Heart Lab 1

2nd Order Linear PDEs

• The general form is given by $Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + Fu = G$, where A, B, C, D, E, F, and G are functions of (x, y).

Classification of PDEs

• The discriminant, $\Delta = B^2 - AC$, determines the classification.
• If $\Delta > 0$, the PDE is hyperbolic, similar to the wave equation.
• If $\Delta = 0$, the PDE is parabolic, similar to the heat equation.
• If $\Delta < 0$, the PDE is elliptic, similar to the Laplace equation.

Canonical Forms for Hyperbolic PDEs ($\Delta > 0$)

• Use variables $\eta = y - r_2x$ and $\xi = y - r_1x$.
• $r_{1,2}$ are calculated as $r_{1,2} = \frac{-B \pm \sqrt{B^2 - AC}}{A}$.
• The transformed PDE is $u_{\xi\eta} = F_1(\xi, \eta, u, u_\xi, u_\eta)$.

Canonical Forms for Parabolic PDEs ($\Delta = 0$)

• Use variables $\xi = x$ and $\eta = Ay + Bx$.
• The transformed PDE is $u_{\eta\eta} = F_2(\xi, \eta, u, u_\xi, u_\eta)$.

Canonical Forms for Elliptic PDEs ($\Delta < 0$)

• Use variables $\xi = \frac{1}{2}(y - r_1x + y - r_2x) = y + \frac{B}{A}x$ and $\eta = \frac{1}{2i}(y - r_1x - (y - r_2x)) = \frac{\sqrt{-\Delta}}{A}x$.
• The transformed PDE is $u_{\xi\xi} + u_{\eta\eta} = F_3(\xi, \eta, u, u_\xi, u_\eta)$.

1st Law of Thermodynamics

• The change in internal energy equals heat added minus work done: $\Delta U = Q - W$.
• This expresses conservation of energy.
• Internal energy ($U$) is a state function; heat ($Q$) and work ($W$) are path-dependent.
• For an ideal gas, internal energy is $U = \frac{3}{2}nRT$.
• Work done by a gas is $W = \int{PdV}$, and at constant pressure, $W = P\Delta V$.
• Heat capacity is given by $Q = mc\Delta T$, where $c$ is specific heat capacity.

Heat Capacity for Ideal Gases

• Gases have $c_p$ at constant pressure and $c_v$ at constant volume.
• For monatomic ideal gases:
  • $c_v = \frac{3}{2}R$
  • $c_p = \frac{5}{2}R$
• For diatomic ideal gases:
  • $c_v = \frac{5}{2}R$
  • $c_p = \frac{7}{2}R$

Thermodynamic Processes

• Isobaric: constant pressure, $W = P\Delta V$, $\Delta U = Q - W$
• Isochoric (isovolumetric): constant volume, $W = 0$, $\Delta U = Q$
• Isothermal: constant temperature, $\Delta U = 0$, $Q = W$, $W = nRT\ln(\frac{V_2}{V_1})$
• Adiabatic: no heat exchange, $Q = 0$, $\Delta U = -W$, $PV^\gamma = \text{constant}$, $TV^{\gamma-1} = \text{constant}$, where $\gamma = \frac{c_p}{c_v}$

Heat Engines

• Heat engines convert heat into work.
• Work done is $W = Q_H - Q_C$.
• Efficiency is $e = \frac{W}{Q_H} = 1 - \frac{Q_C}{Q_H}$.
• Carnot engine: a theoretical engine providing maximum possible efficiency.
• Carnot efficiency: $e_c = 1 - \frac{T_C}{T_H}$.

Refrigerators

• Refrigerators transfer heat from a cold reservoir to a hot reservoir.
• Coefficient of performance is $COP = \frac{Q_C}{W} = \frac{Q_C}{Q_H - Q_C}$.
• Carnot refrigerator: a theoretical refrigerator providing maximum possible COP.
• Carnot COP: $COP_c = \frac{T_C}{T_H - T_C}$.

2nd Law of Thermodynamics

• The entropy of an isolated system always increases or remains constant.
• Entropy is a measure of disorder.
• The change in entropy is $\Delta S = \frac{Q}{T}$.
• For a reversible process, $\Delta S = 0$.
• For an irreversible process, $\Delta S > 0$.
• The 2nd Law implies heat engines can not be 100% efficient.

Fourier Transform: Linearity

• $a f(t) + b g(t) \leftrightarrow aF(f) + bG(f)$

Fourier Transform: Time Shifting

• $f(t - t_0) \leftrightarrow e^{-j2\pi f t_0}F(f)$

Fourier Transform: Frequency Shifting

• $e^{j2\pi f_0 t} f(t) \leftrightarrow F(f - f_0)$

Fourier Transform: Scaling

• $f(at) \leftrightarrow \frac{1}{|a|} F(\frac{f}{a})$

Fourier Transform: Conjugation

• $f^(t) \leftrightarrow F^(-f)$

Fourier Transform: Duality

• $F(t) \leftrightarrow f(-f)$

Fourier Transform: Differentiation

• $\frac{d}{dt} f(t) \leftrightarrow j2\pi f F(f)$

Fourier Transform: Integration

• $\int_{-\infty}^{t} f(\tau) d\tau \leftrightarrow \frac{1}{j2\pi f} F(f) + \frac{1}{2}F(0)\delta(f)$

Fourier Transform: Multiplication

• $f(t)g(t) \leftrightarrow \int_{-\infty}^{\infty} F(\theta)G(f-\theta)d\theta$

Fourier Transform: Convolution

• $f(t) * g(t) = \int_{-\infty}^{\infty} f(\tau)g(t-\tau)d\tau \leftrightarrow F(f)G(f)$