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Heat equation: $\frac{\partial u}{\partial t} - \alpha \frac{\partial^2 u}{\partial x^2} = 0$ , $u(0, t) = u(L, t) = 0$ , $u(x, 0) = f(x)$

Discretize time and space using a grid. Apply explicit or implicit methods to approximate $u(x,t)$ . Stability and convergence are based on the chosen time and space step sizes.

Poisson's equation: $-\nabla^2 u = f(x,y)$ , with Dirichlet boundary conditions

Create a grid over the domain, and use a finite difference approximation for the Laplacian. Solve the resulting linear system for $u(x,y)$ .

Burgers' equation: $\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = \nu\frac{\partial^2 u}{\partial x^2}$ , $u(0,t)=u(L,t)=0$ , $u(x, 0) = f(x)$

A combined hyperbolic and parabolic PDE. Use finite differences for time and space, applying explicit/implicit methods and a nonlinear solver for the convective term.

Diffusion equation with variable coefficients: $\frac{\partial u}{\partial t} = \nabla\cdot(D(x,y)\nabla u)$ , with mixed boundary conditions

Use finite differences for spatial discretization. Handle variable $D(x,y)$ by averaging at cell edges and apply an appropriate time-stepping method.

Wave equation: $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$ , $u(0, t) = u(L, t) = 0$ , $u(x, 0) = g(x)$ , $\frac{\partial u}{\partial t}(x, 0) = h(x)$

Use a central difference approximation for both time and space. Solve for the wave height at each time step, ensuring stability with proper step sizes.

Laplace's equation: $\nabla^2 u = 0$ , with Neumann boundary conditions

Construct a grid and replace the Laplacian with a finite difference counterpart. The Neumann conditions imply derivatives on boundaries, adjust the finite differences accordingly.

Convection-diffusion equation: $\frac{\partial u}{\partial t} + \vec{v} \cdot \nabla u = \kappa \nabla^2 u$ , with Dirichlet and Neumann boundary conditions

Discretize the domain and apply finite difference to both convection and diffusion terms. Use a mixed approach for boundary conditions, applying Dirichlet and Neumann as specified.

Nonlinear Schrödinger equation: $i\frac{\partial \Psi}{\partial t} = -\frac{1}{2}\nabla^2 \Psi + V(x)\Psi + g|\Psi|^2\Psi$ , with absorbing boundary conditions

Grid the space-time domain, discretize the Laplacian and nonlinear terms, and apply an appropriate method to deal with absorption at the boundaries.

Advection equation: $\frac{\partial u}{\partial t} + c \frac{\partial u}{\partial x} = 0$ , with periodic boundary conditions

Use a temporal and spatial discretization. For periodic boundaries, wrap the ends of the domain. Upwind schemes can help with stability.

Telegraph equation: $\frac{\partial^2 u}{\partial t^2} + a\frac{\partial u}{\partial t} = c^2 \frac{\partial^2 u}{\partial x^2}$ , with fixed endpoints

Discretize time and space. Use a time-stepping scheme that handles the second derivative in time, and apply boundary conditions to the spatial discretization.

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