Novel numerical methods for stochastic ordinary and partial differential equations in modelling complex systems : A thesis submitted in partial fulfilment of the requirements for the Degree of Doctor of Philosophy at Lincoln University

Abstract

Many natural and engineered systems are complex due to inherent uncertainty. Stochastic Differential Equations (SDEs) and Stochastic Partial Differential equations (SPDEs) provide a rigorous mathematical foundation for modelling these systems. Understanding the dynamics of complex systems under stochastic influences is crucial for predicting system behaviour. Numerical techniques often struggle to handle the complexity and stochastic nature of these equations. This research focuses on adapting and enhancing numerical methods to provide efficient and reliable solutions. The numerical accuracy and stability of these methods are assessed through simulations and examples. This study introduces the synthesis of stochastic spectral methods to solve complex systems by representing random variables as a sum of orthogonal polynomials. We applied Polynomial Chaos Expansion (PCE) methods to contaminant transport problem and to differential equations with random forcing term. We compute the Wick exponentials and show that Wick product coincides with the ordinary product for deterministic functions. We use Malliavin calculus to find the derivatives of a stochastic quantity and are visualised through graphs. We discuss numerical challenges associated with the PCE methods and their solution strategies. In all examples, our chosen method does better and allows us to lead the way in developing robust and efficient strategies to deal with randomness, ultimately enhancing the reliability and resilience of complex systems across various scientific and engineering domains.