Partial Differential Equations Commons^™

Optimal Control Of Stochastic Systems: A Numerical Study Of The Stochastic Linear Quadratic Regulator Framework, Gülşen Orucova Büyüköz, Yaprak Güldoğan Dericioğlu, Tuğçem Partal

Mathematical Modelling and Numerical Simulation with Applications

Optimal control of stochastic linear systems is fundamental in control theory, with applications in robotics, finance, and engineering. The Stochastic Linear Quadratic Regulator (SLQR) derives optimal feedback laws via the Riccati equation but requires numerical discretization of the resulting stochastic dynamics. Despite extensive studies on numerical methods for stochastic differential equations, their performance within the SLQR framework remains insufficiently explored. This study compares two predictor–corrector schemes of different orders: the Order 1.0 Predictor-Corrector (PC) method and the Order 2.0 Weak PC method. A one-dimensional linear quadratic problem with a closed-form solution enables precise error evaluation against the analytical trajectory. Convergence …

Advanced Soliton Structures And Elliptic Wave Patterns In A Sixth-Order Nonlinear Schrödinger Equation Using Improved Modified Extended Tanh Function Method, Mina Fahim, Hamdy Mohamed Ahmed, Islam Samir, Kamal Dib

Basic Science Engineering

In this work, a sixth–order extension of the nonlinear Schrödinger equation (NLSE) within its integrable hierarchy is investigated to model higher–order nonlinear and dispersive effects relevant to optical fiber systems and nonlinear wave propagation. By employing the Improved Modified Extended Tanh Function Method, a comprehensive family of exact analytical solutions is derived, encompassing bright and dark solitons, singular soliton structures, and singular periodic solutions. In addition, solution families expressed in terms of Jacobi elliptic functions, Weierstrass doubly periodic elliptic functions, and exponential profiles are obtained. The novelty of this study lies in extending the analytical framework of the NLSE hierarchy …

On Sobolev Spaces And The Existence Of Weak Solutions To Boundary Value Problems, Skye X. Paul

Master's Theses

Many boundary value problems that arise in mathematical models have close connections to second order elliptic partial differential equations. This thesis introduces the idea of weak derivatives and Sobolev Spaces to generalize possible solutions. Using functional analysis centered around the Lax-Milgram theorem, we show the existence of these generalized solutions to boundary value problems including Laplace's Equation, 2nd order linear ODEs, and ultimately a general second order elliptic PDE. The work cumulates with recovering a number of central theorems of functional analysis in the context of Sobolev Spaces, creating a new perspective on the solvability of these boundary value problems.

Computational Models For Pre-Lens Tear Film Drug Concentration Dynamics With Drug Supply From A Contact Lens And Drug Exchange During Blinking, Mazen A. Althobaiti

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Mathematical Model Of Ovarian Cancer Tumor Growth In Mice, Jessica A. Hoffman

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Can We End The Hiv Epidemic In The U.S.? Linking Clinical And National Surveillance Data Through Multiscale Modeling From Patients To Populations, Necibe Tuncer

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Quantification Of Parameters To Predict The Rupture Of Intracranial Saccular Aneurysms Using Physics Informed Neural Networks, Alonso Gabriel Ogueda, Padmanabhan Seshaiyer

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Meshless Discrete Velocity Boltzmann Model For Porous Media Flow, Amandine Maidenberg

Doctoral Dissertations and Master's Theses

This dissertation explores the combination of two sophisticated techniques for addressing computational fluid dynamics: the discrete velocity Boltzmann equation (DVBE) and the localized collocation meshless model with upwinding (U-LCMM). The DVBE is a high-level model that describes the foundations of transport phenomena by addressing the microscale motions of particles themselves and the effect of their aggregate behaviors on continuum principles. This equation integrates multiple scales of phenomena; while it can be used for fluid flow at Navier-Stokes scales, it can also resolve fine features that can only be described at the molecular level. This type of model is necessary for …

(Si15-093) Dynamics Of The Generalized Nonlinear Schrödinger Equation With A Source Using An Analytical Method, Manish Raghav, Manoj .

