Cubic spline method for highly oscillatory integrals

Archives of Computational Methods in Engineering, 2021

This work develops formulas for numerical integration with spline interpolation. The new formulas are shown to be alternatives to the Newton-Cotes integration formulas. These methods have important application in integration of tables or for discrete functions with constant steps. An error analysis of the technique was conducted. A new type of spline interpolation is proposed in which a polynomial passes through more than two tabulated points. The results show that the proposed formulas for numerical integration methods have high precision and absolute stability. The obtained methods can be used for the integration of stiff equations. This paper opens a new field of research on numerical integration formulas using splines.

Numerical Algorithms, 1993

In this paper product quadratures based on quasi-interpolating splines are proposed for the numerical evaluation of integrals with anL 1-kernel and of Cauchy Principal Value integrals.

Numerische Mathematik, 1990

In this paper the convergence of product integration rules, based on cubic spline interpolation at equally spaced nodes, with "not-a-knot" end condition, is investigated for integrand functions with a interior or endpoint singularity in the integration interval.

Carpathian Journal of Mathematics

In this paper we have considered the asymptotic expressions for remainder term of quadrature formulas of the interpolator type. We derive some corrected versions of the quadrature formulas of interpolatory type, which provide a better approximation accuracy than the original rules. A method to improve the degree of exactness of the quadrature formulas is also considered. A numerical example of the proposed method is given.

2006

In this paper, we find numerical solution of xðtÞ þ k

SpringerPlus, 2016

In the last two decades, have constructed a direct cubic spline that fits the first derivatives at the knots together with the value of the function and its second derivative at the beginning of the interval. They used it for the solution quadrature formula. El Tarazi and have constructed five types of even degree splines ( j = 2k, k = 1, 2, 3, 4, 5) that match the derivatives up to the order k at the knots of a uniform partition for each k = 1, 2, 3, 4, and 5. These splines are also applied to quadrature. Recently, Rathod et al. ( ) presented a formulation and study of an interpolatory cubic spline (named Subbotin cubic spline) to compute the integration over curved domains in the Cartesian two space and the integral approximations (quadrature). In this work, we construct a twelfth degree spline which interpolates the derivatives up to the order 6 of a given function at the knots and its value at the beginning of the interval. We obtain a direct simple formula for the proposed spline. Error bounds for the function is derived in the sense of the Hermite interpolation. Also, a mistakes in the literature was corrected. Finally, numerical examples and comparison with other available methods are presented to illustrate the usefullness of proposed method. We construct here a class of interpolating splines of degree 12. Error estimates for this spline is also represented.

2011

Kampus Binawidya Pekanbaru (28293) we discuss and do some analysis o., .ruorllilrrr"to*"" based on interpolation, midpoint, trapezoidal rule and Simpson rule. We end up with some new formulas, which are not mentioned in numerical analysis textbooks. The strategy we discuss, in terms of pedagogy, illuminate how research on mathematics can be carried out.

BIT Numerical Mathematics, 2013

In this paper we consider the space generated by the scaled translates of the trivariate C 2 quartic box spline B defined by a set X of seven directions, that forms a regular partition of the space into tetrahedra. Then, we construct new cubature rules for 3D integrals, based on spline quasi-interpolants expressed as linear combinations of scaled translates of B and local linear functionals. We give weights and nodes of the above rules and we analyse their properties. Finally, some numerical tests and comparisons with other known integration formulas are presented.

Applied Mathematics and Computation, 2006

Years ago, after I presented a paper on spline functions at Oxford University, Professor Powell criticized us for using, most of the time, the function values rather than the integral values on constructing of the spline functions. His comments and his request became the main motivation for this work. In this paper, we assume that, on each subinterval of the spline interval [a, b], the integral value of the function y = y(x) is known. By using these values, rather than the function values at the knots, we introduce a class of new types of interpolatory cubic splines to approximate the function y = y(x). The selection of the end conditions for our integro cubic splines will be discussed. The numerical examples and computational results, illustrate and guarantee a higher accuracy for this approximation.

Journal of Scientific Computing, 2012

In this paper we revisit some quadrature methods for highly oscillatory integrals of the form 1 −1 f (x)e iωx dx, ω > 0. Exponentially Fitted (EF) rules depend on frequency dependent nodes which start off at the Gauss-Legendre nodes when the frequency is zero and end up at the endpoints of the integral when the frequency tends to infinity. This makes the rules well suited for small as well as for large frequencies. However, the computation of the EF nodes is expensive due to iteration and ill-conditioning. This issue can be resolved by making the connection with Filon-type rules. By introducing some S-shaped functions, we show how Gauss-type rules with frequency dependent nodes can be constructed, which have an optimal asymptotic rate of decay of the error with increasing frequency and which are effective also for small or moderate frequencies. These frequencydependent nodes can also be included into Filon-Clenshaw-Curtis rules to form a class of methods which is particularly well suited to be implemented in an automatic software package.