{ "cells": [ { "cell_type": "markdown", "id": "b1f8295b", "metadata": {}, "source": [ "# Solving ODEs Symbolically\n", "\n", "## Overview, Objectives, and Key Terms\n", "\n", "L26 gave us a family of balance laws. Some balances stayed algebraic. Some reduced to ordinary differential equations. Some stayed as partial differential equations until we made one more assumption.\n", "\n", "Today the organizing idea is simple:\n", "\n", "> reduce the balance to algebra when you can, then solve and check the result.\n", "\n", "SymPy is useful here because it lets us keep parameters visible. A numerical solver can tell us one answer for one set of numbers. A symbolic expression can show the time scale, the equilibrium value, the boundary influence, or the mode that controls the physics.\n", "\n", "By the end of this lesson, you should be able to:\n", "\n", "- define SymPy symbols and functions without confusing them,\n", "- solve simple linear IVPs and BVPs with `dsolve`,\n", "- apply initial and boundary data to symbolic constants,\n", "- check a symbolic solution by substitution,\n", "- use an ansatz to turn a differential equation into algebra,\n", "- recognize separation of variables as a symbolic reduction of a PDE.\n", "\n", "**Key terms.** symbolic function, initial value problem, boundary value problem, ansatz, characteristic equation, separation of variables, eigenvalue, mode, residual." ] }, { "cell_type": "code", "execution_count": null, "id": "436e8e1e", "metadata": { "execution": { "iopub.execute_input": "2026-07-07T21:25:18.417737Z", "iopub.status.busy": "2026-07-07T21:25:18.417641Z", "iopub.status.idle": "2026-07-07T21:25:19.309787Z", "shell.execute_reply": "2026-07-07T21:25:19.309124Z" } }, "outputs": [], "source": [ "import numpy as np\n", "import sympy as sy\n", "import matplotlib.pyplot as plt\n", "\n", "sy.init_printing()\n", "\n", "COLORS = {\n", " \"ink\": \"#1f2937\",\n", " \"muted\": \"#64748b\",\n", " \"grid\": \"#cbd5e1\",\n", " \"panel\": \"#f8fafc\",\n", " \"purple\": \"#512888\",\n", " \"orange\": \"#D55E00\",\n", " \"blue\": \"#2563eb\",\n", " \"green\": \"#059669\",\n", " \"red\": \"#dc2626\",\n", "}\n", "\n", "def style_axes(ax):\n", " ax.spines[\"top\"].set_visible(False)\n", " ax.spines[\"right\"].set_visible(False)\n", " ax.spines[\"left\"].set_color(COLORS[\"muted\"])\n", " ax.spines[\"bottom\"].set_color(COLORS[\"muted\"])\n", " ax.tick_params(colors=COLORS[\"muted\"])\n", " ax.xaxis.label.set_color(COLORS[\"ink\"])\n", " ax.yaxis.label.set_color(COLORS[\"ink\"])\n", " ax.title.set_color(COLORS[\"ink\"])\n", " ax.grid(True, color=COLORS[\"grid\"], alpha=0.45, linewidth=0.7)\n" ] }, { "cell_type": "markdown", "id": "6f063885", "metadata": {}, "source": [ "## From L26 to L27\n", "\n", "The one-dimensional ADR balance from L26 was\n", "\n", "$$\n", "u_t+w u_x=D u_{xx}+s-\\Sigma u.