Applications and Applied Mathematics: An International Journal (AAM)

In the present article, an analytical method is used to obtain hyperbolic, trigonometric, and rational solutions of the generalized nonlinear Schrödinger (GNLS) equation with a source. The ability of solitons to preserve their shapes during propagation makes them suitable for optical fiber communication. Solutions to the generalized nonlinear Schrödinger equation with a source can also describe solitons, and understanding their dynamics helps to design communication systems based on solitons. The analytical method used is compelling and effective for finding exact solutions to various nonlinear evolution equations (NLEEs). To further understand the phenomena, we create 3−D, contour, and 2−D graphs of …

(Si15-113) Augmenting Cryptographic Security Through Inventive Application Of The Kharrat-Toma Transform Algorithm, Prabakaran Raghavendran, Tharmalingam Gunasekar, K. Sakthivel, Kamalendra Kumar, Shalini Gupta

Applications and Applied Mathematics: An International Journal (AAM)

This paper introduces a cryptographic technique combining the Kharrat-Toma Transform and congruence modulo operators to improve the security of message encryption. The proposed model uses the mathematical properties of the Kharrat-Toma Transform and its inverse for direct scrambling and unscrambling processes while embedding sufficient complexity to resist modern cryptanalytic attacks. The model is subjected to experimental tests, including encryption quality analysis, Shannon entropy, and NIST randomness tests, in order to prove the strength of the model. Through encryption quality analysis, symbol frequencies in the ciphertext are masked heavily from having much correlation between plaintext and ciphertext. Entropy values indicate near-theoretical …

Stabilized Weak-Gradient Discontinuous Finite Elements With Optimal Error Estimates For Second-Order Elliptic Pdes, Aymen Laadhari

Mathematical Modelling and Numerical Simulation with Applications

This work introduces an accurate finite element approach employing a new stabilized discrete weak gradient, designed for second-order elliptic problems on arbitrary conforming meshes. We formulate the approach within a discontinuous Galerkin framework and derive a consistent and coercive bilinear form. Appropriate error analysis on a model problem confirms optimal convergence. Building on the core analysis, we extend the method to more challenging settings, including time-dependent heterogeneous scenarios and a biophysically realistic optimal-control model of photobleaching in the budding yeast cell. We further illustrate the versatility of the weak-gradient construction by applying it to an unsteady level-set equation relevant to …

Solution Of Fractional Order Diffusion Equations With Clique Neural Network, Merve Zeynep Kaya, Mesut Karabacak, Ercan Çelik

Mathematical Modelling and Numerical Simulation with Applications

In this paper, the clique artificial neural network method is used to solve the fractional diffusion equation, which is a subclass of partial differential equations. The clique neural network architecture is constructed using input, hidden, and output layers. Several degrees of clique polynomials were used as activation functions, and the output layer was obtained by multiplying them with weight coefficients. Subsequently, the optimization equation was derived, and the exact solution, numerical solution, and error function graphs were obtained using a specialized algorithm. Analysis of the results demonstrates that the clique artificial neural network method provides quicker and more accurate results …

Accurate Temporal Integration Schemes For Nonlinear Adsorption Problems, Evan D. Butterworth

All Dissertations

We consider a nonlinear transport problem to model the chromatography process of high-capacity multimodal membranes. Robust and efficient algorithms that simulate these bioseparation processes are critical to developing therapeutics for various chronic illnesses and infectious diseases. However, much of the current methodology focuses on stabilization and linearization techniques, often implementing low-order time-discretizations and linearized adsorption, resulting in inefficiencies and inaccuracies in the numerical solution. Utilizing Rothe's method, we develop various time-discretization schemes coupled with the finite element method to solve the fully implicit problems. Stability and solvability results are presented for several methods. Through multiple high-level software implementations paired with …

