Preface
This section gives an introduction to the homotopy pertubation method, first proposed by Ji-Huan He in 1998.
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Glossary
Homotopy method
\[
f(x) =0
\]
in the interval I = [a, b], we consider the homotopy
\[
H(x,p) = p\, f(x) + \left( 1- p \right) g(x) , \qquad x\in I, \quad p \in [0,1],
\]
which is a mapping I × [0,1] ↦ I. Here g(x) is an auxiliary function for which we know the solution g(x) = 0, and p is an embedding parameter. Usually, the function g is chosen in two forms:
- g(x) is a linear function, quadratic function, and so on;
- g(x) is the part function of f(x).
\[
H(x,0) = g(x) = 0, \qquad H(x,1) = f(x) = 0, [0,1],
\]
and the changing process of p
from 0 to 1 corresponds to moving H(x, p) from g(x) to f(x). In topology, this is called deformation.
Assuming that p is a small parameter from the interval [0,1], we can assume that the zero of the function H(x, p) can be expressed as a series in p:
\[
x= x_0 + p\,x_1 + p^2 x_2 + p^3 x_3 + \cdots .
\]
When p &maps; 1, we obtain (hopefully) the true solution of f(x) = 0, that is,
\[
x^{\ast} = \lim_{p\to 1} x = x_0 + x_1 + x_2 + x_3 + \cdots .
\]
To obtain its approximate solution of H(x, p) = 0, we first expand it into Taylor series
\begin{align*}
f(x) &= f(x_0 ) + f' (x_0 ) \left( p\,x_1 + p^2 x_2 + p^3 x_3 + \cdots \right) + \frac{1}{2!}\,f'' \left( x_0 \right) \left( p\,x_1 + p^2 x_2 + p^3 x_3 + \cdots \right) + \cdots ,
\\
g(x) &= g\left( x_0 \right) + g' \left( x_0 \right) \left( p\,x_1 + p^2 x_2 + p^3 x_3 + \cdots \right) + \frac{1}{2!}\,g'' \left( x_0 \right) \left( p\,x_1 + p^2 x_2 + p^3 x_3 + \cdots \right) + \cdots .
\end{align*}
Substituting these expansions into the equation H(x, p) = 0 and equating the coefficients of like powers of p, we get
\begin{align*}
p^0 \,&: \ g\left( x_0 \right) =0;
\\
p^1 \,&: \ g'\left( x_0 \right) x_1 + f\left( x_0 \right) - g\left( x_0 \right) =0;
\\
p^2 \,&: \ g'\left( x_0 \right) x_2 + \left[ f'\left( x_0 \right) - g'\left( x_0 \right) \right] x_1 + \frac{1}{2}\, g'' \left( x_0 \right) x_1^2 =0;
\\
p^3 \,&: \ g'\left( x_0 \right) x_3 + \left[ f'\left( x_0 \right) - g'\left( x_0 \right) \right] x_2 + \frac{1}{2}\left[ f'' \left( x_0 \right) - g'' \left( x_0 \right) \right] x_1 + \frac{1}{3!}\, g''' \left( x_0 \right) x_1^3 =0;
\end{align*}
and so on. If \( g'\left( x_0 \right) \ne 0 , \) then the above equations can be solved to obtain
\begin{align*}
x_0 & \ \mbox{ is exact null of the auxiliary function } \ g(x) = 0;
\\
x_1 &= - \frac{f\left( x_0 \right)}{g'\left( x_0 \right)} ;
\\
x_2 &= \frac{\left[ g'\left( x_0 \right) - f'\left( x_0 \right) \right] x_1 - \frac{1}{2}\, g'' \left( x_0 \right) x_1^2}{g'\left( x_0 \right)} ;
\\
x_3 &= \frac{\frac{1}{2} \left[ g'' \left( x_0 \right) - f'' \left( x_0 \right) \right] x_1 - \left[ f'\left( x_0 \right) - g'\left( x_0 \right) \right] x_2 - g'' \left( x_0 \right) x_1 x_2 - \frac{1}{3!}\, g''' \left( x_0 \right) x_1^3}{g'\left( x_0 \right)} .
\end{align*}
Particularly, if we choose the function \( g(x) = f(x) - f\left( x_0 \right) , \) where x0 is an initial approximation, then
\begin{align*}
x_1 &= - \frac{f\left( x_0 \right)}{g'\left( x_0 \right)} ;
\\
x_2 &= - \frac{1}{2!}\,\frac{f'' \left( x_0 \right)}{f'\left( x_0 \right)} \, x_1^2 = - \frac{1}{2!}\,\frac{f'' \left( x_0 \right)}{f'\left( x_0 \right)} \left\{ \frac{f\left( x_0 \right)}{g'\left( x_0 \right)} \right\}^2 ;
\\
x_3 &= - \frac{1}{2!}\,\frac{f'' \left( x_0 \right)}{f'\left( x_0 \right)} \, x_1 x_2 - \frac{1}{3!} \, \frac{f''' \left( x_0 \right)}{f'\left( x_0 \right)} \, x_1^3 .
