On the zero stability of the variable order variable stepsize BDF-Formulas

1990 ◽  
Vol 57 (1) ◽  
pp. 39-50 ◽  
Author(s):  
M. Calvo ◽  
T. Grande ◽  
R. D. Grigorieff
Keyword(s):  
Variable Order ◽  
Variable Stepsize ◽  
Order Variable

scholarly journals Variable order variable stepsize algorithm for solving nonlinear Duffing oscillator

2017 ◽  
Vol 890 ◽  
pp. 012045 ◽  
Author(s):  
Ahmad Fadly Nurullah Rasedee ◽  
Norizarina Ishak ◽  
Siti Raihana Hamzah ◽  
Hazizah Mohd Ijam ◽  
Mohamed Suleiman ◽  
...  
Keyword(s):  
Duffing Oscillator ◽  
Variable Order ◽  
Variable Stepsize ◽  
Order Variable

scholarly journals The solution of Riccati type differential equation by means of variable order variable stepsize backward difference method

2016 ◽  
Vol 2016 (1) ◽  
pp. 35-42
Author(s):  
Ahmad Fadly Nurullah b. Rasedee ◽  
Mohamed b. Suleiman ◽  
Ali Ahmadian ◽  
Zarina Bibi bt. Ibrahim ◽  
Khairil Iskandar b. Othman ◽  
...  
Keyword(s):  
Differential Equation ◽  
Difference Method ◽  
Variable Order ◽  
Variable Stepsize ◽  
Type Differential Equation ◽  
Order Variable

A Variable Stepsize, Variable Order Family of Low Complexity

2021 ◽  
Vol 43 (3) ◽  
pp. A2130-A2160
Author(s):  
Victor DeCaria ◽  
Ahmet Guzel ◽  
William Layton ◽  
Yi Li
Keyword(s):  
Low Complexity ◽  
Variable Order ◽  
Variable Stepsize

On the Cole–Hopf transformation and integration by parts formulae in computational methods within fractional differential equations and fractional optimal control theory

2021 ◽  
pp. 107754632110310
Author(s):  
Ahmed S Hendy ◽  
Mahmoud A Zaky ◽  
José A Tenreiro Machado
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Heat Equation ◽  
Fractional Differential Equations ◽  
Burgers Equation ◽  
Variable Order ◽  
Integer Order ◽  
Order Variable ◽  
Fractional Differential ◽  
Distributed Order

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.

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