Modeling with diffirential equations and numerical methods for solving inverse problems

我们针对生物降解问题中常见的数学问题，如数学模型、反问题的分析和数值技术的发展，在生物降解问题中，高分子化合物被微生物的作用和水锤产生的水蒸气两相流降解。压力波动产生于供水钢管等管道中的液体流动，以及水锤作用下的压降。针对高分子化合物的生物降解，建立数学模型，结合实验结果进行分析，并通过数值模拟验证模型的正确性。微生物解聚过程分为外源型和内源型，外源型降解限制在分子末端，单体单元逐步分离，内源型分子在任意位置分离。我们进行了一次…更多的分析是基于外源解聚过程和内源解聚过程的模型，在这些过程中，降解率仅取决于分子量。另一方面，在微生物种群和温度等临时因素影响生物降解的情况下，有必要将这些时间因素引入降解率。在假设降解速率是时间和分子量函数的乘积的假设下，我们证明了通过变量的变化将模型转化为降解与时间无关的模型，并且利用先前发展的分析技术可以得到与分子量有关的因子。此外，还得到了与时间因素有关的结果。我们将实验结果引入到分析中，数值求解反问题，并模拟了重量分布随相对分子质量t的变化，以验证这种模型作为一般模型的合理性。较少

We aimed at objects of mathematical problems such as mathematical models, analysis of inverse problems, and development of numerical techniques common to problems concerning biodegradation in which high molecular compounds are degraded by actions of microorganisms and water-vapor two-phase flow generated by waterhammer. pressure surge generated in liquid flow in pipelines such as steel pipes of water supply, and by pressure drop in reaction to waterhammer. For biodegradation of high molecular compounds, we aimed at establishment of mathematical techniques of analysis such as proposals of mathematical models, analysis with introduction of experimental results, and verification of propriety of model by numerical simulation. Microbial depolymerization processes are classified into exogenous type in which degradation is restricted to terminals of molecules and monomer units are separated stepwise, and endogenous type in which molecules are, separated at arbitrary positions. We conducted an … More alyses based on models for exogenous depolymerization processes and endogenous depolymerization processes, in which degradation rates depend only on molecular weight. On the other hand, in case temporary factors such as microbial population and temperature affect biodegradation, it becomes necessary to introduce those temporal factors into a degradation rate. Under the assumption that a degradation rate is a product of a function of time and a function of molecular weight, we showed that a model was transformed by a change of variable to one in which the degradation is independent of time, and that the factor dependent of molecular weight can be obtained with previously developed techniques of analysis. Results concerning temporal factor were also obtained. We introduced experimental results into analysis, solved inverse problems numerically, and simulated transitions of weight distribution with respect to molecular weigh t in order to verify that such models are appropriate as general models. Less