomes imperative. Before an iterative method can be employed to solve a linear system of equations there must be a guaranty that the process of solution will converge. This guaranty, which must be determined a priori, involve the use of some criterion expressible in terms of the entries of the coefficient matrix. It is, therefore, logical that the convergence criterion should depend implicitly on the algebraic structure of such a method. However, in deference to this view is the practice of conducting convergence analysis for Gauss-Seidel iteration on a criterion formulated based on the algebraic structure of Jacobi iteration. To remedy this anomaly, the Gauss-Seidel iteration was studied for its algebraic structure and contrary to the usual assumption, it was discovered that some property of the iteration matrix of Gauss-Seidel method is only diagonally dominant in its first row while the other rows do not satisfy diagonal dominance. With the aid of this structure we herein fashion out an improved version of Gauss-Seidel iteration with the prospect of enhancing convergence and robustness of the method. A numerical section is included to demonstrate the validity of the theoretical results obtained for the improved Gauss-Seidel method.

Keywords: linear algebraic system, Gauss-Seidel iteration, algebraic structure, convergence

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Authors: Desmond Adair, Kairat Ismailov, Martin Jaeger

Abstract:

Modelling of Timoshenko beams on elastic foundations has been widely used in the analysis of buildings, geotechnical problems, and, railway and aerospace structures. For the elastic foundation, the most widely used models are one-parameter mechanical models or two-parameter models to include continuity and cohesion of typical foundations, with the two-parameter usually considered the better of the two. Knowledge of free vibration characteristics of beams on an elastic foundation is considered necessary for optimal design solutions in many engineering applications, and in this work, the efficient and accurate variational iteration method is developed and used to calculate natural frequencies of a Timoshenko beam on a two-parameter foundation. The variational iteration method is a technique capable of dealing with some linear and non-linear problems in an easy and efficient way. The calculations are compared with those using a finite-element method and other analytical solutions, and it is shown that the results are accurate and are obtained efficiently. It is found that the effect of the presence of the two-parameter foundation is to increase the beam’s natural frequencies and this is thought to be because of the shear-layer stiffness, which has an effect on the elastic stiffness. By setting the two-parameter model’s stiffness parameter to zero, it is possible to obtain a one-parameter foundation model, and so, comparison between the two foundation models is also made.

Keywords: Timoshenko beam, variational iteration method, two-parameter elastic foundation model

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Authors: R. B. Ogunrinde, C. C. Jibunoh

Abstract:

In this paper, a spectral decomposition method is developed for the direct integration of stiff and nonstiff homogeneous linear (ODE) systems with linear, constant, or zero right hand sides (RHSs). The method does not require iteration but obtains solutions at any random points of t, by direct evaluation, in the interval of integration. All the numerical solutions obtained for the class of systems coincide with the exact theoretical solutions. In particular, solutions of homogeneous linear systems, i.e. with zero RHS, conform to the exact analytical solutions of the systems in terms of t.

Keywords: spectral decomposition, linear RHS, homogeneous linear systems, eigenvalues of the Jacobian

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Authors: Soyoon Bak, Sunyoung Bu, Philsu Kim

Abstract:

In this paper, a backward semi-Lagrangian scheme combined with the second-order backward difference formula is designed to calculate the numerical solutions of nonlinear advection-diffusion equations. The primary aims of this paper are to remove any iteration process and to get an efficient algorithm with the convergence order of accuracy 2 in time. In order to achieve these objects, we use the second-order central finite difference and the B-spline approximations of degree 2 and 3 in order to approximate the diffusion term and the spatial discretization, respectively. For the temporal discretization, the second order backward difference formula is applied. To calculate the numerical solution of the starting point of the characteristic curves, we use the error correction methodology developed by the authors recently. The proposed algorithm turns out to be completely iteration-free, which resolves the main weakness of the conventional backward semi-Lagrangian method. Also, the adaptability of the proposed method is indicated by numerical simulations for Burgers’ equations. Throughout these numerical simulations, it is shown that the numerical results are in good agreement with the analytic solution and the present scheme offer better accuracy in comparison with other existing numerical schemes. Semi-Lagrangian method, iteration-free method, nonlinear advection-diffusion equation, second-order backward difference formula

Keywords: Semi-Lagrangian method, iteration free method, nonlinear advection-diffusion equation, second-order backward difference formula

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Authors: O. Acan, Y. Keskin

Abstract:

In this study, we will carry out a comparative study between the reduced differential transform method, the adomian decomposition method, the variational iteration method and the homotopy analysis method. These methods are used in many fields of engineering. This is been achieved by handling a kind of 2-Dimensional and forth-order partial differential equations called the Kuramoto–Sivashinsky equations. Three numerical examples have also been carried out to validate and demonstrate efficiency of the four methods. Furthermost, it is shown that the reduced differential transform method has advantage over other methods. This method is very effective and simple and could be applied for nonlinear problems which used in engineering.

