Papers by Hossein Behforooz
Applied Mathematics and Computation, Mar 1, 2010
Recently, Behforooz [1], has introduced a new approach to construct cubic splines by using the in... more Recently, Behforooz [1], has introduced a new approach to construct cubic splines by using the integral values, rather than the usual function values at the knots. Also he has established different sets of end conditions for cubic and quintic splines by using the integral values, see Behforooz [2-4]. In this paper, we will use the same techniques of [1] to construct integro quintic splines. Although by using the integral values we expected to face a more complicated process for our construction, it turned out that the matrix of the system of linear equations that produces the parameters became a diagonally dominant matrix and the process became very simple. The selection of the required end conditions for our integro quintic splines will be discussed. The numerical examples and computational results illustrate and guarantee a higher accuracy for this approximation.
Chaos, Solitons & Fractals, 2017
This paper develops a technique for the approximate solution of a class of variable-order fractio... more This paper develops a technique for the approximate solution of a class of variable-order fractional differential equations useful in the area of fluid dynamics. The method adopts a piecewise integro quadratic spline interpolation and is used in the study of the variable-order fractional Bagley-Torvik and Basset equations. The accuracy of the proposed algorithm is verified by means of illustrative examples.
Communications in Numerical Analysis, 2018
In this paper, we obtain approximate inverses of popular tri-diagonal and penta-diagonal matrices... more In this paper, we obtain approximate inverses of popular tri-diagonal and penta-diagonal matrices which are used to construct local (or a discrete quasi-interpolant) interpolatory and integro splines.
Trouble Bipolaire a L Adolescence Reflexions Cliniques
Elsevier, 1998
Thinning Out the Harmonic Series
Mathematics Magazine, 1995
k=i is close to log n; this shows that the harmonic series diverges very, very slowly. For exampl... more k=i is close to log n; this shows that the harmonic series diverges very, very slowly. For example, it takes more than 1.5 x 1043 terms for its partial sums to reach 100; see [3], [4], and [5]. In this paper, we refine and thin out the harmonic series to show that the divergence of this series depends on some of its specific terms and without those terms the remaining subseries is convergent. Then, an upper bound and an approximate value will be given to the value of the convergent subseries. Finally, by use of the derived results, it will be shown that the divergence of the Euler series E: (over prime numbers p) depends on specific prime numbers.
Interpolation of Fuzzy Data by Using Fuzzy Splines
International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 2008
In this paper we define a new set of spline functions called “Fuzzy Splines” to interpolate fuzzy... more In this paper we define a new set of spline functions called “Fuzzy Splines” to interpolate fuzzy data. Numerical examples will be presented to illustrate the differences between of using our spline and other interpolations that have been studied before.
The use of spline-on-spline for the approximation of Cauchy principal value integrals
Applied Mathematics and Computation, 1996
In this paper the spline-on-spline interpolatory polynomials are used to approximate Cauchy princ... more In this paper the spline-on-spline interpolatory polynomials are used to approximate Cauchy principal value integrals and their derivatives. This technique yields better results than using the derivatives of the traditional single spline functions, which has been considered by Orsi. Improved order of convergence for these approximations are discussed.
Applied Mathematics and Computation, 2006
Years ago, after I presented a paper on spline functions at Oxford University, Professor Powell c... more 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.
A comparison of theE(3) and not-a-knot cubic splines
Applied Mathematics and Computation, 1995
Abstract The algorithms and schemes of the cubic spline interpolation with two end conditions whi... more Abstract The algorithms and schemes of the cubic spline interpolation with two end conditions which do not require thederivative information at the end points are of great practical importance and have been included in several general purpose software libraries. The cubic spline with not-a-knot end conditions is one of them and it is implemented in the widely used IMSL and NAG software libraries. In this paper we compare theE(3) cubic spline with the not-a-knot cubic spline and we will show that theE(3) cubic spline is more accurate than the second one, and also, it has superconvergence properties which the not-a-knot cubic spline does not have. These superconvergence properties of theE(3) cubic spline can be used in different fields for better approximation.
