TY - JOUR
T1 - Dual approach as empirical reliability for fractional differential equations
AU - Dubovski, P. B.
AU - Slepoi, J. A.
N1 - Publisher Copyright:
© 2021 Institute of Physics Publishing. All rights reserved.
PY - 2021/12/13
Y1 - 2021/12/13
N2 - Computational methods for fractional differential equations exhibit essential instability. Even a minor modification of the coefficients or other entry data may switch good results to the divergent. The goal of this paper is to suggest the reliable dual approach which fixes this inconsistency. We suggest to use two parallel methods based on the transformation of fractional derivatives through integration by parts or by means of substitution. We introduce the method of substitution and choose the proper discretization scheme that fits the grid points for the by-parts method. The solution is reliable only if both methods produce the same results. As an additional control tool, the Taylor series expansion allows to estimate the approximation errors for fractional derivatives. In order to demonstrate the proposed dual approach, we apply it to linear, quasilinear and semilinear equations and obtain very good precision of the results. The provided examples and counterexamples support the necessity to use the dual approach because either method, used separately, may produce incorrect results. The order of the exactness is close to the exactness of fractional derivatives approximations.
AB - Computational methods for fractional differential equations exhibit essential instability. Even a minor modification of the coefficients or other entry data may switch good results to the divergent. The goal of this paper is to suggest the reliable dual approach which fixes this inconsistency. We suggest to use two parallel methods based on the transformation of fractional derivatives through integration by parts or by means of substitution. We introduce the method of substitution and choose the proper discretization scheme that fits the grid points for the by-parts method. The solution is reliable only if both methods produce the same results. As an additional control tool, the Taylor series expansion allows to estimate the approximation errors for fractional derivatives. In order to demonstrate the proposed dual approach, we apply it to linear, quasilinear and semilinear equations and obtain very good precision of the results. The provided examples and counterexamples support the necessity to use the dual approach because either method, used separately, may produce incorrect results. The order of the exactness is close to the exactness of fractional derivatives approximations.
UR - http://www.scopus.com/inward/record.url?scp=85119059825&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85119059825&partnerID=8YFLogxK
U2 - 10.1088/1742-6596/2099/1/012004
DO - 10.1088/1742-6596/2099/1/012004
M3 - Conference article
AN - SCOPUS:85119059825
SN - 1742-6588
VL - 2099
JO - Journal of Physics: Conference Series
JF - Journal of Physics: Conference Series
IS - 1
M1 - 012004
T2 - International Conference on Marchuk Scientific Readings 2021, MSR 2021
Y2 - 4 October 2021 through 8 October 2021
ER -