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1.
R. P. Tewarson 《BIT Numerical Mathematics》1980,20(2):223-232
Cubic splines on splines and quintic spline interpolations are used to approximate the derivative terms in a highly accurate scheme for the numerical solution of two-point boundary value problems. The storage requirement is essentially the same as for the usual trapezoidal rule but the local accuracy is improved fromO(h
3) to eitherO(h
6) orO(h
7), whereh is the net size. The use of splines leads to solutions that reflect the smoothness of the slopes of the differential equations. 相似文献
2.
The problems of determining the B–spline form of a C 2 Pythagorean–hodograph (PH) quintic spline curve interpolating given points, and of using this form to make local modifications, are addressed. To achieve the correct order of continuity, a quintic B–spline basis constructed on a knot sequence in which each (interior) knot is of multiplicity 3 is required. C 2 quintic bases on uniform triple knots are constructed for both open and closed C 2 curves, and are used to derive simple explicit formulae for the B–spline control points of C 2 PH quintic spline curves. These B-spline control points are verified, and generalized to the case of non–uniform knots, by applying a knot removal scheme to the Bézier control points of the individual PH quintic spline segments, associated with a set of six–fold knots. Based on the B–spline form, a scheme for the local modification of planar PH quintic splines, in response to a control point displacement, is proposed. Only two contiguous spline segments are modified, but to preserve the PH nature of the modified segments, the continuity between modified and unmodified segments must be relaxed from C 2 to C 1. A number of computed examples are presented, to compare the shape quality of PH quintic and “ordinary” cubic splines subject to control point modifications. 相似文献
3.
Arshad Khan 《PAMM》2007,7(1):2020133-2020134
In this paper a fourth-order variable coefficient parabolic partial differential equation, that governs the behaviour of a vibrating beam, is solved by using a three level method based on non-polynomial quintic spline in space and finite difference discretization in time. We also obtain two new high accuracy schemes of O (k4, h6) and O (k4, h8) and two new schemes which are analogues of Jain's formula for the non-homogeneous case. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
4.
A high‐accuracy numerical approach for a nonhomogeneous time‐fractional diffusion equation with Neumann and Dirichlet boundary conditions is described in this paper. The time‐fractional derivative is described in the sense of Riemann‐Liouville and discretized by the backward Euler scheme. A fourth‐order optimal cubic B‐spline collocation (OCBSC) method is used to discretize the space variable. The stability analysis with respect to time discretization is carried out, and it is shown that the method is unconditionally stable. Convergence analysis of the method is performed. Two numerical examples are considered to demonstrate the performance of the method and validate the theoretical results. It is shown that the proposed method is of order O(Δx4 + Δt2 ? α) convergence, where α ∈ (0,1) . Moreover, the impact of fractional‐order derivative on the solution profile is investigated. Numerical results obtained by the present method are compared with those obtained by the method based on standard cubic B‐spline collocation method. The CPU time for present numerical method and the method based on cubic B‐spline collocation method are provided. 相似文献
5.
Seluk Kutluay Melike Karta Nuri M. Yamurlu 《Numerical Methods for Partial Differential Equations》2019,35(6):2221-2235
In this article, the generalized Rosenau–KdV equation is split into two subequations such that one is linear and the other is nonlinear. The resulting subequations with the prescribed initial and boundary conditions are numerically solved by the first order Lie–Trotter and the second‐order Strang time‐splitting techniques combined with the quintic B‐spline collocation by the help of the fourth order Runge–Kutta (RK‐4) method. To show the accuracy and reliability of the proposed techniques, two test problems having exact solutions are considered. The computed error norms L2 and L∞ with the conservative properties of the discrete mass Q(t) and energy E(t) are compared with those available in the literature. The convergence orders of both techniques have also been calculated. Moreover, the stability analyses of the numerical schemes are investigated. 相似文献
6.
W. K. Zahra 《Numerical Algorithms》2009,52(4):561-573
In this paper, we developed numerical methods of order O(h
2) and O(h
4) based on exponential spline function for the numerical solution of class of two point boundary value problems over a Semi-infinite
range. The present approach gives better approximations over all the existing finite difference methods. Properties of the
infinite linear system are established. Convergence analysis and a bound on the approximate solution are discussed. Test problem
with various kinds of boundary conditions is included to illustrate the practical usefulness and superiority of our methods. 相似文献
7.
The boundary element spline collocation method is studied for the time-fractional diffusion equation in a bounded two-dimensional
domain. We represent the solution as the single layer potential which leads to a Volterra integral equation of the first kind.
We discretize the boundary integral equation with the spline collocation method on uniform meshes both in spatial and time
variables. In the stability analysis we utilize the Fourier analysis technique developed for anisotropic pseudodifferential
equations. We prove that the collocation solution is quasi-optimal under some stability condition for the mesh parameters.
