scholarly journals Linear functionals of foliage angle density

1984 ◽  
Vol 25 (4) ◽  
pp. 431-442 ◽  
Author(s):  
R. S. Anderssen ◽  
D. R. Jackett
Keyword(s):  
Integral Equation ◽  
Environmental Monitoring ◽  
Fredholm Integral Equation ◽  
Inversion Formula ◽  
Canopy Reflectance ◽  
Linear Functionals ◽  
General Transformation ◽  
Area Index ◽  
Contact Frequency ◽  
Special Case

AbstractKnowledge about the foliage angle density g(α) of the leaves in the canopy of trees is crucial in foresty mangement, modelling canopy reflectance, and environmental monitoring. It is usually determined from observations of the contact frequency f(β) by solving a version of the first kind Fredholm integral equation derived by Reeve (Appendix in Warren Wilson [22]). However, for inference purposes, the practitioner uses functionals defined on g(α), such as the leaf area index F, rather than g(α) itself. Miller [12] has shown that F can be computed directly from f(β) without solving the integral equation. In this paper, we show that his result is a special case of a general transformation for linear functionals defined on g(α). The key is the existence of an alternative inversion formula for the integral equation to that derived by Miller [11].

Linear Functionals of the Foliage Angle Distribution as Tools to Study the Structure of Plant Canopies

1984 ◽  
Vol 32 (2) ◽  
pp. 147 ◽  
Author(s):  
RS Anderssen ◽  
DR Jackett ◽  
DLB Jupp
Keyword(s):  
Synthetic Data ◽  
Angle Distribution ◽  
Linear Functionals ◽  
General Transformation ◽  
Area Index ◽  
Plant Canopies ◽  
Contact Frequency ◽  
Numerical Experimentation ◽  
Actual Evaluation ◽  
Foliage Density

In 1967, Miller showed how average foliage density could be computed from contact frequency data. It formalized mathematically the idea posed earlier by Warren Wilson of estimating the leaf area index as a linear combination of measured values of the contact frequency. Recently, it has been shown that Miller's result is a special case of a general transformation that allows linear functionals defined on the (generally unknown) foliage angle distribution (foliage angle functionals) to be evaluated as linear functionals defined on the (measured) contact frequency (contact frequency functionals). This result has important consequences for the use of foliage angle functionals in the study of the structure of plant canopies. For example, it allows Warren Wilson's idea to be extended to the evaluation of such functionals, and thereby simplifies greatly their actual evaluation. In this paper, we first motivate and review the use of foliage angle functionals in the study of plant canopies; then we introduce new functionals (the segmented foliage density and the moments); and finally, we use numerical experimentation with synthetic data to illustrate the advantages of having formulas for the foliage angle functionals of interest that are defined explicitly in terms of the (measured) contact frequency.

scholarly journals A note on dual integral equations involving inverse associated Weber-Orr transforms

1996 ◽  
Vol 19 (1) ◽  
pp. 161-169
Author(s):  
Nanigopal Mandal ◽  
B. N. Mandal
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Dual Integral Equations ◽  
Elementary Methods ◽  
Special Case

We consider dual integral equations involving inverse associated Weber-Orr transforms. Elementary methods have been used to reduce dual integral equations to a Fredholm integral equation of second kind. Some known results are obtained as special case.

Convergence and error estimation of homotopy analysis method for some type of nonlinear and linear integral equations

2015 ◽  
Vol 23 (4) ◽  
Author(s):  
Edyta Hetmaniok ◽  
Iwona Nowak ◽  
Damian Słota ◽  
Roman Wituła
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Homotopy Analysis Method ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Linear Integral ◽  
Special Case ◽  
Linear Integral Equations

AbstractIn this paper an application of the homotopy analysis method for some type of nonlinear and linear integral equations of the second kind is presented. A special case of considered equation is the Volterra- Fredholm integral equation. In homotopy analysis method a series is created. It has shown that if the series is convergent, its sum is the solution of the considered equation. It has been also shown that under proper assumptions the considered equation possesses a unique solution and the series obtained in homotopy analysis method is convergent. The error of the approximate solution was estimated. This approximate solution is obtained when we limit to the partial sum of the series.Application of the method is illustrated with examples.

Simultaneous Wiener–Hopf equations

1980 ◽  
Vol 58 (3) ◽  
pp. 420-428 ◽  
Author(s):  
A. D. Rawlins
Keyword(s):  
Integral Equation ◽  
Half Plane ◽  
Fredholm Integral Equation ◽  
Coupled System ◽  
Algebraic Equations ◽  
Branch Points ◽  
Linear Algebraic Equations ◽  
Upper Half Plane ◽  
Infinite Set ◽  
Special Case

In Noble's book The Wiener Hopf Technique, Pergamon, 1958, he considers the coupled system of Wiener–Hopf equations (§4.4, pp. 153–154)[Formula: see text]He shows that provided the functions L(α), M(α), Q(α), and R(α) have only simple pole singularities the solution can be reduced to two sets of infinite simultaneous linear algebraic equations. In this article a different approach is used which gives the solution in the form of a Fredholm integral equation of the second kind. This Fredholm integral equation can be reduced to infinite sets of simultaneous linear algebraic equations under the less restrictive conditions that either (i) L(α)/M(α) has no branch points in the lower α-half plane: Im(α) < τ+; or (ii) Q(α)/R(α) has no branch points in the upper α-half plane: Im(α) > τ−. In the special case considered by Noble if L(α)/M(α) (or Q(α)/R(α)) only have simple poles in the lower (upper) half plane then the Fredholm integral equation reduces to one infinite set of simultaneous equations. This extends the Wiener–Hopf technique to yet a larger class of boundary value problems, and simplifies the numerical computations.

On Improved Approximate Solution of the Fredholm Integral Equation

2006 ◽  
Vol 6 (3) ◽  
pp. 264-268
Author(s):  
G. Berikelashvili ◽  
G. Karkarashvili
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Fredholm Integral Equation ◽  
Algebraic Equations ◽  
Order Convergence ◽  
One Dimensional ◽  
Averaging Operator ◽  
Linear Algebraic Equations ◽  
Convergence Solution

AbstractA method of approximate solution of the linear one-dimensional Fredholm integral equation of the second kind is constructed. With the help of the Steklov averaging operator the integral equation is approximated by a system of linear algebraic equations. On the basis of the approximation used an increased order convergence solution has been obtained.

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