\end{align*}
Therefore, we can obtain
\[
x= x_0 + x_1 \quad\mbox{with first order approximation},
\]
and
\[
x= x_0 + x_1 + x_2 \quad\mbox{with second order approximation},
\]
and
\[
x= x_0 + x_1 + x_2 + x_3 \quad\mbox{with third order approximation},
\]
From above, we can write down the iteration forms:
\begin{align*}
x_{n+1} &= x_n - \frac{f\left( x_n \right)}{f'\left( x_n \right)} ;
\\
x_{n+1} &= x_n - \frac{f\left( x_n \right)}{f'\left( x_n \right)} - \frac{f'' \left( x_n \right)}{2\,f'\left( x_n \right)} \left\{ \frac{f\left( x_n \right)}{f'\left( x_n \right)} \right\}^2 ;
\\
x_{n+1} &= x_n - \frac{f\left( x_n \right)}{f'\left( x_n \right)} - \frac{f'' \left( x_n \right)}{2\,f'\left( x_n \right)} \left\{ \frac{f\left( x_n \right)}{f'\left( x_n \right)} \right\}^2 +
\left\{ \frac{f''' \left( x_n \right)}{6\, f'\left( x_n \right)} - \frac{1}{2} \left[ \frac{f'' \left( x_n \right)}{f' \left( x_n \right)} \right]^2 \right\} \left\{ \frac{f\left( x_n \right)}{f'\left( x_n \right)} \right\}^3 .
\end{align*}
Obviously, the first iteration formula is the well-known Newton's iteration formula.
In order to avoid computing derivatives, we replace
\( f'\left( x_n \right) \) and
\( f'' \left( x_n \right) \) by the centered differences
\[
\frac{f \left( x_n +h \right) -f \left( x_n -h \right) }{2h} \qquad\mbox{and} \qquad \frac{f \left( x_n +h \right) + f \left( x_n -h \right) - 2\,f \left( x_n \right)}{h^2} ,
\]
respectively, where \( h = \alpha\, f\left( x_n \right) \) is a step size and α ≠ 0 is a parameter. So we get the following second-order iterative method:
\[
x_{n+1} = x_n - \frac{2h\,f\left( x_n \right) }{f\left( x_n +h \right) - f\left( x_n -h \right)}
\]
and the third order iterative method
\[
x_{n+1} = x_n - \frac{2h\,f\left( x_n \right) }{f\left( x_n +h \right) - f\left( x_n -h \right)} - \frac{4h\,f^2 \left( x_n \right) \left[ f\left( x_n +h \right) + f\left( x_n -h \right) -2\,f\left( x_n \right) \right]}{\left[ f\left( x_n +h\right) - f\left( x_n -h \right) \right]^3} ,
\]
respectively.
Example: Suppose that we need to solve the rtanscendent equation
\[
e^{\cos x} -1-x =0 , \qquad x\in [0,1].
\]
Of course, Mathematica can find its root:
FindRoot[Exp[Cos[x]] == 1 + x, {x, 1}]
{x -> 0.884511}
which is an approximation to 0.8845106161658526.
■ Example: Solve the polynomial equation
\[
x^9 = 1+x \quad\mbox{in the interval} \quad [1,2].
\]
FindRoot[x^9 == 1 + x, {x, 1}]
{x -> 1.08507}
which is an approximation to 1.085070245491451.
■
- Chun, C., Ham, Y.M., Some fourth-order modifications of Newton’s method, Applied Mathematics and Computation, 2008, Vol. 197, pp. 654--658.
- Feng, X.L., He, Y.N., High order iterative methods without derivatives for solving nonlinear equations, Applied Mathematics and Computation, 2007, Vol. 186, pp. 1617--1623.
- He, J.H., A new iterative method for solving algebraic equations, Applied Mathematics and Computation, 2003, Vol. 135, pp. 81--84. https://doi.org/10.1016/S0096-3003(01)00313-7
- Ide, N.A.D., Some new iterative algorithms by using homotopy pertubation method for solving nonlinear algebraic equations, Asean Journal of Mathematics and Compute Research, No 2395-4205 (online), Vol. 5, Issue 3. 2015.
- Noor, M.A., Some iterative methods for solving nonlinear equations using homotopy perturbation method, International Journal of Computer Mathematics, Vol. 87, No. 1, January 2010, pp. 141–149
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