Constructing The Soliton Wave Structure And Stability Analysis To Generalized Calogero–Bogoyavlenskii–Schiff Equation Using Improved Simple Equation Method, Mina Fahim, Hamdy Mohamed Ahmed, Islam Samir Soliman, Mohamed Elsaid, Kamal Hassan Eldib

Basic Science Engineering

In this work, we investigated the (3+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation, which models long wave propagation in shallow water and plays a significant role in fluid mechanics and plasma physics. Using the improved simple equations method, we obtained various solutions, including dark, bright, and singular solitons, and combinations of singular periodic solutions and exponential rational solutions. Additionally, we performed a linear stability analysis to examine the stability properties of these wave solutions. To further illustrate their characteristics during propagation, we provided 3D and contour plots for some opted wave solutions.

A Modified Sir Model Used To Investigate The Relationship Between Congenital And Adult Syphilis, Kaleesta R. Waysman

Honors Thesis

A complex SIR model integrating the relationship between adult and congenital syphilis was developed. The goal of the project was to determine the specific population(s) or control strategies that should be enforced, altered, or removed to decrease the number of children experiencing clinical sequelae due to congenital syphilis. Early clinical sequelae include hydrops fetalis, preterm birth, central nervous system infection, hepatosplenomegaly, hyperbilirubinemia, cholestasis, hemolytic anemia, snuffles, osteochondritis, and lesions or rashes in the palms and soles. Late clinical sequelae include interstitial keratitis, hearing loss, Hutchinson teeth, saber shins, Clutton joints, mulberry molars, and saddle nose. After implementing real-world data into …

Globally Adaptive Exponential Integrators For Stiff Systems Of Odes, Anzhelika Vasilyeva

Honors Theses

This thesis introduces a novel method for solving systems of Ordinary Differential Equations (ODEs) resulting from the spatial discretization of Partial Differential Equations (PDEs). The proposed approach builds upon an existing technique that employs Krylov projection, which requires evaluating a matrix function at each timestep. The innovation of the new method lies in its reuse strategy, which shifts the perspective from direct matrix function evaluation to polynomial interpolation. Numerical experiments conducted on constant and variable coefficient heat equations, with both smooth and discontinuous initial data, demonstrate the computational time advantage of the new approach. The results indicate that this method …

Robust And Efficient Solvers For Physics-Based Pde’S, Elizabeth Hawkins

All Dissertations

This work was partially supported by the U.S. Department of Energy under award DE- SC0025292, by NSF grant DMS 2152623, and by NSF grant DMS 2011490.

This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Mathematical Multifaceted Integrated Capability Centers (MMICCs) program, under Field Work Proposal 22-025291 (Multifaceted Math- ematics for Predictive Digital Twins (M2dt)), Field Work Proposal 23-020467, and Computing and Information Sciences (CIS) investment area in the Laboratory Directed Research and Development program at Sandia National Laboratories. This written work is authored by an employee …

Domain Decomposition For Coupled Systems Of Fluid-Structure Interaction And Numerical Modeling For Thin Film Polymers, Amy De Castro

All Dissertations

We consider two primary areas of physical application in this work: fluid interaction systems with either linear elastic structures or with poroelastic structures, and thin film polymers, where the majority of the work focuses on the fluid-structure interaction systems.

In the first chapter, we present a strongly coupled partitioned method for fluid structure interaction (FSI) problems based on a monolithic formulation of the system which employs a Lagrange multiplier (LM). We prove that both the semi-discrete and fully discrete formulations are well-posed. To derive the partitioned scheme, a Schur complement equation, which implicitly expresses the Lagrange multiplier and the fluid …

Properties Of Eigenvalues Of The Fractal Laplacian, Eric Stachura, Andrew Chincea

Symposium of Student Scholars

We investigate the properties of the eigenvalues of the fractal Laplacian. We begin by defining the fractal Laplacian operator in one dimension and formulate the corresponding Dirichlet eigenvalue problem. Analytical solutions are obtained for specific fractal parameters, and computational results illustrate the structure of eigenvalues and their associated eigenfunctions. We extend our analysis to two dimensions using separation of variables. Our findings contribute to a deeper understanding of how fractal geometry affects the spectral characteristics of differential operators.

