Graph theory, irreducibility, and structural analysis of differential-algebraic equation systems

The $\Sigma$-method for structural analysis of a differential-algebraic equation (DAE) system produces offset vectors from which the sparsity pattern of a system Jacobian is derived. This pattern implies a block-triangular form (BTF) of the DAE that can be exploited to speed up numerical solution. The paper compares this fine BTF with the usually coarser BTF derived from the sparsity pattern of the \sigmx. It defines a Fine-Block Graph with weighted edges, which gives insight into the relation between coarse and fine blocks, and the permitted ordering of blocks to achieve BTF. It also illuminates the structure of the set of normalised offset vectors of the DAE, e.g.\ this set is finite if and only if there is just one coarse block

How AD Can Help Solve Differential-Algebraic Equations

A characteristic feature of differential-algebraic equations is that one needs to find derivatives of some of their equations with respect to time, as part of so called index reduction or regularisation, to prepare them for numerical solution. This is often done with the help of a computer algebra system. We show in two significant cases that it can be done efficiently by pure algorithmic differentiation. The first is the Dummy Derivatives method, here we give a mainly theoretical description, with tutorial examples. The second is the solution of a mechanical system directly from its Lagrangian formulation. Here we outline the theory and show several non-trivial examples of using the "Lagrangian facility" of the Nedialkov-Pryce initial-value solver DAETS, namely: a spring-mass-multipendulum system, a prescribed-trajectory control problem, and long-time integration of a model of the outer planets of the solar system, taken from the DETEST testing package for ODE solvers

DAESA—A Matlab tool for structural analysis of differential-algebraic equations: theory

DAESA, Differential-Algebraic Equations Structural Analyzer, is a MATLAB tool for structural analysis of differential-algebraic equations (DAEs). It allows convenient translation of a DAE system into MATLAB and provides a small set of easy-to-use functions. DAESA can analyze systems that are fully nonlinear, highindex, and of any order. It determines structural index, number of degrees of freedom, constraints, variables to be initialized, and suggests a solution scheme. The structure of a DAE can be readily visualized by this tool. It also can construct a block-triangular form of the DAE, which can be exploited to solve it efficiently in a block-wise manner. This paper describes the theory and algorithms underlying the code

Conversion methods for improving structural analysis of differential-algebraic equation systems

Differential-algebraic equation systems (DAEs) are generated routinely by simulation and modeling environments. Before a simulation starts and a numerical method is applied, some kind of structural analysis (SA) is used to determine which equations to be differentiated, and how many times. Both Pantelides's algorithm and Pryce's Σ-method are equivalent: if one of them finds correct structural information, the other does also. Nonsingularity of the Jacobian produced by SA indicates a success, which occurs on many problems of interest. However, these methods can fail on simple, solvable DAEs and give incorrect structural information including the index. This article investigates Σ-method's failures and presents two conversion methods for fixing them. Both methods convert a DAE on which the Σ-method fails to an equivalent problem on which this SA is more likely to succeed

Algorithm 948: DAESA — a Matlab tool for structural analysis of differential-algebraic equations: Software

DAESA, Differential-Algebraic Equations Structural Analyzer, is a MATLAB tool for structural analysis of differential-algebraic equations (DAEs). It allows convenient translation of a DAE system into MATLAB and provides a small set of easy-to-use functions. DAESA can analyze systems that are fully nonlinear, high- index, and of any order. It determines structural index, number of degrees of freedom, constraints, variables to be initialized, and suggests a solution scheme. The structure of a DAE can be readily visualized by this tool. It can also construct a block-triangular form of the DAE, which can be exploited to solve it efficiently in a block-wise manner

Investigation into actions of Xiebai and Zengye decoction on cough sensitivity, airway inflammation and gut microbiota in the rat model of post-infectious cough

This work was undertaken to observe therapeutic effect of Xiebai and Zengye (XBZY) decoction on post-infectious cough (PIC) in rats, as well as its effect on gut microbiota and the exploration of the intestinal microecological mechanisms of XBZY decoction in the treatment of PIC. Using a random number table, the rats that were successfully modelled were assigned to the PIC, XBZY group (14.8 g/kg/d), and montelukast sodium treatment (MAS) group (1 mg/kg/d). The cough sensitivity of rats and changes in fecal water content were assessed, and serum interleukin-8 (IL-8) and tumor necrosis factor-alpha (TNF-α) levels were determined by ELISA. The histopathological changes in the bronchus and colon tissues were observed under the microscope after hematoxylin-eosin staining. Short-chain fatty acids (SCFAs) in fecal samples were measured by gas chromatography, and changes in gut microbiota were observed using 16S rRNA sequencing. The PIC rats showed decreased fecal water content, increased cough sensitivity, elevated serum TNF-α and IL-8 levels, and higher bronchitis scores comparing to normal control group. The PIC rats showed reductions in SCFAs and significant changes in the structure of gut microbiota. XBZY decoction intervention led to increased fecal water content in rats, reduced cough sensitivity, decreased serum IL-8 and TNF-α levels, decreased bronchitis scores, and alleviated inflammatory cell infiltration was observed in the colonic mucosa. Additionally, elevated SCFAs levels were observed in the PIC rats. XBZY decoction intervention improved alpha-/beta-diversity, and corrected microbiota imbalance in PIC rats. SCFAs, TNF-α and IL-8, acetic acid was revealed to be positively associated with Allobaculum but inversely correlated with unclassified_f_Oscillospiraceae; propanoic acid was positively associated with Lactobacilli but negatively associated with Romboutsia; butanoic acid exhibited positive correlations with Akkermansia and Lactobacilli, but negative correlations with unclassified_f_Oscillospiraceae, Eubacterium_xylanophilum_group, and Lachnospiraceae_NK4A136_group; Additionally, TNF-α was inversely linked to Allobaculum, while IL-8 was positively related to Romboutsia and Turicibacter. In conclusion, XBZY decoction significantly reduced cough sensitivity and airway inflammation in PIC rats while ameliorating stool dryness and colonic inflammation. The protective effects of XBZY decoction could be linked to modulat gut microbiota in PIC rats, and regulat SCFAs contents in PIC rats, while the regulator mechanisms of XBZY decoction in gut microbiota still requires further in-depth investigation