How do you find a Power Series solution of a partial differential equation?

Question

Christopher Allers · Accepted Answer

Let us solve #y''+y=0# by Power Series Method. Let #y=sum_{n=0}^inftyc_nx^n#, where #c_n# is to be determined. By taking derivatives, #y'=sum_{n=1}^inftync_nx^{n-1} Rightarrow y''=sum_{n=2}^inftyn(n-1)c_nx^{n-2}# We can rewrite #y''+y=0# as #sum_{n=2}^inftyn(n-1)c_nx^{n-2}+sum_{n=0}^inftyc_nx^n=0# by shifting the indices of the first summation by 2, #Rightarrow sum_{n=0}^infty(n+2)(n+1)c_{n+2}x^n+sum_{n=0}^inftyc_nx^n=0# by combining the summations, #Rightarrow sum_{n=0}^infty[(n+2)(n+1)c_{n+2}+c_n]x^n=0#, #Rightarrow (n+2)(n+1)c_{n+2}+c_n=0# #Rightarrow c_{n+2}=-{c_n}/{(n+2)(n+1)}# Let us look at even coefficients. #c_2={-c_0}/{2cdot1}=-c_0/{2!}#
#c_4={-c_2}/{4cdot3}={-1}/{4cdot3}cdot{-c_0}/{2!}=c_0/{4!}#
#c_6={-c_4}/{6cdot5}={-1}/{6cdot5}cdot c_0/{4!}=-{c_0}/{6!}#
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#c_{2n}=(-1)^n{c_0}/{(2n)!}# Let us look at odd coefficients. #c_3={-c_1}/{3cdot2}=-{c_1}/{3!}#
#c_5={-c_3}/{5cdot4}={-1}/{5cdot4}cdot{-c_1}/{3!}={c_1}/{5!}#
#c_7={-c_5}/{7cdot6}={-1}/{7cdot6}cdot{c_1}/{5!}=-{c_1}/{7!}#
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#c_{2n+1}=(-1)^n{c_1}/{(2n+1)!}# Hence, the solution can be written as: #y=sum_{n=0}^inftyc_nx^n# by splitting into even terms and odd terms, #=sum_{n=0}^inftyc_{2n}x^{2n}+sum_{n=0}^inftyc_{2n+1}x^{2n+1}# by pluggin in the formulas for #c_{2n}# and #c_{2n+1}# we found above, #=c_0sum_{n=0}^infty(-1)^n{x^{2n}}/{(2n)!}+c_1 sum_{n=0}^infty(-1)^n{x^{2n+1}}/{(2n+1)!}# By recognizing the power series, #y=c_0 cosx+c_1 sinx# I hope that this was helpful.

Christopher Allers · Answer

undefined To find a power series solution of a partial differential equation (PDE), follow these steps:

1. Express the solution as a power series: Assume that the solution $ u(x, y) $ can be written as a power series in $ x $ and $ y $:

$$ u(x, y) = \sum_{n=0}^\infty \sum_{m=0}^\infty a_{nm}x^n y^m $$

2. Substitute the power series into the PDE: Replace $ u(x, y) $ and its derivatives in the PDE with their corresponding power series expansions.

3. Equate coefficients of like powers of $ x $ and $ y $: Match coefficients of like powers of $ x $ and $ y $ on both sides of the PDE.

4. Solve the resulting system of algebraic equations: Solve the system of equations obtained by equating coefficients.

5. Determine the convergence of the solution: Analyze the convergence of the power series solution, considering the radius of convergence.

6. Verify the solution: Substitute the power series solution back into the original PDE to ensure it satisfies the equation.

7. Apply boundary or initial conditions: If boundary or initial conditions are given, apply them to determine the values of the coefficients in the power series solution.

8. Check for uniqueness: Ensure that the power series solution is unique within the given boundary or initial conditions.

By following these steps, you can find a power series solution of a partial differential equation.