Find the First Five Terms of the Power Series Solution

Ordinary Differential Equations

George B. Arfken , ... Frank E. Harris , in Mathematical Methods for Physicists (Seventh Edition), 2013

Fuchs' Theorem

The answer to the basic question as to when the method of series substitution can be expected to work is given by Fuchs' theorem, which asserts that we can always obtain at least one power-series solution, provided that we are expanding about a point which is an ordinary point or at worst a regular singular point.

If we attempt an expansion about an irregular or essential singularity, our method may fail as it did for Eqs. (7.51) and (7.53). Fortunately, the more important equations of mathematical physics, listed in Section 7.4, have no irregular singularities in the finite plane. Further discussion of Fuchs' theorem appears in Section 7.6.

From Table 7.1, Section 7.4, infinity is seen to be a singular point for all the equations considered. As a further illustration of Fuchs' theorem, Legendre's equation (with infinity as a regular singularity) has a convergent series solution in negative powers of the argument (Section 15.6). In contrast, Bessel's equation (with an irregular singularity at infinity) yields asymptotic series ( Sections 12.6 and 14.6 Section 12.6 Section 14.6 ). Although only asymptotic, these solutions are nevertheless extremely useful.

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Differential Transform Method

L. Zheng , X. Zhang , in Modeling and Analysis of Modern Fluid Problems, 2017

5.1 Introduction

The differential transformation method (DTM) was developed by Zhao (1986). It originates from the Taylor expansion of the function, but the fundamental idea is to avoid directly seeking values of higher-order derivatives or partial derivatives of an unknown function. By performing differential transformation operations on differential equations and subjected initial or boundary conditions, algebraic relations between higher-order derivatives of the adjacent order are obtained for ordinary differential equations or partial differential equations; then an iterative procedure is established to obtain the analytical solution in terms of the polynomial form.

DTM does not need to calculate derivatives or partial derivatives. Its main advantage is that not only does it require a smaller amount of computation than the traditional Taylor series method, but it can be applied directly to nonlinear differential equations of physics or mathematics. However, for differential equations in unbounded domains, results obtained by DTM are effective only in a small subregion of the problem; the result is not correct in a whole region of existing solution (Boyd, 1997). The reason is that the solution obtained by DTM is divergent when the independent variable is infinite. To overcome this problem, two new methods for solving differential equations of the infinite region have been proposed, DTM-Padé and DTM-basic function (BF), proposed by Professor Zheng's doctoral student, Xiaohong Su, while pursing a doctorate at the University of Science and Technology Beijing. The basic ideas of the two methods are introduced as follows.

5.1.1 Ideas of Differential Transform–Padé and Differential Transform–Basic Function

Many scholars have applied DTM to solve differential equations arising in different fields (Abazari and Borhanifar, 2010; Arikoglu, 2005; Arikoglu and Özkol, 2009; Ayaz, 2004; Jang et al., 2010).

To solve BVPs of differential equations in unbounded domains by DTM, we usually need to introduce some assumed initial parameters such that the initial value problems (IVPs) of original differential equations with introduced initial parameters can be solved first. Then we need to determine the initial parameters by pertinent boundary conditions at infinity. Two methods can be used to determine the introduced initial parameters: one is to use DTM coupled with Padé's approximation (DTM-Padé) (Rashidi, 2009; Su et al., 2012a,b); another is to use DTM coupled with the BF method (DTM-BF).

The DTM-Padé approximation method has some problems. The solutions to algebraic equations with undetermined parameters obtained by DTM-Padé are usually not unique; in some cases the obtained algebraic equations have multiple solutions. Thus, to determine the correct initial parameter values, continuous-order Padé approximation is necessary and the initial parameters can be determined according to the convergence trend of continuous Padé approximation. In this process, according to the complexity of the equation, it is difficult to judge the correct value of the parameter; sometimes it is impossible, and one can only determine the range of the parameters. Only when the introduced initial parameters of the problem are exactly determined can solutions be obtained with high accuracy.

To overcome the difficulties in applying DTM-Padé, a novel method, i.e., DTM-BF, was developed by Su and Zheng (2011) and Su et al. (2012a,b) . The proposed method can be used to solve some complex and coupled nonlinear BVPs with an infinite boundary. A few items can obtain good accuracy and fewer initial parameters are needed to be introduced for BVPs. Another advantage is that in many cases, the algebraic equations with introduced initial parameters obtained by DTM-BF have unique solutions. DTM-BF can also be used as the basis for determining the parameters of the algebraic equations obtained by DTM-Padé. The basic idea is to use DTM to give a power series solution of the IVP, according to the characteristics of the problem, and then, by performing a linear combination of the BFs, to represent the solution of BVP. The special solutions are:

1.

First, initial parameters are introduced according to the BVP, such that we can solve the IVP with introduced initial parameters using DTM. For the introduced initial parameters, in a neighboring region of the initial point, the power series solution of the IVP is obtained and the Taylor series in this region is convergent.

2.

Second, according to the physical characteristics of the differential equation and boundary conditions, the appropriate basis function is chosen and the linear combination is performed to approximate the solution of the problem.

3.

Third, Taylor expansion is carried out for the solution series combination by the BF, and then the power series form of the power series is organized. Furthermore, the power series solution of the corresponding IVP is matched with the same power. The approximate solution of the problem is obtained using the BF series.

5.1.2 Definition of Differential Transformation Method and Formula

5.1.2.1 Differential Transformation for Function of One Variable

Supposing that a function w(t) has k order derivatives for variable t, the differential transform of function w(t) is defined as Zhao (1986)

(5.1) $W\left(k\right)=M\left(k\right){\left[\frac{d^{k}q\left(t\right)w\left(t\right)}{d{t}^k}\right]}_{t=t_0},k=\mathrm{0,1,2,3},\cdots$

where w(t) and W(k) are called the original function and differential transform function, respectively. The inverse differential transform function of W(k) is defined as

(5.2) $w\left(t\right)=\frac{1}{q\left(t\right)}\sum _0^{\infty }\frac{W\left(k\right)}{M\left(k\right)k!}{\left(t-t_0\right)}^k$

where M(k)   ≠   0, displayed for the transformation of the known function of the independent variable to be proportional to the integer. q(t)   ≠   0 is the kernel of the transformation of a known function. If q(t)   =   1, the proportional function M(k)   = H ^k or $M\left(k\right)=\frac{H^k}{k!}$ (H is a proportional constant). When $M\left(k\right)=\frac{H^k}{k!}$ , the product operation of the transformation is relatively simple, so the scaling function is generally used in this form, and in the following discussion, this definition is used.

5.1.2.2 Differential Transformation for Functions of Several Variables

Supposing that the function w(x,t) has a continuous partial derivative, the differential transformation of the function w(x,t) is defined.

