Series Solutions: Power, Bessel, Legendre & Fourier

TAKEAWAYS:

• Power series solutions convert ODEs into algebraic equations via Frobenius method for regular/irregular singular points.
• Bessel’s equation (and its functions $J_n(x)$) models vibrations, heat conduction, and wave propagation in cylindrical systems.
• Legendre polynomials (Rodrigues’ formula) solve Laplace’s equation in spherical coordinates and appear in quantum mechanics.
• Fourier series decomposes periodic functions into sine/cosine sums, critical for signal processing (eSewa, Ncell) and heat transfer.
• Orthogonality of trigonometric/series functions enables efficient approximations (e.g., compressing audio in WhatsApp calls).
• Convergence tests (ratio, interval of convergence) ensure solutions are valid and physically meaningful.

1. Power Series Solutions to ODEs

1.1 Definition and Frobenius Method

A power series solution assumes a solution of the form: $$y(x)=\sum _{n=0}^{\infty }{a}_n{x}^{n+r}$$ where $r$ is determined by the indicial equation. This method works for linear ODEs with polynomial coefficients.

Key Steps:

1. Assume $y(x)=\sum a_n{x}^{n+r}$.
2. Substitute into the ODE to find a recurrence relation for $a_n$.
3. Solve for $a_n$ and determine $r$ from the indicial equation.
4. Classify the singular point (regular/irregular) to check validity.

1.2 Worked Example: Solve $y^{\prime \prime }-4x{y}^{\prime }+(4x^2-2)y=0$

Assumption: Regular singular point at $x=0$. Assume: $$y(x)=\sum _{n=0}^{\infty }{a}_n{x}^{n+r}$$ Step 1: Compute derivatives: $$y^{\prime }=\sum (n+r)a_n{x}^{n+r-1},y^{\prime \prime }=\sum (n+r)(n+r-1)a_n{x}^{n+r-2}$$ Step 2: Substitute into the ODE: $$\sum (n+r)(n+r-1)a_n{x}^{n+r-2}-4x\sum (n+r)a_n{x}^{n+r-1}+(4x^2-2)\sum a_n{x}^{n+r}=0$$ Step 3: Collect terms with $x^{n+r-2}$: $$\sum \left[(n+r)(n+r-1)a_n-4(n+r)a_{n-1}+(4x^2-2)a_{n-2}\right]x^{n+r-2}=0$$ Step 4: Set coefficients of like powers to zero:

• For $n=0$ (indicial equation): $$r(r-1)a_0-2a_0=0⟹r^2-r-2=0⟹r=2\text{ or }r=-1$$
• For $n=1$: $$(r+1)r{a}_1-4r{a}_0+4x^2{a}_{-1}-2a_0=0⟹a_1=0\text{ (if }r=2\text{)}$$
• For $n\ge 2$: $$a_n=\frac{4(n+r-2)a_{n-2}}{n(n+r-1)}$$ Solution: Two independent solutions: $$y_1(x)=1+2x^2+\frac{2}{3}{x}^4+\cdots ,y_2(x)=x^{-1}-2x+\frac{2}{3}{x}^3+\cdots$$ Visual: The recurrence relation forms a tree of coefficients (see

1.3 Regular vs. Irregular Singular Points

2. Bessel’s Equation and Functions

2.1 Bessel’s Equation

[ x^2 y'' + x y' + (x^2 - n^2) y = 0 ] Solutions: (J_n(x)) (Bessel function of first kind) and (Y_n(x)) (second kind). Applications:

• Vibrating membranes (drumheads).
• Heat conduction in cylindrical rods.
• Wave propagation in optical fibers.

2.2 Worked Example: Solve (x^2 y'' + x y' + (x^2 - 4) y = 0)

This is Bessel’s equation with (n = 2). The general solution is: [ y(x) = c_1 J_2(x) + c_2 Y_2(x) ] Graph of (J_2(x)):

2.3 Real-World: Ncell’s Antenna Design

Ncell uses Bessel functions to model antenna radiation patterns in its 4G/5G base stations. The shape of the antenna’s electromagnetic field follows (J_n(x)), ensuring optimal signal coverage in Kathmandu’s hilly terrain.

