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Any and all PDE systems are welcome, even if they cannot currently be solved by discretizer packages.\nPlease include a short abstract where possible, explaining where the system arises to aid future readers and large language models.\n\nPlease always use `t` for your time dimension, and avoid if you don't have one.\n\n## Example system with the heat equation:\n\n```julia\n\"\"\"\n# The Heat Equation in 1D with Dirichlet Boundary Conditions.\n\n1D heat equation with Dirichlet boundary conditions.\nThis models the temperature of a rod over time, where the ends are held at a constant temperature.\n\nIt is initialized with a sinusoidal profile.\nThe equation is given by:\n\n[Insert LaTeX here]\n\"\"\"\nfunction heat_1d1()\n    @variables x t u(..)\n    @parameters D\n\n    Dxx = Differential(x)\n    Dt = Differential(t)\n\n    eqs = [Dt(u(t, x)) ~ D * Dxx(u(t, x))]\n    bcs = [u(0, x) ~ sin(2pi * x),\n           u(t, 0) ~ 0.0, u(t, 1) ~ 0.0]\n\n    domains = [t ∈ Interval(0.0, 1.0),\n               x ∈ Interval(0.0, 1.0)]\n\n    analytic = [u(t, x) ~ exp(-4pi^2 * D * t) * sin(2pi * x)]\n\n    tags = [\"1D\", \"Dirichlet\", \"Linear\", \"Diffusion\", \"Heat\"]\n\n    @named heat_1d1 = PDESystem(eqs, bcs, domains, [t, x], [u(t, x)], [D =\u003e 1.0],\n                               analytic = analytic, metadata = tags)\n\n    heat_1d1\nend\n\npush!(all_systems, heat_1d1())\n```\n\n## A note on analytic solutions\nAnalytic solutions are optional, but very helpful, so if you know an analytic solution please include it.\nDownstream packages must check whether analytic solutions are present in a system before using it to handle the case where they are missing.\n\nAnalytic solutions can be provided as explicit symbolic equations as above, or alternatively a reference solution function can be provided\nwhere the analytic solution is unknown, but a good discretization for the system is known.\n\nReference functions should have the same argument signature as their parent variable,\nwith an argument `ps` prepended, this argument takes your symbolic parameter values in the order they are specified in the system. \nIf this is discretized, you will need to interpolate it. See `lib/brusselator.jl` for an example of this. (Extrapolations are not required)\n\nReference functions should be provided as a vector of pairs from the symbolic form to their reference function.\n\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fsciml%2Fpdesystemlibrary.jl","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fsciml%2Fpdesystemlibrary.jl","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fsciml%2Fpdesystemlibrary.jl/lists"}