Partial Differential Equations Commons^™

Asymptotic Expansion Of A Maier-Saupe Type Potential Near The Nematic-Isotropic Transition Point In Liquid Crystals, Ryan P. Oneill, Giangvuthanh Nguyen, Xiang Xu

OUR Journal: ODU Undergraduate Research Journal

In this paper we study a Maier-Saupe type bulk potential (Maier & Saupe, 1959) in the Landau-de Gennes free energy in the Q-tensor theory modeling nematic liquid crystal configurations. This potential was originally introduced in Katriel et al. (1986), which is considered as a natural enforcement of a physical constraint on the eigenvalues of symmetric, traceless Q-tensors. More specifically, we present a rigorous derivation of the asymptotic expansion of this singular potential near the nematic-isotropic transition point up to the 4-th order.

An Efficient Fourier Caching Algorithm For Walk On Spheres, Zihong Zhou

Dartmouth College Master’s Theses

Walk on Spheres (WoS) is a grid-free Monte Carlo method for solving elliptic partial differential equations (PDEs).
Rather than discretizing the domain, WoS leverages the mean-value principle to obtain Monte Carlo estimates by recursively averaging the solution over the largest contained sphere, terminating upon reaching the boundary.
Unfortunately, WoS requires many independent estimates to achieve noise-free results.

We propose an acceleration technique for WoS, inspired by irradiance caching methods, that computes the solution at a sparse set of locations, and extrapolates these cached values to local neighborhoods. A key insight is that WoS can be extended to compute not only …

Solving Fractional Differential Equations On A Quantum Computer: A Variational Approach, Fong Yew Leong, Dax Enshan Koh, Jian Feng Kong, Siong Thye Goh, Jun Yong Khoo, Wei Bin Ewe, Hongying Li, Jayne Thompson, Dario Poletti

Research Collection School Of Computing and Information Systems

We introduce an efficient variational hybrid quantum-classical algorithm designed for solving Caputo time-fractional partial differential equations. Our method employs an iterable cost function incorporating a linear combination of overlap history states. The proposed algorithm is not only efficient in terms of time complexity but also has lower memory costs compared to classical methods. Our results indicate that solution fidelity is insensitive to the fractional index and that gradient evaluation costs scale economically with the number of time steps. As a proof of concept, we apply our algorithm to solve a range of fractional partial differential equations commonly encountered in engineering …

On Weak Solutions And The Navier-Stokes Equations, Aryan Prabhudesai

Mathematical Sciences Undergraduate Honors Theses

In this paper, I will discuss a partial differential equation that has solutions that are discontinuous. This example motivates the need for distribution theory, which will provide an interpretation of what it means for a discontinuous function to be a “solution” to a PDE. Then I will give a detailed foundation of distributions, including the definition of the derivative of a distribution. Then I will introduce and give background on the Navier-Stokes equations. Following that, I will explain the Millennium Problem concerning global regularity for the Navier-Stokes equations and share mathematical results regarding weak solutions. Finally, I will go over …

Scalable Solution Of Time-Dependent Pdes Through Component-Wise Exponential Integrator, Chelsea Drum

Dissertations

Exponential integrators, such as exponential Runge-Kutta or Rosenbrock methods, are designed specifically for the time integration of stiff systems of ordinary differential equations (ODEs) and allow the use of larger time steps than other general-purpose ODE solvers. However, these methods rely on computing matrix function-vector products that are traditionally computed using a Krylov projection, such as Lanczos or Arnoldi iteration, that involves substantial computational expense at high spatial resolution. Krylov Subspace Spectral (KSS) methods' frequency-dependent approach, designed to circumvent stiffness in linear problems, computes these products with greater scalability. We propose the combination of such KSS methods with exponential integrators …

Delta-Shaped Approximation Based Homotopy Analysis Method For Nonlinear Poisson-Type Partial Differential Equations, Cyril Ocloo

Dissertations

This research aims to solve nonlinear Poisson-type partial differential equations (PDEs) by the approach of the homotopy analysis method (HAM) incorporated with approximate particular solutions (APS) using Delta-shaped basis (DSB) approximations.

With the inclusion of the h auxiliary parameters, we tackle nonlinear problems by studying the mathematical characteristics of the h curve. This is to ensure the numerical convergence of the HAM.

In the solution process, we use the homotopy analysis method to convert a nonlinear PDE into linear inhomogeneous PDEs, which are solved using the method of approximate particular solutions with DSB.

A proper value of the h is …

Resonant Solutions Of The Non-Linear Schrödinger Equation With Periodic Potential, Arein Duaibes, Yulia Karpeshina

Mathematics Faculty Publications

The goal is construction of stationary solutions close to non-trivial combinations of two plane waves at high energies for a periodic non-linear Schrödinger Equation in dimension two. The corresponding isoenergetic surface is described for any sufficiently large energy k2. It is shown that the isoenergetic surface corresponding to k2 is essentially different from that for the zero potential even for small potentials. We use a combination of the perturbative results obtained earlier for the linear case and a method of successive approximation.