(5.3) $W\left(k,h\right)=M\left(k\right)N\left(h\right){\left[\frac{{\partial }^{k+h}q\left(t\right)p\left(x\right)w\left(x,t\right)}{\partial x^h\partial t^k}\right]}_{x=x_0,t=t_0},k=\mathrm{0,1,2,3},\cdots \text{,}$

where w(t,x) and W(k,h) are called the original function and differential transform function, respectively. The inverse differential transform function of W(k,h) is defined as

(5.4) $w\left(t,x\right)=\frac{1}{q\left(t\right)p\left(x\right)}\sum _0^{\infty }\frac{W\left(k,h\right)}{M\left(k\right)N\left(h\right)h!k!}{\left(t-t_0\right)}^k{\left(x-x_0\right)}^h\text{.}$

It is analogous to the case of one variable function, M(k)   ≠   0 and N(h)   ≠   0, displayed for the transformation of the known function of the independent variable to be proportional to the integer; q(t)   ≠   0 and p(x)   ≠   0 are the respective kernels of the transformation of a known function. If q(t)   =   1, p(x)   =   1, the proportional functions $M\left(k\right)=H_1^k$ and $N\left(h\right)=H_2^h$ or $M\left(k\right)=\frac{H_1^k}{k!}$ and $N\left(h\right)=\frac{H_2^h}{h!}$ [H _i   (i  =   1, 2) are called proportional constants]. When $M\left(k\right)=\frac{H_1^k}{k!}$ and $N\left(h\right)=\frac{H_2^h}{h!}$ , the product operations of the transformation are relatively simple. If q(t)   = p(x)   =   1 is chosen, the differential transform of the two-variable function of w(t,x) can be written as

(5.5) $W\left(k,h\right)=\frac{1}{k!h!}{\left[\frac{{\partial }^{k+h}w\left(x,t\right)}{\partial x^h\partial t^k}\right]}_{x=x_{0}t=t_0},k=\mathrm{0,1,2,3},\cdots \text{.}$

The inverse differential transform of function W(k,h) is written as

(5.6) $w\left(t,x\right)=\sum _0^{\infty }W\left(k,h\right){\left(t-t_0\right)}^k{\left(x-x_0\right)}^h\text{,}$

In the same way, we can define the differential transform of three variables or more.

5.1.2.3 Differential Transformation Formula

According to the definition of differential transformation, the formula for calculating differential transformation with a continuous derivative or partial derivative can be derived. Tables 5.1 and 5.2 show the basic operation principle of the differential transformation of a function.

Table 5.1. Basic Operation Principle of Differential Transformation of Single Variable Function

Original Function	Differential Transform
w(t)   = αw 1(t)   + βw 2(t)	W(k)   = αW 1(k)   + βW 2(k), α and β are constants
$w\left(t\right)=w_1^{\left(n\right)}\left(t\right)$	W(k)   =   (k  +   1)(k  +   2) ⋯ (k  + n)W 1(k  + n)
w(t)   = w 1(t)w 2(t)	$W\left(k\right)=\sum _{i=0}^k{W}_1\left(i\right)W_2\left(k-i\right)$
w(t)   =   (t  − t 0) m	$W\left(k\right)=\delta \left(k-m\right)=\{\begin{array}{l}1,k=m,\\ 0,k\ne m.\end{array}$

Table 5.2. Basic Operation Principle of Differential Transformation of Two-Variable Function

Original Function	Differential Transform
w(x,t)   = αw 1(x,t)   + βw 2(x,t)	W(k)   = αW 1(k,h)   + βW 2(k,h), α, β are constants
$w(x,t)=\frac{{\partial }^n{w}_1(x,t)}{\partial x^n}$	W(k,h)   =   (k  +   1)(k  +   2) ⋯ (k  + n)W 1(k  + n,h)
$w(x,t)=\frac{{\partial }^m{w}_1(x,t)}{\partial t^m}$	W(k,h)   =   (h  +   1)(h  +   2) ⋯ (h  + m)W 1(k,h  + m)
$w(x,t)=\frac{{\partial }^{i+j}{w}_1(x,t)}{\partial x^i\partial t^j}$	W(k,h)   =   (k  +   1) ⋯ (k  + i)(h  +   1) ⋯ (h  + j)W 1(k  + i,h  + j)
w(x,t)   = w 1(x,t)w 2(x,t)	$W\left(k,h\right)=\sum _{j=0}^k\sum _{i=0}^h{W}_1\left(j,h-i\right)W_2\left(k-j,i\right)$
$w(x,t)=\frac{\partial w_1(x,t)}{\partial x}\frac{\partial w_2(x,t)}{\partial x}$	$W\left(k,h\right)=\sum _{j=0}^k\sum _{i=0}^h\left(j+1\right)\left(k-j+1\right)W_1\left(j+1,h-i\right)W_2\left(k-j+1,i\right)\text{,}$

In this chapter we first apply the DTM-Padé to study the magnetohydrodynamic (MHD) Falkner–Skan boundary layer flow present in the magnetic effects over a permeable wall. Then we apply DTM-BF to study mixed convective and radiation heat transfer of MHD fluid over the stretching wedge surface and MHD nanofluids in porous media with variable surface heat flux and chemical reactions.

The reliability and validity of the method are verified by numerical comparison for all results.

5.1.3 Magnetohydrodynamic Boundary Layer Problem

MHD fluid has received much attention by scholars owing to its wide applications in engineering and technology. Many mathematical models have been proposed to characterize MHD flow and heat transfer under different conditions (Abbasbandy and Hayat, 2009a,b; Abel and Nandeppanavar, 2009; Hayat et al., 2008; Ishak et al., 2009; Parand et al., 2011; Prasad et al., 2009; Robert and Vajravelu, 2010; Soundalgekar et al., 1981; Sutton and Sherman, 1965).

Soundalgekar et al. (1981) studied the Falkner–Skan boundary layer flow and heat transfer of MHD fluid. Suitable similarity transformation was introduced to reduce the governing partial differential equations to nonlinear ordinary differential equations and then to solve them numerically. Abbasbandy and Hayat (2009a) studied the MHD Falkner–Skan boundary layer flow using the Hankel-Padé and homotopy analytical methods (Abbasbandy and Hayat, 2009b), respectively. Parand et al. (2011) investigated MHD flow around a surface of infiltration that was stationary without a wedge; the approximate solution was obtained using the spectrum of Hermite function fitting method. Robert and Vajravelu (2010) discussed the existence and uniqueness of solutions of MHD Falkner–Skan flow. Professor Zheng and coauthors carried out a series of investigations using DTM-Padé or DTM-BF, perturbation methods, homotopy analysis methods, and numerical methods on boundary layer flow and heat transfer of MHD fluid over a continuous stretching permeable sheet embedded in viscous electrically conducting fluid (Jiao et al., 2015; Li et al., 2016; Si et al., 2016; Su and Zheng, 2011; Su et al., 2012a,b; Zhang et al., 2015; Zhu et al., 2010).

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Higher Order Equations

Martha L. Abell , James P. Braselton , in Introductory Differential Equations (Fourth Edition), 2014

Exercises 4.8

In Exercises 1-3, determine the singular points of the equations. Use these points to find an upper bound on the radius of convergence of a series solution about x ₀.

