Equations of Mathematical Physics and Separation of Variables Notes

Course Overview and Introduction to Mathematical Physics Equations

• Content Summary: The course covers four primary methods, three fundamental equations, and two special functions used in mathematical physics.     - Four Methods: Separation variable method, Traveling wave method, integral-transform method, and Green’s function method.     - Three Equations: Wave equation, Heat equation, and Laplace equation.     - Two Special Functions: Bessel Function and Legendre Function.
• Definition of Mathematical Equations: Using mathematical equations to describe physical phenomena.
• Chapter 2 Focus: Separation Variable Method, applied specifically to:     - A. The free vibration of a finite string.     - B. Heat conduction on a finite rod.     - C. Laplace’s Equation.     - D. Solution of non-homogeneous Partial Differential Equations (PDE).     - E. Treatment of non-homogeneous boundary conditions.

The Separation of Variables Method

• Main Concept: The core idea is to assume that the solution to a PDE, such as $u(\alpha, \beta, \gamma)$, can be decomposed into a product of three independent functions: $u(\alpha, \beta, \gamma) = f(\alpha)g(\beta)h(\gamma)$.
• Procedure:     - 1. Determine three ordinary differential equations (ODE) that each function ($f$, $g$, $h$) must satisfy.     - 2. Solve each ODE independently (these are usually familiar, elementary equations).     - 3. "Paste" the solutions together to reconstruct the full PDE solution.
• Formal Steps:     - Step 1: Find particular solutions (satisfying boundary or initial conditions) in variable separation form.     - Step 2: Use the superposition principle to create a linear combination of these solutions.     - Step 3: Use the remaining initial or boundary conditions to determine the superposition coefficients.
• Scope of Application: Wave equations, heat conduction problems, stable field (steady-state) problems, etc.
• Characteristics:     - Physical Guarantee: The superposition principle ensures physical validity.     - Mathematical Guarantee: The uniqueness of the solution ensures correctness.     - Simplification: Transforms complex PDEs into manageable ODEs.

Free Vibration of a Finite String

• Problem Definition: Vibration behavior of a string with two ends fixed.
• General PDE Formulation:     - Equation: $\frac{\partial^2 u}{\partial t^2} = a^2 \frac{\partial^2 u}{\partial x^2}$ where $0 < x < l$ and $t > 0$.     - Boundary Conditions (BCs): $u(0, t) = 0$ and $u(l, t) = 0$ (ends are fixed).     - Initial Conditions (ICs): $u(x, 0) = \phi(x)$ and $\frac{\partial u}{\partial t} |_{t=0} = \psi(x)$.
• Derivation via Separation of Variables:     - Let $u(x, t) = X(x)T(t)$.     - Substitute into the initial equation: $X(x)T''(t) = a^2 X''(x)T(t)$.     - Rearrange to separate variables: $\frac{X''(x)}{X(x)} = \frac{T''(t)}{a^2 T(t)} = -\lambda$.     - Resulting ODEs:         - $X''(x) + \lambda X(x) = 0$         - $T''(t) + \lambda a^2 T(t) = 0$     - Boundary conditions for $X(x)$: $X(0) = 0$ and $X(l) = 0$.

