Power series approach

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A power series is a solution for differential equations . The function you are looking for is represented as a power series with unknown coefficients and then inserted into the differential equation. The solution can be found by comparing coefficients and in some cases can be expressed again using elementary functions .

In the general case, if the coefficient functions are meromorphic as in the Fuchs differential equation (to which the hypergeometric differential equation belongs), the differential equation must always be considered in the complex ( Riemann number sphere ). There are generalized power series solutions (see Frobenius method ) for differential equations of the Fuchs type (with exclusively liftable singularities also in infinity) and the fundamental solutions of the differential equation given locally as power series solutions are connected by considering analytical extensions around the singular points of the coefficient functions via monodromic matrices .

The exponential function as a motivating example

As a simple example we consider the following question: Which function is derived as a multiple of this function? As an equation:

${\displaystyle {\frac {dy(x)}{dx}}=\lambda \,y(x)}$

This ordinary differential equation of the first order can be solved uniquely if an initial condition is specified:

${\displaystyle y(0)=y_{0}}$

For we put on a power series: ${\displaystyle y}$

${\displaystyle y(x)=\sum _{k=0}^{\infty }a_{k}x^{k}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\ldots }$

The initial condition translates to because . ${\displaystyle y_{0}=a_{0}}$ ${\displaystyle y(0)=a_{0}+a_{1}0+a_{2}0^{2}+...=a_{0}}$

The derivation of is therefore: ${\displaystyle y(x)}$

${\displaystyle {\frac {dy(x)}{dx}}={\frac {d}{dx}}\sum _{k=0}^{\infty }a_{k}x^{k}=\sum _{k=0}^{\infty }{\frac {d}{dx}}a_{k}x^{k}=\sum _{k=1}^{\infty }a_{k}\,k\,x^{k-1}=a_{1}+2a_{2}x+3a_{3}x^{2}+\ldots }$

Inserted into the above differential equation this means:

${\displaystyle a_{1}+2a_{2}x+3a_{3}x^{2}+\ldots =\lambda a_{0}+\lambda a_{1}x+\lambda a_{2}x^{2}+\lambda a_{3}x^{3}+\ldots }$

Since this is supposed to apply to everyone , the coefficients in front etc. must be the same. Is thus: etc. This can be switched and insert: , , . General is: ${\displaystyle x}$ ${\displaystyle x_{0},x_{1},x_{2}}$ ${\displaystyle a_{1}=\lambda a_{0},2a_{2}=\lambda a_{1},3a_{3}=\lambda a_{2}}$ ${\displaystyle a_{1}={\tfrac {\lambda \,a_{0}}{1}}}$ ${\displaystyle a_{2}={\tfrac {\lambda \,a_{1}}{2}}={\tfrac {\lambda ^{2}\,a_{0}}{2\cdot 1}}}$ ${\displaystyle a_{3}={\tfrac {\lambda \,a_{2}}{3}}={\tfrac {\lambda ^{3}\,a_{0}}{3\cdot 2\cdot 1}}}$

${\displaystyle \sum _{k=1}^{\infty }a_{k}\,k\,x^{k-1}=\sum _{k=0}^{\infty }\lambda \,a_{k}\,x^{k}}$and therefore for everyone . ${\displaystyle a_{k}={\frac {\lambda }{k}}\,a_{k-1}}$ ${\displaystyle k\geq 1}$

This is a recursion equation for the coefficients and we get: ${\displaystyle a_{k}}$

${\displaystyle a_{k}={\frac {\lambda ^{k}}{k!}}\,a_{0}}$.

Inserted into the power series this means:

${\displaystyle y(x)=a_{0}\sum _{k=0}^{\infty }{\frac {\lambda ^{k}}{k!}}x^{k}=a_{0}\left(1+\lambda x+{\frac {1}{2}}\lambda ^{2}x^{2}+{\frac {1}{3!}}\lambda ^{3}x^{3}+\ldots \right)}$.

If we recognize the power series of the exponential function in it, the solution can be written even more compactly than:

${\displaystyle y(x)=y_{0}\,e^{\lambda \,x}}$.

Theoretical justification

For the theoretical justification of this procedure, one should already know in advance that there is a holomorphic solution , i.e. a solution that can be developed into a power series.

Of course, one can simply assume that, based on this assumption, as in the introductory example, construct a solution and then test it by inserting it. However, if one cannot resolve the recursion of the coefficients and one can only calculate a few coefficients, one has a polynomial as an approximation of a possible solution, but this only makes sense if the existence of a holomorphic solution is certain. The following sentence delivers this:

• Theorem : Be as well as given and holomorphic, where and . Then there is exactly one holomorphic solution of the initial value problem ${\displaystyle z_{0},y_{0}\in \mathbb {C} }$ ${\displaystyle a,b>0}$ ${\displaystyle F:U\rightarrow \mathbb {C} }$ ${\displaystyle U\supset U_{0}:=\{(z,y)\in \mathbb {C} ^{2};|z-z_{0}|\leq a,|y-y_{0}|\leq b\}}$ ${\displaystyle M:=\sup\{|F(z,y)|;(z,y)\in U_{0}\}}$ ${\displaystyle f}$

${\displaystyle f'(z)=F(z,f(z)),\quad f(z_{0})=y_{0}}$,

at least on the open circle . ${\displaystyle \textstyle \{z\in \mathbb {C} ;|z-z_{0}|<\min(a,{\frac {b}{M}})\}}$

In the above example is and . For is ${\displaystyle z_{0}=0}$ ${\displaystyle F(z,y)=\lambda y}$ ${\displaystyle a,b>0}$

${\displaystyle M=\sup\{|F(z,y)|;|z|\leq a,|y-y_{0}|\leq b\}=|\lambda |\sup\{|y|;|y-y_{0}|\leq b\}=|\lambda |(|y_{0}|+b)}$.