We have to assume that the mesh parameter in time satisfies
(ht=c h\frac2a)(h_t=c h^{\frac{2}{\alpha}}), where (h) is the spatial mesh parameter. 相似文献
8.
B. Bialecki M. Ganesh K. Mustapha 《Numerical Methods for Partial Differential Equations》2005,21(5):918-937
We propose and analyze an application of a fully discrete C2 spline quadrature Petrov‐Galerkin method for spatial discretization of semi‐linear parabolic initial‐boundary value problems on rectangular domains. We prove second order in time and optimal order H1 norm convergence in space for the extrapolated Crank‐Nicolson quadrature Petrov‐Galerkin scheme. We demonstrate numerically both L2 and H1 norm optimal order convergence of the scheme even if the nonlinear source term is not smooth. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005. 相似文献
9.
Matrix decomposition algorithms for the finite element Galerkin method with piecewise Hermite cubics
Bernard Bialecki Graeme Fairweather David B. Knudson D. Abram Lipman Que N. Nguyen Weiwei Sun Gadalia M. Weinberg 《Numerical Algorithms》2009,52(1):1-23
Matrix decomposition algorithms (MDAs) employing fast Fourier transforms are developed for the solution of the systems of
linear algebraic equations arising when the finite element Galerkin method with piecewise Hermite bicubics is used to solve
Poisson’s equation on the unit square. Like their orthogonal spline collocation counterparts, these MDAs, which require O(N
2logN) operations on an N×N uniform partition, are based on knowledge of the solution of a generalized eigenvalue problem associated with the corresponding
discretization of a two-point boundary value problem. The eigenvalues and eigenfunctions are determined for various choices
of boundary conditions, and numerical results are presented to demonstrate the efficacy of the MDAs.
Weiwei Sun was supported in part by a grant from City University of Hong Kong (Project No. CityU 7002110). 相似文献
10.
A sixth-order numerical scheme is developed for general nonlinear fifth-order two point boundary-value problems. The standard
sextic spline for the solution of fifth order two point boundary-value problems gives only O(h
2) accuracy and leads to non-optimal approximations. In order to derive higher orders of accuracy, high order perturbations
of the problem are generated and applied to construct the numerical algorithm. O(h
6) global error estimates obtained for these problems. The convergence properties of the method is studied. This scheme has
been applied to the system of nonlinear fifth order two-point boundary value problem too. Numerical results are given to illustrate
the efficiency of the proposed method computationally. Results from the numerical experiments, verify the theoretical behavior
of the orders of convergence. 相似文献
11.
A numerical algorithm is developed for the approximation of the solution to certain boundary value problems involving the third-order ordinary differential equation associated with draining and coating flows. The authors show that the approximate solutions obtained by the numerical algorithm developed by using nonpolynomial quintic spline functions are better than those produced by other spline and domain decomposition methods. The algorithm is tested on two problems associated with draining and coating flows to demonstrate the practical usefulness of the approach. 相似文献
12.
M.A. Ramadan I.F. Lashien W.K. Zahra 《Communications in Nonlinear Science & Numerical Simulation》2009,14(4):1105-1114
In this paper, we develop quintic nonpolynomial spline methods for the numerical solution of fourth order two-point boundary value problems. Using this spline function a few consistency relations are derived for computing approximations to the solution of the problem. The present approach gives better approximations and generalizes all the existing polynomial spline methods up to order four. This approach has less computational cost. Convergence analysis of these methods is discussed. Two numerical examples are included to illustrate the practical usefulness of our methods. 相似文献
13.
In this article we use linear spline approximation of a non-linear Riemann–Hilbert problem on the unit disk. The boundary condition for the holomorphic function is reformulated as a non-linear singular integral equation A(u) = 0, where A : H 1(Γ) → H 1(Γ) is defined via a Nemytski operator. We approximate A by A n : H 1(Γ) → H 1(Γ) using spline collocation and show that this defines a Fredholm quasi-ruled mapping. Following the results of (A.I. ?nirel'man, The degree of quasi-ruled mapping and a nonlinear Hilbert problem, Math. USSR-Sbornik 18 (1972), pp. 373–396; M.A. Efendiev, On a property of the conjugate integral and a nonlinear Hilbert problem, Soviet Math. Dokl. 35 (1987), pp. 535–539; M.A. Efendiev, W.L. Wendland, Nonlinear Riemann–Hilbert problems for multiply connected domains, Nonlinear Anal. 27 (1996), pp. 37–58; Nonlinear Riemann–Hilbert problems without transversality. Math. Nachr. 183 (1997), pp. 73–89; Nonlinear Riemann–Hilbert problems for doubly connected domains and closed boundary data, Topol. Methods Nonlinear Anal. 17 (2001), pp. 111–124; Nonlinear Riemann–Hilbert problems with Lipschitz, continuous boundary data without transversality, Nonlinear Anal. 47 (2001), pp. 457–466; Nonlinear Riemann–Hilbert problems with Lipschitz-continuous boundary data: Doubly connected domains, Proc. Roy. Soc. London Ser. A 459 (2003), pp. 945–955.), we define a degree of mapping and show the existence of the spline solutions of the fully discrete equations A n (u) = 0, for n large enough. We conclude this article by discussing the solvability of the non-linear collocation method, where we shall need an additional uniform strong ellipticity condition for employing the spline approximation. 相似文献
14.