1.

x ² y″ − 2xy′ + 7y = 0, x ₀ = 1

2.

(x − 2)y″ + y′− y = 0, x ₀ = −2

3.

(x ² − 4)y″ + 16(x + 2)y′− y = 0, x ₀ = 1

In Exercises 4-13, solve the differential equation with a power series expansion about x = 0. Write out at least the first five nonzero terms of each series.

4.

y″ + 3y′− 18y = 0

5.

y″ − 11y′ + 30y = 0

6.

y″ + y = 0

7.

y″ − y′− 2y = e^−x

8.

(−2 − 2x)y″ + 2y′ + 4y = 0

9.

(2 + 3x)y″ + 3xy′ = 0

10.

(1 + 3x)y″ − 3y′− 2y = 0

11.

(2 − x ²)y″ + 2(x − 1)y′ + 4y = 0

12.

y″ − xy′ + 4y = 0

13.

(2 + 2x ²)y″ + 2xy′− 3y = 0

14.

(3 − 2x)y″ + 2y′− 2y = 0, y(0) = 3, y′(0) = −2

15.

y″ − 4x ² y = 0, y(0) = 1, y′(0) = 0

16.

(2x ² − 1)y″ + 2xy′− 3y = 0, y(0) = −2, y′(0) = 2

In Exercises 14-16, determine at least the first five nonzero terms in a power series about x = 0 for the solution of each IVP.

The English physicist John William Strutt "Lord Rayleigh" (1842-1919) won the Nobel prize in 1904 for his discovery of the inert gas argon in 1895. He donated the award money to the University of Cambridge to extend the Cavendish laboratories. His theory on traveling waves laid the foundations for the development of soliton theory.

In 1873, the French mathematician Charles Hermite (1822-1901) became the first person to prove that e is a transcendental number. "Analysis takes back with one hand what it gives with the other. I recoil in fear and loathing from that deplorable evil: continuous functions with no derivatives." Interestingly, Hermite married Louise Bertrand, Joseph Bertrand's sister. One of Hermite's daughters married Emile Picard so Hermite was Picard's father-in-law.

In Exercises 17 and 18, solve the equation with a power series expansion about t = 0 by making the indicated change of variables.

17.

4xy″ + y′ = 0, x = t + 1

18.

4x ² y″ + (x + 1)y′ = 0, x = t + 2

In Exercises 19 and 20, determine at least the first five nonzero terms in a power series expansion about x = 0 of the solution to each nonhomogeneous IVP.

19.

$y^{″}+x{y}^{\prime }=sinx$ , y(0) = 1, y′(0) = 0

20.

$y^{″}+y^{\prime }+xy=cosx$ , y(0) = 0, y′(0) = 1

21.

(a) If y = a ₀ + a ₁ x + a ₂ x ² + a ₃ x ³ + ⋯, what are the first three nonzero terms of the power series for y ² ? (b) Use this series to find the first three nonzero terms in the power series solution about x = 0 to Van-der-Pol's equation,

$y^{″}+(y^2-1)y^{\prime }+y=0$

if y(0) = 0 and y′(0) = 1.

22.

Use a method similar to that in Exercise 21 to find the first three nonzero terms of the power series solution about x = 0 to Rayleigh's equation,

$y^{″}+\left(\frac{1}{3}{(y^{\prime })}^2-1\right)y^{\prime }+y=0$

if y(0) = 1 and y′(0) = 0, an equation that arises in the study of the motion of a violin string.

23.

Hermite's equation is given by

$y^{″}-2x{y}^{\prime }+2ky,k\ge 0.$

Using a power series expansion about the ordinary point x = 0, obtain a general solution of this equation for (a) k = 1 and (b) k = 3. Show that if k is a nonnegative integer, then one of the solutions is a polynomial of degree k.

24.

Chebyshev's equation is given by

$(1-x^2)y^{″}-x{y}^{\prime }+k^{2}y=0,k\ge 0.$

Using a power series expansion about the ordinary point x = 0, obtain a general solution of this equation for (a) k = 1 and (b) k = 3. Show that if k is a nonnegative integer, then one of the solutions is a polynomial of degree k.

In addition to his work with orthogonal polynomials, the Russian mathematician Pafnuty Lvovich Chebyshev (1821-1894) is also known for his contributions to number theory. Chebyshev was rich and never married but financially supported a daughter who he never recognized, although he did socialize with her relatively frequently, especially after she married.

25.

(a)

Show that Legendre's equation can be written as $\frac{d}{\text{d}x}\left[\left(1-x^2\right)y^{\prime }\right]+k(k-1)y=0$ .

(b)

Using the previous result, verify that the Legendre polynomial given in Table 4.5 satisfy the orthogonality condition

$\underset{-1}{\overset{1}{\int }}{P}_m(x)P_n(x)\text{d}x=0,m\ne n.$

(c)

Evaluate

$\underset{-1}{\overset{1}{\int }}{\left[P_n(x)\right]}^2\text{d}x$

for n = 0, …, 5. How do these values compare to the value of 2/(2n + 1), for n = 0, …, 5? Hint: P _m (x) and P _n (x) satisfy the differential equations $\frac{d}{\text{d}x}\left[\left(1-x^2\right)P_n^{\prime }(x)\right]+n(n+1)P_n(x)=0$ and $\frac{d}{\text{d}x}\left[\left(1-x^2\right)P_m^{\prime }(x)\right]+$ m(m + 1)P _m (x) = 0, respectively. Multiply the first equation by P _m (x) and the second by P _n (x) and subtract the results. Then integrate from − 1 to 1.

26.

Consider the IVP y″ + f(x)y′ + y = 0, y(0) = 1, y′(0) = −1 where $f(x)=\left\{\begin{array}{c}\frac{sinx}{x},\text{if}x\ne 0\\ 1,\text{if}x=0\end{array}\right..$ (a) Show that x = 0 is an ordinary point of the equation. (b) Find a power series solution of the equation and graph an approximation of the solution on an interval. (c) Generate a numerical solution of the equation. Explain any unexpected results.

27.

(a)

Use a power series to solve $y^{″}-ycosx=sinx$ , y(0) = 1, y′(0) = 0.

(b)

Compare the polynomial approximations of degree 4, 7, 10, and 13 to the numerical solution obtained with a computer algebra system.

Although the English scientist George Biddell Airy (1801-1892) made major contributions to mathematics and astronomy, by many he was considered to be a sarcastic snob. According to Eggen, "Airy was not a great scientist, but he made great science possible." Airy's son wrote "The life of Airy was essentially that of a hard-working business man, and differed from that of other hard-working people only in the quality and variety of his work. It was not an exciting life, but it was full of interest."

28.

The differential equation y″ − xy = 0 is called Airy's equation and arises in electromagnetic theory and quantum mechanics. Two linearly independent solutions to Airy's equation, denoted by Ai(x) and Bi(x) are called the Airy functions. The function Ai(x) → 0 as $x\to \infty$ while $Bi(x)\to \infty$ as $x\to \infty$ . (Most computer algebra systems contain built-in definitions of the Airy functions.)

