Top PDF I. Numerical solution of exact pair equations. II. Numerical solution of first-order pair equations

I. Numerical solution of exact pair equations. II. Numerical solution of first-order pair equations

I. Numerical solution of exact pair equations. II. Numerical solution of first-order pair equations

The results for the ground states are superior to previously reported values~ The coupled equations resulting from the partial wave expansion of the exact helium atom wavefunction were s[r]

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Numerical Solution of First Order Ordinary Differential Equations

Numerical Solution of First Order Ordinary Differential Equations

Picard’s and Taylor’s series methods are powerful mathematical tools for solving linear and nonlinear differential equations. It is concluded that Picard’s and Taylor’s series methods gives more accurate solutions, which are much closer to exact solutions, for solving first order differential equations arising in some applications of sciences and engineering.

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Numerical solution of the first order linear fuzzy differential equations using He$’$s variational iteration method

Numerical solution of the first order linear fuzzy differential equations using He$’$s variational iteration method

In this section, the exact solutions and approximated solutions obtained by He’s variational iteration method and Leapfrog method. To show the efficiency of the He’s variational it- eration method, we have considered the following problem taken from C. Duraisamy and B. Usha [4] and T.Jayakumar, D.Maheskumar and K.Kanagarajan [14], with step size r = 0.1 along with the exact solutions.

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Numerical Solution of Sixth Order Differential Equations Arising in Astrophysics by Neural Network

Numerical Solution of Sixth Order Differential Equations Arising in Astrophysics by Neural Network

Lee & Kang [24] initially introduced a method to solve first order differential equation using Hopfield neural network models. Then, another approach by Meade & Fernandez [25, 26] was proposed for both linear and non-linear differential equations using Splines and feed forward neural network. Artificial neural networks based on Broyden-Fletcher-Goldfarb-Shanno (BF GS) optimization technique for solving ordinary and partial differential equations have been excellently presented by Lagaris et al. [27]. Furthermore, Lagaris et al. [28] investigated neural network meth- ods for boundary value problems with irregular boundaries. Parisi et al. [29] presented unsupervised feed forward neural networks for the solution of differential equations. The potential of the hybrid and optimization technique to deal with differential equation of lower order as well as higher order has been presented by Malek & Shekari Beidokhti [30]. Choi & Lee [31] discussed comparison of generalizing ability on solving differential equation using back propagation and reformulated radial basis function network. Yazdi et al. [32] used unsupervised kernel least mean square algorithm for solving ordinary differential equations. A new algorithm for solving matrix Riccati differential equations has been developed and employed by Selvaraju & Abdul Samant [33]. He et al. [34] investigated a class of partial differential equations using multi- layer neural network. Kumar & Yadav [35] surveyed multilayer perceptrons and radial basis function neural network methods for the solution of differential equations. Tsoulos et al. [36] solved differential equations with neural networks using a scheme that worked on the basis of grammatical evolution. Numerical solution of elliptic partial differential equation using radial basis function neural networks has been presented by Jianyu et al. [37]. Shirvany et al. [38] proposed multilayer perceptron and radial basis function (RBF) neural networks with a new unsupervised training method for numerical solution of partial differential equations. Mai-Duy & Tran-Cong [39] discussed numerical solution of differential equations using multiquadric radial basis function networks. Fuzzy linguistic model in neural network to solve differential equations is presented by Leephakpreeda [40]. Franke & Schaback [41] solved partial differential equations by collocation using radial basis functions. Smaoui & Al-Enezi [42] presented the dynamics of two non-linear partial differential equations using artificial neural networks. Differential equations with genetic programming have been analyzed by Tsoulos & Lagaris [43]. McFall & Mahan [44] used artificial neural network for solution of boundary value problems with exact satisfaction of arbitrary boundary conditions. Hoda & Nagla [45] solved mixed boundary value problems using multilayer perceptron neural network method.
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A Numerical Solution of First order Simultaneous Fuzzy Differential Equations by Sixth Order Runge Kutta Method

A Numerical Solution of First order Simultaneous Fuzzy Differential Equations by Sixth Order Runge Kutta Method

 , t  I, 0 <   I provided the equation defines fuzzy number as in [11]. Similarly, let I be a real interval. A mapping z: IE is called a fuzzy process and its  - level set is denoted by   z   t    z  t , y , z   , z t , y , z   , t  I, 0 <   I. For u, v  E and   , the u + v and the product u can be defined by [u + v]  = [u]  + [v]  and [u]  = [u]  , where   [0, 1] and [u]  + [v]  means the addition of two intervals of  and
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The Adomian decomposition method for numerical solution of first-order differential equations

The Adomian decomposition method for numerical solution of first-order differential equations

The main advantage of this method is that it can be applied directly to all types of differential and integral equations, linear or non-linear, homogeneous or inhomogeneous, with constant or variable coefficients. Another important advantage is that, the method is capable of greatly reducing the size of computational work while still maintaining high accuracy of the numerical solution [2]. The ADM decomposes a solution into an infinite series which converges rapidly to the exact solution. The convergence of the ADM has been investigated by a number of authors [3, 4].
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Numerical Solution of First Order Ordinary Differential Equations

Numerical Solution of First Order Ordinary Differential Equations

Picard’s and Taylor’s series methods are powerful mathematical tools for solving linear and nonlinear differential equations. It is concluded that Picard’s and Taylor’s series methods gives more accurate solutions, which are much closer to exact solutions, for solving first order differential equations arising in some applications of sciences and engineering.