The radius of convergence of assured by the sentence can therefore be smaller than the actual radius of convergence of the solution, which is known to be infinite in the present example. The identity theorem for holomorphic functions then shows that the solution found also solves the initial value problem outside of the convergence radius, as long as one can still form the convergence circle in a coherent environment. ${\displaystyle \textstyle \min(a,{\frac {b}{M}})}$ ${\displaystyle F(z,f(z))}$

In particular, this theorem shows that the power series approach leads to success in the case of holomorphic right-hand side of the initial value problem.

Another example: Hermitian differential equation

We are looking for the solution to Hermit's differential equation

${\displaystyle {\frac {d^{2}y(x)}{dx^{2}}}-2\,x\,{\frac {dy(x)}{dx}}+2\,\lambda \,y(x)=0}$

The solution is applied as a power series:

${\displaystyle y(x)=\sum _{k=0}^{\infty }{\frac {a_{k}}{k!}}x^{k}}$

In order to make the further calculation easier, a factor was introduced in this approach compared to the last example . ${\displaystyle {\tfrac {1}{k!}}}$

Hence:

${\displaystyle 2\,\lambda \,y(x)=2\,\lambda a_{0}+\sum _{k=1}^{\infty }{\frac {2\,\lambda \,a_{k}}{k!}}x^{k}}$

${\displaystyle -2x\,{\frac {dy(x)}{dx}}=-2x\,\sum _{k=1}^{\infty }{\frac {a_{k}}{k!}}k\,x^{k-1}=\sum _{k=1}^{\infty }{\frac {-2a_{k}}{(k-1)!}}\,x^{k}}$

${\displaystyle {\frac {d^{2}y(x)}{dx^{2}}}=\sum _{k=2}^{\infty }{\frac {a_{k}}{k!}}k\,(k-1)\,x^{k-2}=\sum _{k=2}^{\infty }{\frac {a_{k}}{(k-2)!}}\,x^{k-2}=\sum _{k=0}^{\infty }{\frac {a_{k+2}}{k!}}\,x^{k}=a_{2}+\sum _{k=1}^{\infty }{\frac {a_{k+2}}{k!}}\,x^{k}}$

Inserted into the differential equation this means:

${\displaystyle a_{2}+\sum _{k=1}^{\infty }{\frac {a_{k+2}}{k!}}\,x^{k}\;+\;\sum _{k=1}^{\infty }{\frac {-2a_{k}}{(k-1)!}}\,x^{k}\;+\;2\,\lambda a_{0}+\sum _{k=1}^{\infty }{\frac {2\,\lambda \,a_{k}}{k!}}x^{k}=0}$

${\displaystyle a_{2}+2\,\lambda a_{0}+\sum _{k=1}^{\infty }\left({\frac {a_{k+2}}{k!}}+{\frac {-2a_{k}}{(k-1)!}}+{\frac {2\,\lambda \,a_{k}}{k!}}\right)x^{k}=0}$

The coefficient comparison results for the constant terms ( ): and for all other ( ): ${\displaystyle k=0}$ ${\displaystyle 2\,\lambda a_{0}+a_{2}=0}$ ${\displaystyle k>0}$

${\displaystyle {\frac {a_{k+2}}{k!}}+{\frac {-2a_{k}}{(k-1)!}}+{\frac {2\,\lambda \,a_{k}}{k!}}=0}$.

Multiplication with gives: ${\displaystyle k!}$

${\displaystyle a_{k+2}-2a_{k}\,k+2\,\lambda \,a_{k}=0}$, d. H.

${\displaystyle a_{k+2}=(2k-2\lambda )a_{k}}$.

If the coefficients and, for example, from the initial conditions are known, then all further coefficients can be calculated and, if necessary, combined as a series. The analytical solution of the differential equation is thus: ${\displaystyle a_{0}}$ ${\displaystyle a_{1}}$ ${\displaystyle a_{k}}$

${\displaystyle {\begin{aligned}y(x)&=a_{0}\left(1-{\frac {2\lambda }{2!}}x^{2}-{\frac {2\lambda (4-2\lambda )}{4!}}x^{4}-{\frac {2\lambda (4-2\lambda )(8-2\lambda )}{6!}}x^{6}-\ldots \right)\\&\qquad +a_{1}\left(x+{\frac {(2-2\lambda )}{3!}}x^{3}+{\frac {(2-2\lambda )(6-2\lambda )}{5!}}x^{5}+{\frac {(2-2\lambda )(6-2\lambda )(10-2\lambda )}{7!}}x^{7}+\ldots \right)\end{aligned}}}$.

literature

• Earl A. Coddington, Norman Levinson: Theory of Ordinary Differential Equations . McGraw – Hill, New York 1955.
• Einar Hille: Ordinary Differential Equations in the Complex Plane, Dover Publications, Mineola, New York, 1976
• Gerald Teschl Ordinary Differential Equations and Dynamical Systems, publisher = American Mathematical Society, Providence, Rhode Island, 2012, ISBN 978-0-8218-8328-0 , pdf

Web links

• Frobenius Method (MathWorld)

Individual evidence

1. ^ W. Walter: Ordinary Differential Equations , Springer-Verlag (1986), ISBN 3-540-16143-0 , Chapter I, §8, Sentence II
2. ↑ Harro Heuser: Ordinary differential equations: Introduction to teaching and use . Teubner, 3rd edition, 1995, ISBN 3-519-22227-2 , p. 262.