The increasing concern over global warming has driven interest in clean energy solutions, with piezoelectricity emerging as a promising alternative. Piezoelectric materials generate electric voltage under external mechanical forces, offering an innovative method for energy harvesting. This work derives a system of partial differential equations (PDEs) and their accompanying boundary conditions that describe the coupled elastic-electric behavior of an Euler-Bernoulli piezoelectric beam. Under the quasi-static approximation for the electrical field, the assumptions of Euler-Bernoulli beam theory, and the constitutive relations for the 3-1 piezoelectric coupling mode (i.e., voltage is generated in a direction perpendicular to external mechanical force), we develop a Hamilton’s variational principle to derive the governing equations and boundary conditions for the piezoelectric Euler-Bernoulli beam. The obtained equations consist of Gauss’s law of electrostatics and the Euler-Bernoulli beam equation that are coupled due to the piezoelectric effect: apparent electric charges that depend on elastic deflection appear in Gauss’s law, while apparent mechanical forces and moments that depend on the electric potential appear in the Euler-Bernoulli beam equation and its boundary conditions. The derivation from first principles, as well as the study of the governing equations constitutes a fundamental framework for analyzing piezoelectric beam behavior, with implications to the improvement of design of piezoelectric energy harvesters.

Chimeric Antigen Receptor (CAR) T-cell therapy has revolutionized immunotherapy for blood cancers, achieving unprecedented outcomes for many patients. However, variability in treatment responses—ranging from complete remission to relapse or severe side effects—remains a critical challenge. Mathematical and computational models that have been calibrated to experimental data can help to predict treatment efficacy and inform personalized therapeutic strategies. Working with Dr. Konstantinos Mamis (UW Applied Mathematics) and Dr. Katherine Owens (Fred Hutchinson Cancer Center), Rohan Pandey and Ray Chen (UW ACMS Department) employ models consisting of systems of ordinary differential equations (ODEs)- to simulate tumor and CAR T-cell dynamics. Though several prior mathematical models analyzing the interactions between CAR T-cells, tumor cells, and effector cells under varying treatment conditions exist, there has not been a systematic comparison of models representing competing mechanistic hypotheses against data from patients undergoing CAR T-cell treatment and/or chemotherapy. For two existing mathematical models, we explore the practical identifiability of model parameters using synthetic data and a population approach with nonlinear mixed effects implemented in Monolix. Furthermore, we calibrate the model parameters to real data from 10 patients with B-cell acute Lymphoblastic Leukemia (B-ALL) and identify the most accurate and parsimonious of the existing models. Finally, we determine and study the effect of key variables that largely influence patient responses to therapy, including those associated with sustained remission or relapse. This computational oncology work has the potential to inform strategies for optimal CAR T-cell therapy, improve patient outcomes, and further innovation in cancer treatment.

Nonlinear dynamics and its related stochastic phenomena described by linear partial differential equations are exceptionally useful for modeling climate processes. In particular, climate often exhibits bistability: the system under one forcing can exist in two stable states. With changing parameters and fluctuations, the state may transition through a bifurcation (tipping point) or spontaneous pre-tipping transition. One example is Snowball Earth. Earth is thought to have a bistable ice-covered and ice-free climate and once transitioned away from ice-covered. With a time dependent parameter, we expect a bistable system will eventually undergo bifurcation. Methods have been used to predict the time until tipping. However, these predictions do not include the possibility of transition prior to the bifurcation. I calculate this probability by modeling the Snowball Earth state as an Ornstein–Uhlenbeck process through a saddle-node bifurcation. As the system approaches the bifurcation, we expect the variance of the OU process about the stable state to increase. While my model shows it increasing for most time towards the tipping point, there is an unexpected decrease near the end. I found this corresponds to a change in symmetry of the OU process distribution. To check the significance of this, I will compare with a numerical simulation of the Fokker-Planck equation for the OU process. I will also show a probability distribution for transition over time, by modeling the state as a continuous-time Markov chain that depends on the varying shape of the “barrier” in the potential function. Ultimately, Snowball Earth is a paradigm for the exchange between stochastic and partial differential equations that describes many systems. We can compare it with other systems to characterize similarities among them, as well as features making the climate system unique. This is also important as critical transition in Earth systems is a growing concern under climate change.

In this work, I propose an improved remeshing approach for particle method approximations. Particle approximation methods are a flexible tool for approximating solutions to nonlinear continuity equations, and are especially useful for aggregation-diffusion equations, which have important applications in fields ranging from modeling physical processes to neural networks. They work by decomposing functions into constituent parts, called particles. By tracking the motion and mass associated with each of these particles over time, we then use these to construct a high-resolution approximation to a desired solution. However, particle methods suffer from accuracy decay over time, necessitating remeshing (resetting particle positions) to maintain a useful approximation. It's important that the techniques used for this remeshing preserve existing structures, so that our approximation exhibits the same qualities as the true solution of the underlying equation. For instance, existing remeshing techniques often preserve conservation of mass, but not entropy decay. By combining remeshing techniques to periodically merge clustered particles and introduce new particles, I'm developing a method that maintains approximation accuracy and preserves structural properties. I present the results of the numerical analysis done using Python, as well as an implementation of the method using a finite-difference approach, which examines the approximation at various steps through time. This approach is expected to preserve the structure of the true solution within the particle method approximation, contributing to the development of robust particle methods for a broad class of partial differential equations.

Halofun is an object-oriented program using Chebfun software in MATLAB that employs low rank spectral approximation methods to efficiently compute and store functions on annuli. The computational cost of numerical techniques are often a major constraint in computing and working with large and complex mathematical problems. Low rank representations of functions can reduce computational costs; low rank function approximation has relations and advantages similar to that of the singular value decomposition in terms of isolating the most important features of the problem. Similar to writing a matrix representation as a sum of rank 1 matrices, we can write a function as a sum of k rank 1 functions. By using a representation based on Chebyshev/Fourier basis functions, we can make use of fast, FFT-based transforms and other fast algorithms to compute with functions on annuli. Halofun expands on previously developed low rank spectral methods in rectangular, circular, disk and spherical domains, so that a user-friendly software for fast and spectrally accurate computations now exists for ring-shaped domains.