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# Partial Differential Equations ## What are PDEs? * Equation involving an unknown function of $\geq 2$ variables and its derivatives * General form: $F(x,y,u,u_x,u_y,u_{xx},u_{yy},u_{xy}) = 0$ * u is the dependent variable (unknown) * x,y are independent variables ### Examples *...

# Partial Differential Equations ## What are PDEs? * Equation involving an unknown function of $\geq 2$ variables and its derivatives * General form: $F(x,y,u,u_x,u_y,u_{xx},u_{yy},u_{xy}) = 0$ * u is the dependent variable (unknown) * x,y are independent variables ### Examples * $u_x + u_y = 0$ * $xu_x + yu_y = u$ * $u_t + uu_x = 0$ * $u_{tt} - c^2u_{xx} = 0$ * $u_t - ku_{xx} = 0$ * $u_{xx} + u_{yy} = 0$ * $u_{xx} + u_{yy} = f(x,y)$ ### Notation $u_x = \frac{\partial u}{\partial x}$, $u_{xx} = \frac{\partial^2 u}{\partial x^2}$, $u_{xy} = \frac{\partial^2 u}{\partial x \partial y}$ ### Order Highest order derivative in the equation. ### Examples * $u_x + u_y = 0$ (1st order) * $u_{tt} - c^2u_{xx} = 0$ (2nd order) * $u_{xxt} + u_x = y^2$ (3rd order) ### Linear PDE A PDE is linear if $F$ is a linear function of $u$ and its derivatives. ### Examples #### Linear * $u_x + u_y = 0$ * $xu_x + yu_y = u$ * $u_{tt} - c^2u_{xx} = 0$ * $u_t - ku_{xx} = 0$ * $u_{xx} + u_{yy} = 0$ * $u_{xx} + u_{yy} = f(x,y)$ #### Non-linear * $u_t + uu_x = 0$ * $u_x^2 + u_y^2 = 1$ ### Superposition Principle If $u_1$ and $u_2$ are solutions of a homogeneous linear PDE, then $c_1u_1 + c_2u_2$ is also a solution for any constants $c_1$ and $c_2$. ### Initial and Boundary Conditions To obtain a unique solution to a PDE, we need to specify initial and boundary conditions. * Initial conditions: specify the value of the unknown function and its derivatives at an initial time. * Boundary conditions: specify the value of the unknown function and its derivatives on the boundary of the domain. ### Examples * $u_{tt} = c^2u_{xx}$, $0 < x < L$, $t > 0$ $u(x,0) = f(x)$, $u_t(x,0) = g(x)$, $0 \leq x \leq L$ $u(0,t) = 0$, $u(L,t) = 0$, $t > 0$ * $u_t = ku_{xx}$, $0 < x < L$, $t > 0$ $u(x,0) = f(x)$, $0 \leq x \leq L$ $u(0,t) = T_1$, $u(L,t) = T_2$, $t > 0$ * $u_{xx} + u_{yy} = 0$, $0 < x < a$, $0 < y < b$ $u(x,0) = f(x)$, $u(x,b) = g(x)$, $0 \leq x \leq a$ $u(0,y) = h(y)$, $u(a,y) = l(y)$, $0 \leq y \leq b$ ### Well-Posed Problem A PDE problem is well-posed if: 1. A solution exists. 2. The solution is unique. 3. The solution depends continuously on the initial and boundary data. ### 2nd Order Linear PDEs $Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + Fu = G$ where $A, B, C, D, E, F, G$ are functions of $x$ and $y$. #### Classifications * $B^2 - AC < 0$ : Elliptic (e.g., Laplace equation, Poisson equation) * $B^2 - AC = 0$ : Parabolic (e.g., Heat equation) * $B^2 - AC > 0$ : Hyperbolic (e.g., Wave equation) **Note:** The classification can vary from point to point if $A, B, C$ are functions of $x$ and $y$. ### Examples * $u_{xx} + u_{yy} = 0$ : Elliptic, $A = 1$, $B = 0$, $C = 1$, $B^2 - AC = -1 < 0$ * $u_t = ku_{xx}$ : Parabolic, $-ku_{xx} + u_t = 0$, $A = -k$, $B = 0$, $C = 0$, $B^2 - AC = 0$ * $u_{tt} = c^2u_{xx}$ : Hyperbolic, $u_{tt} - c^2u_{xx} = 0$, $A = -c^2$, $B = 0$, $C = 1$, $B^2 - AC = c^2 > 0$