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The numerical solution of boundary value problems in partial differential equations

The numerical solution of boundary value problems in partial differential equations

where u Is the vector of values u(x,t) at the nodes x. « £ , (i ■ 0,1,.. • .,N; Nh « 1), ln is a vector Involving the boundary conditions at t = nk, and U is a matrix of onir N«d. It la well known (e.g. 3j' that the systen (17) of first order ordinary differential equations, is asymptotically stable (in the Liapunov sense) if and only if 1° is bounded as n -* «, and the matrix U is positive seoi-definite; and that the aero vector, £ , is a stable solution (i.e. all solutions tend to sero as t tends to If and only If ln-» 0, aa n •* «, and U la positive definite. Thus, whatever difference replacement is used for -|~a, the resulting matrix U will be positive semi-definita only when the solutions of the differential equation are bounded. If some implicit formula is used now, to replaoe the time
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Numerical solution of two-dimensional integral equations of the first kind by multi-step methods

Numerical solution of two-dimensional integral equations of the first kind by multi-step methods

In general, the integral equations of the first kind are ill-posed problems, that is, a small perturbation in the given data makes a large perturbation in the solution [5]. These equations in two-dimensional case have many interesting applications, for example in mechanic, physics and other applied sciences [11]. But many of these equations can not be usually solved analytically and they should be solved by nu- merical methods. Therefore, giving suitable numerical methods for these equations is very worthwhile. Recently, many researchers have studied two-dimensional inte- gral equations. For example, homotopy analysis method, the method based on the piecewise approximation by Chebyshev polynomials and wavelet method have been presented in [1], [6] and [13], respectively. In [4], the nonlinear Volterra-Fredholm integral equations have been solved by collocation methods based on polynomials of spline spaces. In [15], an Euler-type method has been presented for 2D-VIEs of the first kind. In [12], an adaptive multi-scale moment method has been proposed for solving two-dimensional Fredholm integral equations (2D-FIEs) of the first kind. The authors of [17, 18] have developed the well-known Tau method to solve linear and
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Numerical solution of multi-order fractional differential equations via the sinc collocation method

Numerical solution of multi-order fractional differential equations via the sinc collocation method

In this paper, the sinc collocation method is proposed for solving linear and nonlinear multi-order fractional differential equations based on the new definition of fractional derivative which is recently presented by Khalil, R., Al Horani, M., Yousef, A. and Sababeh, M. in A new definition of fractional derivative, J. Comput. Appl. Math. 264 (2014), 65–70. The properties of sinc functions are used to reduce the fractional differential equation to a system of algebraic equations. Several numerical examples are provided to illustrate the accuracy and effectiveness of the presented method.
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Numerical Solution of Linear Ordinary Differential Equations of Higher Order by Differential Transformation Method

Numerical Solution of Linear Ordinary Differential Equations of Higher Order by Differential Transformation Method

The comparison in between the exact solution and its approximate solution in Examples 4.1,4.2, 4.3, 4.4 obtained with the help of Method of variation of parameters and DTM. From the numerical results, it is clear that the DTM is efficient and accurate. By increasing the order of approximation more accuracy can be obtained. The results are also expressed graphically in Figures. The Blue line represents the curve corresponding to the exact solution whereas the Red line corresponds to the approximate solution.

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Numerical solution of fractional partial differential equations by numerical Laplace inversion technique

Numerical solution of fractional partial differential equations by numerical Laplace inversion technique

based on Laguerre polynomial series expansion of the inverse function under the assump- tion that the Laplace transform is known on the real axis only. The main contribution of the paper is to provide computable estimates of truncation, discretization, conditioning and roundoff errors introduced by numerical computations. In the present work, we apply the Stehfest [] algorithm for numerical inversion of Laplace transform.