\n", "$$\n", "\n", "Different assumptions give different symbolic problems:\n", "\n", "| Assumptions | Reduced model | What we solve |\n", "|---|---|---|\n", "| no space, transient | $u'=-\\Sigma u+s_0$ | first-order IVP |\n", "| steady, no diffusion | $w u' + \\Sigma u=s_0$ | first-order ODE with inflow data |\n", "| steady, no advection | $-D u''+\\Sigma u=s_0$ | second-order BVP |\n", "| no source, no advection | $u_t=D u_{xx}-\\Sigma u$ | PDE, often separated into modes |\n", "\n", "The trick is not to memorize a different solver for each row. The trick is to write the reduced balance clearly, state the data, and check the answer." ] }, { "cell_type": "code", "execution_count": null, "id": "caaf1236", "metadata": { "execution": { "iopub.execute_input": "2026-07-07T21:25:19.311375Z", "iopub.status.busy": "2026-07-07T21:25:19.311227Z", "iopub.status.idle": "2026-07-07T21:25:19.313742Z", "shell.execute_reply": "2026-07-07T21:25:19.313219Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Input\n", "Each row starts from the L26 balance and adds assumptions.\n", "Output\n", "well-mixed reaction -> u' = -Sigma*u + s0 (linear IVP)\n", "steady 1-D flow -> w*u' + Sigma*u = s0 (inflow ODE)\n", "steady slab diffusion -> -D*u'' + Sigma*u = s0 (two-point BVP)\n", "modal diffusion-reaction -> u_t = D*u_xx - Sigma*u (separation / eigenvalue)\n" ] } ], "source": [ "# Input: several L26 balances and their L27 symbolic form.\n", "reductions = [\n", " (\"well-mixed reaction\", \"u' = -Sigma*u + s0\", \"linear IVP\"),\n", " (\"steady 1-D flow\", \"w*u' + Sigma*u = s0\", \"inflow ODE\"),\n", " (\"steady slab diffusion\", \"-D*u'' + Sigma*u = s0\", \"two-point BVP\"),\n", " (\"modal diffusion-reaction\", \"u_t = D*u_xx - Sigma*u\", \"separation / eigenvalue\"),\n", "]\n", "\n", "print(\"Input\")\n", "print(\"Each row starts from the L26 balance and adds assumptions.\")\n", "print(\"Output\")\n", "for name, model, kind in reductions:\n", " print(f\"{name:27s} -> {model:27s} ({kind})\")" ] }, { "cell_type": "markdown", "id": "c961bc61", "metadata": {}, "source": [ "## Symbols Are Not Functions\n", "\n", "This is the most common SymPy mistake in this lesson. A symbol like `t` is an independent variable. A symbolic function like `u(t)` must be created with `sy.Function`.