Keywords: reduced differential transform method, adomian decomposition method, variational iteration method, homotopy analysis method

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Authors: Alemayehu Geremew Geremew

Abstract:

We prove the weak convergence of Mann iteration for a hybrid pair of maps to a common fixed point of a selfmap f and a multivalued f nonexpansive mapping T in Banach space E.

Keywords: common fixed point, Mann iteration, multivalued mapping, weak convergence

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Authors: Yixun Shi

Abstract:

Many interior point methods for convex programming solve an (n+m)x(n+m)linear system in each iteration. Many implementations solve this system in each iteration by considering an equivalent mXm system (4) as listed in the paper, and thus the job is reduced into solving the system (4). However, the system(4) has to be solved exactly since otherwise the error would be entirely passed onto the last m equations of the original system. Often the Cholesky factorization is computed to obtain the exact solution of (4). One Cholesky factorization is to be done in every iteration, resulting in higher computational costs. In this paper, two iterative methods for solving linear systems using vector division are combined together and embedded into interior point methods. Instead of computing one Cholesky factorization in each iteration, it requires only one Cholesky factorization in the entire procedure, thus significantly reduces the amount of computation needed for solving the problem. Based on that, a hybrid algorithm for solving convex programming problems is proposed.

Keywords: convex programming, interior point method, linear systems, vector division

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Authors: Olayiwola Moruf Oyedunsi

Abstract:

This paper discusses the Modified Variational Iteration Method (MVIM) for the solution of nonlinear Burgers’ equation arising in longitudinal dispersion phenomena in fluid flow through porous media. The method is an elegant combination of Taylor’s series and the variational iteration method (VIM). Using Maple 18 for implementation, it is observed that the procedure provides rapidly convergent approximation with less computational efforts. The result shows that the concentration C(x,t) of the contaminated water decreases as distance x increases for the given time t.

Keywords: modified variational iteration method, Burger’s equation, porous media, partial differential equation

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Authors: David C. Ni

Abstract:

In the previous efforts, we constructed N-Body Systems by an extended Blaschke product (EBP), which represents a non-temporal and nonlinear extension of Lorentz transformation. In this construction, we rely only on two parameters, nonlinear degree, and relative momentum to characterize the systems. We further explored root computation via iteration with an algorithm extended from Jenkins-Traub method. The solution sets demonstrate a form of σ+ i [-t, t], where σ and t are the real numbers, and the [-t, t] shows various canonical distributions. In this paper, we correlate the convergent sets in the original domain with solution sets, which demonstrating large-deviation distributions in the codomain. We proceed to compare our approach with the formula or principles, such as Donsker-Varadhan and Wentzell-Freidlin theories. The deterministic model based on this construction allows us to explore applications in the areas of finance and statistical mechanics.

Keywords: nonlinear Lorentz transformation, Blaschke equation, iteration solutions, root computation, large deviation distribution, deterministic model

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Authors: Taysir Soliman

Abstract:

One of the current popular clustering techniques is Spectral Clustering (SC) because of its advantages over conventional approaches such as hierarchical clustering, k-means, etc. and other techniques as well. However, one of the disadvantages of SC is the time consuming process because it requires computing the eigenvectors. In the past to overcome this disadvantage, a number of attempts have been proposed such as the Power Iteration Clustering (PIC) technique, which is one of versions from SC; some of PIC advantages are: 1) its scalability and efficiency, 2) finding one pseudo-eigenvectors instead of computing eigenvectors, and 3) linear combination of the eigenvectors in linear time. However, its worst disadvantage is an inter-class collision problem because it used only one pseudo-eigenvectors which is not enough. Previous researchers developed Deflation-based Power Iteration Clustering (DPIC) to overcome problems of PIC technique on inter-class collision with the same efficiency of PIC. In this paper, we developed Parallel DPIC (PDPIC) to improve the time and memory complexity which is run on apache spark framework using sparse matrix. To test the performance of PDPIC, we compared it to SC, ESCG, ESCALG algorithms on four small graph benchmark datasets and nine large graph benchmark datasets, where PDPIC proved higher accuracy and better time consuming than other compared algorithms.

