Comparison of Single Term Walsh Series Technique and Extended RK Methods Based on Variety of Means to Solve Stiff Non-linear Systems

This paper presents a comparison of Single Term Walsh Series (STWS) technique and the extended Runge-Kutta (RK) methods based on variety of means such as Arithmetic Mean (AM), Harmonic Mean (HaM), Centroidal Mean (CeM) and Contraharmonic Mean (CoM) to solve stiff non-linear systems of initial value problems (IVPs). Numerical solutions of some stiff non-linear systems are investigated for their stiffness. The discrete solutions obtained through STWS technique are compared with that of the RK methods based on variety of means. The applicability of the STWS technique has been demonstrated. The results show that the STWS technique is more suitable to solve stiff non-linear systems including highly stiff problems

An Application of STWS Technique in Solving Stiff Non-linear System: 'High Irradiance Responses' (HIRES) of Photomorphogenesis

This paper illustrates an application of the Single Term Walsh Series (STWS) technique in solving stiff non-linear system: ‘High Irradiance RESponses’ (HIRES) of Photomorphogenesis from plant physiology. The chemical reaction scheme of HIRES problem has been modelled into system of stiff non-linear differential equations. This stiff system has been solved using the STWS technique. The STWS solutions are compared with the results obtained by the well-known solvers, namely, VODE and RADAU5. The applicability of the STWS technique has been tested

The Theory of Functional Connections: A Journey from Theory to Application

The Theory of Functional Connections (TFC) is a general methodology for functional interpolation that can embed a set of user-specified linear constraints. The functionals derived from this method, called "constrained expressions," analytically satisfy the imposed constraints and can be leveraged to transform constrained optimization problems to unconstrained ones. By simplifying the optimization problem, this technique has been shown to produce a numerical scheme that is faster, more accurate, and robust to poor initialization. The content of this dissertation details the complete development of the Theory of Functional Connections. First, the seminal paper on the Theory of Functional Connections is discussed and motivates the discovery of a more general formulation of the constrained expressions. Leveraging this formulation, a rigorous structure of the constrained expression is produced with associated mathematical definitions, claims, and proofs. Furthermore, the second part of this dissertation explains how this technique can be used to solve ordinary differential equations providing a wide variety of examples compared to the state-of-the-art. The final part of this work focuses on unitizing the techniques and algorithms produced in the prior sections to explore the feasibility of using the Theory of Functional Connections to solve real-time optimal control problems, namely optimal landing problems