Unfitted Finite Element Methods For Shape Optimization And Liquid Crystals, Jeremy T. Shahan

LSU Doctoral Dissertations

We present an approach to shape optimization problems that uses an unfitted finite element method (FEM). The domain geometry is represented, and optimized, using a (dis- crete) level set function and we consider objective functionals that are defined over bulk domains. For a discrete objective functional, defined in the unfitted FEM framework, we show that the exact discrete shape derivative essentially matches the shape derivative at the continuous level. In other words, our approach has the benefits of both optimize-then- discretize and discretize-then-optimize approaches.

Specifically, we establish the shape Fréchet differentiability of discrete (unfitted) bulk shape functionals using both the …

(Si14-15) Heat Source/Sink And Chemical Reaction Effects On Micropolar Mhd Nano Fluid Flow In Stretching/Shrinking Sheet, Tejal Nagar, Harshad Patel, Akhil Mittal, Vikas Kori

Applications and Applied Mathematics: An International Journal (AAM)

This paper deals with the effects of a non-uniform heat source/sink and chemical reaction on micropolar nanofluid flow in a stretching and shrinking sheet. The flow is considered as a laminar mixed convective two-dimensional steady flow. In this flow, water is considered as a base fluid, whereas iron oxide is considered to be a conventional fluid. The governing non-linear system of PDEs are transformed into a system of ODEs using the similarity transformation, and HAM is employed for obtaining solutions. For more understanding of the effects of various physical conditions, approximate results are obtained, and expressed graphically. From the results, …

(Si14-11) Analysing Co-Current Imbibition Phenomenon In Heterogeneous Reservoir Using Multistep Hybrid Differential Transform Finite Difference Method, Aruna Sharma, Amit Parikh

Applications and Applied Mathematics: An International Journal (AAM)

The primary focus of this study is to analyse the co-current imbibition phenomenon in an inclined heterogeneous reservoir. This phenomenon occurs during the secondary oil recovery process. Capillary force is responsible for the displacement of a non-wetting phase by a wetting phase, and this phenomenon is called spontaneous imbibition. Imbibition is of two types and can be differentiated based on the direction in which the wetting phase (water) and non-wetting phase (oil) move. If the two phases flow in the same direction, it is called co-current imbibition, and if they flow in the opposite direction, it is called counter-current imbibition. …

Analyzing And Comparing Refinement Indicators For Rbf-Fd Adaptive Algorithms, Anders R. Johnson

Theses and Dissertations

Recent progress has been made in the development of collocation-based iterative algorithms that approximate solutions to PDEs. These algorithms rely on the ability to identify regions within a domain where a finer discretization is required. Such iterative algorithms are beneficial particularly when solution functions have highly localized behavior. This thesis proposes an indicator for node refinement that is constructed by approximating the forward error. This proposed indicator also helps to establish confidence in the accuracy of a given solution estimate. The proposed error estimator is theoretically examined and compared with contemporary refinement indicators. It is shown that an iterative algorithm, …

(Si14-06) Influence Of Hall Effects On Rotating Mhd Casson Fluid Flow Over A Vertical Plate, Zankhana Mali, Akhil Mittal, Harshad Patel

Applications and Applied Mathematics: An International Journal (AAM)

Studies of Hall effects regarding motion of fluid due to some external forces in MHD transport of reacting Casson fluid with heat generation over an impulsively emerging vertical plate are considered in this work. This theory proposes that because of a sudden rise in temperature and an accompanying surface concentration profile, which shows an elevation with time, the boundary plate has endured rapid expansion. In a rotational environment, this characteristic occurs homogeneously inside a porous uniform material. It applies the Laplace transform method for determining the fundamental equations subject to imposed starting and side conditions. Under isothermal conditions, accurate formulae …

