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Applied Mathematics

West Chester University

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Full-Text Articles in Physical Sciences and Mathematics

Bioheat Equation Analysis, Johnathan Makar Apr 2024

Bioheat Equation Analysis, Johnathan Makar

Mathematics Student Work

In our research, we are investigating Pennes Bioheat equation, which is used for simulating the propagation of heat energy in human tissues. This equation was proposed by Pennes in 1948 based on his experiments of measuring the radial temperature distribution in the forearm of nine subjects. Pennes' equation provides the theoretical basis for studying heat transfer in perfused tissue and has been widely studied since then. However, Pennes' equation has been criticized for various reasons, including the fact that his experimental data did not seem to match the model. One of the objectives of our work is to find the …

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An Augmented Matched Interface And Boundary (Amib) Method For Solving Problems On Irregular 2d Domains, Benjamin Pentecost Apr 2024

An Augmented Matched Interface And Boundary (Amib) Method For Solving Problems On Irregular 2d Domains, Benjamin Pentecost

Mathematics Student Work

A new method called Augmented Matched Interface and Boundary (AMIB) has been developed to solve partial differential equation models, such as the heat equation, over irregular two-dimensional domains. The original AMIB method features unique numerical treatments to solve problems with various boundary conditions and shapes, resulting in highly accurate and efficient numerical solutions. However, recent numerical experiments have revealed that the original AMIB method can fail when dealing with sharply curved boundaries. To address this issue, new numerical techniques have been introduced in our latest work to enhance the robustness of the AMIB method. These techniques have been numerically verified …

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Solving The Heat Equation With Interfaces, Michael Bauer, Rex Llewellyn, Shauna Frank Apr 2021

Solving The Heat Equation With Interfaces, Michael Bauer, Rex Llewellyn, Shauna Frank

Mathematics Student Work

When modeling systems made up of two materials with different thermodynamic properties, a physical interface can be introduced to account for the border where the materials meet. This interface separates our model’s standard grid into two regions, each with its unique physical properties. At these interfaces, boundary conditions can be imposed to represent the difference in heat and in heat flux between the different materials so that their interaction may be modeled accurately. Because standard finite difference methods are inadequate to deal with interfaces, a Matched Interface and Boundary (MIB) technique is investigated in this work to solve the heat …

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Parallel Computation Of Action Potentials In The Hodgkin-Huxley Model Via The Parareal Algorithm, Eric Boerman, Khanh Pham, Katie Peltier Apr 2021

Parallel Computation Of Action Potentials In The Hodgkin-Huxley Model Via The Parareal Algorithm, Eric Boerman, Khanh Pham, Katie Peltier

Mathematics Student Work

The Hodgkin-Huxley model is a system of differential equations that describe the membrane voltage of an axon as it fires the basic signal of the nervous system: the action potential. When charge-carrying ions such as sodium, potassium, and others are enabled to cross a selectively permeable membrane, the resulting current propagates along the length of the axon as a wave of altered ionic potential. However, the degree to which the membrane is permeable to sodium and potassium is itself gated by voltage; therefore, voltage depends on permeability and permeability depends on voltage. This interdependent cellular system is expressed as a …

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On Approximately Controlled Systems, Nazim I. Mahmudov, Mark A. Mckibben Aug 2016

On Approximately Controlled Systems, Nazim I. Mahmudov, Mark A. Mckibben

Mathematics Faculty Publications

No abstract provided.

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General Existence Results For Abstract Mckean-Vlasov Stochastic Equations With Variable Delay, Mark A. Mckibben May 2016

General Existence Results For Abstract Mckean-Vlasov Stochastic Equations With Variable Delay, Mark A. Mckibben

Mathematics Faculty Publications

Results concerning the global existence and uniqueness of mild solutions for a class of first-order abstract stochastic integro-differential equations with variable delay in a real separable Hilbert space in which we allow the nonlinearities at a given time t to depend not only on the state of the solution at time t, but also on the corresponding probability distribution at time t are established. The classical Lipschitz is replaced by a weaker so-called Caratheodory condition under which we still maintain uniqueness. The time-dependent case is discussed, as well as an extension of the theory to the case of a nonlocal …

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Controllability Of Neutral Stochastic Integro-Differential Evolution Equations Driven By A Fractional Brownian Motion, El Hassan Lakhel, Mark A. Mckibben Jan 2016

Controllability Of Neutral Stochastic Integro-Differential Evolution Equations Driven By A Fractional Brownian Motion, El Hassan Lakhel, Mark A. Mckibben

Mathematics Faculty Publications

We establish sufficient conditions for the controllability of a certain class of neutral stochastic functional integro-differential evolution equations in Hilbert spaces. The results are obtained using semigroup theory, resolvent operators and a fixed-point technique. An application to neutral partial integro-differential stochastic equations perturbed by fractional Brownian motion is given.

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A Discontinuous Galerkin Method For Unsteady Two-Dimensional Convective Flows, Andreas C. Aristotelous, N. C. Papanicolaou Jan 2016

A Discontinuous Galerkin Method For Unsteady Two-Dimensional Convective Flows, Andreas C. Aristotelous, N. C. Papanicolaou

Mathematics Faculty Publications

We develop a High-Order Symmetric Interior Penalty (SIP) Discontinuous Galerkin (DG) Finite Element Method (FEM) to investigate two-dimensional in space natural convective flows in a vertical cavity. The physical problem is modeled by a coupled nonlinear system of partial differential equations and admits various solutions including stable and unstable modes in the form of traveling and/or standing waves, depending on the governing parameters. These flows are characterized by steep boundary and internal layers which evolve with time and can be well-resolved by high-order methods that also are adept to adaptive meshing. The standard no-slip boundary conditions which apply on the …

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Abstract Functional Stochastic Evolution Equations Driven By Fractional Brownian Motion, Mark A. Mckibben, Micah Webster Jan 2014

Abstract Functional Stochastic Evolution Equations Driven By Fractional Brownian Motion, Mark A. Mckibben, Micah Webster

Mathematics Faculty Publications

We investigate a class of abstract functional stochastic evolution equations driven by a fractional Brownianmotion in a real separable Hilbert space.Global existence results concerningmild solutions are formulated under various growth and compactness conditions. Continuous dependence estimates and convergence results are also established. Analysis of three stochastic partial differential equations, including a second-order stochastic evolution equation arising in the modeling of wave phenomena and a nonlinear diffusion equation, is provided to illustrate the applicability of the general theory.