(a) If your computer algebra system contains built-in definitions of the Airy functions, graph each on the interval [−15, 5]. (b) Find a series solution of Airy's equation and obtain formulas for both Ai(x) and Bi(x). (c) Graph the polynomial approximation of degree n of Ai(x) for n = 6, 15, 30, and 45 on the interval [−15, 5]. Compare your results to (a) if applicable. (d) Graph the polynomial approximation of degree n of Bi(x) for n = 6, 15, 30, and 45 on the interval [−15, 5]. Compare your results to (a) if applicable.

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The Hydrogen-Like System

Valerio Magnasco , in Elementary Methods of Molecular Quantum Mechanics, 2007

4.5.2 Solution of the Θ-Equation

If we put:

(68) $\begin{array}{cc}x=\text{cos}\theta & -1\le x\le 1\end{array}$

we obtain the differential equation in x (Problem 4.8):

(69) $\left(1-x^2\right)\frac{d^2\Theta }{d{x}^2}-2x\frac{d\Theta }{dx}+\left(\lambda -\frac{m^2}{1-x^2}\right)\Theta =0$

which shows regular singularities at |x| = 1.

The asymptotic behaviour of the function Θ(x) at |x| = 1 can be obtained by considering the approximate equation:

(70) $\left(1-x^2\right)\frac{d^2\Theta }{d{x}^2}-2x\frac{d\Theta }{dx}\approx \frac{m^2}{1-x^2}\Theta .$

It can be easily shown (Problem 4.9) that the solution of this asymptotic equation is:

(71) $\begin{array}{cc}\Theta \left(x\right)={\left(1-x^2\right)}^{\frac{m}{2}}& m=\left|m\right|\ge 0\end{array}$

which is regular for |x| = 1. The complete solution Θ(x) must hence have the form:

(72) $\Theta \left(x\right)={\left(1-x^2\right)}^{\frac{m}{2}}G\left(x\right),$

where G(x) has to be determined, with its regularity conditions, over the whole interval |x| ≤ 1. The differential equation for G(x) is (Problem 4.10):

(73) $\left(1-x^2\right)\frac{d^{2}G}{d{x}^2}-2\left(m-1\right)x\frac{dG}{dx}+\left[\lambda -m\left(m+1\right)\right]G=0.$

For λ = l(l + 1) (l ≥ 0 integer) this is nothing but the differential equation for the associated Legendre functions $P_l^m\left(x\right)$ , well known to mathematical physicists in potential theory. We will try again the power series solution for G(x) using the expansion:

(74) $\begin{array}{c}G\left(x\right)=\sum _{k=0}^{\infty }{a}_k{x}^k\\ G^{\prime }\left(x\right)=\sum _{k=1}^{\infty }k{a}_k{x}^{k-1}\\ G^{″}\left(x\right)=\sum _{k=2}^{\infty }k\left(k-1\right)a_k{x}^{k-2}.\end{array}$

Proceeding as for the radial solution, we substitute expansions (74) in (73), thereby obtaining:

(75) $\sum _{k}k\left(k-1\right)a_k{x}^{k-2}-\sum _{k}k\left(k-1\right)a_k{x}^k-2\left(m+1\right)\sum _{k}k{a}_k{x}^k+\left(\lambda -m\left(m+1\right)\right)\sum _k{a}_k{x}^k=0,$

where:

Coefficient of x^k :

(76) $\left(k+1\right)\left(k+2\right)a_{k+2}-\left(k-1\right)k{a}_k-2\left(m+1\right)k{a}_k+\left(\lambda -m\left(m+1\right)\right)a_k=0,$

so that we obtain the 2-term recursion formula for the coefficients:

(77) $\begin{array}{cc}{a}_{k+2}=\frac{\left(k+m\right)\left(k+m+1\right)-\lambda }{\left(k+1\right)\left(k+2\right)}{a}_k& k=\mathrm{0,1,2},\cdots \end{array}$

According to this recursion formula, we shall obtain this time an even series (k = 0, 2, 4, · · ·) and an odd series (k = 1, 3, 5, · · ·), which are characteristic of trigonometric functions. We have now to study the convergence of the series (74) when k → ∞. The ratio test shows that this series has the same asymptotic behaviour of the geometrical series of reason x ² :

(78) $S=\sum _{k=0}^{\infty }{x}^k$

with:

(79) $S_n=1+x+x^2+\cdots +x^{n-1}=\frac{1-x^n}{1-x}.$

Now, lim _n→∞ S_n does exist if |x| < 1, but the series will diverge at |x| = 1, since:

(80) $\underset{n\to \infty }{\text{lim}}{S}_n=\frac{1}{1-x}.$

Going back to our series (74), the ratio test says that:

(81) $\underset{n\to \infty }{\text{lim}}\frac{a_{k+2}{x}^{k+2}}{a_k{x}^k}=x^2\underset{n\to \infty }{\text{lim}}\frac{a_{k+2}}{a_k}=x^2$

so that:

$\begin{array}{l}\text{for}\left|x\right|<1\text{the series is absolutely convergent}\\ \text{for}\left|x\right|=1\text{the series is divergent}.\end{array}$

To get a physically acceptable solution of equation (69) even for |x| = 1, the series must reduce to a polynomial, so that:

(82) $\begin{array}{cc}{a}_k\ne 0,& a_{k+2}=a_{k+4}=\cdots =0\end{array}$

and we obtain the relation:

(83) $\left(k+m\right)\left(k+m+1\right)-\lambda =0$

(84) $\begin{array}{cc}\lambda =\left(k+m\right)\left(k+m+1\right)& k,m=\mathrm{0,1,2},\cdots \end{array}$

Put:

(85) $\begin{array}{cc}k+m=l& \mathrm{a}\text{non}-\text{negative}\text{integer}\left(l\ge 0\right)\end{array}$

(86) $l=m,m+1,m+2,\dots$

(87) $\begin{array}{cc}l\ge \left|m\right|& -l\le m\le l\end{array}$

and we recover the remaining relation between angular quantum numbers l and m. Hence we get for the eigenvalue of ${\hat{L}}^2$ :

(88) $\lambda =l\left(l+1\right)$

(89) $l=\mathrm{0,1,2,3},\cdots \left(n-1\right)$

(90) $\begin{array}{cc}m=0,\pm 1,\pm 2,\cdots \pm l& \left(2l+1\right)\text{values}\text{of}m.\end{array}$

The infinite series (74) reduces to a polynomial whose degree is at most:

(91) $\begin{array}{cc}{k}_{\text{max}}=l-m& \left(\ge 0\right)\end{array}$

giving as complete solution for the angular equation:

(92)

where we still use m = |m| for short. The recursion relation for the coefficients is then:

(93) $\begin{array}{cc}{a}_{k+2}=\frac{\left(k+m\right)\left(k+m+1\right)-l\left(l+1\right)}{\left(k+1\right)\left(k+2\right)}{a}_k& k=\mathrm{0,1,2},\cdots \end{array}$

The explicit expressions for the first few angular functions Θ _lm (x) have been derived in Problem 4.11, and tested as correct solutions of the differential equation (69) in Problem 4.12. Our simple polynomial solution Θ(x) gives a result which is seen to differ from the conventional associated Legendre polynomials $P_l^m\left(x\right)$ (Problem 4.13) by a constant factor irrelevant from the standpoint of the differential equation. If [· · ·] stands for "integer part of", we can state that:

(94) $P_l^m\left(x\right)={\left(-1\right)}^{m+{\left[\frac{l+m}{2}\right]}_{\Theta lm\left(x\right)}},$

where it must be noted that $P_l^m\left(x\right)$ is normalized to $\frac{2}{2l+1}\frac{\left(l+m\right)!}{\left(l+m\right)!}$ and not to 1.

