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Partial Differential Equations and Boundary Value Problems
Published in George F. Simmons, Differential Equations with Applications and Historical Notes, 2016
In general (but not always) the adjoint equation is just as difficult to solve as the original equation. Find the adjoint equation in each of the following cases: Legendre’s equation: (1 − x2)y″ − 2xy′ + p(p + 1)y = 0;Bessel’s equation: x2y″ + xy′ + (x2 − p2)y = 0;Chebyshev’s equation: (1 − x2)y″ − xy′ + p2y = 0;Hermite’s equation: y″ − 2xy′ + 2py = 0;Airy’s equation: y″ + xy = 0;Laguerre’s equation: xy″ + (1 − x)y′ + py = 0.
Virtual Stackelberg game coupled with the adjoint method for aerodynamic shape optimization
Published in Engineering Optimization, 2018
Jing Wang, Fangfang Xie, Yao Zheng, Jifa Zhang, Bowei Yang, Tingwei Ji
Above are the principles of the adjoint method in general terms (Brezillon and Gauger 2004). Both the governing equation (2) and the adjoint equation (5) are partial differential equations. So the same iterative methods that are efficient for the governing equation can be applied to solve the adjoint equation. For the problems in this article, the objective functions are all in integral form and a general formulation of the adjoint method with the detailed gradients is given in Appendix B. For more details, see Carpentieri (2007), Palacios (2014) and Palacios (2013).
Stress-constrained shell-lattice infill structural optimisation for additive manufacturing
Published in Virtual and Physical Prototyping, 2020
Huangchao Yu, Jiaqi Huang, Bin Zou, Wen Shao, Jikai Liu
Put Equations (11–15) into Equation (10). Collect the terms including and the sum is naturally equivalent to zero. Then, collect the terms including and make the sum equivalent to zero, where the adjoint equation is derived as demonstrated in Equation (16). The adjoint variable can be derived by solving the adjoint equation.
Necessary and sufficient conditions for near-optimality of stochastic delay systems
Published in International Journal of Control, 2018
As is well known, the study of adjoint equation plays a key role in deriving the necessary and sufficient conditions of optimality. The adjoint equation is a kind of ABSDE with the terminal conditions given. This section is mainly devoted to investigating certain continuity property of the adjoint processes with respect to the metic d.