Keywords: spectral clustering, power iteration clustering, deflation-based power iteration clustering, Apache spark, large graph

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Authors: Nicholas D. Assimakis

Abstract:

In linear estimation, the traditional Kalman filter uses the Kalman filter gain in order to produce estimation and prediction of the n-dimensional state vector using the m-dimensional measurement vector. The computation of the Kalman filter gain requires the inversion of an m x m matrix in every iteration. In this paper, a variation of the Kalman filter eliminating the Kalman filter gain is proposed. In the time varying case, the elimination of the Kalman filter gain requires the inversion of an n x n matrix and the inversion of an m x m matrix in every iteration. In the time invariant case, the elimination of the Kalman filter gain requires the inversion of an n x n matrix in every iteration. The proposed Kalman filter gain elimination algorithm may be faster than the conventional Kalman filter, depending on the model dimensions.

Keywords: discrete time, estimation, Kalman filter, Kalman filter gain

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

Abstract:

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: Muhammad Younis

Abstract:

In this article, the novel (G′/G)-expansion method is successfully applied to construct the abundant travelling wave solutions to the modified Burgers’ equation with the aid of computation. The method is reliable and useful, which gives more general exact travelling wave solutions than the existing methods. These obtained solutions are in the form of hyperbolic, trigonometric and rational functions including solitary, singular and periodic solutions which have many potential applications in physical science and engineering. Some of these solutions are new and some have already been constructed. Additionally, the constraint conditions, for the existence of the solutions are also listed.

Keywords: traveling wave solutions, NLPDE, computation, integrability

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Authors: Magdy G. Asaad

Abstract:

Solitons are among the most beneficial solutions for science and technology for their applicability in physical applications including plasma, energy transport along protein molecules, wave transport along poly-acetylene molecules, ocean waves, constructing optical communication systems, transmission of information through optical fibers and Josephson junctions. In this talk, we will apply the bilinear technique to generate a class of soliton solutions to the (3+1)-dimensional nonlinear soliton equation of Jimbo-Miwa type. Examples of the resulting soliton solutions are computed and a few solutions are plotted.

Keywords: Pfaffian solutions, N-soliton solutions, soliton equations, Jimbo-Miwa

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Authors: Aydin Teymourifar, Gurkan Ozturk, Ozan Bahadir

Abstract:

NSGA-II is one of the most well-known and most widely used evolutionary algorithms. In addition to its new versions, such as NSGA-III, there are several modified types of this algorithm in the literature. In this paper, a hybrid NSGA-II algorithm has been suggested for solving the multi-objective flexible job shop scheduling problem. For a better search, new neighborhood-based crossover and mutation operators are defined. To create new generations, the neighbors of the selected individuals by the tournament selection are constructed. Also, at the end of each iteration, before sorting, neighbors of a certain number of good solutions are derived, except for solutions protected by elitism. The neighbors are generated using a constraint-based neural network that uses various constructs. The non-dominated sorting and crowding distance operators are same as the classic NSGA-II. A comparison based on some multi-objective benchmarks from the literature shows the efficiency of the algorithm.

Keywords: flexible job shop scheduling problem, multi-objective optimization, NSGA-II algorithm, neighborhood structures

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Authors: A. Zerarka, W. Djoudi

Abstract:

We use some fractional transformations to obtain many types of new exact solutions of nonlinear lattice equation. These solutions include rational solutions, periodic wave solutions, and doubly periodic wave solutions.

Keywords: fractional transformations, nonlinear equation, travelling wave solutions, lattice equation

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Authors: Chao-Qing Dai

Abstract:

Two families of exact travelling solutions for the discrete sine-Gordon equation are constructed based on the variable-coefficient Jacobian elliptic function method and different transformations. When the modulus of Jacobian elliptic function solutions tends to 1, soliton solutions can be obtained. Some soliton solutions degenerate into the known solutions in literatures. Moreover, dynamical properties of exact solutions are investigated. Our analysis and results may have potential values for certain applications in modern nonlinear science and textile engineering.

Keywords: exact solutions, variable-coefficient Jacobian elliptic function method, discrete sine-Gordon equation, dynamical behaviors

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Authors: Edamana Krishnan, Khalil Al-Ghafri

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In this paper, modified K(n,n) and K(n+1,n+1) equations have been solved using mapping methods which give a variety of solutions in terms of Jacobi elliptic functions. The solutions when m approaches 0 and 1, with m as the modulus of the JEFs have also been deduced. The role of constraint conditions has been discussed.

Keywords: travelling wave solutions, solitary wave solutions, compactons, Jacobi elliptic functions, mapping methods

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Authors: Armin Ardekani, Mohammad Akbari

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We present a method of generating series solutions to large classes of nonlinear differential equations. The method is well suited to be adapted in mathematical software and unlike the available commercial solvers, we are capable of generating solutions to boundary value ODEs and PDEs. Many of the generated solutions converge to closed form solutions. Our method can also be applied to systems of ODEs or PDEs, providing all the solutions efficiently. As examples, we present results to many difficult differential equations in engineering fields.