\n", "\n", "Use this pattern:\n", "\n", "```python\n", "t = sy.symbols(\"t\")\n", "u = sy.Function(\"u\")\n", "u(t)\n", "```\n", "\n", "Do not create `u` with `sy.symbols` and then try to call `u(t)`. A symbol is not callable." ] }, { "cell_type": "code", "execution_count": null, "id": "e3554dcd", "metadata": { "execution": { "iopub.execute_input": "2026-07-07T21:25:19.314684Z", "iopub.status.busy": "2026-07-07T21:25:19.314600Z", "iopub.status.idle": "2026-07-07T21:25:19.773585Z", "shell.execute_reply": "2026-07-07T21:25:19.773147Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Input\n", "t is a Symbol\n", "u is a UndefinedFunction\n", "Output\n", "The symbolic unknown is u(t):\n" ] }, { "data": { "image/png": "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", "text/latex": [ "$\\displaystyle u{\\left(t \\right)}$" ], "text/plain": [ "u(t)" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Input: independent variable and unknown symbolic function.\n", "t = sy.symbols(\"t\", real=True)\n", "u = sy.Function(\"u\")\n", "\n", "print(\"Input\")\n", "print(f\"t is a {type(t).__name__}\")\n", "print(f\"u is a {type(u).__name__}\")\n", "print(\"Output\")\n", "print(\"The symbolic unknown is u(t):\")\n", "u(t)" ] }, { "cell_type": "markdown", "id": "e7390feb", "metadata": {}, "source": [ "## A Reusable Symbolic Scaffold\n", "\n", "For the rest of your ODE-solving career, a safe symbolic workflow is:\n", "\n", "1. define the independent variable and unknown function,\n", "2. write the differential equation as `sy.Eq(left, right)`,\n", "3. solve with `sy.dsolve` or solve for constants by hand,\n", "4. check the solution by substitution,\n", "5. turn the expression into a numerical function with `sy.lambdify` when you want values or plots.\n", "\n", "The checking step matters. It catches sign errors, missing constants, and boundary conditions that were not actually applied." ] }, { "cell_type": "markdown", "id": "9a443ed1", "metadata": {}, "source": [ "## Example 1: Pure Reaction with a Constant Source\n", "\n", "Start with the L26 reduction for a well-mixed amount or density:\n", "\n", "$$\n", "\\frac{du}{dt}=-\\Sigma u+s_0,\n", "\\qquad u(0)=u_0.\n", "$$\n", "\n", "This is the balance statement\n", "\n", "$$\n", "\\text{accumulation}=\\text{source}-\\text{removal}.\n", "$$\n", "\n", "The equilibrium value should be $u=s_0/\\Sigma$. The symbolic solution should make that visible." ] }, { "cell_type": "code", "execution_count": null, "id": "93f1430b", "metadata": { "execution": { "iopub.execute_input": "2026-07-07T21:25:19.774794Z", "iopub.status.busy": "2026-07-07T21:25:19.774697Z", "iopub.status.idle": "2026-07-07T21:25:20.205132Z", "shell.execute_reply": "2026-07-07T21:25:20.204688Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Input\n", "du/dt = -Sigma*u + s0, u(0)=u0\n", "Output\n", "ODE residual : 0\n", "initial residual : 0\n", "long-time value : s_0/Sigma\n" ] }, { "data": { "image/png": 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"text/latex": [ "$\\displaystyle \\frac{\\left(\\Sigma u_{0} + s_{0} e^{\\Sigma t} - s_{0}\\right) e^{- \\Sigma t}}{\\Sigma}$" ], "text/plain": [ "⎛ Σ⋅t ⎞ -Σ⋅t\n", "⎝Σ⋅u₀ + s₀⋅ℯ - s₀⎠⋅ℯ \n", "───────────────────────────\n", " Σ " ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Input block: symbols, unknown function, and pure reaction balance.