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# Partial Differential Equations ## Linear Second-Order PDEs ### General Form $A(x, y) \frac{\partial^2 u}{\partial x^2} + B(x, y) \frac{\partial^2 u}{\partial x \partial y} + C(x, y) \frac{\partial^2 u}{\partial y^2} + D(x, y) \frac{\partial u}{\partial x} + E(x, y) \frac{\partial u}{\partial y}...

# Partial Differential Equations ## Linear Second-Order PDEs ### General Form $A(x, y) \frac{\partial^2 u}{\partial x^2} + B(x, y) \frac{\partial^2 u}{\partial x \partial y} + C(x, y) \frac{\partial^2 u}{\partial y^2} + D(x, y) \frac{\partial u}{\partial x} + E(x, y) \frac{\partial u}{\partial y} + F(x, y) u = G(x, y)$ Where A, B, C, D, E, F, and G are functions of x and y. ### Classification The discriminant $\Delta = B^2 - 4AC$ determines the type of PDE: * If $\Delta > 0$: Hyperbolic PDE (e.g., Wave Equation) * If $\Delta = 0$: Parabolic PDE (e.g., Heat Equation) * If $\Delta < 0$: Elliptic PDE (e.g., Laplace Equation) ### Examples 1. **Wave Equation:** $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$ (Hyperbolic) 2. **Heat Equation:** $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$ (Parabolic) 3. **Laplace Equation:** $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ (Elliptic) ## Common PDEs in Physics | Equation | Type | Context | | :---------------- | :--------- | :----------------------------------------- | | Wave Equation | Hyperbolic | Wave propagation in various media | | Heat Equation | Parabolic | Heat transfer and diffusion processes | | Laplace Equation | Elliptic | Steady-state temperature, electrostatics | | Poisson Equation | Elliptic | Electrostatics with charge distribution | | Schrödinger Eq. | Parabolic | Quantum mechanics | | Burgers' Equation | Parabolic/Hyperbolic | Fluid dynamics, traffic flow | ## Boundary Conditions * **Dirichlet:** $u(x, y) = f(x, y)$ on the boundary * **Neumann:** $\frac{\partial u}{\partial n} = g(x, y)$ on the boundary (n is the normal direction) * **Robin:** $a u + b \frac{\partial u}{\partial n} = h(x, y)$ on the boundary ## Solution Techniques 1. **Separation of Variables:** * Assume a solution of the form $u(x, t) = X(x)T(t)$ * Substitute into PDE and separate variables * Solve resulting ODEs * Apply boundary/initial conditions to determine constants 2. **Fourier Series/Transforms:** * Represent functions as a sum of sine and cosine functions * Useful for solving PDEs on bounded domains 3. **Green's Functions:** * Represent solution as an integral involving Green's function * Useful for solving inhomogeneous PDEs 4. **Numerical Methods:** * Finite Difference Method (FDM) * Finite Element Method (FEM) ## Example: Solving the Heat Equation Consider the 1D heat equation: $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$ With boundary conditions: $u(0, t) = 0, \quad u(L, t) = 0$ And initial condition: $u(x, 0) = f(x)$ ### Separation of Variables Let $u(x, t) = X(x)T(t)$. Substituting into the heat equation, we get: $X(x)T'(t) = \alpha X''(x)T(t)$ Divide by $X(x)T(t)$: $\frac{T'(t)}{\alpha T(t)} = \frac{X''(x)}{X(x)} = -\lambda$ Now we have two ODEs: $T'(t) + \alpha \lambda T(t) = 0$ $X''(x) + \lambda X(x) = 0$ The solutions for X(x) are: $X(x) = A \cos(\sqrt{\lambda}x) + B \sin(\sqrt{\lambda}x)$ Applying the boundary conditions: $X(0) = 0 \Rightarrow A = 0$ $X(L) = 0 \Rightarrow \sqrt{\lambda}L = n\pi, \quad n = 1, 2, 3,...$ So, $\lambda_n = \left(\frac{n\pi}{L}\right)^2$ and $X_n(x) = B_n \sin\left(\frac{n\pi x}{L}\right)$ The solutions for $T(t)$ are: $T_n(t) = C_n e^{-\alpha \left(\frac{n\pi}{L}\right)^2 t}$ Thus, the general solution is: $u(x, t) = \sum_{n=1}^{\infty} D_n \sin\left(\frac{n\pi x}{L}\right) e^{-\alpha \left(\frac{n\pi}{L}\right)^2 t}$ Where $D_n$ are determined by the initial condition $u(x, 0) = f(x)$: $D_n = \frac{2}{L} \int_0^L f(x) \sin\left(\frac{n\pi x}{L}\right) dx$