Investigation Of Node Refinement Methods In Local Adaptive Kernel Based Approximation, Shelby W. Woodrum

Theses and Dissertations

This thesis explores computational efficiency and accuracy of six node refinement methods for local adaptive kernel-based approximations of solutions to the two-dimensional Poisson equation. Using an adaptive kernel-based approximation algorithm, this research investigates performance of Delaunay triangulation-based methods (shifted barycenters and edge midpoints), refinement via approximate Fekete and discrete Leja points, and a meshless predefined shift refinement method across two domains with varying complexities. Computational experiments reveal that Delaunay triangulation-based methods achieve a practical balance between accuracy and efficiency, particularly in square domains. Refinement via approximate Fekete and discrete Leja points produce accurate results but incur greater computational costs, making …

Mathematical Modelling Of Hybrid Photonic Structures For Holographic Sensors, Jack Lyons

Doctoral

This thesis outlines a mathematical framework for modelling the formation of holographic gratings in hybrid photopolymer based nanocomposites with the aim of optimising their holographic recording properties for optical sensing applications. Thus, the second aim of the work is to model the change in optical properties of the grating in response to exposure to a target analyte. This work has been a collaborative research project between the School of Mathematics & Statistics at Technological University Dublin and the Centre for Industrial and Engineering Optics that have done extensive experimental work with holographic gratings recorded in photopolymer materials.

In recent years, …

Long-Time Asymptotics For The Kadomtsev–Petviashvili I Equation With Small Initial Data, Samir Donmazov

Theses and Dissertations--Mathematics

We study the initial value problem for the Kadomtsev--Petviashvili I (KP I) equation (u_t + 6uu_x + u_xxx)_x = 3u_yy with small initial data belonging to a subspace of the energy space for the KP I equation. We establish the long-time asymptotics for solutions of the KP I equation using the inverse scattering transform formalism developed by Zhou. Within this framework, the inverse problem for the KP I equation is formulated as a nonlocal Riemann--Hilbert problem (RHP) in two spatial dimensions. As part of the asymptotic analysis, we determine the long-time behavior of the …

Ergodic Switching Control For Markov-Feller Processes Ii, Jose L. Menaldi, Maurice Robin

Mathematics Faculty Research Publications

This is the continuation of Part I [14], where we considered control problems with long term average (or ergodic) cost for Markov switching processes (zt , nt ), nt being a discrete component with values in a finite set N . The control acts only on this discrete component and consists of immediate switching actions. We solve the ergodic problem in several situations extending previous works, mainly when zt is a reflected diffusion with or without jumps and when the set of control values is strictly smaller than N . In this Part II, we conclude our theoretical analysis with …

Ergodic Switching Control For Markov-Feller Processes I, Jose L. Menaldi, Maurice Robin

Mathematics Faculty Research Publications

We consider control problems with long term average (or ergodic) cost for Markov switching processes (zt, nt ), nt being a discrete component with values in a finite set N . The control acts only on this discrete component and consists of immediate switching actions. We solve the ergodic problem in several situations extending previous works, mainly when zt is a reflected diffusion with or without jumps and when the set of control values is strictly smaller than N .

Mathematical Contributions To The Study Of Chemotaxis And Cell Signaling, Hajr Zam

Graduate Theses, Dissertations, and Problem Reports (ETD)

This dissertation presents results from two mathematical projects concerned with the biology of cells. Chapter 1 provides biological background and places the two mathematical problems in the context of cell signaling. The larger project, with Prof. H. Hattori on a chemotaxis model is presented in Chapters 3 and 4. Work with Prof. \'{A}. Hal\'{a}sz on a chemical reaction network system with linear multimers and two types of labels is presented in Chapter 2. The chemotaxis system describes the one-dimensional dynamics of a species of cells with two chemical species, a chemo-attractant and chemo-repellent. The goal is to analyze the behavior …