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| In [[mathematics]], the '''power series method''' is used to seek a [[power series]] solution to certain [[differential equation]]s. In general, such a solution assumes a [[power series]] with unknown coefficients, then substitutes that solution into the differential equation to find a [[recurrence relation]] for the coefficients.
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| == Method ==
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| Consider the second-order linear differential equation
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| : <math>a_2(z)f''(z)+a_1(z)f'(z)+a_0(z)f(z)=0.\;\!</math>
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| Suppose ''a''<sub>2</sub> is nonzero for all ''z''. Then we can divide throughout to obtain
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| : <math>f''+{a_1(z)\over a_2(z)}f'+{a_0(z)\over a_2(z)}f=0.</math>
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| Suppose further that ''a''<sub>1</sub>/''a''<sub>2</sub> and ''a''<sub>0</sub>/''a''<sub>2</sub> are [[analytic function]]s.
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| The power series method calls for the construction of a power series solution
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| :<math>f=\sum_{k=0}^\infty A_kz^k.</math>
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| If ''a''<sub>2</sub> is zero for some ''z'', then the [[Frobenius method]], a variation on this method, is suited to deal with so called ''singular points''. The method works analogously for higher order equations as well as for systems.
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| == Example usage ==
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| Let us look at the [[Hermite differential equation]],
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| : <math>f''-2zf'+\lambda f=0;\;\lambda=1</math>
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| We can try to construct a series solution
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| : <math>f=\sum_{k=0}^\infty A_kz^k</math>
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| : <math>f'=\sum_{k=0}^\infty kA_kz^{k-1}</math>
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| : <math>f''=\sum_{k=0}^\infty k(k-1)A_kz^{k-2}</math>
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| Substituting these in the differential equation
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| : <math>
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| \begin{align}
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| & {} \quad \sum_{k=0}^\infty k(k-1)A_kz^{k-2}-2z\sum_{k=0}^\infty kA_kz^{k-1}+\sum_{k=0}^\infty A_kz^k=0 \\
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| & =\sum_{k=0}^\infty k(k-1)A_kz^{k-2}-\sum_{k=0}^\infty 2kA_kz^k+\sum_{k=0}^\infty A_kz^k
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| \end{align}
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| </math>
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| Making a shift on the first sum
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| : <math>
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| \begin{align}
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| & = \sum_{k+2=0}^\infty (k+2)((k+2)-1)A_{k+2}z^{(k+2)-2}-\sum_{k=0}^\infty 2kA_kz^k+\sum_{k=0}^\infty A_kz^k \\
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| & =\sum_{k=-2}^\infty (k+2)(k+1)A_{k+2}z^k-\sum_{k=0}^\infty 2kA_kz^k+\sum_{k=0}^\infty A_kz^k \\
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| & =(0)(-1)A_0 z^{-2} + (1)(0)A_{1}z^{-1}+\sum_{k=0}^\infty (k+2)(k+1)A_{k+2}z^k-\sum_{k=0}^\infty 2kA_kz^k+\sum_{k=0}^\infty A_kz^k \\
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| & =\sum_{k=0}^\infty (k+2)(k+1)A_{k+2}z^k-\sum_{k=0}^\infty 2kA_kz^k+\sum_{k=0}^\infty A_kz^k \\
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| & =\sum_{k=0}^\infty \left((k+2)(k+1)A_{k+2}+(-2k+1)A_k\right)z^k
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| \end{align}
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| </math>
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| If this series is a solution, then all these coefficients must be zero, so:
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| : <math>(k+2)(k+1)A_{k+2}+(-2k+1)A_k=0\;\!</math>
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| We can rearrange this to get a [[recurrence relation]] for ''A''<sub>''k''+2</sub>.
