scholarly journals Asymptotic Solution of the Perturbed First Boundary Value Problem with a Non-Smooth Coefficient

2020 ◽  
Vol 12 (3) ◽  
pp. 41-47
Author(s):  
D.A. Tursunov ◽  
◽  
M.O. Orozov ◽  
Keyword(s):  
Boundary Value Problem ◽  
Asymptotic Solution ◽  
Boundary Value ◽  
Smooth Coefficient

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.

A SUM-DIFFERENCE METHOD FOR CONSTRUCTING AN ASYMPTOTIC SOLUTION TO A BOUNDARY VALUE PROBLEM OF A NONLINEAR DIFFERENCE EQUATION WITH A SMALL PARAMETER

2019 ◽  
Vol 19 (4) ◽  
pp. 255-259
Author(s):  
Asangul Alymbaev ◽  
Keyword(s):  
Finite Difference ◽  
Difference Equation ◽  
Boundary Value Problem ◽  
Asymptotic Solution ◽  
Small Parameter ◽  
Difference Equations ◽  
Difference Method ◽  
Financial Analysis ◽  
Boundary Value ◽  
Nonlinear Difference Equation

Finite-difference equations proved to be a convenient mathematical model on describing impulse systems, combinatorial analysis problems, discrete analogues of mathematical physics equations, financial analysis tasks, etc. Oneshould point out that difference equations are encountered in the numerical solution of various classes of differential and integro- differential ones using the finite difference method. The article deals with methods of constructing an asymptotic solution to the boundary value problem of a system of a nonlinear difference equation with a small parameter. The problem is solved by reducing the boundary-value problem to the Cauchy problem for a system of total-difference equations with a small parameter. The efficiency of the method algorithm for the asymptotic expansion of the task of a boundary value problem in a definite example is shown.

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