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Measure-Dependent Stochastic Nonlinear Beam Equations Driven By Fractional Brownian Motion, Mark A. Mckibben Jan 2013

Measure-Dependent Stochastic Nonlinear Beam Equations Driven By Fractional Brownian Motion, Mark A. Mckibben

Mathematics Faculty Publications

We study a class of nonlinear stochastic partial differential equations arising in themathematicalmodeling of the transverse motion of an extensible beam in the plane. Nonlinear forcing terms of functional-type and those dependent upon a family of probability measures are incorporated into the initial-boundary value problem (IBVP), and noise is incorporated into the mathematical description of the phenomenon via a fractional Brownian motion process. The IBVP is subsequently reformulated as an abstract second-order stochastic evolution equation driven by a fractional Brownian motion (fBm) dependent upon a family of probability measures in a real separableHilbert space and is studied using the tools …

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On A Class Of Backward Mckean-Vlasov Stochastic Equations In Hilbert Space: Existence And Convergence Properties, Nazim I. Mahmudov, Mark A. Mckibben Dec 2007

On A Class Of Backward Mckean-Vlasov Stochastic Equations In Hilbert Space: Existence And Convergence Properties, Nazim I. Mahmudov, Mark A. Mckibben

Mathematics Faculty Publications

This investigation is devoted to the study of a class of abstract first-order backward McKean-Vlasov stochastic evolution equations in a Hilbert space. Results concerning the existence and uniqueness of solutions and the convergence of an approximating sequence of solutions (and corresponding probability measures) are established. Examples that illustrate the abstract theory are also provided.

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On Backward Stochastic Evolution Equations In Hilbert Space And Optimal Control, Nazim I. Mahmudov, Mark A. Mckibben Aug 2007

On Backward Stochastic Evolution Equations In Hilbert Space And Optimal Control, Nazim I. Mahmudov, Mark A. Mckibben

Mathematics Faculty Publications

In this paper a new result on the existence and uniqueness of the adapted solution to a backward stochastic evolution equation in Hilbert spaces under non Lipschitz condition is established. The applicability of this result is then illustrated in a discussion of some concrete backward stochastic partial differential equation. Furthermore, stochastic maximum principle for optimal control problems of stochastic systems governed by backward stochastic evolution equations in Hilbert spaces is obtained.

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Abstract Semilinear Itó-Volterra Integro-Differential Stochastic Evolution Equations, David N. Keck, Mark A. Mckibben Nov 2006

Abstract Semilinear Itó-Volterra Integro-Differential Stochastic Evolution Equations, David N. Keck, Mark A. Mckibben

Mathematics Faculty Publications

We consider a class of abstract semilinear stochastic Volterra integrodifferential equations in a real separable Hilbert space. The global existence and uniqueness of a mild solution, as well as a perturbation result, are established under the so-called Caratheodory growth conditions on the nonlinearities. An approximation result is then established, followed by an analogous result concerning a so-called McKean-Vlasov integrodi fferential equation, and then a brief commentary on the extension of the main results to the time-dependent case. The paper ends with a discussion of some concrete examples to illustrate the abstract theory.

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Abstract Second-Order Damped Mckean-Vlasov Stochastic Evolution Equations, N. I. Mahmudov, Mark A. Mckibben Apr 2006

Abstract Second-Order Damped Mckean-Vlasov Stochastic Evolution Equations, N. I. Mahmudov, Mark A. Mckibben

Mathematics Faculty Publications

We establish results concerning the global existence, uniqueness, approximate and exact controllability of mild solutions for a class of abstract second-order stochastic evolution equations in a real separable Hilbert space in which we allow the nonlinearities at a given time t to depend not only on the state of the solution at time t, but also on the corresponding probability distribution at time t. First-order equations of McKean-Vlasov type were first analyzed in the finite dimensional setting when studying diffusion processes, and then subsequently extended to the Hilbert space setting. The current manuscript provides a formulation of such …

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On State-Dependent Delay Partial Neutral Functional–Differential Equations, Eduardo Hernandez M., Mark A. Mckibben Jan 2006

On State-Dependent Delay Partial Neutral Functional–Differential Equations, Eduardo Hernandez M., Mark A. Mckibben

Mathematics Faculty Publications

No abstract provided.

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Some Comments On: Existence Of Solutions Of Abstract Nonlinear Second-Order Neutral Functional Integrodifferential Equations, Eduardo Hernandez, Mark A. Mckibben Jan 2005

Some Comments On: Existence Of Solutions Of Abstract Nonlinear Second-Order Neutral Functional Integrodifferential Equations, Eduardo Hernandez, Mark A. Mckibben

Mathematics Faculty Publications

We establish the existence of mild solutions for a class of abstract second-order partial neutral functional integro-differential equations with infinite delay in a Banach space using the theory of cosine families of bounded linear operators and Schaefer's fixed-point theorem.

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