6,309 research outputs found
A restricted Runge-Kutta method
No full textConsider the differential system dy[subscript r]/dx = f[subscript r](x, y���, y���, ..., y[subscript m]), y[subscript r](x���) = r[subscript r0] (r = 1, 2, ..., m). The Runge-Kutte method applies to all functions f[subscript r](x, y���, y���, ..., y[subscript m]), of suitable differentiability. By restricting the class of functions to g[subscript r](x) + r[subscript r1]y��� + ... + c[subscript rm]y[subscript m] where g[subscript r](x) are arbitrary functions of x and c[subscript rj] arbitrary constants, the nth order of this restricted Runge-Kutte method for the explicit case can be defined as [y bar][subscript r1] = y[subscript r0] + [sigma q i=1] R[subscript i]k[subscript ri]. ..
Stemflow chemistry and epiphytic lichen diversity in dieback-affected spruce forest of the Harz Mountains, Germany
No full textIn the German Harz Mountains, epiphytic lichen diversity was found to be higher in a Picea abies forest affected by pollution-caused dieback than in a comparable healthy stand. Although amount and chemical composition of incident precipitation did not differ between the stands, element concentrations of S, H+, K, Fe, Mn, and Al in stemflow were significantly lower in the dieback-affected plot than in the healthy one. These lower concentrations are attributed to reduced interception and reduced leaching due to needle loss. Cover of Hypogymnia physodes decreased with increasing concentrations of many elements in stemflow and bark. Among these parameters, S concentration of stemflow is considered to influence directly H. physodes. Cover of the extremely toxitolerant Lecanora conizaeoides was less affected by chemical variables, but a significant dependence of cover on S concentration of stemflow, resulting in an optimum regression curve, could be established in this case. Total number of lichen species per sample tree decreased as concentrations of several elements increased, indicating that most lichen species had similar habitat requirements as H. physodes
Order conditions of stochastic Runge-Kutta methods by B-series
No full textIn this paper, general order conditions and a global convergence proof are given for stochastic Runge-Kutta methods applied to stochastic ordinary differential equations (SODEs) of Stratonovich type. This work generalizes the ideas of B-series as applied to deterministic ordinary differential equations (ODEs) to the stochastic case and allows a completely general formalism for constructing high order stochastic methods, either explicit or implicit. Some numerical results will be given to illustrate this theory
Optimum runge-kutta formulas of third-, fourth-, and fifth-orders
No full textIn considering explicit Runge-Kutta methods to find starting values for linear multi-step methods in solving numerically first-order differential equations, the minimization of truncation error is of utmost importance. The criterion chosen to minimize the truncation error is minimizing the sum of the magnitudes of the coefficients of the principal error function. The parameters in the Runge-Kutta formulas permit this. Further, for fourth- and fifth-order Runge-Kutta methods one must choose between working with a single differential equation or a system because the principal error functions are different. In this dissertation the choice is to work with systems of differential equations. For both third- and fourth-order methods the optimum formula is found. The third-order formula is not new, but the fourth-order formula is. But for fifth-order Runge-Kutta, the optimum formula is found for only the classes of formulas resulting from Lobatto, Newton-Cotes, Radau, and Legendre-Gauss quadrature numbers. The optimum formula is of Lobatto type. The restriction on the classes of formulas considered is made because otherwise the number of parameters seems to make the error function unmanageable
High-order Explicit Runge-Kutta Methods Using M-Symmetry
No full textThe Runge-Kutta equations of condition are reformulated. The concept of m-symmetry is defined. It is shown that any m-symmetric method is of order m. The equations of condition for a twelfth-order explicit Runge-Kutta method with twenty-five stages are solved using m-symmetry. The method contains an embedded tenth-order method that can be used to estimate the local truncation errors and thus to vary the stepsize. Numerical experiments demonstrate that the method compares favorably with other high-order methods, especially for those problems requiring highly accurate solutions
Theorema runge
No full textDi dalam pembahasan theorema RUnge diperlihatkan
bahwa suatu fungsi rasional di dalamisuatu daerah ter¬tentu dapat dapat didekati oleh suatu fungi analitik.