Some of the constant factors occurring between Θ _lm and $P_l^m$ are given in Problem 4.14.

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Lie symmetry analysis and exact explicit solutions for general Burgers' equation

Hanze Liu , ... Quanxin Zhang , in Journal of Computational and Applied Mathematics, 2009

Thus, for arbitrary chosen $c_0=\eta$ , then the other terms of the sequence ${\left\{c_n\right\}}_{n=1}^{\infty }$ can be determined successively from (50) in a unique manner. Furthermore, by induction method, we have

$c_n={(-1)}^n\frac{a^n}{b^n}{\eta }^{n+1}\text{,}n=0,1,2,\dots \text{.}$

This implies that for Eq. (36) , there exists a power series solution:

(51) $f\left(\xi \right)=\frac{b}{2a}log\left|\xi \right|+\beta +\sum _{n=0}^{\infty }{(-1)}^n\frac{a^n}{(n+1)b^n}{\eta }^{n+1}{\xi }^{n+1}\text{.}$

Note that in terms of the above example, we can write the approximate form of (51) as follows:

$f\left(\xi \right)=\frac{b}{2a}log\left|\xi \right|+\beta +\eta \xi -\frac{a}{2b}{\eta }^2{\xi }^2+\frac{a^2}{3b^2}{\eta }^3{\xi }^3-\frac{a^3}{4b^3}{\eta }^4{\xi }^4+\frac{a^4}{5b^4}{\eta }^5{\xi }^5-\frac{a^5}{6b^5}{\eta }^6{\xi }^6+\cdots \text{.}$

Thus, we obtain the power series solution of Eq. (1):

(52) $u(x,t)=\beta -\frac{b}{2a}logt-\frac{x^2}{4at}+\sum _{n=0}^{\infty }{(-1)}^n\frac{a^n}{(n+1)b^n}{\eta }^{n+1}{\left(x{t}^{-1}\right)}^{n+1}\text{,}$

where $\beta$ and $\eta$ are arbitrary constant numbers.

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Reduction of parameters in Finite Unified Theories and the MSSM

Sven Heinemeyer , ... George Zoupanos , in Nuclear Physics B, 2018

2 Unification of couplings by the RGI method

In this section we will briefly outline the method of reduction of couplings. Any RGI relation among couplings (which does not depend on the renormalization scale μ explicitly) can be expressed in the implicit form $\mathrm{\Phi }(g_1,\cdots ,g_A)=\text{const.}$ , which has to satisfy the partial differential equation (PDE)

(1) $\mu \frac{d\mathrm{\Phi }}{d\mu }=\overrightarrow{\mathrm{\nabla }}\cdot \overrightarrow{\beta }=\sum _{a=1}^A{\beta }_a\frac{\partial \mathrm{\Phi }}{\partial g_a}=0,$

where ${\beta }_a$ is the β-function of $g_a$ . This PDE is equivalent to a set of ordinary differential equations, the so-called reduction equations (REs) [1],

(2) ${\beta }_g\frac{d{g}_a}{dg}={\beta }_a,a=1,\cdots ,A,$

where g and ${\beta }_g$ are the primary coupling and its β-function, and the counting on a does not include g. Since maximally ( $A-1$ ) independent RGI "constraints" in the A-dimensional space of couplings can be imposed by the ${\mathrm{\Phi }}_a$ 's, one could in principle express all the couplings in terms of a single coupling g . The strongest requirement is to demand power series solutions to the REs,

(3) $g_a=\sum _n{\rho }_a^{(n)}{g}^{2n+1},$

which formally preserve perturbative renormalizability. Remarkably, the uniqueness of such power series solutions can be decided already at the one-loop level [1]. To illustrate this, let us assume that the β-functions have the form

(4) $\begin{array}{ccc}\hfill {\beta }_a& \hfill =\hfill & \frac{1}{16{\pi }^2}[\sum _{b,c,d\ne g}{\beta }_a^{(1)bcd}{g}_b{g}_c{g}_d+\sum _{b\ne g}{\beta }_a^{(1)b}{g}_b{g}^2]+\cdots ,\hfill \\ \hfill {\beta }_g& \hfill =\hfill & \frac{1}{16{\pi }^2}{\beta }_g^{(1)}{g}^3+\cdots ,\hfill \end{array}$

where ⋯ stands for higher order terms, and ${\beta }_a^{(1)bcd}$ 's are symmetric in $b,c,d$ . We then assume that the ${\rho }_a^{(n)}$ 's with $n\le r$ have been uniquely determined. To obtain ${\rho }_a^{(r+1)}$ 's, we insert the power series (3) into the REs (2) and collect terms of $\mathcal{O}(g^{2r+3})$ and find

$\sum _{d\ne g}M{(r)}_a^d{\rho }_d^{(r+1)}=\text{lower order quantities},$

where the r.h.s. is known by assumption, and

(5) $M{(r)}_a^d=3\sum _{b,c\ne g}{\beta }_a^{(1)bcd}{\rho }_b^{(1)}{\rho }_c^{(1)}+{\beta }_a^{(1)d}-(2r+1){\beta }_g^{(1)}{\delta }_a^d,$

(6) $0=\sum _{b,c,d\ne g}{\beta }_a^{(1)bcd}{\rho }_b^{(1)}{\rho }_c^{(1)}{\rho }_d^{(1)}+\sum _{d\ne g}{\beta }_a^{(1)d}{\rho }_d^{(1)}-{\beta }_g^{(1)}{\rho }_a^{(1)}.$

Therefore, the ${\rho }_a^{(n)}$ 's for all $n>1$ for a given set of ${\rho }_a^{(1)}$ 's can be uniquely determined if $\det M{(n)}_a^d\ne 0$ for all $n\ge 0$ .

As it will be clear later by examining specific examples, the various couplings in supersymmetric theories have easily the same asymptotic behavior. Therefore searching for a power series solution of the form (3) to the REs (2) is justified. This is not the case in non-supersymmetric theories, although the deeper reason for this fact is not fully understood.

The possibility of coupling unification described in this section is without any doubt attractive because the "completely reduced" theory contains only one independent coupling, but it can be unrealistic. Therefore, one often would like to impose fewer RGI constraints, and this is the idea of partial reduction [29].

