Homotopy analysis method

The homotopy analysis method (HAM) is a semi-analytical technique to solve nonlinear ordinary/partial differential equations.

The capability of the HAM to naturally show convergence of the series solution is unusual in analytical and semi-analytic approaches to nonlinear partial differential equations.

It provides great freedom to choose the basis functions of the desired solution and the corresponding auxiliary linear operator of the homotopy.

Finally, unlike the other analytic approximation techniques, the HAM provides a simple way to ensure the convergence of the solution series.

In the last twenty years, the HAM has been applied to solve a growing number of nonlinear ordinary/partial differential equations in science, finance, and engineering.

denote an auxiliary linear operator, u0(x) an initial guess of u(x), and c0 a constant (called the convergence-control parameter), respectively.

for k > 1, and the right-hand side Rm is dependent only upon the known results u0, u1, ..., um − 1 and can be obtained easily using computer algebra software.

Since the HAM is based on a homotopy, one has great freedom to choose the initial guess u0(x), the auxiliary linear operator

Thus, the HAM provides the mathematician freedom to choose the equation-type of the high-order deformation equation and the base functions of its solution.

The optimal value of the convergence-control parameter c0 is determined by the minimum of the squared residual error of governing equations and/or boundary conditions after the general form has been solved for the chosen initial guess and linear operator.

Thus, the convergence-control parameter c0 is a simple way to guarantee the convergence of the homotopy series solution and differentiates the HAM from other analytic approximation methods.

[8] Another HAM-based Mathematica code, APOh, has been produced to solve for an explicit analytic approximation of the optimal exercise boundary of American put option, which is also available online [5].