On the asymptotic formulas of solutions to the boundary value problem without initial condition for Schrödinger systems in domain with conical points

2016 ◽  
Vol 130 ◽  
pp. 18-30
Author(s):  
Nguyen Manh Hung ◽  
Nguyen Thi Lien
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Asymptotic Formulas ◽  
Initial Condition ◽  
Schrödinger Systems ◽  
Conical Points

Spectral Problem for a Polynomial Pencil of the Sturm-Liouville Equations

2020 ◽  
pp. 163-181
Author(s):  
Anar Adiloğlu-Nabiev
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Spectral Problem ◽  
Boundary Value ◽  
Asymptotic Formulas ◽  
Complex Numbers ◽  
Arbitrary Complex ◽  
Sturm Liouville ◽  
Complex Valued

A boundary value problem for the second order differential equation -y′′+∑_{m=0}N−1λ^{m}q_{m}(x)y=λ2Ny with two boundary conditions a_{i1}y(0)+a_{i2}y′(0)+a_{i3}y(π)+a_{i4}y′(π)=0, i=1,2 is considered. Here n>1, λ is a complex parameter, q0(x),q1(x),...,q_{n-1}(x) are summable complex-valued functions, a_{ik} (i=1,2; k=1,2,3,4) are arbitrary complex numbers. It is proved that the system of eigenfunctions and associated eigenfunctions is complete in the space and using elementary asymptotical metods asymptotic formulas for the eigenvalues are obtained.

scholarly journals Asymptotic Solution of a Boundary Value Problem for a Spring–Mass Model of Legged Locomotion

2020 ◽  
Vol 30 (6) ◽  
pp. 2971-2988
Author(s):  
Hanna Okrasińska-Płociniczak ◽  
Łukasz Płociniczak
Keyword(s):  
Boundary Value Problem ◽  
Asymptotic Solution ◽  
Asymptotic Expansions ◽  
Boundary Value ◽  
Legged Locomotion ◽  
Asymptotic Formulas ◽  
Numerical Verification ◽  
Mass Model ◽  
Leg Stiffness ◽  
Elastic Pendulum

Abstract Running is the basic mode of fast locomotion for legged animals. One of the most successful mathematical descriptions of this gait is the so-called spring–mass model constructed upon an inverted elastic pendulum. In the description of the grounded phase of the step, an interesting boundary value problem arises where one has to determine the leg stiffness. In this paper, we find asymptotic expansions of the stiffness. These are conducted perturbatively: once with respect to small angles of attack, and once for large velocities. Our findings are in agreement with previous results and numerical simulations. In particular, we show that the leg stiffness is inversely proportional to the square of the attack angle for its small values, and proportional to the velocity for large speeds. We give exact asymptotic formulas to several orders and conclude the paper with a numerical verification.

On solutions to the heat equation with the initial condition in the Orlicz—Slobodetskii space

2014 ◽  
Vol 144 (4) ◽  
pp. 787-807 ◽  
Author(s):  
Agnieszka Kałamajska ◽  
Mirosłav Krbec
Keyword(s):  
Sobolev Space ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Boundary Value ◽  
Orlicz Function ◽  
Initial Condition ◽  
Lipschitz Boundary

We study the boundary-value problem ũt = Δxũ(x,t), ũ(x, 0) = u(x), where x ∈ Ω, t ∈ (0,T), Ω ⊆ ℝn−1 is a bounded Lipschitz boundary domain, u belongs to a certain Orlicz–Slobodetskii space YR,R(Ω). Under certain assumptions on the Orlicz function R, we prove that the solution u belongs to the Orlicz–Sobolev space W1,R(Ω × (0,T)).

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