On the Cole–Hopf transformation and integration by parts formulae in computational methods within fractional differential equations and fractional optimal control theory
Hendy A.S. Zaky M.A. Tenreiro Machado J.A.
November 2022SAGE Publications Inc.
JVC/Journal of Vibration and Control
2022#28Issue 21-223364 - 3370 pp.
The treatment of fractional differential equations and fractional optimal control problems is more difficult to tackle than the standard integer-order counterpart and may pose problems to non-specialists. Due to this reason, the analytical and numerical methods proposed in the literature may be applied incorrectly. Often, such methods were established for the classical integer-order operators and are then applied directly without having in mind the restrictions posed by their fractional-order versions. It was recently reported that the Cole–Hopf transformation can be used to convert the time-fractional nonlinear Burgers’ equation into the time-fractional linear heat equation. In this article, we show that, unlike integer-order differential equations, employing the Cole–Hopf transformation for reducing the nonlinear time-fractional Burgers’ equation into the time-fractional heat equation is wrong from two different perspectives. Indeed, such a reduction is accomplished using incorrect transcripts of the Leibniz or chain rules. Hence, providing numerical or analytical schemes based on the Cole–Hopf transformation leads to erroneous results for the nonlinear time-fractional Burgers’ equation. Regarding constant-order, variable-order, and distributed-order Caputo fractional optimal problems, we note an inconsistency in the necessary optimality conditions derived in the literature. The transversality conditions were introduced as identical to those for the integer-order case, with a vanishing multiplier at the terminal of the interval. The correct condition should involve a constant-order, variable-order, or distributed-order fractional integral operator. We also deduce that if the control system is defined with a Caputo derivative, then the adjoint equations should be expressed in the Riemann–Liouville sense and vice versa. In fact, neglecting some terms in the integration by parts formulae, during the derivation of the optimality conditions, causes some confusion in the literature.
chain rule , Cole–Hopf transformation , fractional Burgers’ equation , fractional optimal control problems , Leibniz rule , variable-order fractional derivative
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Department of Computational Mathematics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, Russian Federation
Department of Mathematics, Faculty of Science, Benha University, Egypt
Department of Applied Mathematics, National Research Centre, Egypt
Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
Institute of Engineering, Polytechnic of Porto, Portugal
Department of Computational Mathematics and Computer Science
Department of Mathematics
Department of Applied Mathematics
Institute of Mathematics and Mathematical Modeling
Institute of Engineering
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