Piecewise Quartic, **Quintic** and higher degree **spline** are popular for smooth and best approximation see Deboor [1]. Interpolation by lower degree **spline** are widely used in the method of Piecewise Polynomial approximation to represent a non analytic funciton. Interpolation by **Quintic** and higher degree **spline**, the maximum error between a function and its interpolant can be controlled by mesh spacing and such function have no corner at the joint of two pieces and therefore no more data are required than lower order method to get desire accuracy. Therefore, **quintic** and higher degree **spline** are useful than lower degree **spline** for best approximation See [1]. In the direction of more higher degree **spline** Jianzhone and Hung [9] have obtained optimal error bounds for quartic and **quintic** interpolatory splines (see also Howell and Verma) [2] Wong and Agrawal [5], Gmelling Meling [6] have obtained explicit error bounds for **quintic** and biquintic interpolatory **spline** Dubey [10] has obtained existence, uniqueness and error bound for higher degree **spline** (See also Rana and Dubey) [8]. In the present paper we shall obtain precise error estimate of deficient **quintic** **spline** interpolation matching the given function values at mesh points and its derivative at two interior points of the interval.

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This paper presents a novel approach to numerical solution of a class of fourth-order time fractional partial diﬀerential equations (PDEs). The ﬁnite diﬀerence formulation has been used for temporal discretization, whereas the space discretization is achieved by means of non-polynomial **quintic** **spline** method. The proposed algorithm is proved to be stable and convergent. In order to corroborate this work, some test problems have been considered, and the computational outcomes are compared with those found in the exiting literature. It is revealed that the presented scheme is more accurate as compared to current variants on the topic.

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The purpose of this article is to present a technique for the numerical solution of Caputo time fractional superdiﬀusion equation. The central diﬀerence approximation is used to discretize the time derivative, while non-polynomial **quintic** **spline** is employed as an interpolating function in the spatial direction. The proposed method is shown to be unconditionally stable and O(h 4 + t 2 ) accurate. In order to check the feasibility of the proposed technique, some test examples have been considered and the simulation results are compared with those available in the existing literature.

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In this paper parametric **quintic** **spline** method employed for finding the extremum of a functional over the specified domain. The main pur- pose is to find the solution of boundary value problems which arise from the variational problems. The parametric **quintic** **spline** method reduce the computation of boundary value problems to some algebraic equations. The proposed scheme is simple and computationally attractive. Applica- tions are demonstrated through illustrative examples.

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We applied our method and compared the results with those obtained in the **quintic** **spline** method at grid points [5,8], the quartic **spline** method at o-step points [6] and the nite dierence method [9]. The results in Table 2 show that our method is giving better accuracy.

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order convergent for arbitrary , , p, r and s. In this article, rst, the consistency relation of our non-polynomial **quintic** **spline** in [17] is used for the solution of Equations 1. Then, the methods and the development of boundary conditions are described and classes of the methods are discussed. Following that a new approach for convergence analysis is presented. Here, we obtained the restriction on function g only. Finally, some numerical evidence is included to show the practical applicability and superiority of our meth- ods.

In this paper we used a nonpolynomial **Quintic** **spline** function to develop numerical algorithms of system of nonlinear third order boundary value problems. Here the result obtained by our algorithm is better than that ob- tained by some other method as compared in Tables 1

The phase plot in Figure 4 exhibits the behavior of the oscillator when 1, 1, 1 and 1 . it is quasi-periodic. Five equilibrium points are observed. There are asymptotically stable spirals at 0. 786, 0 and 1.272, 0 while a saddle exists at (0, 0). As men- tioned above, this situation is possible due to the addition of the **quintic** term. This situation also cannot occur in the cubic Duffing oscillator.

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Each relative quadratic extension of a **quintic** ﬁeld will be given by second degree polynomial with coeﬃcients in the subﬁeld. To construct such polynomials we make call to the geometry of number methods which allow us to ﬁnd bounds for the coeﬃcients of the searched polynomi- als. The basic tool allowing us to construct explicitly all

It is known that isochronism of a center of a planar polynomial system is equivalent to the existence of an analytic transversal system commuting with a given system in a neighborhood of[r]

The geometric interpretation of the tangent cone gives a method for classifying singularities. The first coarse classification of singularities that arise in **quintic** three- folds is to classify singularties by describing their tangent cones. The higher the degree of the tangent cone the more complicated the singularity. Unfortunately clas- sifying tangent cones is not enough to completely describe the singularities that arise in **quintic** threefolds. The decomposition (2.2) can also give some finer data about intersection multiplicities. As stated earlier, the zero locus of f i give the set of points

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The partial diﬀerential equation which is the **quintic** nonlinear equation of motion is in- teresting. The nonlinear vibration of Euler-Bernoulli beam has been explained by Zhang et al. [], whereas the governing equation which was studied by Sedighi et al. [] is studied in this paper. A hinged-hinged ﬂexible beam model with length subjected to constant axial load is simpliﬁed to the **quintic** nonlinear equation of motion. This model is studied only in the fundamental transverse mode. The interaction between transverse and longitudinal vibrations is neglected to be an assumption.