3. Legendre Polynomials

3.1 Rodrigues’ Formula

[ P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} \left( (x^2 - 1)^n \right) ] Orthogonality: [ \int_{-1}^1 P_m(x) P_n(x) , dx = \frac{2}{2n+1} \delta_{mn} ] Applications:

• Spherical harmonics in quantum mechanics.
• Solving Laplace’s equation in spherical coordinates.

3.2 Worked Example: Express (x^2 - 4x + 2) as a Legendre Polynomial

Using the first 3 Legendre polynomials: [ P_0(x) = 1, \quad P_1(x) = x, \quad P_2(x) = \frac{1}{2}(3x^2 - 1) ] We write: [ x^2 - 4x + 2 = a P_0 + b P_1 + c P_2 ] Solving for coefficients: [ a = 2, \quad b = -4, \quad c = \frac{8}{3} ] Thus: [ x^2 - 4x + 2 = 2 P_0 - 4 P_1 + \frac{8}{3} P_2 ]

3.3 Real-World: Google Maps’ Terrain Rendering

Google Maps uses Legendre polynomials to compress terrain data (elevation models). The orthogonality property reduces storage needs for global topography, enabling real-time 3D rendering.

4. Fourier Series

4.1 Definition

For a periodic function (f(x)) with period (2L): [ f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos \frac{n \pi x}{L} + b_n \sin \frac{n \pi x}{L} \right) ] where: [ a_n = \frac{1}{L} \int_{-L}^L f(x) \cos \frac{n \pi x}{L} , dx, \quad b_n = \frac{1}{L} \int_{-L}^L f(x) \sin \frac{n \pi x}{L} , dx ]

4.2 Worked Example: Fourier Series of (f(x) = |x|) on ([-2, 2])

Step 1: Compute (a_0): [ a_0 = \frac{1}{2} \int_{-2}^2 |x| , dx = \frac{1}{2} \left[ \int_{-2}^0 -x , dx + \int_0^2 x , dx \right] = 2 ] Step 2: Compute (a_n) (even function, (b_n = 0)): [ a_n = \frac{1}{2} \int_{-2}^2 |x| \cos \frac{n \pi x}{2} , dx = \frac{4}{n \pi} \sin \frac{2n \pi}{2} = \frac{4}{n \pi} (-1)^{n+1} ] Solution: [ f(x) = 1 + \sum_{n=1}^{\infty} \frac{4}{n \pi} (-1)^{n+1} \cos \frac{n \pi x}{2} ] Graph of Partial Sums:

4.3 Real-World: eSewa’s Payment Signal Compression

eSewa compresses transaction data (e.g., payment waveforms) using Fourier series. The periodic nature of payment spikes (e.g., during Dashain) is decomposed into sine/cosine terms, reducing bandwidth for secure transmission.

5. Half-Range Fourier Series

5.1 Cosine and Sine Series

For even extensions (cosine series): [ f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos \frac{n \pi x}{L}, \quad a_n = \frac{2}{L} \int_0^L f(x) \cos \frac{n \pi x}{L} , dx ] For odd extensions (sine series): [ f(x) = \sum_{n=1}^{\infty} b_n \sin \frac{n \pi x}{L}, \quad b_n = \frac{2}{L} \int_0^L f(x) \sin \frac{n \pi x}{L} , dx ]

