Numerical Solution of Some Uncertain Diffusion Problems

Abstract

Diffusion is an important phenomenon in various fields of science and engineering. These problems depend on various parameters viz. diffusion coefficients, geometry, material properties, initial and boundary conditions etc. Governing differential equations with deterministic parameters have been well studied. But, in real practice these parameters may not be crisp (exact) rather it involves vague, imprecise and incomplete information about the system variables and parameters. Uncertainties occur due to error in measurements, observations, experiments, applying different operating conditions or it may be due to maintenance induced errors, etc. As such, it is an important concern to model these type of uncertainties. Traditionally uncertain problems are modelled through probabilistic approach. But probabilistic methods may not able to deliver reliable results at the required precision without sufficient data. In this context, interval and fuzzy theory may be used to manage such uncertainties. Accordingly, the system parameters and variables are represented here as interval and fuzzy numbers. Generally, we get interval or fuzzy system of equations for uncertain steady state problems with interval or fuzzy parameters whereas interval or fuzzy eigenvalue problems may be obtained for unsteady state. This thesis redefined interval or fuzzy arithmetic in order to handle the uncertain problems. The proposed arithmetic has been used to solve fuzzy and interval system of equations and eigenvalue problems. Various numerical methods viz. Finite Element Method (FEM), Wavelet Method (WM), Euler Maruyama and Milstein Methods are studied by introducing interval or fuzzy theory. The proposed arithmetic has been combined with FEM and WM to develop Interval or Fuzzy Finite Element Method (I/FFEM) and Interval or Fuzzy Wavelet Method (I/FWM). Further, it may be pointed out that sometimes systems may possess uncertainties due to randomness and fuzziness of the parameters. As such, here we have hybridized the concept of fuzziness as well as stochasticity to develop numerical fuzzy stochastic methods viz. interval or Fuzzy Euler Maruyama and Interval/Fuzzy Milstein. These methods are also been used to solve various diffusion problems. Numerical examples and different application problems are solved to demonstrate the efficiency and capabilities of the developed methods. In this respect, imprecisely defined diffusion problems such as heat conduction and conjugate heat transfer in rod, homogeneous and non-homogeneous fin and plate, along with one group, multi group and point kinetic neutron diffusion with interval or fuzzy uncertainties have been investigated. The convergence of the field variables have been investigated with respect to the number of element discretization of the domain in case of I/FEM. Accordingly, convergence of the proposed interval or fuzzy FEM has been studied for unsteady heat conduction in a cylindrical rod. For conjugate heat transfer problems, the convergence of uncertain temperature distributions with respect to the number of element discretizations has also been studied. Further, various combinations of uncertain parameters are considered and the sensitivity of these parameters has been reported. Next, one group and two group problems have been solved and the sensitivity of the uncertain parameters in the context of fast and thermal neutrons are presented. The hybrid fuzzy stochastic methods have also been used to investigate uncertain stochastic point kinetic neutron diffusion problem. Uncertain variation of neutron populations are analysed by considering two random samples. Developed interval or fuzzy WM has also been used to solve uncertain differential equation. Finally obtained results for the said problems are compared in special cases for the validation of proposed methods.