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  "body": "\n## Definition\n\nLet $d \\ge 2$, let $\\Omega \\subseteq \\Z^d$, and let $a_1, \\dots, a_r \\in \\Z^d$ be fixed shift vectors.\nLet $F \\colon \\Omega \\times \\R^{r} \\to \\R$ be a given [[function|0EPAT51DXQ]].\nA *partial difference equation* for an unknown function $u \\colon \\Z^d \\to \\R$ is a condition of the form\n$$F\\big(m, u(m + a_1), u(m + a_2), \\dots, u(m + a_r)\\big) = 0 \\quad \\text{for all } m \\in \\Omega.$$\nA *solution* is a function $u$ that satisfies it for every $m \\in \\Omega$.\n\n## Remarks\n\nExample: the explicit scheme for the heat equation,\n$$u(j, n+1) = u(j, n) + \\lambda \\big( u(j+1, n) - 2u(j, n) + u(j-1, n) \\big), \\qquad (j, n) \\in \\Z \\times \\N,$$\nwith a constant $\\lambda \u003e 0$, is a partial difference equation with $d = 2$ and shifts $(0, 1), (0, 0), (1, 0), (-1, 0)$.\n\nNon-example: $u(n+2) = u(n+1) + u(n)$ for $u \\colon \\Z \\to \\R$ involves only one integer variable, so $d = 1$ and it is an ordinary difference equation.\n\nOne may also take $u$ defined only on a subset of $\\Z^d$ that contains all the points $m + a_i$, as in the heat scheme on $\\Z \\times \\N$.\nFor example, the binomial coefficients $c(n, k) = \\binom{n}{k}$ on $\\N^2$ satisfy $c(n+1, k+1) = c(n, k) + c(n, k+1)$.\nThe values may also lie in another field or ring instead of $\\R$.\n",
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