Open Access. Powered by Scholars. Published by Universities.®

Physical Sciences and Mathematics Commons™

Open Access. Powered by Scholars. Published by Universities.®

Applied Mathematics

Variational methods

Articles 1 - 8 of 8

Full-Text Articles in Physical Sciences and Mathematics

Swirling Fluid Flow In Flexible, Expandable Elastic Tubes: Variational Approach, Reductions And Integrability, Rossen Ivanov, Vakhtang Putkaradze Jan 2020

Swirling Fluid Flow In Flexible, Expandable Elastic Tubes: Variational Approach, Reductions And Integrability, Rossen Ivanov, Vakhtang Putkaradze

Articles

Many engineering and physiological applications deal with situations when a fluid is moving in flexible tubes with elastic walls. In real-life applications like blood flow, a swirl in the fluid often plays an important role, presenting an additional complexity not described by previous theoretical models. We present a theory for the dynamics of the interaction between elastic tubes and swirling fluid flow. The equations are derived using a variational principle, with the incompressibility constraint of the fluid giving rise to a pressure-like term. In order to connect this work with the previous literature, we consider the case of inextensible and …

Go to article

A Non-Autonomous Second Order Boundary Value Problem On The Half-Line, Gregory S. Spradlin Oct 2010

A Non-Autonomous Second Order Boundary Value Problem On The Half-Line, Gregory S. Spradlin

Greg S. Spradlin Ph.D.

By variational arguments, the existence of a solution to a nonautonomous second-order boundary problem on the half-line is proven. The corresponding autonomous problem has no solution, revealing significant differences between the autonomous and the non-autonomous case.

Go to article

Heteroclinic Solutions To An Asymptotically Autonomous Second-Order Equation, Gregory S. Spradlin Dec 2009

Heteroclinic Solutions To An Asymptotically Autonomous Second-Order Equation, Gregory S. Spradlin

Gregory S. Spradlin

We study the differential equation ¨x(t) = a(t)V' (x(t)), where V is a double-well potential with minima at x = ±1 and a(t) → l > 0 as |t| → ∞. It is proven that under certain additional assumptions on a, there exists a heteroclinic solution x to the differential equation with x(t) → −1 as t → −∞ and x(t) → 1 as t → ∞. The assumptions allow l − a(t) to change sign for arbitrarily large values of |t|, and do not restrict the decay rate of |l −a(t)| as |t| → ∞.

Go to article

An Elliptic Equation With No Monotonicity Condition On The Nonlinearity, Gregory S. Spradlin Oct 2006

An Elliptic Equation With No Monotonicity Condition On The Nonlinearity, Gregory S. Spradlin

Greg S. Spradlin Ph.D.

An elliptic PDE is studied which is a perturbation of an autonomous equation. The existence of a nontrivial solution is proven via variational methods. The domain of the equation is unbounded, which imposes a lack of compactness on the variational problem. In addition, a popular monotonicity condition on the nonlinearity is not assumed. In an earlier paper with this assumption, a solution was obtained using a simple application of topological (Brouwer) degree. Here, a more subtle degree theory argument must be used.

Go to article

Interacting Near-Solutions To A Hamiltonian System, Gregory S. Spradlin Mar 2004

Interacting Near-Solutions To A Hamiltonian System, Gregory S. Spradlin

Gregory S. Spradlin

A Hamiltonian system with a superquadratic potential is examined. The system is asymptotic to an autonomous system. The difference between the Hamiltonian system and the “problem at infinity,” the autonomous system, may be large, but decays exponientially. The existence of a nontrivial solution homoclinic to zero is proven. Many results of this type rely on a monotonicity condition on the nonlinearity, not assumed here, which makes the problem resemble in some sense the special case of homogeneous (power) nonlinearity. The proof employs variational, minimax arguments. In some similar results requiring the monotonicity condition, solutions inhabit a manifold homeomorphic to the …

Go to article

Existence Of Solutions To A Hamiltonian System Without Convexity Condition On The Nonlinearity, Gregory S. Spradlin Dec 2003

Existence Of Solutions To A Hamiltonian System Without Convexity Condition On The Nonlinearity, Gregory S. Spradlin

Gregory S. Spradlin

We study a Hamiltonian system that has a superquadratic potential and is asymptotic to an autonomous system. In particular, we show the existence of a nontrivial solution homoclinic to zero. Many results of this type rely on a convexity condition on the nonlinearity, which makes the problem resemble in some sense the special case of homogeneous (power) nonlinearity. This paper replaces that condition with a different condition, which is automatically satisfied when the autonomous system is radially symmetric. Our proof employs variational and mountain-pass arguments. In some similar results requiring the convexity condition, solutions inhabit a submanifold homeomorphic to the …

Go to article

Interfering Solutions Of A Nonhomogeneous Hamiltonian System, Gregory S. Spradlin Jan 2001

Interfering Solutions Of A Nonhomogeneous Hamiltonian System, Gregory S. Spradlin

Greg S. Spradlin Ph.D.

A Hamiltonian system is studied which has a term approaching a constant at an exponential rate at infinity. A minimax argument is used to show that the equation has a positive homoclinic solution. The proof employs the interaction between translated solutions of the corresponding homogeneous equation. What distinguishes this result from its few predecessors is that the equation has a nonhomogeneous nonlinearity.

Go to article

An Elliptic Equation With Spike Solutions Concentrating At Local Minima Of The Laplacian Of The Potential, Gregory S. Spradlin Dec 1999

An Elliptic Equation With Spike Solutions Concentrating At Local Minima Of The Laplacian Of The Potential, Gregory S. Spradlin

Gregory S. Spradlin

We consider a singularly perturbed elliptic PDE that arises in the study of nonlinear Schrodinger equations. We seek solutions that are positive on the entirety of Euclidean space and that vanish at infinity. Under the assumption that the nonlinear term of the PDE satisfies super-linear and sub-critical growth conditions, we show that for small values of the epsilon parameter in the PDE, there solutions that concentrate near local minima of V (a coefficient function in the PDE) . The local minima may occur in unbounded components, as long as the Laplacian of V achieves a strict local minimum along such …

Go to article