Applied Mathematics Commons^™

Performance Of Numerical Methods Applied To The Black–Scholes Model, Scott Cameron Williams

UNF Graduate Theses and Dissertations

We compare five numerical approaches for approximating solutions to the Black–Scholes partial differential equation for pricing European call options: FTCS, BTCS, Crank– Nicolson, Monte Carlo simulation, and a physics–informed neural network (PINN). These methods span finite difference techniques, probabilistic simulation, and machine learning. Performance is evaluated based on computational efficiency and accuracy relative to the analytical Black–Scholes solution.

Among the methods, Crank–Nicolson and the PINN demonstrated the strongest overall performance. Crank–Nicolson achieved the highest accuracy but exhibited increased runtime as the number of underlying stock price grid points grew. In contrast, the PINN produced slightly less accurate results but with …

A Reconstruction Method For The Anisotropic Electrical Conductivity In A Hybrid Planar Inverse Problem, Athanasios Dimitriadis

Graduate Studies Theses and Dissertations 2026

Some single-physics medical imaging methods fail to reliably discriminate benign from malignant biological tissue. In partial response, a new class of inverse problems employ multi-physics phenomena, coupling methods of high resolution with methods of high contrast. In this thesis we are concerned with one such problem in which an anisotropic electrical conductivity is to be recovered from some internal knowledge of the current density field generated by maintaining a fixed boundary voltage. This internal data is obtained by the method of Current Density Impedance Imaging (CDII), which combines Maxwell's equations with Magnetic Resonance Imaging (MRI) measurements. The anisotropy class is …

Microgravity-Induced Alterations In Left Atrial Hemodynamics And Thrombogenic Risk: Insights From Healthy And Atrial Fibrillation Models, Grace M. Hoeppner

Dissertations, Master's Theses and Master's Reports

Background: Microgravity exposure alters cardiovascular loading, yet its impact on left atrial flow dynamics and thrombotic risk remains poorly understood. This study investigates how spaceflight-relevant microgravity-induced changes in cardiac outflow affect left atrial hemodynamics in healthy individuals and patients with atrial fibrillation.

Methods: Patient-specific left atrial models were generated for three healthy individuals and three AF patients. Computational fluid dynamics (CFD) simulations were performed using each patient’s baseline mitral outflow waveform and two modified waveforms representing short- and long-duration post-flight cardiac loading changes derived from echocardiographic observations. Hemodynamic metrics included left atrial velocity, time averaged wall shear stress, oscillatory shear …

(R2151) Error Estimates Of Barycentric Lagrange Interpolation, Alvira Yawar, Swarnima Bahadur

Applications and Applied Mathematics: An International Journal (AAM)

Barycentric interpolation, which comes from Lagrange interpolation, is a useful method in numerical analysis. In this research paper, we explain how the barycentric interpolation formula is derived and discuss its features. We compare its stability and performance with the traditional Lagrange formula. First, we show how to get the barycentric formula from the Lagrange polynomial and present it as a rational function. We also provide an estimate of the error. Then, we use numerical examples to show that the barycentric formula is more stable and works better, especially when the degree of interpolation is high. Our results show that the …

Analysis And Numerical Investigation Of A Breast Cancer Treatment Model Incorporating Ketogenic Diet And Immune Boosters Via Optimal Control Theory, Kunnisai Muniroh, Ummu Habibah, Wuryansari Muharini Kusumawinahyu, Nur’Izzati Hamdan

Mathematical Modelling and Numerical Simulation with Applications

This paper develops a mathematical model to investigate breast cancer dynamics by incorporating tumor–immune interactions, ketogenic diet effects, and medical treatment. The model is formulated as a system of nonlinear ordinary differential equations and analyzed within an optimal control framework. Time-dependent control variables are introduced to represent treatment strategies aimed at minimizing tumor progression while reducing therapeutic costs. The model’s well-posedness is established through positivity and boundedness analysis. The necessary conditions for optimality are derived using Pontryagin’s Minimum Principle, resulting in a coupled system of state and adjoint equations. Numerical solutions are obtained using the fourth-order Runge–Kutta method combined with …

Mathematical Analysis And Numerical Simulation Of A Fractional-Order Sir-Si Model For Malaria Transmission Dynamics, Gassan A. M. O. Farah, Abdulaziz Y. A. Mukhtar, Kailash C. Patidar

Mathematical Modelling and Numerical Simulation with Applications

This study investigates the complex transmission dynamics of malaria, a critical global health challenge, with a focus on the African continent. We introduce a novel approach that employs Fractional Differential Equations (FDEs) to advance the understanding of malaria spread and control. Specifically, we develop a new SIR-SI model using the Caputo fractional operator, which captures the memory effects and time-delay characteristics inherent in real-world epidemiological systems. A detailed analysis of the model's solvability and uniqueness is conducted using fixed-point theory. To obtain an analytical solution, the system is solved via the Laplace transform method, with solutions expressed in closed form …