\n", "t = sy.symbols(\"t\", real=True)\n", "Sigma, s0, u0 = sy.symbols(\"Sigma s_0 u_0\", positive=True)\n", "u = sy.Function(\"u\")\n", "\n", "reaction_ode = sy.Eq(sy.diff(u(t), t), -Sigma*u(t) + s0)\n", "reaction_solution = sy.dsolve(reaction_ode, ics={u(0): u0})\n", "u_reaction = sy.simplify(reaction_solution.rhs)\n", "\n", "reaction_residual = sy.simplify(sy.diff(u_reaction, t) + Sigma*u_reaction - s0)\n", "initial_residual = sy.simplify(u_reaction.subs(t, 0) - u0)\n", "\n", "equilibrium = sy.simplify(sy.limit(u_reaction, t, sy.oo))\n", "\n", "print(\"Input\")\n", "print(\"du/dt = -Sigma*u + s0, u(0)=u0\")\n", "print(\"Output\")\n", "print(f\"ODE residual : {reaction_residual}\")\n", "print(f\"initial residual : {initial_residual}\")\n", "print(f\"long-time value : {equilibrium}\")\n", "u_reaction" ] }, { "cell_type": "code", "execution_count": null, "id": "d98fbf38", "metadata": { "execution": { "iopub.execute_input": "2026-07-07T21:25:20.206226Z", "iopub.status.busy": "2026-07-07T21:25:20.206131Z", "iopub.status.idle": "2026-07-07T21:25:20.390736Z", "shell.execute_reply": "2026-07-07T21:25:20.389886Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Input\n", "Sigma: 0.35\n", "s_0: 2.0\n", "u_0: 0.5\n", "Output\n", "equilibrium density: 5.714\n", "u(18 h): 5.705\n" ] }, { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Input block: numerical values for the symbolic reaction solution.\n", "reaction_values = {\n", " Sigma: 0.35, # 1/h\n", " s0: 2.0, # density/h\n", " u0: 0.5, # initial density\n", "}\n", "\n", "tu = np.linspace(0.0, 18.0, 250)\n", "u_fun = sy.lambdify((t, Sigma, s0, u0), u_reaction, \"numpy\")\n", "u_vals = u_fun(tu, reaction_values[Sigma], reaction_values[s0], reaction_values[u0])\n", "ueq = float(equilibrium.subs(reaction_values))\n", "\n", "print(\"Input\")\n", "for key, value in reaction_values.items():\n", " print(f\"{key}: {value}\")\n", "print(\"Output\")\n", "print(f\"equilibrium density: {ueq:.3f}\")\n", "print(f\"u(18 h): {u_vals[-1]:.3f}\")\n", "\n", "fig, ax = plt.subplots(figsize=(7.2, 3.6), constrained_layout=True)\n", "ax.plot(tu, u_vals, color=COLORS[\"purple\"], lw=2.5, label=\"$u(t)$\")\n", "ax.axhline(ueq, color=COLORS[\"orange\"], ls=\"--\", lw=2, label=\"$s_0/\\\\Sigma$\")\n", "ax.set_xlabel(\"time [h]\")\n", "ax.set_ylabel(\"density\")\n", "ax.set_title(\"pure reaction approaches its source/removal equilibrium\")\n", "ax.legend(frameon=False)\n", "style_axes(ax)\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "5b4467cf", "metadata": {}, "source": [ "## Example 2: Steady Advection-Reaction in a Flow\n", "\n", "Now keep one spatial coordinate, but make the problem steady and neglect diffusion:\n", "\n", "$$\n", "w\\frac{du}{dx}+\\Sigma u=s_0,\n", "\\qquad u(0)=u_{\\mathrm{in}},\n", "\\qquad w>0.