Approximate Approximations and a Boundary Point Method for the Linearized Stokes System

The method of approximate approximations, introduced by Maz'ya [1], can also be used for the numerical solution of boundary integral equations. In this case, the matrix of the resulting algebraic system to compute an approximate source density depends only on the position of a finite number of boundary points and on the direction of the normal vector in these points (Boundary Point Method). We investigate this approach for the Stokes problem in the whole space and for the Stokes boundary value problem in a bounded convex domain G subset R^2, where the second part consists of three steps: In a first step the unknown potential density is replaced by a linear combination of exponentially decreasing basis functions concentrated near the boundary points. In a second step, integration over the boundary partial G is replaced by integration over the tangents at the boundary points such that even analytical expressions for the potential approximations can be obtained. In a third step, finally, the linear algebraic system is solved to determine an approximate density function and the resulting solution of the Stokes boundary value problem. Even not convergent the method leads to an efficient approximation of the form O(h^2) + epsilon, where epsilon can be chosen arbitrarily small.

Zur Konvergenz der Randpunktmethode

Das von Maz'ya eingeführte Approximationsverfahren, die Methode der näherungsweisen Näherungen (Approximate Approximations), kann auch zur numerischen Lösung von Randintegralgleichungen verwendet werden (Randpunktmethode). In diesem Fall hängen die Komponenten der Matrix des resultierenden Gleichungssystems zur Berechnung der Näherung für die Dichte nur von der Position der Randpunkte und der Richtung der äußeren Einheitsnormalen in diesen Punkten ab. Dieses numerisches Verfahren wird am Beispiel des Dirichlet Problems für die Laplace Gleichung und die Stokes Gleichungen in einem beschränkten zweidimensionalem Gebiet untersucht. Die Randpunktmethode umfasst drei Schritte: Im ersten Schritt wird die unbekannte Dichte durch eine Linearkombination von radialen, exponentiell abklingenden Basisfunktionen approximiert. Im zweiten Schritt wird die Integration über den Rand durch die Integration über die Tangenten in Randpunkten ersetzt. Für die auftretende Näherungspotentiale können sogar analytische Ausdrücke gewonnen werden. Im dritten Schritt wird das lineare Gleichungssystem gelöst, und eine Näherung für die unbekannte Dichte und damit auch für die Lösung der Randwertaufgabe konstruiert. Die Konvergenz dieses Verfahrens wird für glatte konvexe Gebiete nachgewiesen.

Nitrogen in combination with Desmodium intortum effectively suppress Striga asiatica in a sorghum‒Desmodium intercropping system

It is well known that the parasitic weed Striga asiatica (L.) Kuntze can be suppressed by Striga-tolerant sorghum (Sorghum bicolor L. Moench) cultivars, Desmodium intortum (Mill.) Urb. (greanleaf desmodium), and by fertilization with nitrogen. The study objective was the assessment of Striga control provided by integration of Desmodium density, timing of sorghum-Desmodium intercrop establishment, and nitrogen fertilization. Growth responses and yield of three sorghum cultivars were measured in three pot experiments. A soil naturally infested with Striga was used, and that part of the soil which served as uninfested control was chemically sterilised. Striga numbers and growth were affected significantly by sorghum cultivars, sorghum-Desmodium intercrop ratios, timing of the sorghum-Desmodium association, as well as by their interactions. Desmodium caused 100% suppression of Striga emergence when Desmodium was established in the 1:3 sorghum-Desmodium ratio at seeding of sorghum. Total control of Striga was also achieved with the 1:1 sorghum-Desmodium ratio when Desmodium was transplanted 30 days before sorghum seeding. However, these two treatments also caused significant reductions in sorghum yield. In contrast, 100% Striga control and a dramatic increase in sorghum yield were achieved with 100 kg N ha^{-1} in the 1:1 sorghum-Desmodium intercrop. Compatibility of sorghum and Desmodium was evident at the 1:1 sorghum-Desmodium intercrop established at sorghum seeding. Overall, the Ethiopian cultivars Meko and Abshir showed better agronomic performance and higher tolerance to Striga than the South African cultivar PAN 8564. It is recommended that the N × Desmodium × sorghum interaction be investigated under field conditions.