# nLab homotopy analysis method

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

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

(1)$q \cdot N_x u(x,q) \;+\; (1-q) \cdot L_x\big(u(x,q) - u_0(x,q)\big) \;=\; 0 \,, {\phantom{AAA}} for\; q \in [0,1]$

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:

• Cheng-shi Liu, The essence of the homotopy analysis method, Applied Mathematics and Computation Volume 216, Issue 4, 15 April 2010, Pages 1299-1303 (doi:10.1016/j.amc.2010.02.022)

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)