\n", "$$\n", "\n", "This is the same source-removal balance, but the independent variable is now distance. The inflow value replaces the initial value. The symbolic solution should relax toward $s_0/\\Sigma$ over a length scale $w/\\Sigma$." ] }, { "cell_type": "code", "execution_count": null, "id": "4e78eae1", "metadata": { "execution": { "iopub.execute_input": "2026-07-07T21:25:20.392362Z", "iopub.status.busy": "2026-07-07T21:25:20.392159Z", "iopub.status.idle": "2026-07-07T21:25:20.756976Z", "shell.execute_reply": "2026-07-07T21:25:20.756602Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Input\n", "w*dU/dx + Sigma*U = s0, U(0)=u_in\n", "Output\n", "ODE residual : 0\n", "inflow residual : 0\n", "length scale : w/Sigma\n" ] }, { "data": { "image/png": "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", "text/latex": [ "$\\displaystyle \\frac{\\left(\\Sigma u_{in} + s_{0} e^{\\frac{\\Sigma x}{w}} - s_{0}\\right) e^{- \\frac{\\Sigma x}{w}}}{\\Sigma}$" ], "text/plain": [ "⎛ Σ⋅x ⎞ -Σ⋅x \n", "⎜ ─── ⎟ ─────\n", "⎜ w ⎟ w \n", "⎝Σ⋅uᵢₙ + s₀⋅ℯ - s₀⎠⋅ℯ \n", "─────────────────────────────\n", " Σ " ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Input block: steady advection-reaction balance.\n", "x = sy.symbols(\"x\", real=True)\n", "w, Sigma_x, s0_x, u_in = sy.symbols(\"w Sigma s_0 u_in\", positive=True)\n", "U = sy.Function(\"U\")\n", "\n", "adv_reaction_ode = sy.Eq(w*sy.diff(U(x), x) + Sigma_x*U(x), s0_x)\n", "adv_reaction_solution = sy.dsolve(adv_reaction_ode, ics={U(0): u_in})\n", "u_adv = sy.simplify(adv_reaction_solution.rhs)\n", "\n", "adv_residual = sy.simplify(w*sy.diff(u_adv, x) + Sigma_x*u_adv - s0_x)\n", "inflow_residual = sy.simplify(u_adv.subs(x, 0) - u_in)\n", "length_scale = sy.simplify(w/Sigma_x)\n", "\n", "print(\"Input\")\n", "print(\"w*dU/dx + Sigma*U = s0, U(0)=u_in\")\n", "print(\"Output\")\n", "print(f\"ODE residual : {adv_residual}\")\n", "print(f\"inflow residual : {inflow_residual}\")\n", "print(f\"length scale : {length_scale}\")\n", "u_adv" ] }, { "cell_type": "code", "execution_count": null, "id": "dc63b536", "metadata": { "execution": { "iopub.execute_input": "2026-07-07T21:25:20.758208Z", "iopub.status.busy": "2026-07-07T21:25:20.758115Z", "iopub.status.idle": "2026-07-07T21:25:20.828826Z", "shell.execute_reply": "2026-07-07T21:25:20.828308Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Input\n", "w: 1.2\n", "Sigma: 0.4\n", "s_0: 1.0\n", "u_in: 5.5\n", "Output\n", "equilibrium density: 2.500\n", "relaxation length : 3.000 cm\n", "U(12 cm): 2.555\n" ] }, { "data": { "image/png": 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", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Input block: numerical values for the spatial balance.