Partial Differential Equations Commons^™

On Solutions Of First Order Pde With Two-Dimensional Dirac Delta Forcing Terms, Ian Robinson

Rose-Hulman Undergraduate Mathematics Journal

We provide solutions of a first order, linear partial differential equation of two variables where the nonhomogeneous term is a two-dimensional Dirac delta function. Our results are achieved by applying the unilateral Laplace Transform, solving the subsequently transformed PDE, and reverting back to the original space-time domain. A discussion of existence and uniqueness of solutions, a derivation of solutions of the PDE coupled with a boundary and initial condition, as well as a few worked examples are provided.

(R1966) Semi Analytical Approach To Study Mathematical Model Of Atmospheric Internal Waves Phenomenon, Patel Yogeshwari, Jayesh M. Dhodiya

Applications and Applied Mathematics: An International Journal (AAM)

This research aims to study atmospheric internal waves which occur within the fluid rather than on the surface. The mathematical model of the shallow fluid hypothesis leads to a coupled nonlinear system of partial differential equations. In the shallow flow model, the primary assumption is that vertical size is smaller than horizontal size. This model can precisely replicate atmospheric internal waves because waves are dispersed over a vast horizontal area. A semi-analytical approach, namely modified differential transform, is applied successfully in this research. The proposed method obtains an approximate analytical solution in the form of convergent series without any linearization, …

(R2052) Flow Patterns For Newtonian And Non-Newtonian Fluids In A Cylindrical Pipe, Erick Sanchez, Dambaru Bhatta

Applications and Applied Mathematics: An International Journal (AAM)

A fully developed laminar steady flow of an incompressible, viscous fluid in a horizontal cylindrical pipe is considered here. Flow patterns for an incompressible, viscous fluid for both Newtonian and non-Newtonian fluids such as shear-thinning, shear-thickening and Bingham plastic fluids are analyzed in this study. Assuming that the flow is only due to the wall shear stress and the pressure drop, the velocity component in the axial direction for these cases is derived. Computational results of the velocity profiles for various cases are obtained using MATLAB and presented in graphical forms. It is observed that the velocity profile is parabolic …

(R2028) A Brief Note On Space Time Fractional Order Thermoelastic Response In A Layer, Navneet Lamba, Jyoti Verma, Kishor Deshmukh

Applications and Applied Mathematics: An International Journal (AAM)

In this study, a one-dimensional layer of a solid is used to investigate the exact analytical solution of the heat conduction equation with space-time fractional order derivatives and to analyze its associated thermoelastic response using a quasi-static approach. The assumed thermoelastic problem was subjected to certain initial and boundary conditions at the initial and final ends of the layer. The memory effects and long-range interaction were discussed with the help of the Caputo-type fractional-order derivative and finite Riesz fractional derivative. Laplace transform and Fourier transform techniques for spatial coordinates were used to investigate the solution of the temperature distribution and …

Solving The Cable Equation, A Second-Order Time Dependent Pde For Non-Ideal Cables With Action Potentials In The Mammalian Brain Using Kss Methods, Nirmohi Charbe

Master's Theses

In this thesis we shall perform the comparisons of a Krylov Subspace Spectral method with Forward Euler, Backward Euler and Crank-Nicolson to solve the Cable Equation. The Cable Equation measures action potentials in axons in a mammalian brain treated as an ideal cable in the first part of the study. We shall subject this problem to the further assumption of a non-ideal cable. Assume a non-uniform cross section area along the longitudinal axis. At the present time, the effects of torsion, curvature and material capacitance are ignored. There is particular interest to generalize the application of the PDEs including and …

Pde Model For Protocell Evolution And The Origin Of Chromosomes Via Multilevel Selection, Daniel B. Cooney, Fernando W. Rossine, Dylan H. Morris, Simon A. Levin

Biology and Medicine Through Mathematics Conference

No abstract provided.