In this paper, we will consider the interpolation of fuzzy data by fuzzy-valued E(3) splines. Num... more In this paper, we will consider the interpolation of fuzzy data by fuzzy-valued E(3) splines. Numerical examples will be presented to illustrate the differences between of using E(3) spline and other interpolations that have been studied before. AMS Subject Classification: 94D05, 26E50
International Journal of Nonlinear Analysis and Applications, 2017
In this paper, a new set of spline functions called ``Flat End Fuzzy Spline" is defined to i... more In this paper, a new set of spline functions called ``Flat End Fuzzy Spline" is defined to interpolate given fuzzy data. Some important theorems on these splines together with their existence and uniqueness properties are discussed. Then numerical examples are presented to illustrate the differences between of using our spline and other interpolations that have been studied before.
Interpolation of Fuzzy Data
In this paper, we will consider the interpolation of fuzzy data by fuzzy-valued E(3) splines. Num... more In this paper, we will consider the interpolation of fuzzy data by fuzzy-valued E(3) splines. Numerical examples will be presented to illustrate the differences between of using E(3) spline and other interpolations that have been studied before.
A new approach to spline functions
Applied Numerical Mathematics, 1993
Abstract Suppose that the values of the integral of the function y = y ( x ) are known on the end... more Abstract Suppose that the values of the integral of the function y = y ( x ) are known on the end subintervals of the spline interval [ a , b ]. By employing these values, the required end conditions for the cubic and quintic interpolatory splines will be derived. The order of convergence of the interpolatory splines with these end conditions are O( h 4 ) for the cubic spline, and O( h 6 ) for the quintic spline.
Communications in Numerical Analysis, 2016
In this paper, we show that the integro quintic splines can locally be constructed without solvin... more In this paper, we show that the integro quintic splines can locally be constructed without solving any systems of equations. The new construction does not require any additional end conditions. By virtue of these advantages the proposed algorithm is easy to implement and effective. At the same time, the local integro quintic splines possess as good approximation properties as the integro quintic splines. In this paper, we have proved that our local integro quintic spline has superconvergence properties at the knots for the first and third derivatives. The orders of convergence at the knots are six (not five) for the first derivative and four (not three) for the third derivative.
Mirror magic squares from Latin Squares
Mathematical Gazette, 2007
The not-a-knot piecewise interpolatory cubic polynomial
Amc, 1992
Manifold subbotin spline
Amc, 1990
Consistency relations of the spline functions derived from a Pascal-like triangle
Applied Mathematics and Computation, 1996
... spline S( x): 4 1) + 111)1 + 111+)2 + 1+)3 = (Yi 3Yi+l + 3Y+2 + Y,+3) 12 2) + 11 2)1 + 112+)2... more ... spline S( x): 4 1) + 111)1 + 111+)2 + 1+)3 = (Yi 3Yi+l + 3Y+2 + Y,+3) 12 2) + 11 2)1 + 112+)2 + 2+)3 = ( Yi Y+I Yi+2 + Yi+3) 24 3) + 113+)1 + 11i3+)2 + 3+)3 = .(Y + 3yi+1 3Yi+2 "4 Yi+3) n = 5, quintic spline 5( x): 1) + 261+) + 661+)z + 261+)3 + 1+)4 5 = (Yi IOy+I + lOyi+3 + Yi+4) 2 ...
Mathematical Sciences, 2012
Purpose: In this paper, we will consider the interpolation of fuzzy data by using the fuzzy-value... more Purpose: In this paper, we will consider the interpolation of fuzzy data by using the fuzzy-valued piecewise quartic polynomials Q y 0 ,y 1 ,..., y n (x) induced from E(3) cubic spline functions. Method: It has been many years since researchers have attended to the problem of interpolation of fuzzy data. Here, for Lagrange interpolation of fuzzy data, we will use the piecewise quartic polynomial induced from E(3) cubic spline functions to interpolate the fuzzy data. To do this, we will apply the extension principle to construct the membership function of Q y 0 ,y 1 ,..., y n (x). Results: By using piecewise quartic polynomials, a new set of fuzzy spline functions was defined to interpolate given fuzzy data. Conclusions: In our previous study, we used E(3) cubic spline to construct E(3) fuzzy cubic spline. In this article, we added one extra term to this spline to compute the piecewise quartic polynomials and hence the fuzzy-valued piecewise quartic polynomials.
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Papers by Hossein Behforooz