Mak& dapat dikatakan bahwa fungsi ra5ional akan konver-. gen ke suatu fungsi analitik atau limit dati fungsi ra-sional adalah fungsi analitik.
Salah satu kegunaan dari theorema Runge adalah di
pakai dalam pembuktian theorema Mittag-Leffler.
Maka akan dibahas sedikit tentang kegunaan theorema Runge t ers ebut.
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Analysis of Runge-Kutta methods using Butcher tableaus
No full textThis Bachelor thesis provides an analysis of Runge-Kutta methods using Butcher tableaus. Runge-Kutta method are numerical methods used for approximating initial value problems. A Runge-Kutta method can be classified as either an explicit or an implicit method. A special kind of implicit methods are diagonally implicit methods. The type of method can be recognised by the Butcher tableau. Using the entries of the Butcher tableau, one can compute the amplification factor of a Runge-Kutta method. The amplification factor can then be used to compute the order of the local truncation error and the stability region. Examples of these computations are given for seven methods. Furthermore, this thesis provides an algorithm to perform time steps for each of the three types of Runge-Kutta methods. Finally, in order to analyse the global truncation error of the seven methods, the algorithm to perform time steps is used with different step sizes.Applied Mathematic
The significance of stemflow chemistry for epiphytic lichen diversity in a dieback-affected spruce forest on Mt Brocken, northern Germany
No full textEpiphytic lichen diversity in a boggy stand of Norway spruce (Picea abies) was studied in the eastern Harz Mountains, northern Germany. Spruce trees at wet sites were affected by forest dieback, whereas trees on drier sites remained unaffected. Lichen diversity was higher on dieback-affected trees than on healthy ones. The foliose lichen Hypogymnia physodes was significantly more frequent on dead trees, whereas the crustose, extremely toxitolerant Lecanora conizaeoides occurred more frequently on healthy trees. Stemflow concentrations of NH4+ NO3-, PO3-, and SO42- were lower on affected trees. This is attributed to reduced interception from the atmosphere due to needle loss. Cover of H. physodes decreased with increasing mean SO42- concentration in stemflow. The total of lichen species per sample tree also decreased with increasing SO42- concentration in stemflow, indicating that most species reacted in a similar way as H. physodes. Cover of L. conizaeoides increased with increasing SO42- concentration, but decreased at higher SO42- concentrations. Bark chemistry had a minor influence on lichen diversity. (C) 2002 The British Lichen Society. Published by Elsevier Science Ltd. All rights reserved
Contrast-Enhanced Clinical Magnetic Resonance Imaging
No full textIn Contrast-Enhanced Clinical Magnetic Resonance Imaging, Val M. Runge and other leading experts present an overview of the basic principles regarding MR contrast media, a review of clinical applications in the head, spine, and body, and a look at future developments. Their focus is on clinical applications, with extensive illustrations to demonstrate the use of MR in each anatomic area and to aid in film interpretation.
Val M. Runge is Rosenbaum Professor of Radiology and director of the Magnetic Resonance Imaging and Spectroscopy Center at the University of Kentucky Chandler Medical Center.https://uknowledge.uky.edu/upk_medicine_and_health_sciences/1006/thumbnail.jp
Relevance of element content of bark for the distribution of epiphytic lichens in a montane spruce forest affected by forest dieback
No full textElement content in the bark of Norway spruce (Picea abies) was measured in a montane forest heavily affected by forest dieback and compared to that in a nearby intact stand. Bark contained less S, K, Fe, Mn. Pb, Cu, and H+ and more N, Ca, Mg, and Zn in the dieback-affected stand than in the intact one. Diversity of epiphytic lichen vegetation was higher in the dieback-affected stand than in the intact one. Cover of the foliose lichen Hypogymnia physodes was negatively correlated with Mn and Cu content of bark. Cover of the extremely acidophytic species Lecanora conizaeoides decreased with increasing Mg and increased with increasing Cu content of bark. The measurements support the hypothesis that chemical site factors are decisive for the high lichen diversity in dieback-affected montane spruce forests. (C) 2001 Elsevier Science Ltd. All rights reserved
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