The Eigenvalue Problem

• Definition: The set of nonzero values $\lambda$ is called the Eigenvalue, and the corresponding non-zero functions $X(x)$ are the Eigenfunctions.
• Case Analysis for $\lambda$:     - Case 1: $\lambda < 0$:         - Let $\lambda = -\beta^2$. The general solution is $X(x) = Ae^{\beta x} + Be^{-\beta x}$.         - Applying $X(0) = 0$: $A + B = 0 \rightarrow B = -A$.         - Applying $X(l) = 0$: $A(e^{\beta l} - e^{-\beta l}) = 0$. Since $\beta l \neq 0$, then $A = 0$.         - Trivial solution: $X(x) = 0$.     - Case 2: $\lambda = 0$:         - The general solution is $X(x) = Ax + B$.         - Applying $X(0) = 0 \rightarrow B = 0$.         - Applying $X(l) = 0 \rightarrow Al = 0 \rightarrow A = 0$.         - Trivial solution: $X(x) = 0$.     - Case 3: $\lambda > 0$:         - Let $\lambda = \beta^2$. The general solution is X(x) = A \cos(\beta x) + B \sin(̢\beta x).         - Applying $X(0) = 0 \rightarrow A = 0$.         - Applying $X(l) = 0 \rightarrow B \sin(\beta l) = 0$. For non-zero solution, $\sin(\beta l) = 0$.         - Eigenvalues: $\beta = \frac{n\pi}{l}$, so $\lambda_n = ( \frac{n\pi}{l} )^2$ for $n = 1, 2, 3, \dots$.         - Eigenfunctions: $X_n(x) = B_n \sin( \frac{n\pi x}{l} )$.

General Solution and Coefficients

• Time-dependent Solution: For $\lambda_n$, the time equation is $T''_n(t) + ( \frac{n\pi a}{l} )^2 T_n(t) = 0$.     - Solution: $T_n(t) = C'_n \cos( \frac{n\pi a t}{l} ) + D'_n \sin( \frac{n\pi a t}{l} )$.
• Superposition: The general solution is the sum of particular solutions:     - $u(x, t) = \sum_{n=1}^{\infty} [ C_n \cos( \frac{n\pi a t}{l} ) + D_n \sin( \frac{n\pi a t}{l} ) ] \sin( \frac{n\pi x}{l} )$.
• Determining Coefficients ($C_n, D_n$):     - Use orthogonality of trigonometric functions: $\int_0^{l} \sin(\frac{n\pi x}{l}) \sin(\frac{m\pi x}{l}) \, dx = \begin{cases} 0 &amp; n \neq m \ \frac{l}{2} &amp; n = m \end{cases}$.     - From Initial Condition $u(x, 0) = \phi(x)$: $C_n = \frac{2}{l} \int_0^l \phi(x) \sin( \frac{n\pi x}{l} ) \, dx$.     - From Initial Condition $u_t(x, 0) = \psi(x)$: $D_n = \frac{2}{n\pi a} \int_0^l \psi(x) \sin( \frac{n\pi x}{l} ) \, dx$.

Characteristics of Vibration Solutions

• Standing Wave Method: A particular solution can be expressed as $u_n(x, t) = A_n \sin( \frac{n\pi x}{l} ) \cos( \omega_n t - \theta_n )$.     - Amplitude: $A_n = \sqrt{C_n^2 + D_n^2}$.     - Frequency: $\omega_n = \frac{n\pi a}{l}$.     - Initial Phase: $\theta_n = \arctan( \frac{D_n}{C_n} )$.
• Nodes: Points where the displacement is always zero ($\sin( \frac{n\pi x}{l} ) = 0$).
• Antinodes: Points where the displacement amplitude is maximal.
• Harmonics:     - $N=1$: Fundamental mode (1st harmonic).     - $N=2, 3, \dots$: Overtones (2nd, 3rd harmonics).
• Visual Concepts: The material includes diagrams of the 1st through 5th harmonics and mentions application to musical instruments.

Detailed Example 1: Finite String Vibration

• Parameters: Length $l = 10$, fixed ends. Initial speed $\psi(x) = 0$. Initial displacement $\phi(x) = \frac{x(10-x)}{1000}$. $a = 100$.
• Equation: $\frac{\partial^2 u}{\partial t^2} = 100^2 \frac{\partial^2 u}{\partial x^2}$.
• Separation Result:     - Eigenvalues: $\lambda_n = ( \frac{n\pi}{10} )^2$.     - Eigenfunctions: $X_n(x) = \sin( \frac{n\pi x}{10} )$.     - Time solution: $T_n(t) = C'_n \cos( 10n\pi t ) + D'_n \sin( 10n\pi t )$.
• Coefficient Solution:     - Since velocity is 0, $D_n = 0$.     - $C_n = \frac{2}{10} \int_0^{10} [ \frac{x(10-x)}{1000} ] \sin( \frac{n\pi x}{10} ) \, dx$.     - Calculation result: $C_n = \frac{1}{5000} \int_0^{10} (10x - x^2) \sin( \frac{n\pi x}{10} ) \, dx = \frac{4}{5(n\pi)^3} (1 - \cos(n\pi))$.     - If $n$ is even, $C_n = 0$. If $n$ is odd, $C_n = \frac{8}{5n^3\pi^3}$.
• Final Series Solution: $u(x, t) = \sum_{n=1, 3, 5 \dots} \frac{8}{5n^3\pi^3} \cos( 10n\pi t ) \sin( \frac{n\pi x}{10} )$.