Lakshmi Chandrasekharan Nair Ashish Awasthi 《Numerical Methods for Partial Differential Equations》2019,35(3):1269-1289
This paper presents a numerical method based on quintic trigonometric B‐splines for solving modified Burgers' equation (MBE). Here, the MBE is first discretized in time by Crank–Nicolson scheme and the resulting scheme is solved by quintic trigonometric B‐splines. The proposed method tackles nonlinearity by using a linearization process known as quasilinearization. A rigorous analysis of the stability and convergence of the proposed method are carried out, which proves that the method is unconditionally stable and has order of convergence O(h4 + k2). Numerical results presented are very much in accordance with the exact solution, which is established by the negligible values of L2 and L∞ errors. Computational efficiency of the scheme is proved by small values of CPU time. The method furnishes results better than those obtained by using most of the existing methods for solving MBE. 相似文献
15.
Riaz A. Usmani 《分析论及其应用》1996,12(3):1-9
In this paper we develop periodic quartic spline inter polation theory which, in general, gives better fits to continuous
functions than does the existing quintic spline inter polation theory. The main theorem of the paper is to establish that
⋎s(r)-y(r)⋎=O(h6−r), r=0,1,2,3. Also, the nonperiodic cases cannot be constructed empolying the methodology of this paper because that will
involve several other end conditions entirely different than (1.10). 相似文献
16.
Vikas Gupta Mohan K. Kadalbajoo 《Numerical Methods for Partial Differential Equations》2011,27(5):1143-1164
In this article, we develop a parameter uniform numerical method for a class of singularly perturbed parabolic equations with a multiple boundary turning point on a rectangular domain. The coefficient of the first derivative with respect to x is given by the formula a0(x, t)xp, where a0(x, t) ≥ α > 0 and the parameter p ∈ [1,∞) takes the arbitrary value. For small values of the parameter ε, the solution of this particular class of problem exhibits the parabolic boundary layer in a neighborhood of the boundary x = 0 of the domain. We use the implicit Euler method to discretize the temporal variable on uniform mesh and a B‐spline collocation method defined on piecewise uniform Shishkin mesh to discretize the spatial variable. Asymptotic bounds for the derivatives of the solution are established by decomposing the solution into smooth and singular component. These bounds are applied in the convergence analysis of the proposed scheme on Shishkin mesh. The resulting method is boundary layer resolving and has been shown almost second‐order accurate in space and first‐order accurate in time. It is also shown that the proposed method is uniformly convergent with respect to the singular perturbation parameter ε. Some numerical results are given to confirm the predicted theory and comparison of numerical results made with a scheme consisting of a standard upwind finite difference operator on a piecewise uniform Shishkin mesh. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1143–1164, 2011 相似文献
17.
In this paper, we present a numerical method for solving a class of nonlinear, singularly perturbed two-point boundary-value problems with a boundary layer on the left end of the underlying interval. The original second-order problem is reduced to an asymptotically equivalent first-order problem and is solved by a numerical method using a fourth-order cubic spline in the inner region. The method has been analyzed for convergence and is shown to yield anO(h
4) approximation to the solution. Some test examples have been solved to demonstrate the efficiency of the method.The authors thank the referee for his helpful comments. 相似文献
18.
In this paper we develop a non-polynomial quintic spline function to approximate the solution of third order linear and non-linear
boundary value problems associated with odd-order obstacle problems. Such problems arise in physical oceanography and can
be studied in the framework of variational inequality theory. The class of methods are second and fourth order convergent.
End equations of the splines are derived and truncation error is obtained. Two numerical examples are given to illustrate
the applicability and efficiency of proposed method. It is shown that the new method gives approximations, which are better
than those produced by other methods. 相似文献
19.
《Journal of Computational and Applied Mathematics》1999,103(2):221-237
The construction of range restricted univariate and bivariate interpolants to gridded data is considered. We apply Gregory's rational cubic C1 splines as well as related rational quintic C2 splines. Assume that the lower and upper obstacles are compatible with the data set. Then the tension parameters occurring in the mentioned spline classes can be always determined in such a way that range restricted interpolation is successful. 相似文献