A Parallel Interval Modeling Framework For Nonlinear Systems: Application To A Modified Duffing Oscillator, Roman Voliansky, Nina Volianska

Northeast Journal of Complex Systems (NEJCS)

The paper presents a mathematical framework for converting nonlinear dynamical systems into parallel forms. This framework replaces the exact system motion equations with interval equations, enabling the representation of nonlinear functions over piecewise linear domains. Such representation enables the description of system motions using linear-like differential equations, which can be analyzed and manipulated using well-known control methods. One such method is eigenvalue analysis, a powerful tool in classical control theory since many techniques rely on the system’s characteristic polynomial and its eigenvalues. We apply this method to define interval system eigenvalues and track their variation during system operation. These eigenvalues …

(R2134) Continuous Shooting Approach With Improved Shooting Slope For Solving Higher Integer Order Boundary Value Problem, Razaq Adekola Oderinu, Adebowale Niyi Aderibigbe, Saheed Alao, Ahmed Adeyi Yahaya

Applications and Applied Mathematics: An International Journal (AAM)

This study presents a semi-analytical shooting method for solving nonlinear higher-order boundary value problems by integrating the Adomian Decomposition Method into the shooting technique, enabling series-form solutions. To enhance convergence, new higher-order shooting slopes and their corresponding supplementary equation formulas were introduced. Three numerical examples demonstrated the method’s accuracy: for the first two, absolute errors were computed using available exact solutions, while the third was compared with reference literature due to the absence of an exact solution. The method achieved very small absolute errors in the first two cases, and results from the third closely matched the literature. Tolerance values—defined …

Role Of Competition In Avoiding Social Collapse, Gabriel M. Tonks

All Graduate Reports and Creative Projects, Fall 2023 to Present

Historical examples suggest that isolated, resource-scarce societies are prone to increased hostility and social disasters. The research in this report explores the role of intrasocietal competition in avoiding resource collapse. Two resource-consumer models are proposed with competition which depends on the level of available resources. One of these models is selected for in-depth analysis, and the region in the parameter space where saddle-node bifurcations emerge is computed numerically. The effects of environmental noise on resource growth are simulated, showing that the increased noise usually has negative long-term effects which might be mitigated via increased consumer competition.

Modeling And Analysis Of Electric Signal In Neurons, Kevin J. Roberts

All Graduate Reports and Creative Projects, Fall 2023 to Present

Neurons in humans and other species transmit information by sending electric signals via axons. This process relies on the generation and propagation of action potentials—rapid changes in the membrane potential of the axon. Understanding the mechanisms of action potentials, including how they are generated and influenced by the axon geometry and material parameters, is crucial for gaining insight into neurological diseases such as Alzheimer’s and Multiple Sclerosis (diseases highly correlated to demyelination). In this work, we review and summarize mathematical models for signal transmission–including the classical Hodgkin-Huxley model, the Single Cable (SC) model, and the Double Cable (DC) model. We …

Coupled Machine Learning Models: Combining Observations And Numerical Analysis In A Physics-Regularized Approach, Austin B. Schmidt

LSU New Orleans Theses and Dissertations

This dissertation investigates surrogate modeling for fixed-location environmental forecasting using novel data-combination techniques. The work surveys the landscape of observational measurements and numerically generated data, identifying similar research and gaps in current methodologies. The ratio-coupled training framework is introduced to combine two data sources per predicted feature through a tunable parameter that weights training signal strength. An optimization scheme is developed to simultaneously tune surrogate weights and the coupled signal ratio, allowing relative influence between signals to act as an explicit regularizer. Three case studies demonstrate the methodology and approach in a variety of contexts. The first study is based …

(R2135) System Dynamical Analysis For Ann-Based Numerical Solutions Of A Compartmental Model: A Bio-Mathematical Model Of Drug Diffusion Through The Compartments Of Blood And Tissue, Rakesh Kumar, Sudarshan Dhua

Applications and Applied Mathematics: An International Journal (AAM)

This paper provides a considerably efficient numerical approach to acquire the solutions of a biomathematical model administrating oral and intravenous distribution of pharmaceuticals in the human body. The proposed numerical approach based on an artificial neural network is employed to extract numerical solutions for a detailed set of ordinary differential equations and analyze the change in concentration of drug diffusion via the compartments of blood and tissue medium. We primarily focus on analyzing three different models established on the diffusion process, exercising laws of mass action and Fick’s principle. In this work, the existing model is reformulated as an optimization …

Universal Systems Simulation Via Constraint Hypergraphs With Applications To Digital Twins, John Morris

All Dissertations

The characterization of systems encompasses a variety of modeling frameworks designed to capture specific behaviors and components of various system domains. Whatever the framework, the core elements of a system representation are the information of the system and a description of how that information is related. The relations in deterministic systems are functions, which, when composed to form executable processes, can be used to simulate system data. A declarative modeling framework is one that encodes mechanisms for preparing these simulations within the model structure, allowing an external agent to form the execution processes required for a given context. To date, …

The Odds Don’T Lie: Mathematical Reasoning And Societal Ignorance In Don’T Look Up, Nysa Vedwan, Shane Carey

LASER Journal

In Adam McKay’s 2021 satirical sci-fi movie Don’t Look Up, two astronomers discover a comet heading directly toward Earth. Despite overwhelming evidence and near-certainty of global extinction, their warnings are ignored and ridiculed. This paper discusses the mathematical and scientific foundations of the movie’s social and political reception, and specifically focuses on orbital prediction and probabilistic modeling as they relate to public understanding of risk. This paper shows how data is often undermined by political and social dynamics, by connecting the fictional events of the movie with real-world crises like the COVID-19 pandemic and the climate emergency. In Don’t Look …

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.