Keywords: computational mathematics, differential equations, engineering, series

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Authors: Hancy Issac, Gerben Keijzers, Ian Yang, Clint Moloney, Jackie Lea, Melissa Taylor

Abstract:

Introduction: Chronic obstructive pulmonary disease guideline non-adherence is associated with a reduction in health-related quality of life in patients (HRQoL). Improving guideline adherence has the potential to mitigate fragmented care thereby sustaining pulmonary function, preventing acute exacerbations, reducing economic health burdens, and enhancing HRQoL. The development of an electronic proforma stemming from expert consensus, including digital guideline resources and direct interdisciplinary referrals is hypothesised to improve guideline adherence and patient outcomes for emergency department (ED) patients with COPD. Aim: The aim of this study was to develop consensus among ED and respiratory staff for the correct composition of a COPD electronic proforma that aids in guideline adherence and management in the ED. Methods: This study adopted a mixed-method design to develop the most important indicators of care in the ED. The study involved three phases: (1) a systematic literature review and qualitative interdisciplinary staff interviews to assess barriers and solutions for guideline adherence and qualitative interdisciplinary staff interviews, (2) a modified Delphi panel to select interventions for the proforma, and (3) a consensus process through three rounds of scoring through a quantitative survey (ED and Respiratory consensus) and qualitative thematic analysis on each indicator. Results: The electronic proforma achieved acceptable and good internal consistency through all iterations from national emergency department and respiratory department interdisciplinary experts. Cronbach’s alpha score for internal consistency (α) in iteration 1 emergency department cohort (EDC) (α = 0.80 [CI = 0.89%]), respiratory department cohort (RDC) (α = 0.95 [CI = 0.98%]). Iteration 2 reported EDC (α = 0.85 [CI = 0.97%]) and RDC (α = 0.86 [CI = 0.97%]). Iteration 3 revealed EDC (α = 0.73 [CI = 0.91%]) and RDC (α = 0.86 [CI = 0.95%]), respectively. Conclusion: Electronic proformas have the potential to facilitate direct referrals from the ED leading to reduced hospital admissions, reduced length of hospital stays, holistic care, improved health care and quality of life and improved interdisciplinary guideline adherence.

Keywords: COPD, electronic proforma, modified delphi study, interdisciplinary, guideline adherence, COPD-X plan

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Authors: E. V. Krishnan

Abstract:

In this paper, we employ mapping methods to construct exact travelling wave solutions for a modified Korteweg-de Vries equation. We have derived periodic wave solutions in terms of Jacobi elliptic functions, kink solutions and singular wave solutions in terms of hyperbolic functions.

Keywords: travelling wave solutions, Jacobi elliptic functions, solitary wave solutions, Korteweg-de Vries equation

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Authors: Aziza Altaibayeva, Ertan Güdekli, Ratbay Myrzakulov

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In this letter, we explore exact solutions for the Horava-Lifshitz gravity. We use of an extension of this theory with first order dynamical lapse function. The equations of motion have been derived in a fully consistent scenario. We assume that there are some spherically symmetric families of exact solutions of this extended theory of gravity. We obtain exact solutions and investigate the singularity structures of these solutions. Specially, an exact solution with the regular horizon is found.

Keywords: quantum gravity, Horava-Lifshitz gravity, black hole, spherically symmetric space times

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Authors: Song Ni, Junxiang Xu

Abstract:

This paper concerns quasi-periodic perturbations with parameters of 2-dimensional degenerate systems. If the equilibrium point of the unperturbed system is elliptic-type degenerate. Assume that the perturbation is real analytic quasi-periodic with diophantine frequency. Without imposing any assumption on the perturbation, we can use a path of equilibrium points to tackle with the Melnikov non-resonance condition, then by the Leray-Schauder Continuation Theorem and the Kolmogorov-Arnold-Moser technique, it is proved that the equation has a small response solution for many sufficiently small parameters.

