Applications-of-Partial-Differential-Equations - Math-4

1. Classification of Second-Order Linear PDEs (in two independent variables)

General form:
\[ a(x,y) u_{xx} + 2h(x,y) u_{xy} + b(x,y) u_{yy} + \text{lower order terms} = 0 \]

Discriminant: \( \Delta = h^2 - ab \)

Value of \( \Delta \)	Type	Nature	Example
\( \Delta < 0 \)	Elliptic	Boundary value problems	Laplace: \( u_{xx} + u_{yy} = 0 \)
\( \Delta = 0 \)	Parabolic	Initial-boundary value (diffusion)	Heat: \( u_t = \alpha u_{xx} \)
\( \Delta > 0 \)	Hyperbolic	Initial value (wave propagation)	Wave: \( u_{tt} = c^2 u_{xx} \)

Important standard forms (canonical/normal forms):

Hyperbolic: \( u_{\xi\xi} - u_{\eta\eta} + \cdots = 0 \) or \( u_{\xi\eta} + \cdots = 0 \)
Parabolic: \( u_{\eta\eta} + \cdots = 0 \)
Elliptic: \( u_{\xi\xi} + u_{\eta\eta} + \cdots = 0 \)

2. Method of Separation of Variables (Product Solution)

Assume solution \( u(x,t) = X(x) T(t) \) or \( u(x,y) = X(x) Y(y) \)

Steps:

Plug in → ordinary differential equations for \( X \) and \( Y \) (or \( T \)).
Separation constant = \( -\lambda \) (eigenvalue).
Solve eigenvalue problem → eigenvalues \( \lambda_n \), eigenfunctions \( X_n \).
General solution = linear combination: \( u = \sum c_n X_n(x) T_n(t) \) or \( Y_n(y) \).
Apply initial/boundary conditions → Fourier coefficients.

3. One-Dimensional Wave Equation

\[ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}, \quad 0 < x < l, \, t > 0 \]

Boundary conditions (fixed ends): \( u(0,t)=0, \, u(l,t)=0 \)
Initial conditions: \( u(x,0)=f(x), \, \frac{\partial u}{\partial t}(x,0)=g(x) \)

Solution by separation:
Assume \( u(x,t) = X(x) T(t) \Rightarrow \frac{T''}{c^2 T} = \frac{X''}{X} = -\lambda \)
→ \( X'' + \lambda X = 0 \), BCs → \( X(0)=X(l)=0 \)

Eigenvalues: \( \lambda_n = \left( \frac{n\pi}{l} \right)^2 \), Eigenfunctions: \( X_n = \sin \frac{n\pi x}{l} \), \( n=1,2,3,\dots \)
Time part: \( T_n'' + (c \frac{n\pi}{l})^2 T_n = 0 \Rightarrow T_n = A_n \cos \frac{n\pi c t}{l} + B_n \sin \frac{n\pi c t}{l} \)

Complete solution:

\[ u(x,t) = \sum_{n=1}^\infty \left( A_n \cos \frac{n\pi c t}{l} + B_n \sin \frac{n\pi c t}{l} \right) \sin \frac{n\pi x}{l} \]

Coefficients:

\[ A_n = \frac{2}{l} \int_0^l f(x) \sin \frac{n\pi x}{l} \, dx \] \[ B_n = \frac{2}{n\pi c} \int_0^l g(x) \sin \frac{n\pi x}{l} \, dx \]

4. One-Dimensional Heat Conduction Equation

\[ \frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}, \quad 0 < x < l, \, t > 0 \]

BC: \( u(0,t)=0, \, u(l,t)=0 \) (homogeneous)
IC: \( u(x,0)=f(x) \)

Separation: \( u=X(x)T(t) \Rightarrow \frac{T'}{\alpha T} = \frac{X''}{X} = -\lambda \)
Same spatial eigenvalues/functions as wave equation.
Time part: \( T' + \alpha \lambda_n T = 0 \Rightarrow T_n(t) = C_n e^{-\alpha (n\pi/l)^2 t} \)

Solution:

\[ u(x,t) = \sum_{n=1}^\infty C_n e^{-\alpha (n\pi/l)^2 t} \sin \frac{n\pi x}{l} \] \[ C_n = \frac{2}{l} \int_0^l f(x) \sin \frac{n\pi x}{l} \, dx \]