Asymptotic Formula For Scattering Problems Related To Thin Metasurfaces, Zachary Jermain

LSU Doctoral Dissertations

The goal of this work is to develop an asymptotic formula for the behavior of a scattered electromagnetic field in the presence of a thin metamaterial known as a metasurface. By using a carefully chosen Green’s function and the single and double layer potentials we analyze the perturbed scattering problem in the presence of the metamaterial and a background scattering problem. By using Lippman-Schwinger type representation formulas for the two fields we develop the asymptotic formula for the perturbed field. From here we prove the asymptotic formula holds up to a specific error term based on the size of the …

Existence Of Smooth Solutions For The Landau Equation With Hard Potentials, Shelly Ann Taylor

Theses and Dissertations

This dissertation is concerned with the Landau equation, an integro-differential equation that models the particle density of a plasma as it evolves in phase space. The main topic is the (large-data) local existence of classical solutions to the Landau equation in the case of hard potentials (γ ∈ (0, 1]). Solutions have previously been constructed by Chaturvedi [SIAM. J. Math. Anal., 55(5), 5345–5385, 2023] for initial data in an exponentially-weighted Sobolev space of order 10, but it is not a priori clear whether these solutions have more regularity than the initial data. We improve Chaturvedi’s existence result in two ways: …

(R2074) A Comparative Study Of Two Novel Analytical Methods For Solving Time-Fractional Coupled Boussinesq-Burger Equation, Jyoti U. Yadav, Twinkle R. Singh

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, a comparative study between two different methods for solving nonlinear timefractional coupled Boussinesq-Burger equation is conducted. The techniques are denoted as the Natural Transform Decomposition Method (NTDM) and the Variational Iteration Transform Method (VITM). To showcase the efficacy and precision of the proposed approaches, a pair of different numerical examples are presented. The outcomes garnered indicate that both methods exhibit robustness and efficiency, yielding approximations of heightened accuracy and the solutions in a closed form. Nevertheless, the VITM boasts a distinct advantage over the NTDM by addressing nonlinear predicaments without recourse to the application of Adomian polynomials. …

(R2085) Heat And Mass Transport Characteristics In Williamson Fluid Flow Over A Permeable Stretching Cylinder, Amala Olkha, Mukesh Kumar

Applications and Applied Mathematics: An International Journal (AAM)

The intention of this research endeavor is to examine heat and mass transport in Williamson fluid flow induced by a permeable stretching cylinder in a porous medium. Various physical factors (like viscous dissipation, chemical reaction, etc.) affecting the relevant fields (flow, temperature and concentration) are incorporated in the investigation. The governing PDEs are turned into nondimensional ODEs using adequate similarity transformation relations, and then tackled numerically using MATLAB based Bvp4c technique along with shooting method. The impacts of various parameters arising in the problem are exhibited on fluid flow, temperature and concentration distribution by drawing portraits and discussed. Moreover, impressions …

(R2076) New Exact Solution Of Gilson–Pickering Equation In Plasma, Bingnuo Yang, Weinan Wu, Hongfeng Yu, Peng Guo

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we use Paul-Painlev´e approach method, extended rational sine-cosine method and extended rational sinh-cosh method to construct the exact solution of the nonlinear Gilson-Pickering (GP) equation in plasma. The exact solution of GP equation obtained by the above three methods is new, and we use mathematical software to draw the two-dimensional and three-dimensional graphs of the new exact solutions. Through the study of nonlinear equations in plasma, this study will enrich the research and connotation of nonlinear development equations in plasma.

(R2067) Solutions Of Hyperbolic System Of Time Fractional Partial Differential Equations For Heat Propagation, Sagar Sankeshwari, Vinayak Kulkarni

Applications and Applied Mathematics: An International Journal (AAM)

Hyperbolic linear theory of heat propagation has been established in the framework of a Caputo time fractional order derivative. The solution of a system of integer and fractional order initial value problems is achieved by employing the Adomian decomposition approach. The obtained solution is in convergent infinite series form, demonstrating the method’s strengths in solving fractional differential equations. Moreover, the double Laplace transform method is employed to acquire the solution of a system of integer and fractional order boundary conditions in the Laplace domain. An inversion of double Laplace transforms has been achieved numerically by employing the Xiao algorithm in …

A Model Of Oocyte Population Dynamics For Fish Oogenesis, Louis Fostier, Frédérique Clément, Romain Yvinec, Violette Thermes

Biology and Medicine Through Mathematics Conference

No abstract provided.

Reaction-Diffusions System Simulated On Irregular Shapes And Surfaces Model Petal Spot Patterns In Monkeyflower Hybrids, Emily Simmons, Arielle M. Cooley, Joshua R. Puzey, Gregory D. Conradi Smith

Biology and Medicine Through Mathematics Conference

No abstract provided.

Exploring The Evolution Of Altruistic Punishment Using A Pde Model For Multilevel Selection, Daniel Cooney

Biology and Medicine Through Mathematics Conference

No abstract provided.

Identifiability For Pde Models Of Fluorescence Microscopy Experiments, Veronica Ciocanel

Biology and Medicine Through Mathematics Conference

No abstract provided.

Chemoattractant Distribution In The Drosophila Egg Chamber, Lara Scott

Biology and Medicine Through Mathematics Conference

No abstract provided.

Sperm-Egg Interaction For Fertilization Success, Prajakta P. Bedekar

Biology and Medicine Through Mathematics Conference

No abstract provided.

Proof-Of-Concept For Converging Beam Small Animal Irradiator, Benjamin Insley

Dissertations and Theses (Open Access)

The Monte Carlo particle simulator TOPAS, the multiphysics solver COMSOL., and

several analytical radiation transport methods were employed to perform an in-depth proof-ofconcept

for a high dose rate, high precision converging beam small animal irradiation platform.

In the first aim of this work, a novel carbon nanotube-based compact X-ray tube optimized for

high output and high directionality was designed and characterized. In the second aim, an

optimization algorithm was developed to customize a collimator geometry for this unique Xray

source to simultaneously maximize the irradiator’s intensity and precision. Then, a full

converging beam irradiator apparatus was fit with a multitude …