\n", "adv_values = {\n", " w: 1.2, # cm/s\n", " Sigma_x: 0.4, # 1/s\n", " s0_x: 1.0, # density/s\n", " u_in: 5.5, # inflow density\n", "}\n", "\n", "xx = np.linspace(0.0, 12.0, 250)\n", "adv_fun = sy.lambdify((x, w, Sigma_x, s0_x, u_in), u_adv, \"numpy\")\n", "adv_vals = adv_fun(xx, adv_values[w], adv_values[Sigma_x], adv_values[s0_x], adv_values[u_in])\n", "adv_eq = float((s0_x/Sigma_x).subs(adv_values))\n", "ell = float(length_scale.subs(adv_values))\n", "\n", "print(\"Input\")\n", "for key, value in adv_values.items():\n", " print(f\"{key}: {value}\")\n", "print(\"Output\")\n", "print(f\"equilibrium density: {adv_eq:.3f}\")\n", "print(f\"relaxation length : {ell:.3f} cm\")\n", "print(f\"U(12 cm): {adv_vals[-1]:.3f}\")\n", "\n", "fig, ax = plt.subplots(figsize=(7.2, 3.6), constrained_layout=True)\n", "ax.plot(xx, adv_vals, color=COLORS[\"blue\"], lw=2.5, label=\"$U(x)$\")\n", "ax.axhline(adv_eq, color=COLORS[\"orange\"], ls=\"--\", lw=2, label=\"$s_0/\\\\Sigma$\")\n", "ax.axvline(ell, color=COLORS[\"green\"], ls=\":\", lw=2, label=\"$w/\\\\Sigma$\")\n", "ax.set_xlabel(\"distance [cm]\")\n", "ax.set_ylabel(\"density\")\n", "ax.set_title(\"advection-reaction relaxes from inflow data\")\n", "ax.legend(frameon=False)\n", "style_axes(ax)\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "8e5f56f5", "metadata": {}, "source": [ "## Ansatz: Guess a Shape, Reduce to Algebra\n", "\n", "An ansatz is an assumed solution shape. The point is not guessing randomly. The point is choosing a shape that derivatives reproduce.\n", "\n", "For a constant-coefficient homogeneous ODE,\n", "\n", "$$\n", "y''+a y' + b y=0,\n", "$$\n", "\n", "the exponential trial function $y=e^{rt}$ turns differentiation into multiplication by $r$. Substitution reduces the ODE to the characteristic equation\n", "\n", "$$\n", "r^2+a r+b=0.\n", "$$" ] }, { "cell_type": "code", "execution_count": null, "id": "c9a628d6", "metadata": { "execution": { "iopub.execute_input": "2026-07-07T21:25:20.829926Z", "iopub.status.busy": "2026-07-07T21:25:20.829816Z", "iopub.status.idle": "2026-07-07T21:25:20.869110Z", "shell.execute_reply": "2026-07-07T21:25:20.868294Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Input\n", "trial solution y = exp(r*t)\n", "Output\n", "characteristic equation: a*r + b + r**2 = 0\n", "roots: [-a/2 - sqrt(a**2 - 4*b)/2, -a/2 + sqrt(a**2 - 4*b)/2]\n" ] } ], "source": [ "# Input block: characteristic-equation reduction.