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Curvature profile in Figure 2 is almost linearly increasing until 𝑡 = 0.85, where it starts to have a flat value until 𝑡 = 1. This shows that the curve is entering the circular curve of the circle at the end of the curvature. Validation process is made by providing curvature derivative distributions of this spiral curve in Figure 3 to ensure that it has only one sign from 𝑡 = 0 to 𝑡 = 1. For Sections 4, 5, and 6, we will used two pieces of **quintic** trigonometric Bézier spiral curves that were developed earlier to construct another three templates. Therefore, the general equation for two pieces of spiral curves can be defined as follows

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We consider a new class of processes, called LG processes, defined as linear combinations of independent gamma processes. Their distribu- tional and path-wise properties are explored by following their relation to polynomial and Dirichlet (B-) splines. In particular, it is shown that the density of an LG process can be expressed in terms of Dirichlet (B-) splines, introduced independently by Ignatov and Kaishev (1987, 1988, 1989a,b) and Karlin et al. (1986). We further show that the well known variance-gamma (VG) process, introduced by Madan and Seneta (1990), and the Bilateral Gamma (BG) process, recently considered by K¨ uchler and Tappe (2008) are special cases of an LG process. Following this LG interpretation, we derive new (alternative) expressions for the VG and BG densities and consider their numerical properties. The LG process has two sets of parameters, the B-**spline** knots and their multiplicities, and offers further flexibility in controlling the shape of the Levy density, compared to the VG and the BG processes. Such flexibility is often desir- able in practice, which makes LG processes interesting for financial and insurance applications.

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Character 2, Piecewise smooth polynomial surface. Rotating B-**spline** surfaces in each sub-region is on the parameter polynomial surface. Rotating B-**spline** surfaces of k polynomial surface of parameter r in each sub-region [ , r r j j 1 ] . So the rotating B-**spline** surfaces is a segmented polynomial surface. The increase each curves for corresponding to increase the surfaces.

• Numerical results show that our approximations of unknown functions using some the B-**spline** func- tions in collocation method are more accurate than the numerical results of the radial basis function method, also the execution time in the cubic B-**spline** method is faster than the other methods. • Numerical examples also verified the efficiency and accuracy of the method that can be obtained

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The orthonormalization process of the basic **spline** ba- sis is performed with the classical Gram-Schmidt method on each bounded intervals of the initial sequence. The pa- per provided a generalization of the orthonormal **spline** scal- ing and wavelet bases construction whatever the degree of the **spline** function. Our study proves that the scaling and wavelet functions are not, respectively, given by dilating and translating a unique prototype function as in the traditional case. The traditional filter banks are replaced by a set of filters depending on the localization of the samples in the sequence. When the degree of the **spline** function increases, the number of freedom degrees increase oﬀering thus flexibility in the de- sign of the wavelet functions. It is possible to ensure desirable features such as the continuity of the wavelet function and its successive derivatives. The complete process of decomposing and reconstructing a signal irregularly sampled is provided. The orthogonal decomposition, applied to signals irregularly subsampled, shows that the traditional multiresolution anal- ysis behaviour is respected.

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The selection of λ in smoothing **spline** is a simpler procedure than the knots selections procedure for regression **spline** because it only involves an optimization of one parameter in (0, +∞). Even in additive smoothing **spline** models, the optimization is to search over (0, +∞) 1 ×(0, +∞) 2 ×...× (0, +∞) d . In mean smoothing splines, GCV(Generalized Cross Validation) is often used as the criterion to search for the optimal λ because of its computational efficiency. However, in quantile regression, GCV is not motivated. Though later Yuan [2006] proposed GACV(Generalized Approximate Cross-Validation) for quantile regression, the approximation is not working very well when estimating extreme quantiles. Instead, SIC (Schwarz Information Criterion) proposed in Koenker et al. [1994] was widely applied in many VaR estimations. Its definition is given in equation (5.2) while the model dimension R = p λ is the number of interpolated data points,

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Usually,B-splines with order four (degree three) are used in the calculation. Along the x direction we set the knots , ' , … , ) , and the knots at both ends are four-folded.So, the total number of basis B-splines will be * 4 .The univariate **spline** functions are shown in Fig. 1,where ξ ,