Detailed Example 2: Modified Boundary Conditions

• Problem: Solve the wave equation with $u(0, t) = 0$ and $u_x(l, t) = 0$ (one end fixed, one end free).
• Eigenvalue Problem:     - $X''(x) + \lambda X(x) = 0$ with $X(0) = 0$ and $X'(l) = 0$.     - For \lambda > 0, $X(x) = A \cos(\beta x) + B \sin(\beta x)$.     - $X(0) = A = 0$.     - $X'(l) = B\beta \cos(\beta l) = 0 \rightarrow \cos(\beta l) = 0$.     - Eigenvalues: $\beta_n = \frac{(2n-1)\pi}{2l}$, so $\lambda_n = [ \frac{(2n-1)\pi}{2l} ]^2$.
• General Solution: $u(x, t) = \sum_{n=1}^{\infty} [ C_n \cos( \frac{(2n-1)\pi a}{2l} t ) + D_n \sin( \frac{(2n-1)\pi a}{2l} t ) ] \sin( \frac{(2n-1)\pi x}{2l} )$.

Detailed Example 4: Robin Boundary Conditions

• Problem: Lower end fixed ($u(0, t) = 0$), upper end satisfies a convective/spring-like condition ($u_x(l, t) + hu(l, t) = 0$).
• Separation ODEs:     - $X''(x) + \lambda X(x) = 0$     - BCs: $X(0) = 0$ and $X'(l) + hX(l) = 0$.
• Eigenvalue Calculation:     - $X(x) = B \sin(\beta x)$.     - Substitute into second BC: $B\beta \cos(\beta l) + hB \sin(\beta l) = 0$.     - This yields the transcendental equation: $\tan(\beta l) = -\frac{\beta}{h}$.
• Analysis: The eigenvalues $\lambda_n = \beta_n^2$ are found at the intersections of $y = \tan(\beta l)$ and the line $y = -\frac{\beta}{h}$.

Appendices of Supporting Mathematics

• Appendix-I: Fourier Sine/Cosine Series:     - Non-periodic function $f(x)$ on $[0, l]$ can be represented as:     - Cosine Series: $f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos( \frac{n\pi x}{l} )$, where $a_n = \frac{2}{l} \int_0^l f(x) \cos( \frac{n\pi x}{l} ) \, dx$.     - Sine Series: $f(x) = \sum_{n=1}^{\infty} b_n \sin( \frac{n\pi x}{l} )$, where $b_n = \frac{2}{l} \int_0^l f(x) \sin( \frac{n\pi x}{l} ) \, dx$.
• Appendix-II: ODE Calculations:     - General form: $y''(x) + py'(x) + qy(x) = 0$.     - Characteristic equation: $k^2 + pk + q = 0$.     - Roots: $k = \frac{-p \pm \sqrt{p^2 - 4q}}{2}$.     - If $p=0$ and $q=\lambda$:         - $\lambda < 0 \rightarrow y = Ae^{\sqrt{-\lambda}x} + Be^{-\sqrt{-\lambda}x}$.         - $\lambda = 0 \rightarrow y = Ax + B$.         - $\lambda > 0 \rightarrow y = A \cos(\sqrt{\lambda}x) + B \sin(\sqrt{\lambda}x)$.