Reaction-Diffusion System On Irregular Boundaries Reproduces Multiple Generations Of Petal Spot Patterns In Monkeyflower Hybrids, Emily Simmons

Biology and Medicine Through Mathematics Conference

No abstract provided.

Helices In Fluids And Their Applications, Eva M. Strawbridge

Biology and Medicine Through Mathematics Conference

No abstract provided.

U-No: U-Shaped Neural Operators, Md Ashiqur Rahman, Zachary E Ross, Kamyar Azizzadenesheli

Department of Computer Science Faculty Publications

Neural operators generalize classical neural networks to maps between infinite-dimensional spaces, e.g., function spaces. Prior works on neural operators proposed a series of novel methods to learn such maps and demonstrated unprecedented success in learning solution operators of partial differential equations. Due to their close proximity to fully connected architectures, these models mainly suffer from high memory usage and are generally limited to shallow deep learning models. In this paper, we propose U-shaped Neural Operator (U-NO), a U-shaped memory enhanced architecture that allows for deeper neural operators. U-NOs exploit the problem structures in function predictions and demonstrate fast training, data …

Monolithic Multiphysics Simulation Of Hypersonic Aerothermoelasticity Using A Hybridized Discontinuous Galerkin Method, William Paul England

Theses and Dissertations

This work presents implementation of a hybridized discontinuous Galerkin (DG) method for robust simulation of the hypersonic aerothermoelastic multiphysics system. Simulation of hypersonic vehicles requires accurate resolution of complex multiphysics interactions including the effects of high-speed turbulent flow, extreme heating, and vehicle deformation due to considerable pressure loads and thermal stresses. However, the state-of-the-art procedures for hypersonic aerothermoelasticity are comprised of low-fidelity approaches and partitioned coupling schemes. These approaches preclude robust design and analysis of hypersonic vehicles for a number of reasons. First, low-fidelity approaches limit their application to simple geometries and lack the ability to capture small scale flow …

Asymptotic Properties And Separation Rates For Navier-Stokes Flows, Patrick Michael Phelps

Graduate Theses and Dissertations

In this dissertation, we investigate asymptotic properties of local energy solutions to the Navier-Stokes equations and develop an application which controls the separation of non-unique solutions in this class. Specifically, we quantify the rate at which two, possibly unique solutions evolving from the same data may separate pointwise away from a singularity. This is motivated by recent results on non-uniqueness for forced and unforced Navier-Stokes and analytical and numerical evidence suggesting non-uniqueness in the Leray class. Our investigation begins with discretely self-similar solutions known to exist globally in time and to be regular outside a space-time paraboloid. We prove decay …

Advancements In Fluid Simulation Through Enhanced Conservation Schemes, Sean Ingimarson

All Dissertations

To better understand and solve problems involving the natural phenomenon of fluid and air flows, one must understand the Navier-Stokes equations. Branching several different fields including engineering, chemistry, physics, etc., these are among the most important equations in mathematics. However, these equations do not have analytic solutions save for trivial solutions. Hence researchers have striven to make advancements in varieties of numerical models and simulations. With many variations of numerical models of the Navier-Stokes equations, many lose important physical meaningfulness. In particular, many finite element schemes do not conserve energy, momentum, or angular momentum. In this thesis, we will study …

An Integrated Experimental And Modeling Approach To Design Rotating Algae Biofilm Reactors (Rabrs) Via Optimizing Algae Biofilm Productivity, Nutrient Recovery, And Energy Efficiency, Gerald Benjamin Jones

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Microalgae biofilms have been demonstrated to recover nutrients from wastewater and serve as biomass feedstock for bioproducts. However, there is a need to develop a platform to quantitatively describe microalgae biofilm production, which can provide guidance and insights for improving biomass areal productivity and nutrient uptake efficiency. This paper proposes a unified experimental and theoretical framework to investigate algae biofilm growth on a rotating algae biofilm reactor (RABR). The experimental laboratory setups are used to conduct controlled experiments on testing environmental and operational factors for RABRs. We propose a differential-integral equation-based mathematical model for microalgae biofilm cultivation guided by laboratory …

