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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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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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.

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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, 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 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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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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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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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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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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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**.

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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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 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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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.

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.

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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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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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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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.