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Abstract We begin with a treatment of the Caputo time‐fractional diffusion equation, by using the Laplace transform, to obtain a Volterra integro‐differential equation. We derive and utilize a numerical scheme that is derived in parallel to the L1‐method for the time variable and a standard fourth‐order approximation in the spatial variable. The main method derived in this article has a rate of convergence ofO(kα + h4)foru(x,t) ∈ Cα([0,T];C6(Ω)),0 < α < 1, which improves previous regularity assumptions that requireC2[0,T]regularity in the time variable. We also present a novel alternative method for a first‐order approximation in time, under a regularity assumption ofu(x,t) ∈ C1([0,T];C6(Ω)), while exhibiting order of convergence slightly more thanO(k)in time. This allows for a much wider class of functions to be analyzed which was previously not possible under the L1‐method. We present numerical examples demonstrating these results and discuss future improvements and implications by using these techniques.
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Tangle bases: Revisited
Abstract The concept of branch decomposition was first introduced by Robertson and Seymour in their proof of the Graph Minors Theorem, and can be seen as a measure of the global connectivity of a graph. Since then, branch decomposition and branchwidth have been used for computationally solving combinatorial optimization problems modeled on graphs and matroids. General branchwidth is the extension of branchwidth to any symmetric submodular function defined over a finite set. General branchwidth encompasses graphic branchwidth, matroidal branchwidth, and rankwidth. A tangle basis is related to a tangle, a notion also introduced by Robertson and Seymour; however, a tangle basis is more constructive in nature. It was shown in [I. V. Hicks. Graphs, branchwidth, and tangles! Oh my!Networks, 45:55‐60, 2005] that a tangle basis of orderkis coextensive to a tangle of orderk. In this paper, we revisit the construction of tangle bases computationally for other branchwidth parameters and show that the tangle basis approach is still competitive for computing optimal branch decompositions for general branchwidth.
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- Award ID(s):
- 1720225
- PAR ID:
- 10455316
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Networks
- Volume:
- 77
- Issue:
- 1
- ISSN:
- 0028-3045
- Page Range / eLocation ID:
- p. 161-172
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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