TY - JOUR

T1 - Singularly perturbed boundary-value problems

T2 - Sundman-type transformations, test problems, exact solutions, and numerical integration

AU - Shingareva, Inna K.

AU - Polyanin, Andrei D.

N1 - Publisher Copyright:
© 2021 Saint-Petersburg State University. All rights reserved.

PY - 2021

Y1 - 2021

N2 - Solutions of singularly perturbed boundary-value problems with a small parameter are characterized by large gradients in very narrow regions (boundary layers). This circumstance sharply limits the use of standard finite-difference methods with a fixed stepsize in such problems due to significant calculation errors or possible loss of stability. This paper presents an effective method for numerical integration of singularly perturbed boundary-value problems based on replacing the spatial variable with a new independent variable of the Sundman-type, which depends on the derivatives of the unknown function. The use of such non-local transformations, which satisfy a simple asymptotic condition, makes it possible to automatically stretch the boundary-layer region. The resulting problem turns out to be much simpler than the original one in the sense that standard (classical) numerical methods with a fixed stepsize can already be applied to solve it. Several new multiparameter nonlinear and linear singularly perturbed boundary-value problems for second-order reaction-diffusion type ODEs having monotonic and non-monotonic exact or asymptotic solutions, expressed in terms of elementary functions, are constructed. A comparison of numerical solutions with exact and asymptotic solutions is presented. The numerical results show that the method based on Sundman-type transformations for solving boundary-layer problems gives high accuracy. As a result of an extensive analysis of the obtained results, recommendations are given for the choice of regularizing functions that determine the most effective Sundman-type transformations. The difference between regularizing functions in boundary-layer problems and blow-up problems is discussed. The test problems formulated in this paper can be used to estimate the accuracy of any other numerical methods for solving two-point singularly perturbed boundary-value problems with a small parameter.

AB - Solutions of singularly perturbed boundary-value problems with a small parameter are characterized by large gradients in very narrow regions (boundary layers). This circumstance sharply limits the use of standard finite-difference methods with a fixed stepsize in such problems due to significant calculation errors or possible loss of stability. This paper presents an effective method for numerical integration of singularly perturbed boundary-value problems based on replacing the spatial variable with a new independent variable of the Sundman-type, which depends on the derivatives of the unknown function. The use of such non-local transformations, which satisfy a simple asymptotic condition, makes it possible to automatically stretch the boundary-layer region. The resulting problem turns out to be much simpler than the original one in the sense that standard (classical) numerical methods with a fixed stepsize can already be applied to solve it. Several new multiparameter nonlinear and linear singularly perturbed boundary-value problems for second-order reaction-diffusion type ODEs having monotonic and non-monotonic exact or asymptotic solutions, expressed in terms of elementary functions, are constructed. A comparison of numerical solutions with exact and asymptotic solutions is presented. The numerical results show that the method based on Sundman-type transformations for solving boundary-layer problems gives high accuracy. As a result of an extensive analysis of the obtained results, recommendations are given for the choice of regularizing functions that determine the most effective Sundman-type transformations. The difference between regularizing functions in boundary-layer problems and blow-up problems is discussed. The test problems formulated in this paper can be used to estimate the accuracy of any other numerical methods for solving two-point singularly perturbed boundary-value problems with a small parameter.

KW - Sundman-type transformations

KW - boundary layer

KW - exact solutions

KW - multiparameter test problems

KW - non-local transformations

KW - nonlinear ODEs

KW - numerical integration

KW - singularly perturbed boundary-value problems

UR - http://www.scopus.com/inward/record.url?scp=85130582707&partnerID=8YFLogxK

M3 - Artículo

AN - SCOPUS:85130582707

SN - 1817-2172

VL - 2021

SP - 15

EP - 50

JO - Differencialnie Uravnenia i Protsesy Upravlenia

JF - Differencialnie Uravnenia i Protsesy Upravlenia

IS - 4

ER -