Metapopulation Model To Evaluate C.Difficile Potential Vaccine Interventions., Archana Neupane Timsina

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Following Carbon: Pathway And Flux Representations Of Ecosystems, Caner Kazanci

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Exploring Solutions To Food Addiction Challenges Using Mathematical Modeling, Simulation, And Analysis Applying Optimal Control Theory, Dia Bonsu, Padmanabhan Seshaiyer, Alonso Ogueda-Oliva

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Modeling The Cancer Cell Growth Predictions Based On Classical Mathematical Models With Physics-Informed Neural Network, Widodo Samyono

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

[Project Insight] Applications Of Physics Informed Neural Networks For Incorporating Human Behavior Into Epidemiological Models, Alonso Gabriel Ogueda Oliva

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.

Sustainable Insecticide Spraying Strategy For Long-Term Chagas Disease Vector Control, Bismark Oduro

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Understanding The Spread Of Black Sigatoka Disease: A Deterministic And Stochastic Modeling Approach, Bernard Asamoah Afful, Luis F. Gordillo

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

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 …

High-Order Adaptive Solutions For Coupled Fractional Riccati Equations Via Daubechies Redundant Frames, Mutaz Mohammad, Alexander Trounev

Mathematical Modelling and Numerical Simulation with Applications

Coupled fractional Riccati equations play a fundamental role in modeling complex systems with memory effects and anomalous diffusion, frequently arising in engineering, control theory, finance, and quantum mechanics. Their analytical and numerical treatment remains highly challenging due to the nonlocal nature of fractional-order derivatives and the presence of nonlinear coupling terms. This study introduces an adaptive numerical framework that combines the Caputo fractional derivative with redundant Daubechies wavelet frames. The method leverages multi-resolution analysis, compact support, and controlled redundancy to achieve accurate approximation of both localized and global solution features, particularly in scenarios characterized by singular behavior and long-range memory …

The Consideration Of Two Scalarization Methods For The Multi-Objective Nurse-To-Patient Assignment Problem, Ilgın Acar, Steven E. Butt, Aydın Sipahioğlu, İslam Altın

Mathematical Modelling and Numerical Simulation with Applications

In this research, the application of two scalarization methods, namely the conic scalarization method and the $\varepsilon$-constraint method, is investigated within the context of a multi-objective optimization problem. These methods are used to address the challenge of assigning nurses to patients on a hospital unit during a shift. The two objective functions of this assignment problem are based on patient workload metrics and unit-related travel distance measures. The proposed solution approach demonstrates the ability to generate solutions that eluded the previous mathematical programming techniques that relied on simplistic weightings of conflicting objective functions. In addition, it is found that the …

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 …

Some New Three-Term Conjugate Gradient Methods For Riemannian Optimization With Application To The Gough-Stewart Platform, Nasiru Salihu, Poom Kumam, Lin Wang, Sani Salisu

Mathematical Modelling and Numerical Simulation with Applications

Three-term conjugate gradient (TTCG) methods have been extensively studied for optimization within Euclidean geometry, with some inherently satisfying the sufficient descent property, thus enhancing their theoretical superiority. However, other TTCG methods have been developed that incorporate restarting the search direction, simplify the steepest descent approach, or rely on the convexity assumptions of f to establish their convergence results. This paper introduces the Riemannian three-term conjugate gradient (RTTCG) methods. The search direction in these RTTCG methods consistently satisfies the sufficient descent condition, regardless of the line search employed, and does so without requiring a restart mechanism. By utilizing retraction and vector …

(Si15-068) Advanced Numerical Methods For The Solution Of Nonlinear Fisher Equation, Vikash Vimal, Richa Kumari, Ashish Awasthi

Applications and Applied Mathematics: An International Journal (AAM)

This paper examines the use of advanced numerical techniques to approximate solutions of the Fisher equation with higher-order accuracy. This technique integrates the method of lines with a strong stability-preserving Runge–Kutta scheme of orders four and five stages (SSPRK-54) for the numerical formulation. This scheme is then tested on two examples and the results show that it is more efficient than existing methods and requires less computing power. These equations are widely used across scientific and engineering disciplines, with particular relevance in biomedical studies, such as estimating the boundary size of tumors. The difficulties arising from their nonlinear nature are …