5.2 Worked Example: Expand (f(x) = x^2) on ([0, 2]) in Cosine Series

Compute (a_0): [ a_0 = \frac{2}{2} \int_0^2 x^2 , dx = \frac{8}{3} ] Compute (a_n): [ a_n = \frac{2}{2} \int_0^2 x^2 \cos \frac{n \pi x}{2} , dx = 4 \left( \frac{(-1)^n}{n^2 \pi^2} + \frac{2}{n \pi} \sin \frac{n \pi}{2} \right) ] Simplify: [ a_n = \frac{8}{n^2 \pi^2} (-1)^n - \frac{4}{n \pi} \quad \text{(for odd } n\text{)} ] Solution: [ f(x) = \frac{4}{3} + \sum_{n=1}^{\infty} \left( \frac{8}{n^2 \pi^2} (-1)^n - \frac{4}{n \pi} \right) \cos \frac{n \pi x}{2} ]

5.3 Real-World: Daraz’s Order Queue Optimization

Daraz uses Fourier sine series to model order arrival rates during sales (e.g., Dashain). The periodic spikes in orders are approximated to optimize warehouse logistics and delivery routing.

6. Convergence of Series Solutions

6.1 Ratio Test

For a power series (\sum a_n x^n), the radius of convergence (R) is: [ R = \lim_{n \to \infty} \left| \frac{a_n}{a_{n+1}} \right| ] Interval of convergence: (|x| < R).

6.2 Worked Example: Find Convergence for (y = \sum \frac{x^n}{n!})

Apply the ratio test: [ \lim_{n \to \infty} \left| \frac{a_{n+1} x^{n+1}}{a_n x^n} \right| = \lim_{n \to \infty} \left| \frac{x}{n+1} \right| = 0 < 1 \quad \forall x ] Conclusion: Converges for all (x) ((R = \infty)).

In the Real World

1. Ncell’s 5G Base Stations

   • Idea: Bessel functions model antenna radiation patterns.
   • How: The shape of the electromagnetic field emitted by a circular antenna follows (J_n(x)), optimizing coverage in Kathmandu’s uneven terrain.

2. eSewa’s Payment Data Compression

   • Idea: Fourier series decomposes periodic transaction spikes.
   • How: During Dashain, payment volumes spike periodically. Fourier series compresses this data, reducing bandwidth for secure transmission.

3. Google Maps’ Terrain Rendering

   • Idea: Legendre polynomials approximate global elevation data.
   • How: Orthogonality reduces storage needs, enabling real-time 3D terrain rendering worldwide.

4. Pathao’s Traffic Flow Modeling

   • Idea: Fourier series models periodic traffic congestion.
   • How: Rush-hour patterns (e.g., 8 AM and 6 PM) are decomposed into sine/cosine terms to optimize ride-sharing algorithms.

Exam Tip

1. Power Series:

   • Always check for regular/irregular singular points.
   • Write the indicial equation first; solve for (r).
   • For repeated roots, use logarithmic terms in the second solution.

2. Bessel/Legendre:

   • Memorize Rodrigues’ formula and orthogonality conditions.
   • Recognize standard forms (e.g., (x^2 y'' + \cdots)) to identify Bessel/Legendre.

3. Fourier Series:

   • For even/odd functions, simplify integrals using symmetry.
   • Half-range series require extending the function (even/odd).
   • Parseval’s theorem is useful for energy calculations in signals.

4. Convergence:

   • Use the ratio test for power series.
   • Check endpoints separately if the interval is finite.

Common Mistakes to Avoid:

• Forgetting to multiply by (x^r) in the Frobenius method.
• Misapplying boundary conditions in Fourier series.
• Ignoring convergence tests for series solutions.

Visual Summary of Key Ideas:

mindmap
  root((Series Solutions))
    Power Series
      Frobenius Method
      Indicial Equation
      Recurrence Relation
    Bessel Functions
      Jₙ(x), Yₙ(x)
      Applications: Antennas, Vibrations
    Legendre Polynomials
      Rodrigues' Formula
      Orthogonality
      Spherical Harmonics
    Fourier Series
      Full-Range
      Half-Range (Cosine/Sine)
      Parseval's Theorem
    Convergence
      Ratio Test
      Interval of Convergence