Keywords: quasi-periodic systems, KAM-iteration, degenerate equilibrium point, response solution

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Authors: Seyed Amin Vakili, Sahar Sadat Vakili, Seyed Ehsan Vakili, Nader Abdoli Yazdi

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The main purpose of this paper is to present the Modified Newmark Method as a method of non-linear frame analysis by considering the effect of the axial load (second order analysis). The discussion will be restricted to plane frameworks containing a constant cross-section for each element. In addition, it is assumed that the frames are prevented from out-of-plane deflection. This part of the investigation is performed to generalize the established method for the assemblage structures such as frameworks. As explained, the governing differential equations are non-linear and cannot be formulated easily due to unknown axial load of the struts in the frame. By the assumption of constant axial load, the governing equations are changed to linear ones in most methods. Since the modeling and the solutions of the non-linear form of the governing equations are cumbersome, the linear form of the equations would be used in the established method. However, according to the ability of the method to reconsider the minor omitted parameters in modeling during the solution procedure, the axial load in the elements at each stage of the iteration can be computed and applied in the next stage. Therefore, the ability of the method to present an accurate approach to the solutions of non-linear equations will be demonstrated again in this paper.

Keywords: nonlinear, stability, buckling, modified newmark method

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Authors: Wided Batita, Stéphane Roche, Claude Caron

Abstract:

Geodesign is an emergent term related to a new and complex process. Hence, it needs to rethink tools, technologies and platforms in order to efficiently achieve its goals. A few tools have emerged since 2010 such as CommunityViz, GeoPlanner, etc. In the era of Web 2.0 and collaboration, WikiGIS has been proposed as a new category of tools. In this paper, we present WikiGIS functionalities dealing mainly with the iteration and traceability management to support the collaboration of the Geodesign process. Actually, WikiGIS is built on GeoWeb 2.0 technologies —and primarily on wiki— and aims at managing the tracking of participants’ editing. This paper focuses on a simplified simulation to illustrate the strength of WikiGIS in the management of traceability and in the access to history in a Geodesign process. Indeed, a cartographic user interface has been implemented, and then a hypothetical use case has been imagined as proof of concept.

Keywords: geodesign, history, traceability, tracking of participants’ editing, WikiGIS

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Authors: Liang Liu, Xiaopeng Luo

Abstract:

In this paper, we propose an accelerated stochastic gradient method with momentum. The momentum term is the weighted average of generated gradients, and the weights decay inverse proportionally with the iteration times. Stochastic gradient descent with momentum (SGDM) uses weights that decay exponentially with the iteration times to generate the momentum term. Using exponential decay weights, variants of SGDM with inexplicable and complicated formats have been proposed to achieve better performance. However, the momentum update rules of our method are as simple as that of SGDM. We provide theoretical convergence analyses, which show both the exponential decay weights and our inverse proportional decay weights can limit the variance of the parameter moving directly to a region. Experimental results show that our method works well with many practical problems and outperforms SGDM.

Keywords: exponential decay rate weight, gradient descent, inverse proportional decay rate weight, momentum

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Authors: P. K. Singhal, R. Naresh, V. Sharma

Abstract:

This paper presents the three optimization models, namely New Binary Artificial Bee Colony (NBABC) algorithm, NBABC with Local Search (NBABC-LS), and NBABC with Genetic Crossover (NBABC-GC) for solving the Wind-Thermal Unit Commitment (WTUC) problem. The uncertain nature of the wind power is incorporated using the Weibull probability density function, which is used to calculate the overestimation and underestimation costs associated with the wind power fluctuation. The NBABC algorithm utilizes a mechanism based on the dissimilarity measure between binary strings for generating the binary solutions in WTUC problem. In NBABC algorithm, an intelligent scout bee phase is proposed that replaces the abandoned solution with the global best solution. The local search operator exploits the neighboring region of the current solutions, whereas the integration of genetic crossover with the NBABC algorithm increases the diversity in the search space and thus avoids the problem of local trappings encountered with the NBABC algorithm. These models are then used to decide the units on/off status, whereas the lambda iteration method is used to dispatch the hourly load demand among the committed units. The effectiveness of the proposed models is validated on an IEEE 10-unit thermal system combined with a wind farm over the planning period of 24 hours.

Keywords: artificial bee colony algorithm, economic dispatch, unit commitment, wind power

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Authors: Hamdy M. Youssef, Eman A. Al-Lehaibi

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This paper presents an iteration method for the numerical solutions of a one-dimensional problem of generalized thermoelasticity with one relaxation time under given initial and boundary conditions. The thermoelastic material with variable properties as a power functional graded has been considered. Adomian’s decomposition techniques have been applied to the governing equations. The numerical results have been calculated by using the iterations method with a certain algorithm. The numerical results have been represented in figures, and the figures affirm that Adomian’s decomposition method is a successful method for modeling thermoelastic problems. Moreover, the empirical parameter of the functional graded, and the lattice design parameter have significant effects on the temperature increment, the strain, the stress, the displacement.

Keywords: Adomian, decomposition method, generalized thermoelasticity, algorithm

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