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| : <math>(k+2)(k+1)A_{k+2}=-(-2k+1)A_k\;\!</math>
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| : <math>A_{k+2}={(2k-1)\over (k+2)(k+1)}A_k\;\!</math>
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| Now, we have
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| : <math>A_2 = {-1 \over (2)(1)}A_0={-1\over 2}A_0,\, A_3 = {1 \over (3)(2)} A_1={1\over 6}A_1</math>
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| We can determine ''A''<sub>0</sub> and ''A''<sub>1</sub> if there are initial conditions, i.e. if we have an initial value problem.
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| So we have
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| : <math>
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| \begin{align}
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| A_4 & ={1\over 4}A_2 = \left({1\over 4}\right)\left({-1 \over 2}\right)A_0 = {-1 \over 8}A_0 \\[8pt]
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| A_5 & ={1\over 4}A_3 = \left({1\over 4}\right)\left({1 \over 6}\right)A_1 = {1 \over 24}A_1 \\[8pt]
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| A_6 & = {7\over 30}A_4 = \left({7\over 30}\right)\left({-1 \over 8}\right)A_0 = {-7 \over 240}A_0 \\[8pt]
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| A_7 & = {3\over 14}A_5 = \left({3\over 14}\right)\left({1 \over 24}\right)A_1 = {1 \over 112}A_1
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| \end{align}
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| </math>
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| and the series solution is
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| : <math>
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| \begin{align}
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| f & = A_0x^0+A_1x^1+A_2x^2+A_3x^3+A_4x^4+A_5x^5+A_6x^6+A_7x^7+\cdots \\[8pt]
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| & = A_0x^0 + A_1x^1 + {-1\over 2}A_0x^2 + {1\over 6}A_1x^3 + {-1 \over 8}A_0x^4 + {1 \over 24}A_1x^5 + {-7 \over 240}A_0x^6 + {1 \over 112}A_1x^7 + \cdots \\[8pt]
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| & = A_0x^0 + {-1\over 2}A_0x^2 + {-1 \over 8}A_0x^4 + {-7 \over 240}A_0x^6 + A_1x + {1\over 6}A_1x^3 + {1 \over 24}A_1x^5 + {1 \over 112}A_1x^7 + \cdots
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| \end{align}
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| </math>
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| which we can break up into the sum of two linearly independent series solutions:
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| : <math>f=A_0 \left(1+{-1\over 2}x^2+{-1 \over 8}x^4+{-7 \over 240}x^6+\cdots\right) + A_1\left(x+{1\over 6}x^3+{1 \over 24}x^5+{1 \over 112}x^7+\cdots\right)</math>
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| which can be further simplified by the use of [[hypergeometric series]].
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| ==Nonlinear equations==
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| The power series method can be applied to certain [[nonlinear]] differential equations, though with less flexibility. Most nonlinear equations can be solved by using the [[Parker-Sochacki method|Parker-Sochacki]] method, which is a slight variation of the auxiliary variable method. Since the Parker-Sochacki method involves a reformation of the original system of ordinary differential equations, it is not simply referred to as the power series method. Nonetheless, the Parker-Sochacki method (a method that comes before doing the power series method) allows one to easily apply the power series method to most systems of ordinary differential equations. The Parker-Sochacki method allows one to apply the power series method to a larger system, and the exact same coefficients will be produced at the cost of also calculating the coefficients of auxiliary equations.
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| The power series method will give solutions only to [[initial value problem]]s (opposed to [[boundary value problem]]s), this is not an issue when dealing with linear equations since the solution may turn up multiple linearly independent solutions which may be combined (by [[Superposition principle|superposition]]) to solve boundary value problems as well. A further restriction is that the series coefficients will be specified by a nonlinear recurrence (the nonlinearities are inherited from the differential equation).
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| In order for the solution method to work, as in linear equations, it is necessary to express every term in the nonlinear equation as a power series so that all of the terms may be combined into one power series.