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Load Balancing for the Numerical Solution of the Navier-Stokes Equations

Load Balancing for the Numerical Solution of the Navier-Stokes Equations

Abstract. In this paper we simulate the performance of a load balancing scheme. In particular, we study the application of the Extrapolated Diffusion (EDF) method for the efficient parallelization of a simple ‘atmospheric’ model. Our model involves the numerical solution of the steady state Navier-Stokes (NS) equations in the horizontal plane and random load values, corresponding to the “physics” computations, in the vertical plane. For the numerical solution of NS equations we use the Local Modified Successive Overrelaxation (LMSOR) method with local parameters thus avoiding the additional cost caused by the global communication of the involved parameter ω in the classical SOR method. We have implemented an efficient domain decomposition technique by using a larger number of processors in the areas of the domain with heavier work load. Our results show that in certain cases we have a gain as much as approximately 45% in execution time when our load balancing scheme is applied.
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The Numerical Solution of the Fredholm Integral Equations of the Second Kind

The Numerical Solution of the Fredholm Integral Equations of the Second Kind

The analytical methods are the degenerate kernel methods,the Adomain decomposition method, the modified decomposition method and the method of successive approximations. Moreover, we have used the following numerical methods: Projection methods including collocation method and Galerkin method, Degenerate kernel approximation methods and Nystr ̈m methods, for approximating the solution of the Fredholm integral equations. The have presented each numerical method as algorithm and applied these algorithms on the same Freedholm integral equation using Matlab Software; we have found that the numerical solution was approximately as the exact solution. The absolute error has approached zero which was shown that numerical results were acceptable.
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An Algorithm for the Numerical Solution of System of Fractional Differential Equations

An Algorithm for the Numerical Solution of System of Fractional Differential Equations

v  0   ,   1 , where D  u is the derivate of u of order  , D  v is the derivative of v of order  in the sense of Caputo. The algorithm is based on the fractional s s method.

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Numerical Solution of the Navier-Stokes Equations at High Reynolds Numbers

Numerical Solution of the Navier-Stokes Equations at High Reynolds Numbers

We are presenting two efficient numerical schemes for solving the Navier-Stokes in the Stream function-vorticity formulation. The idea of the fixed point iterative method was to work with a symmetric positive definite matrix (matrix A resulting from the discretization of the Laplacian term). This method showed to be robust enough to handle high Reynolds numbers, but computing time was, in general, very large. That is why we seek to reduce computing time, and implemented the second method.

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Analytical and numerical solution of differential equations with generalized fuzzy derivative

Analytical and numerical solution of differential equations with generalized fuzzy derivative

Nowadays, fuzzy differential equations (FDEs) is a popular topic studied by many researchers since it is utilized widely for the purpose of modeling problems in science and engineering. Most of the practical problems require the solution of a FDE which satisfies fuzzy initial or fuzzy boundary conditions, therefore, a fuzzy initial or fuzzy boundary problem should be solved. However, many fuzzy initial or fuzzy boundary value problems could not be solved exactly, sometimes it is even impossible to find their analytical solutions. Thus, considering their approximate solutions is becoming more important [1].
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A Numerical Method -High Accuracy Solution to Singular Differential Equations

A Numerical Method -High Accuracy Solution to Singular Differential Equations

The effectiveness of the proposed method is illustrated by considering two numerical examples. Thehigh degree B-spline basis function is usedin collocation method and comparedwith lessdegree B-spline basis function. The high accuracy is achieved by raising the degree of the basis function very wellwith less interpolating points and performance of the method with high degree basis function at nearby singular points is as ordinary points. Theconvergence rate is high when used low degree B-spline basis function in collocation methodand time is savedbecause more interpolating points are required to get the same level of accuracywhen used lowdegree B-spline basis function in collocation method..This method may be applied to different types of some more singular boundary value problems for its efficiency
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Numerical Solution for Solving a System of Fractional Integro-differential Equations

Numerical Solution for Solving a System of Fractional Integro-differential Equations

We implemented the suggested method with m = 16 and m = 32. The obtained numerical results are shown in Table I and Figs 1-4. In Table I, the absolute error between the exact solution and the approximate solution, at m = 16 (in columns 2,3) and m = 32 (in columns 4,5) respectively, are given. Figs. 1 and 2 show the evolution results for the system of fractional integro differential Eqs. (22) at m = 32 when α = 1. And Figs. 2 and 4 show the behavior of obtained approximate solution for the proposed system (22) at m = 32 with different values of α. From Table I and Figs. 1-2 we can conclude that our approximate solutions are in good agreement with the exact values and with high accuracy in comparison with the approximate solution obtained in [38].
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Numerical Solution of Higher Order Linear Fuzzy Differential Equations using Generalized STWS Technique

Numerical Solution of Higher Order Linear Fuzzy Differential Equations using Generalized STWS Technique

STWS technique was introduced by Rao et al. [10]. Balachandran and Murugesan applied STWS technique to solve first order system of IVPs[11, 12]. Murugesan and Paul Dhayabaran extended STWS technique for solving second order singular system of IVPs[13]. Emimalet al.[14] proposed the generalized STWS Technique to solve system of IVPs of any order ‘n’ with ‘p’ variables.

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