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INVITED: Slow manifold reduction for plasma science

J.W. Burby , T.J. Klotz , in Communications in Nonlinear Science and Numerical Simulation, 2020

5.1 Origins of slow manifolds

The simplest and most direct way to describe the theory of slow manifolds is to work within the context of fast-slow systems developed in Section 4. Therefore suppose that we have been given a plasma model that is formulated as a fast-slow system as in Definition 2. Such a system generally exhibits a pair of disparate timescales, a short O(ϵ) scale owing to the fact that generically $\dot{y}=O(1/ϵ),$ and a longer O(1) scale that characterizes trajectory segments passing within the region where $f_{ϵ}(x,y)=O\left(ϵ\right)$ . Due to this intermingling of fast and slow dynamics a natural question to ask is whether there are special solutions where the fast timescale is completely inactive, or else only excited with a very small amplitude. We will refer to such solutions as slow solutions.

A suggestive mechanical analogy to keep in mind throughout this discussion is a pendulum whose massive end (with mass m ₁) is attached to a second mass (m ₂) by a light, stiff spring. While generic motions of such a system involve both swinging of the pendulum and relatively rapid oscillations of the stiff spring, physical experience suggests that there should be system motions during which the stiff spring is not excited, so that the overall dynamics resembles that of an ordinary pendulum with mass $M=m_1+m_2$ . We leave it as an exercise for the reader to verify that this two-mass system may be formulated as a fast-slow system, where the small parameter ϵ is proportional to the ratio of the pendulum frequency to the spring frequency.

Perhaps the most remarkable feature of fast-slow systems is that slow solutions of such systems organize themselves in a geometrically simple manner. Due to the typical complexity of the phase portrait for a dynamical system one might instead expect that such solutions tend to disperse themselves around phase space in a haphazard fashion. But due to the special structural properties of fast-slow systems these solutions in fact lie along special submanifolds in phase space called slow manifolds that are readily computable using asymptotic methods. We may bring these slow manifolds to light as follows.

First observe, as we did earlier when motivating the definition of fast-slow systems, that if (x _ϵ(t), y _ϵ(t)) is a slow solution of a fast-slow system for each ϵ then the limiting solution (x ₀(t), y ₀(t)) must satisfy Eqs. (48) and (49). In particular the limiting fast variable y ₀(t) must be slaved to the limiting slow variable x ₀(t) according to $y_0\left(t\right)=y_0^{*}\left(x_0\left(t\right)\right),$ where $y_0^{*}\left(x\right)$ is implicitly defined by the formula $f_0(x,y_0^{*}\left(x\right))=0$ . (Recall that the definition of fast-slow systems ensures this equation uniquely determines $y_0^{*},$ at least locally.) Let us refer to the submanifold S ₀ given by the graph of $y_0^{*},$ i.e.

(104) $S_0=\{(x,y)\mid y=y_0^{*}\left(x\right)\},$

as the limiting slow manifold.

Apparently each limiting slow trajectory must be contained in the limiting slow manifold. In fact the limiting slow manifold is the union of all the limiting slow trajectories. If we suppose that slow trajectories exist for finite ϵ this observation leads to the seemingly-reasonable hypothesis that the qualitative organization of slow trajectories when $ϵ=0$ persists when ϵ is small but finite. In particular it suggests that the collection of finite-ϵ slow trajectories form a small deformation of the limiting slow manifold. This deformed submanifold S _ϵ must have the form

(105) $S_{ϵ}=\{(x,y)\mid y=y_{ϵ}^{*}\left(x\right)\},$

where $y_{ϵ}^{*}\left(x\right)$ is some function that tends to $y_0^{*}\left(x\right)$ as ϵ → 0. Moreover, because we expect S _ϵ to be the union of finite-ϵ slow trajectories it is reasonable to suspect that S _ϵ is an invariant manifold for the fast-slow system. (See Section 3 for background on invariant manifolds.) Indeed, if S _ϵ were not an invariant manifold then there would be slow solutions that start in S _ϵ and then eventually leave, contradicting the hypothesis that S _ϵ contains entire slow trajectories.

Let us now test our hypothesis that there is a deformation S _ϵ of the limiting slow manifold S ₀ that (a) contains finite-ϵ slow trajectories, and (b) is invariant under the flow of the (finite-ϵ) fast-slow system. By Proposition 1 from Section 3 with $f(x,y)=f_{ϵ}(x,y)/ϵ$ and $g(x,y)=g_{ϵ}(x,y),$ invariance of S _ϵ implies that the slaving function $y_{ϵ}^{*}\left(x\right)$ must satisfy the (scaled) invariance equation

(106) $ϵD{y}_{ϵ}^{*}\left(x\right)\left[g_{ϵ}(x,y_{ϵ}^{*}\left(x\right))\right]=f_{ϵ}(x,y_{ϵ}^{*}\left(x\right)).$

If there is a $y_{ϵ}^{*}$ that solves this equation then the graph of $y_{ϵ}^{*}$ will necessarily be an invariant manifold. Remarkably, solutions contained in such an invariant manifold are automatically free of the O(ϵ) timescale because for such solutions (a) the timescale of the fast variable $y\left(t\right)=y_{ϵ}^{*}\left(x\left(t\right)\right)$ is determined by x(t), and (b) x(t) is a solution of the equation $\dot{x}=g_{ϵ}(x,y_{ϵ}^{*}\left(x\right))=O\left(1\right)$ . The task of testing our hypothesis therefore reduces to showing that the invariance Eq. (106) admits a solution that is asymptotic to $y_0^{*}$ .

Proving that (106) admits a solution is generally a highly-nontrivial task because first-order systems of partial differential equations usually cannot be solved by hand, and sometimes do not admit solutions at all. Let us therefore attempt to develop a preliminary understanding of solutions of (106) using asymptotic expansions. Specifically let us suppose $y_{ϵ}^{*}\left(x\right)$ has the asymptotic expansion

(107) $y_{ϵ}^{*}\left(x\right)=y_0^{*}\left(x\right)+ϵy_1^{*}\left(x\right)+{ϵ}^2{y}_2^{*}\left(x\right)+\dots ,$

and investigate conditions imposed on the coefficient functions $y_k^{*}$ by the invariance Eq. (106).

Substituting the ansatz (107) into (106) and then collecting O(1) terms leads to

(108) $0=f_0(x,y_0^{*}\left(x\right)),$

which is consistent with our assumption that S _ϵ is a small deformation of the limiting slow manifold S ₀. Collecting the O(ϵ) terms leads to

(109) $D{y}_0^{*}\left[g_0(x,y_0^{*})\right]=D_y{f}_0(x,y_0^{*})\left[y_1^{*}\right]+f_1(x,y_0^{*}),$

which, by the definition of fast-slow systems, can be used to solve for $y_1^{*}$ in terms of quantities already computed at zero'th order. In particular,

(110) $y_1^{*}={\left[D_y{f}_0(x,y_0^{*})\right]}^{-1}(D{y}_0^{*}\left[g_0(x,y_0^{*})\right]-f_1(x,y_0^{*})).$

Going to yet-higher orders the following pattern emerges. Within the collection of O(ϵ ^k ) terms generated by substituting (107) into (106) the only term involving the coefficient $y_k^{*}$ is $D_y{f}_0(x,y_0^{*})\left[y_k^{*}\right]$ . All other terms in the collection involve only the coefficients $y_l^{*}$ with l <k. It follows that the coefficients $y_k^{*}$ may be explicitly computed recursively for all 0 ≤k < ∞.

