This dissertation presents a redefined operator splitting method used in solving semilinear parabolic partial differential equations. As one such form, the reaction-diffusion equation is highly prevalent in mathematical modeling. Besides being physically meaningful as a separation of two distinct physical processes in this equation, operator splitting simplifies the solution method in several ways. The super-linear speed-up of computations is a rewarding simplification as it presents great benefits for large-scale systems. In solving these semilinear equations, we will develop a condition for oscillation-free methods, a condition independent of the usual stability condition. This numerical consideration is important to fully embody our concerns for a method's stability and consistency, and is critical in extending our methods from linear to semilinear problems. The mathematical modeling process will be discussed and demonstrated by developing models from first principles and then reducing them down through simplifying assumptions. These models will be extended to simulate experimental data, using the comparison to real data as validation of the model. Finally, in the face of high data variability, this model validation process will present the need for calibration and further model refinement.
Harwood, R. Corban, "Operator Splitting Method and Applications for Semilinear Parabolic Partial Differential Equations" (2011). Faculty Publications - Department of Mathematics. 12.