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| As an example, consider the initial value problem
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| :<math>F F'' + 2 F'^2 + \eta F' = 0 \quad ; \quad F(1) = 0 \ , \ F'(1) = -\frac{1}{2}</math>
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| which describes a solution to capillary-driven flow in a groove. Note the two nonlinearities: the first and second terms involve products. Note also that the initial values are given at <math>\eta = 1</math>, which hints that the power series must be set up as:
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| :<math>F(\eta) = \sum_{i = 0}^{\infty} c_i (\eta - 1)^i</math>
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| since in this way
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| :<math>\frac{d^n F}{d \eta^n} \Bigg|_{\eta = 1} = n! \ c_n</math>
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| which makes the initial values very easy to evaluate. It is necessary to rewrite the equation slightly in light of the definition of the power series,
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| :<math>F F'' + 2 F'^2 + (\eta - 1) F' + F' = 0 \quad ; \quad F(1) = 0 \ , \ F'(1) = -\frac{1}{2}</math>
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| so that the third term contains the same form <math>\eta - 1</math> that shows in the power series.
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| The last consideration is what to do with the products; substituting the power series in would result in products of power series when it's necessary that each term be its own power series. This is where the [[Cauchy product|identity]]
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| :<math>\left(\sum_{i = 0}^{\infty} a_i x^i\right) \left(\sum_{i = 0}^{\infty} b_i x^i\right) =
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| \sum_{i = 0}^{\infty} x^i \sum_{j = 0}^i a_{i - j} b_j</math>
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| is useful; substituting the power series into the differential equation and applying this identity leads to an equation where every term is a power series. After much rearrangement, the recurrence
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| :<math>
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| \sum_{j = 0}^i \left((j + 1) (j + 2) c_{i - j} c_{j + 2} + 2 (i - j + 1) (j + 1) c_{i - j + 1} c_{j + 1}\right) + i c_i + (i + 1) c_{i + 1} = 0
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| </math> | |
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| is obtained, specifying exact values of the series coefficients. From the initial values, <math>c_0 = 0</math> and <math>c_1 = -1/2</math>, thereafter the above recurrence is used. For example, the next few coefficients:
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| :<math>
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| c_2 = -\frac{1}{6} \quad ; \quad c_3 = -\frac{1}{108} \quad ; \quad c_4 = \frac{7}{3240} \quad ; \quad c_5 = -\frac{19}{48600} \ \dots
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| </math>
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| A limitation of the power series solution shows itself in this example. A numeric solution of the problem shows that the function is smooth and always decreasing to the left of <math>\eta = 1</math>, and zero to the right. At <math>\eta = 1</math>, a slope discontinuity exists, a feature which the power series is incapable of rendering, for this reason the series solution continues decreasing to the right of <math>\eta = 1</math> instead of suddenly becoming zero.
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| == External links ==
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| * {{mathworld|urlname=FrobeniusMethod|title=Frobenius Method}}
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| * [http://math.fullerton.edu/mathews/n2003/FrobeniusSeriesMod.html Module for Frobenius Series Solution]
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| == References ==
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| * {{cite book | last1=Coddington | first1=Earl A. | last2=Levinson | first2=Norman | title=Theory of Ordinary Differential Equations | publisher=[[McGraw-Hill]] | location=New York | year=1955}}
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| * {{cite book
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| | surname = Hille
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| | given = Einar
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| |authorlink=Einar Hille
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| | title = Ordinary Differential Equations in the Complex Plane
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| | publisher=[[Dover Publications]]
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| | place = [[Mineola, New York|Mineola]]
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| | year = 1976}}
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| * {{cite book| last = Teschl| given = Gerald|authorlink=Gerald Teschl| title = Ordinary Differential Equations and Dynamical Systems| publisher=[[American Mathematical Society]]| place = [[Providence, Rhode Island|Providence]]| year = 2012| isbn= 978-0-8218-8328-0| url = http://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}
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| [[Category:Ordinary differential equations]]
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Myrtle Benny is how I'm known as and I really feel comfy when people use the full name. South Dakota is exactly where I've always been living. My working day occupation is a meter reader. Doing ceramics is what her family members and her appreciate.
Also visit my blog post: meal delivery service [relevant web-site]