The conclusion that we draw from this asymptotic analysis is that if a solution $y_{ϵ}^{*}$ of the invariance Eq. (106) exists and is smooth in ϵ then that solution has a unique asymptotic expansion in terms of the $y_k^{*}$ that may be computed recursively for all k. Moreover, even if a true solution does not exist, the asymptotic expansion of such a solution always exists. We summarize this curious result by introducing the notion of a formal slow manifold,

Definition 5 formal slow manifold

Given a fast-slow system $ϵ\dot{y}=f_{ϵ}(x,y),$ $\dot{x}=g_{ϵ}(x,y),$ a formal slow manifold is a formal power series solution $y_{ϵ}^{*}$ of the invariance equation

(111) $ϵD{y}_{ϵ}^{*}\left(x\right)\left[g_{ϵ}(x,y_{ϵ}^{*}\left(x\right))\right]=f_{ϵ}(x,y_{ϵ}^{*}\left(x\right)).$

We then state the following

Theorem 1 Existence and uniqueness of formal slow manifolds

Associated with each fast-slow system is a unique formal slow manifold $y_{ϵ}^{*}=y_0^{*}+ϵy_1^{*}+{ϵ}^2{y}_2^{*}+\dots$ . The coefficients $y_k^{*}$ may be explicitly computed recursively. In particular we have the following low-order formulas:

(112) $0=f_0(x,y_0^{*}\left(x\right))$

(113) $y_1^{*}={\left[D_y{f}_0(x,y_0^{*})\right]}^{-1}(D{y}_0^{*}\left[g_0(x,y_0^{*})\right]-f_1(x,y_0^{*}))$

Example 7

Theorem 1 may be applied directly to compute the formal slow manifold associated with Abraham–Lorentz dynamics in the weak drag regime. (c.f. Section 2.3.) As explained in Example 3 from Section 4.1, when this model is expressed in terms of the fast time τ it becomes a fast-slow system with $x=(\mathit{x},\mathit{v}),$ $y=\mathit{a},$ and

(114) $f_0(x,y)=\frac{3}{2}\mathit{a}-\frac{3}{2}\zeta \mathit{v}\times \mathit{B}\left(\mathit{x}\right)$

(115) $g_0(x,y)=(0,\mathit{a})$

(116) $g_1(x,y)=(\mathit{v},0).$

All higher-order coefficients in the power series expansions of f _ϵ and g _ϵ are zero. Eq. (112) in the statement of Theorem 1 therefore implies that the limiting slow manifold is given as the graph of $y_0^{*}={\mathit{a}}_0^{*},$ where

(117) ${\mathit{a}}_0^{*}(\mathit{x},\mathit{v})=\zeta \mathit{v}\times \mathit{B}\left(\mathit{x}\right).$

In order to compute the first-order correction to the limiting slow manifold from (113) we first note that the derivatives D_yf ₀ and $D{y}_0^{*}$ are given by

(118) $D_y{f}_0\left[\delta y\right]=\frac{3}{2}\delta \mathit{a}$

(119) $D{y}_0^{*}\left(x\right)\left[\delta x\right]=\zeta \delta \mathit{v}\times \mathit{B}+\zeta \mathit{v}\times (\delta \mathit{x}·\nabla \mathit{B}).$

Therefore we have

(120) $D{y}_0^{*}\left[g_0(x,y_0^{*})\right]-f_1(x,y_0^{*})=\zeta {\mathit{a}}_0^{*}\times \mathit{B}=(\mathit{v}\times \mathit{B})\times \mathit{B},$

and $y_1^{*}={\mathit{a}}_1^{*}$ with

(121) ${\mathit{a}}_1^{*}=\frac{2}{3}(\mathit{v}\times \mathit{B})\times \mathit{B}.$

In summary, the formal slow manifold for Abraham–Lorentz dynamics in the weak-drag regime is given by $y_{ϵ}^{*}={\mathit{a}}_{ϵ}^{*},$ where

(122) ${\mathit{a}}_{ϵ}^{*}=\zeta \mathit{v}\times \mathit{B}+ϵ\frac{2}{3}(\mathit{v}\times \mathit{B})\times \mathit{B}+O\left({ϵ}^2\right).$

The first term in this series is nothing more than the usual Lorentz force on a charged particle. The second term is the small-velocity form of the so-called Landau-Lifshitz form of the radiation drag force. We will connect this result to the work of Spohn after introducing the notion of slow manifold reduction in the next subsection.

Example 8

While Theorem 1 cannot be applied directly to Abraham–Lorentz dynamics in the zero-drag regime, it can be applied after changing dependent variables from ( x, v ) to ( x , v _∥, v ₁, v ₂) as in Example 4; recall that $x=(\mathit{x},v_{\parallel })$ and $y=(v_1,v_2)$ comprise a fast-slow split for this system. As for the ingredients required to apply Theorem 1, we have $f_{ϵ}=(f_{ϵ}^{v_1},f_{ϵ}^{v_2}),$ $g_{ϵ}=(g_{ϵ}^{\mathit{x}},g_{ϵ}^{v_{\parallel }}),$

(123) $f_0^{v_1}(x,y)=\zeta \left|\mathit{B}\right|v_2$

(124) $f_0^{v_2}(x,y)=-\zeta \left|\mathit{B}\right|v_1$

(125) $g_0^{v_{\parallel }}(x,y)=\mathit{v}·\nabla \mathit{b}·\mathit{v}$

(126) $g_0^{\mathit{x}}(x,y)=\mathit{v}$

at 0th order, and

(127) $f_1^{v_1}(x,y)=\mathit{v}·\mathit{R}{v}_2-v_{\parallel }\mathit{v}·\nabla \mathit{b}·{\mathit{e}}_1$

(128) $f_1^{v_2}(x,y)=-\mathit{v}·\mathit{R}{v}_1-v_{\parallel }\mathit{v}·\nabla \mathit{b}·{\mathit{e}}_2$

(129) $g_1^{v_{\parallel }}(x,y)=0$

(130) $g_1^{\mathit{x}}(x,y)=0$

at first order. The formal slow manifold associated with this system is therefore of the form $y_{ϵ}^{*}=({\left(v_1\right)}_0^{*},{\left(v_2\right)}_0^{*}),$ i.e. the perpendicular velocity variables are slaved to the particle position x and the parallel velocity v _∥. Because $f_0(x,y)=0$ if and only if $v_1=v_2=0$ the limiting slaving function $y_0^{*}$ vanishes:

