On the relations between finite differences and derivatives of cardinal spline functions

1976 ◽  
pp. 210-219 ◽  
Author(s):  
Hennie ter Morsche
Keyword(s):  
Finite Differences ◽  
Spline Functions ◽  
Cardinal Spline ◽  
Derivatives Of

An Introduction to Finite Differencing

1998 ◽  
Author(s):  
T. N. Krishnamurti ◽  
H. S. Bedi ◽  
V. M. Hardiker
Keyword(s):  
Finite Difference ◽  
Finite Differences ◽  
Single Point ◽  
Finite Differencing ◽  
Finite Difference Approximations ◽  
Difference Approximations ◽  
Student Reading ◽  
Higher Derivatives ◽  
Derivatives Of

This chapter on finite differencing appears oddly placed in the early part of a text on spectral modeling. Finite differences are still traditionally used for vertical differencing and for time differencing. Therefore, we feel that an introduction to finite-differencing methods is quite useful. Furthermore, the student reading this chapter has the opportunity to compare these methods with the spectral method which will be developed in later chapters. One may use Taylor’s expansion of a given function about a single point to approximate the derivative(s) at that point. Derivatives in the equation involving a function are replaced by finite difference approximations. The values of the function are known at discrete points in both space and time. The resulting equation is then solved algebraically with appropriate restrictions. Suppose u is a function of x possessing derivatives of all orders in the interval (x — n∆x, x + n∆x). Then we can obtain the values of u at points x ± n∆ x, where n is any integer, in terms of the value of the function and its derivatives at point x, that is, u(x) and its higher derivatives.

Solvability of cardinal spline interpolation problems

1983 ◽  
Vol 95 (1-2) ◽  
pp. 39-57
Author(s):  
T. N. T. Goodman
Keyword(s):  
Incidence Matrix ◽  
Bounded Solution ◽  
Spline Interpolation ◽  
Piecewise Polynomial ◽  
Piecewise Polynomials ◽  
Cardinal Spline ◽  
Interpolation Problems ◽  
Bounded Data ◽  
Derivatives Of ◽  
Cardinal Spline Interpolation

SynopsisWe consider interpolation by piecewise polynomials, where the interpolation conditions are on certain derivatives of the function at certain points of a periodic vector x, specified by a periodic incidence matrix G. Similarly, we allow discontinuity of certain derivatives of the piecewise polynomial at certain points of x, specified by a periodic incidence matrix H. This generalises the well-known cardinal spline interpolation of Schoenberg. We investigate conditions on G, H and x under which there is a unique bounded solution for any given bounded data.

scholarly journals The Cardinal Spline Methods for the Numerical Solution of Nonlinear Integral Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Xiaoyan Liu ◽  
Jin Xie ◽  
Zhi Liu ◽  
Jiahuan Huang
Keyword(s):  
Integral Equations ◽  
High Accuracy ◽  
Spline Functions ◽  
Sufficient Condition ◽  
Nonlinear Integral Equations ◽  
Effective Technique ◽  
Original Solution ◽  
Cardinal Spline ◽  
The Matrix ◽  
Cardinal Splines

In this study, an effective technique is presented for solving nonlinear Volterra integral equations. The method is based on application of cardinal spline functions on small compact supports. The integral equation is reduced to a system of algebra equations. Since the matrix for the system is triangular, it is relatively straightforward to solve for the unknowns and an approximation of the original solution with high accuracy is accomplished. Several cardinal splines are employed in the paper to enhance the accuracy. The sufficient condition for the existence of the inverse matrix is examined, and the convergence rate is analyzed. We compare our method with other methods proposed in recent papers and demonstrated the advantage of our method with several examples.

Identification of MIMO Hammerstein systems using cardinal spline functions

2006 ◽  
Vol 16 (7) ◽  
pp. 659-670 ◽  
Author(s):  
Kwong Ho Chan ◽  
Jie Bao ◽  
William J. Whiten
Keyword(s):  
Spline Functions ◽  
Hammerstein Systems ◽  
Cardinal Spline
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