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NASA NTRS1999

Generalized Functions for the Fractional Calculus

Previous papers have used two important functions for the solution of fractional order differential equations, the Mittag-Leffler functionE(sub q)[at(exp q)](1903a, 1903b, 1905), and the F-function F(sub q)[a,t] of Hartley & Lorenzo (1998). These functions provided direct solution and important understanding for the fundamental linear fractional order differential equation and for the related initial value problem (Hartley and Lorenzo, 1999). This paper examines related functions and their Laplace transforms. Presented for consideration are two generalized functions, the R-function and the G-function, useful in analysis and as a basis for computation in the fractional calculus. The R-function is unique in that it contains all of the derivatives and integrals of the F-function. The R-function also returns itself on qth order differ-integration. An example application of the R-function is provided. A further generalization of the R-function, called the G-function brings in the effects of repeated and partially repeated fractional poles.

Lorenzo, Carl F.↗

NASA NTRS1974

Effects of prolonged acceleration with or without clinostat rotation on seedlings of Arabidopsis thaliana (L.) Heynh

Three 21-day tests of the effects of chronic centrifugation were carried out on populations of Arabidopsis thaliana. In addition to 1 g the resultant g-forces tested were: 2,4,6,8,16, and 20 g. Observed end points included gross morphological characters such as size of plant organs and, at the other extreme, features of sub-cellular structure and ultrastructure. Plants were grown on banks of clinostats. The acceleration vector was directed either parallel with the plants' axes or transverse to the axes. Plant responses to chronic axial acceleration and to transverse acceleration with clinostated plants were determined. From the data obtained it was possible in some cases: (1) to determine the g-functions of specific plant developmental characters; (2) to extrapolate those functions to the hypothetical value at zero g in order to predict (tentatively) the morphology of a plant grown in space, (3) to describe morphological effects of clinostat rotation, (4) to determine which of those effects was influenced by the prevailing g-force, and (5) to put to direct test the assumption that clinostat rotation nullifies or compensates for the influence of gravity.

Brown, A. H.↗

NASA NTRS1975

Comparison of two methods of predicting characteristics of an organism which develops under the condition of free fall

Five morphological characteristics of Arabidopsis thaliana were measured on plant populations grown under continuous centrifugation. In separate tests different g-levels were used. For each character studied a linear g-function was calculated and extrapolated to zero-g. In other tests Arabidopsis plants were grown on horizontal clinostats after which the same set of characters was measured. Growth on a clinostat might simulate growth at zero-g; but the zero-g predictions by the two methods did not agree consistently. The results were significantly different for three of the five characters for which comparisons were made. Either the extrapolation method or the clinostat method are considered unreliable as a means of predicting plant growth characteristics in the weightless environment of an earth satellite laboratory.

Brown, A. H.↗