(131) $\begin{array}{c}\hfill \left(\begin{array}{c}{\left(v_1\right)}_0^{*}\\ {\left(v_2\right)}_0^{*}\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right).\end{array}$

It follows that the general formula (113) for the first-order slaving function simplifies to

(132) $y_1^{*}=-{\left[D_y{f}_0(x,y_0^{*})\right]}^{-1}\left(f_1(x,y_0^{*})\right).$

And since the derivative matrix $D_y{f}_0(x,y_0^{*})$ is given by

(133) $\begin{array}{c}\hfill D_y{f}_0(x,y_0^{*})\left[\delta y\right]=\left(\begin{array}{cc}0& \zeta \left|\mathit{B}\right|\\ -\zeta \left|\mathit{B}\right|& 0\end{array}\right)\left(\begin{array}{c}\delta v_1\\ \delta v_2\end{array}\right),\end{array}$

Eq. (132) implies

(134) $\begin{array}{c}\hfill \left(\begin{array}{c}{\left(v_1\right)}_1^{*}\\ {\left(v_2\right)}_1^{*}\end{array}\right)=\frac{1}{\zeta \left|\mathit{B}\right|}\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\left(\begin{array}{c}{v}_{\parallel }^2\mathit{b}·\nabla \mathit{b}·{\mathit{e}}_1\\ v_{\parallel }^2\mathit{b}·\nabla \mathit{b}·{\mathit{e}}_2\end{array}\right).\end{array}$

Eq. (134) says that the first-order correction to the zero-drag formal slow manifold is given by the so-called curvature drift velocity from guiding center theory. This connection to guiding center theory may be emphasized further by introducing the notation

(135) ${\mathit{v}}_{\perp ϵ}^{*}={\left(v_1\right)}_{ϵ}^{*}{\mathit{e}}_1+{\left(v_2\right)}_{ϵ}^{*}{\mathit{e}}_2,$

in terms of which (134) becomes

(136) ${\mathit{v}}_{\perp 1}^{*}=-\frac{v_{\parallel }^2(\mathit{b}·\nabla \mathit{b})\times \mathit{b}}{\zeta \left|\mathit{B}\right|}\equiv {\mathit{v}}_c,$

which is the familiar expression for the curvature drift velocity.

We leave it as an exercise for the reader to verify that the second-order slaving function ${\mathit{v}}_{\perp 2}^{*}$ is given by

(137) ${\mathit{v}}_{\perp 2}^{*}=-\frac{v_{\parallel }(\mathit{b}·\nabla {\mathit{v}}_c+{\mathit{v}}_c·\nabla \mathit{b})\times \mathit{b}}{\zeta \left|\mathit{B}\right|}.$

The argument used above to deduce Theorem 1 actually says nothing at all about the existence of a solution $y_{ϵ}^{*}$ of the invariance equation. We therefore cannot infer from that argument that finite-ϵ slow solutions of a fast-slow system comprise an invariant manifold, or even that finite-ϵ slow solutions exist! In fact the formal power series $y_{ϵ}^{*}$ may easily fail to be convergent [25,36,55]. What then, if anything, does a formal slow manifold have to tell us about solutions of a fast-slow system?

While this important question will be taken up in greater detail in Section 6 we may preempt the more complete discussion with the following heuristic picture. Consider the finite truncation

(138) $y_{ϵ}^{*N}=y_0^{*}+ϵy_1^{*}+\dots +{ϵ}^N{y}_N^{*}$

of a formal slow manifold, where N is some fixed non-negative integer. While the function $y_{ϵ}^{*N}$ does not solve the invariance Eq. (111) exactly, it does solve it in an approximate sense: when $y_{ϵ}^{*N}$ is substituted into Eq. (111) the difference between the left- and right-hand sides is $O\left({ϵ}^{N+1}\right)$ . Because N is arbitrary, we say that formal slow manifolds comprise invariant manifolds to all orders in perturbation theory. Referring then to the interpretation of invariant manifolds in terms of tangency given in Proposition 2 the angle between the vector field $U_{ϵ}=(g_{ϵ},f_{ϵ}/ϵ)$ and the graph of the truncation $y_{ϵ}^{*N}$ becomes arbitrarily small as N increases. In this sense finite truncations of formal slow manifolds are almost invariant sets. (See for example [18] for a precise statement of the sense in which a particular class of slow manifolds comprises almost invariant sets.)

Where true invariant sets contain solutions for all time, almost invariant sets generally only keep solutions nearby over some finite time interval. The term "sticky set" is an appropriate descriptor for these objects. This "sticking" time interval is generally O(1) as ϵ tends to zero for truncated formal slow manifolds that do not exhibit normal instabilities. While such an interval certainly significantly exceeds the short O(ϵ) timescale, its length often cannot be increased by increasing N or decreasing ϵ. However, under certain special circumstances, some of which will be explained in Section 6, the sticking time may be much longer, or even infinite. Whatever the case, truncated formal slow manifolds certainly play an important dynamical role on the O(1) timescale. We are therefore motivated to set aside special terminology for truncations of formal slow manifolds.

Definition 6 slow manifold

Given a fast-slow system with formal slow manifold $y_{ϵ}^{*}=y_0^{*}+ϵy_1^{*}+\dots ,$ a slow manifold of order N is a submanifold ${\tilde{S}}_{ϵ}$ in (x, y)-space of the form

(139) ${\tilde{S}}_{ϵ}=\{(x,y)\mid y={\tilde{y}}_{ϵ}\left(x\right)\},$

where ${\tilde{y}}_{ϵ}\left(x\right)$ depends smoothly on x and ϵ and

(140) $y_{ϵ}^{*}-{\tilde{y}}_{ϵ}=O\left({ϵ}^{N+1}\right),$

where the difference is understood in the sense of formal power series.

Remark 1

This definition should be compared with MacKay's definition of slow manifolds from [36].

It is enlightening to compare the qualitative description of slow manifolds just presented with qualitative features of the classical averaging theory due to Kruskal [56]. Recall that this theory forms the mathematical basis for the guiding center description of charged particle motion in strong magnetic fields, amongst other reduced models. Like fast-slow systems theory, Kruskal's theory deals with dynamical systems that exhibit a short O(ϵ) timescale and a longer O(1) timescale. In contrast to the theory of slow manifolds, however, Kruskal's formalism aims to describe system motions during which both timescales are active simultaneously. In order to make progress in such an endeavor Kruskal assumes that the limiting short-timescale dynamics comprise strictly periodic motion and then proceeds to "average" over those rapid periodic oscillations. The averaged equations produced by Kruskal's method generally have a validity time that is O(1) as ϵ tends to zero. This is true in spite of the fact that the guiding center equations of motion are often tacitly assumed to remain valid over arbitrarily-long time intervals. We will argue in Section 6 that understanding the link between many popular reduced models in plasma physics and slow manifold theory exposes an uncomfortable truth: the fallacy of overestimating the validity time of reduced models occurs frequently within plasma physics, even outside of discussions of guiding center theory.

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Find the First Five Terms of the Power Series Solution

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