synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
In ODE/PDE-theory, what is known as the homtopy analysis method (Liao 92) is a method for solving a non-linear differential equation $N_x u(x) = 0$ by considering its convex combination with a suitably chosen linear differential equation $L_x \big(u(x) - u_0(x)\big) = 0$ and then solving the combined equation
in terms of a Taylor series-expansion of the desired solution $u$ in the new parameter $q$ (“homotopy Maclaurin series”). This transforms the original non-linear problem into an infinite sequence of linear differential equations, that can (apparently) often be solved more readily.
The terminology “homotopy analysis method” alludes to regarding the convex combination (1) as a “homotopy of differential equations” from the linear $L$ (at $q = 0$) to the non-linear $N$ (at $q = 1$). But it seems that there is no actual homotopy theory being used in discussion of the “homotopy analysis method”.
Original articles:
Shijun Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, PhD thesis, Shanghai Jiao Tong University (1992)
Shijun Liao, Notes on the homotopy analysis method: Some definitions and theorems, Communications in Nonlinear Science and Numerical Simulation, Volume 14, Issue 4, April 2009, Pages 983-997 (doi:10.1016/j.cnsns.2008.04.013)
Binfeng Pan, Yangyang Ma, Yang Ni, A new fractional homotopy method for solving nonlinear optimal control problems, Acta Astronautica, Volume 161, August 2019, Pages 12-23 (doi:10.1016/j.actaastro.2019.05.005)
But see:
Introduction and review:
Shijun Liao, Beyond Perturbation – Introduction to the Homotopy Analysis Method, CRC Press 2003 (ISBN:9781584884071)
Shijun Liao, Homotopy Analysis Method in Nonlinear Differential Equations Springer 2012 (pdf)
V.G. Gupta, Sumit Gupta, Application of the Homotopy Analysis Method for solving nonlinear Cauchy Problem, Surveys in Mathematics and its Applications, Volume 7 (2012), 105 – 116 (ISSN:1842-6298, pdf)
See also
Further development:
Last revised on May 8, 2021 at 09:45:56. See the history of this page for a list of all contributions to it.