Effects Of Topography On Tornado Paths Using Navier Stokes Equations, Kayleigh Smith

Mathematics Senior Capstone Papers

Tornadoes are relatively common in Louisiana with an average of 55 tornadoes per year. Predicting tornado paths has been extremely challenging due to the many factors that play into the formation of a tornado. According to NASA/ADS topography can have significant influence on tornado direction. The main goal of this research is to analyze tornado patterns and to determine if local topography has an effect on tornadic activity. Navier-Stokes partial differential equations will be used to model the data that is collected from the national weather service and a finite difference method will be used to solve the equations. Data …

Integrable Systems On Symmetric Spaces From A Quadratic Pencil Of Lax Operators, Rossen Ivanov

Conference papers

The article surveys the recent results on integrable systems arising from quadratic pencil of Lax operator L, with values in a Hermitian symmetric space. The counterpart operator M in the Lax pair defines positive, negative and rational flows. The results are illustrated with examples from the A.III symmetric space. The modeling aspect of the arising higher order nonlinear Schrödinger equations is briefly discussed.

Modeling Self-Diffusiophoretic Janus Particles In Fluid, Kausik Das

HMC Senior Theses

We explore spherical Janus particles in which a chemical reaction occurs on one face, depleting a substrate in the suspending fluid, while no reaction occurs on the other face. The steady state concentration field is governed by Laplace’s equation with mixed boundary conditions. We use the collocation method to obtain numerical solutions to the equation in spherical coordinates. The asymmetry of the reaction gives rise to a slip velocity that causes the particle to move spontaneously in the fluid through a process known as self-diffusiophoresis. Using the Lorentz reciprocal theorem, we obtain the swimming velocity of the particle. We extend …

Foundations Of Wave Phenomena: Complete Version, Charles G. Torre

Foundations of Wave Phenomena

This is the complete version of Foundations of Wave Phenomena. Version 8.3.1.

Please click here to explore the components of this work.

Beginner's Analysis Of Financial Stochastic Process Models, David Garcia

HMC Senior Theses

This thesis explores the use of geometric Brownian motion (GBM) as a financial model for predicting stock prices. The model is first introduced and its assumptions and limitations are discussed. Then, it is shown how to simulate GBM in order to predict stock price values. The performance of the GBM model is then evaluated in two different periods of time to determine whether it's accuracy has changed before and after March 23, 2020.

The Lagrangian Formulation For Wave Motion With A Shear Current And Surface Tension, Conor Curtin, Rossen Ivanov

Articles

The Lagrangian formulation for the irrotational wave motion is straightforward and follows from a Lagrangian functional which is the difference between the kinetic and the potential energy of the system. In the case of fluid with constant vorticity, which arises for example when a shear current is present, the separation of the energy into kinetic and potential is not at all obvious and neither is the Lagrangian formulation of the problem. Nevertheless, we use the known Hamiltonian formulation of the problem in this case to obtain the Lagrangian density function, and utilising the Euler-Lagrange equations we proceed to derive some …

Using Modflow To Assess Groundwater Storage Enhancement Via A Floodplain Infiltration Basin, Lindsay Henning

All Master's Theses

Delaying groundwater discharge into rivers until it is critically needed during baseflow conditions provides promise for lowering elevated stream temperatures and improving habitat for aquatic species. Increasing groundwater storage may accomplish this in locations where excess spring runoff can be captured and allowed to infiltrate into the subsurface for later beneficial use, a process known as Managed Aquifer Recharge (MAR). Here, MAR via an infiltration basin is considered at a site along the Teanaway River in central Washington State. The effects of simulated ephemeral ponds of sizes varying from 554 m³ to 2430 m³ (0.449 acre-feet to 1.97 …