5. Two-Dimensional Wave Equation (Vibrating Membrane)

\[ \frac{\partial^2 u}{\partial t^2} = c^2 \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) \]

Rectangular membrane \( 0

Separation: \( u(x,y,t) = X(x) Y(y) T(t) \)
→ \( X''/X = -\lambda \), \( Y''/Y = -\mu \), \( \lambda + \mu = ( \omega/c )^2 \)

Eigenfunctions: \( \sin \frac{m\pi x}{a} \sin \frac{n\pi y}{b} \), \( m,n = 1,2,\dots \)
Frequencies: \( \omega_{mn} = \pi c \sqrt{ (m/a)^2 + (n/b)^2 } \)

Solution:

\[ u(x,y,t) = \sum_{m=1}^\infty \sum_{n=1}^\infty \left( A_{mn} \cos \omega_{mn} t + B_{mn} \sin \omega_{mn} t \right) \sin \frac{m\pi x}{a} \sin \frac{n\pi y}{b} \]

6. Two-Dimensional Heat Equation

\[ \frac{\partial u}{\partial t} = \alpha \nabla^2 u = \alpha \left( u_{xx} + u_{yy} \right) \]

Same separation → same eigenfunctions.
Solution decays exponentially with rate \( \alpha \pi^2 \left( (m/a)^2 + (n/b)^2 \right) \)

7. Two-Dimensional Laplace Equation (2D Potential Problems)

\[ \nabla^2 u = u_{xx} + u_{yy} = 0 \]

Rectangular region \( 0 BCs: e.g., \( u(0,y)=u(a,y)=u(x,0)=0 \), \( u(x,b)=f(x) \) (non-homogeneous on one side)

Assume \( u(x,y) = X(x) Y(y) \Rightarrow \frac{X''}{X} = -\frac{Y''}{Y} = -\lambda \)

\( X(0)=X(a)=0 \) → \( X_m = \sin \frac{m\pi x}{a} \), \( \lambda_m = (m\pi/a)^2 \)
Hyperbolic equation for Y: \( Y'' - (m\pi/a)^2 Y = 0 \Rightarrow Y = A \cosh(k y) + B \sinh(k y) \), \( k = m\pi/a \)

Apply BC at y=0 (usually zero) → use sinh form.

Standard solution:

\[ u(x,y) = \sum_{m=1}^\infty A_m \sinh \left( \frac{m\pi y}{a} \right) \sin \frac{m\pi x}{a} \] \[ A_m = \frac{2}{a \sinh(m\pi b/a)} \int_0^a f(x) \sin \frac{m\pi x}{a} \, dx \]

Polar coordinates (circular region):
Solution involves Bessel functions \( J_0(kr) \), \( Y_0(kr) \), etc.

8. Equations of Transmission Lines (Telegrapher’s Equations)

Voltage \( V(x,t) \) and current \( I(x,t) \) along a transmission line:

Primary constants:

\( R \) = resistance/unit length
\( L \) = inductance/unit length
\( G \) = conductance/unit length
\( C \) = capacitance/unit length

Telegrapher’s equations:

\[ -\frac{\partial V}{\partial x} = R I + L \frac{\partial I}{\partial t} \] \[ -\frac{\partial I}{\partial x} = G V + C \frac{\partial V}{\partial t} \]

Differentiate and combine → both V and I satisfy the same equation:

\[ \frac{\partial^2 V}{\partial x^2} = (RG + \omega^2 LC) V + (RC + LG) \frac{\partial V}{\partial t} + LC \frac{\partial^2 V}{\partial t^2} \quad \text{(general)} \]

Important cases:

Case	Equation for V (and I)	Type
Lossless line (R=G=0)	\( V_{xx} = LC V_{tt} \)	Hyperbolic
Distortionless	\( V_{xx} = LC V_{tt} + (RC + LG) V_t \)	Hyperbolic
Cable (L=0)	\( V_{xx} = RC V_t + RG V \)	Parabolic

Propagation constant: \( \gamma = \sqrt{(R + j\omega L)(G + j\omega C)} \)
Solution for steady-state (sinusoidal): \( V(x) = A e^{-\gamma x} + B e^{\gamma x} \)
Characteristic impedance: \( Z_0 = \sqrt{\frac{R + j\omega L}{G + j\omega C}} \)

This covers the entire Module II syllabus with all standard formulas, solution forms, and methods required for university examinations. Practice boundary value problems extensively using separation of variables.