\n", "r, a, b = sy.symbols(\"r a b\")\n", "y_trial = sy.exp(r*t)\n", "char_residual = sy.simplify(sy.diff(y_trial, t, 2) + a*sy.diff(y_trial, t) + b*y_trial)\n", "char_equation = sy.factor(char_residual / y_trial)\n", "roots = sy.solve(sy.Eq(char_equation, 0), r)\n", "\n", "print(\"Input\")\n", "print(\"trial solution y = exp(r*t)\")\n", "print(\"Output\")\n", "print(f\"characteristic equation: {char_equation} = 0\")\n", "print(f\"roots: {roots}\")" ] }, { "cell_type": "markdown", "id": "135e340c", "metadata": {}, "source": [ "## PDE Example: Separation of Variables\n", "\n", "The same idea helps with PDEs. Consider the source-free, one-dimensional reaction-diffusion model from L26:\n", "\n", "$$\n", "v_t=Dv_{xx}-\\Sigma v,\n", "\\qquad 00$, the time constant is $1/\\lambda$. The spatial part becomes the eigenvalue problem\n", "\n", "$$\n", "-DX''+\\Sigma X=\\lambda X.\n", "$$" ] }, { "cell_type": "code", "execution_count": null, "id": "25686b60", "metadata": { "execution": { "iopub.execute_input": "2026-07-07T21:25:20.870546Z", "iopub.status.busy": "2026-07-07T21:25:20.870380Z", "iopub.status.idle": "2026-07-07T21:25:21.181333Z", "shell.execute_reply": "2026-07-07T21:25:21.180992Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Input\n", "v(x,t) = T(t)*X(x)\n", "Output\n", "residual / (T*X): -D*Derivative(X(x), (x, 2))/X(x) + Sigma + Derivative(T(t), t)/T(t)\n", "time equation solution: Eq(T(t), C1*exp(-lambda*t))\n" ] }, { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPMAAAA4CAYAAAAo2c5IAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAPe0lEQVR4Ae2d7XXcthKG1zoqwDe3g6QDx6rgKh0kcgVOOnCOf9n/dHI7SFSB7HSQpAI76SApwVcd+L4PFgOBIPhN7pK7mHMofA8Hg3cwAEiunnz+/Hl3avT27dtr9empri91faXrvfJ+V1ioaOBkNXB5oj3DeP9F3xRi1P/T9YR0oaKBU9XAxYl27OuoX3jnhyhdokUDJ6mBk/DM8r4/aXRe6fpa8b90/RONFmXfRekSLRpo1ECKpcaKKyw4Fc/8M7rFkGMdK03+jwrLfjlWTIm3aSCLpbYGayk7FWP+VgqtGKwMGI+MIeOp8dpHJ8nBkn8UTWk76obn26iGpaVV0Xdsu+ptcpmtTnGo9VrX37qIv9B1r8uRyv9U5JmuV4qTx7L7v0SORZKDCQU54i3AEHG+FI9vdR21H0ME3kJd6bMVS0v3YSAuWjHwZKlHU17Ib6SMa68QPKcdRKFAiDTeszfAvfIx1m9opwujJe32ywpnI/H+XszYb1sfkPdW+c6gFNIP7m2HbPSD8l8VBlKa2f5K4Y8hc0RE7ZkQHhT+MqJ5aZJoQHq08RuEpagdHBlzcAGBRSjOAxvc52e1q0zESg/Ghdo0YmAxY6ZHkG7Og2yWuvEJs5VhLOxRvlN5xQBchcwf1bM9zQ8UK+2MWeFij57E+71uheJrcqqMfGT5QfHapKQ8BvIPhbX+K38wiQ+Tx38UGoAG8ygN9hqQDkdhSe0wqCtdL+NxUJxHoJ8U8m5DIKXBz28KwySseCsufHlwWoGZIirLYmDRZbZuajNVWAInQv2iOoD8TlcvY1a9G13x6TRes7JfVnpuuhVDM9ogp2Rn1mVWZwXSROzdHWiaKgzMhxc83WQ2sG2pXtXAWCyxyooxuPNYwEDfVW/hUuA/nXy7cAGuwdcnx6H6J4uBi2qd2VO2PG0zNmatp1IGXrqVVAdlccUeEENi1oMHZbOT+HJKznWtOAre+Xv9pLDLqG5UJ8zItJ1Cnhc8F+nrFNm21NbrbzCW1I7x/y3TV8N6row2H5M2XbgA12wjH5J2O+WBpxoGLtKKM6edx9LNK4+MknuYYVaWJkkdl/QdC51TGs9vnplZdkky78oeHxDcKazMzunNVY43t/6lxVPS8Fy6v1PkW31bjQ04GoMlDqFyk7OtznKOi21mfK8+uADXYRWYUWgNA5eZSnNmmaG18cQwIOfx9tHWvxgQBsW+gSUIh0qcZv8dK0zpWYkB1MXSiBXEF7pe9rgBA5wbXNdU/Oi7rUiuFIcneqA/0AfVyQ0oPOGdAxXtCvXTwGAsaTyaxhOsN3nStE0WF+IND1Z64AIcPFMeK1dwUDk8U14NA4sZs25u++XcskOyBHruY708mPjSiVQ5gdnCEYznlS6UG2balnvSN/PouWphmS5+8OXsAN5MVrYvyhkzj+QY+EITNCAdz4Il8cHwMMDcfjknYRYXJo9CPDdbOoy+iWoYuGiqOUO+ga3L8EzgLqOfQaTJLBg0qGufvK+1H2BWDzXSQOGR8fRGTA4MonlbvH9cbvUI4WmyxPklfhwNGNb7YhjDz+LCi49N/NXRlRoGFvPMEsQZqUDbJRQA5tlpxeiVPti3mbpX52Mt1cFT2nNiXt5g5qzInFE+BtnkwT+qfbwa4VQ/7K1U1rYfpx2AKDRAA9LpaEypbRtGzCF14cGkbcMFdZgccisya09Yw8CSxoxArZ2Tgmy/aEYShO1QXqh3iIhkwZB56M+e6FZxJiBkbu2fyhtJfNJJjgMtePehLjD04XF2daTzNoOcog+wnt0vD2UqGZmkWXV1efkaBhYxZgn0zHeiUSAvNMvI3xW3paVvNm8g/ijblDToxwrUFhkxZGd8hLqYFd1jKh9vEpilUKcHFQ+TL8zGynPtFD5kmFMG70JH1oDGB8NjPPrul5G4DRdgYSe+wVEozmPXFAc1DCxizJLFCaQwCISACXFKx2xmS5SkOJ9UfYyLw6Ihr2++V7vBP1agNtyHZ9jOkCOJnIErzd65tqqI6mH0DHaFxI+BoP8cgKEj+LDVoL7Ra6WbeDMrx3WtTQlHaEB6zmJK+Twx4a2/Nl0b1hsdV0akLC58vcp+WfdmFUj9FIM1DFxkbjRH1guYSJBUAPLwaJzEYchjXnF0p8NqW+PdInh8H4wrneUqTcWbmZDB+bfC2oSkPFtJvKJupXE1gYw8ckoJAHDx6l9thlUeZR/SRlHa7a+jdIlO00ANU35ceDTUZsjclYkYquFkn53924QLKgcj9TLwXDuH9RoGLrO3Gpmpm+JtAKdbZvu0cTPQoxz3YrsVDAyZqYYobic54gFhFs4eLqkeclP+XBfyPlfeva6gTF/nTmVGfyqPrYINquUT3utCJykhPxMCRrujrS5WAIDKPT9XPCy5qZMQ7XL3S6qVZE8N5DCFjsO4x3w0NmAEpxCwrjirvweFPFpMnwkru0JNuKASqzG2de48qYVXDQOLf2hR6cKIhDqDwl7rqnzuGHdScZbDEPthFM3guLjKgiErjrGE/a/ii5Puidws1bLAGCqA+AAiDL/zjbmhvM+lvnTXiCmvXyZKMIRxMvFWJnSlJ5PuMxoXXsYaBi4nS7UgA690PFX8iRrG+tJuqzoY8q8KWbbjBTFW6tMOT+uMWWna8TIGe1OWx12zp6rPQtwXcMzlSZm54XlUkv7YhuAd0G9uojLPRQgxJoNWVPtm8/6VDBhyI6ZUTn/ACX2rfc46ozRTcJHFwKqNWYqjwyxhg3dFmUrH4IkPjhgoO4h4afUUmmFjxLCA30GMWffjNVBmUfY+lX4gyBCCh+rDZ66JYcjt07rIgHdBJjfZphVIS1aMgknWjJrsY1IfTCEfjmCxyUd6GYULtWvEwNqN+UYKjfe3AKOiYJSiPCPKHdCVHwxe8fgAzOoeMqQPAHrQyX1GQLYJsxmy9IK+mj4cyNz+MUttWQmZh6BvWR2rDpMxK6ncQeAjw4aY2o6WsYFlH0xxT5wEy+wlaQwuGjFwsaSkU3hLkXhZrtibYQx4OU6bKQukNDNp8NJKr8UT7CQLoOCQy/b2Qe6+Ed82+wMIfXlk6pmOM0XdWZKJ1Q2TK6e+jX1TGYd5lfHq5h5qTJIxcFFEchivLkwFr0wbXaRnJ/F9ENPeuFB9dNyIgTUbMx3lcuQVyowJeJhdd8rDq/DrDtALXfEg4TVWQ5ITTzZ6aU9beKymQ4+C2MqJZ+ZtoLftz2PLA8ckXyemvEgcLn7wcQ5fF9M7Y6qrFy6oR30vVy24rOWsKwOgtH3uiCd+p4sZy4HFx+nFqowZgU6RpG/GwJaLLLezp+yq0/ao7ZCq6cIUsrilrGTm8RAn2cGpHFLQofda/aOpoR0q9ftrQCDl8Qurm16eoY2zeGDI8ONgZ859/Wwytsl/CmWrXWafgnLPrA8cclWWsWfW/6N398mbN2+eSoo/dBH2pdleguh7w1JvvAbkKVk2ct6Q0hc+41NaoDQflNh+OFNczVJd9ssczozyyoeQsSrx6aVmW2ZrMEZ/K3p6aj18j6T/wZ/3qc0sS1jx4ckBB2CNxq+yUS/qTJVR7c8Gl7MdgElpg8F0eMiXO86tAY07Kzr31l0H7+zBWEebycXnhMuyZ54Ml7NnwMFXo0dGOzIonjbwtlihBTUwm2deUMbCeqUakJFiyOyTOfiqkfLx2vZbZ6P20jWmJaNRA8WYI9UIfDymaXwoH1WtRae0rTHbQIb6y6Eae27e/kol5mAtPVAdpdeU8ZbSfTHRt15X34sxew1JoSwFAdxY0DERnM1/aVRf8bTF23r8pMFAPM2Cnc0as5TF8o292rVXJEu9W+W7FyAU4hn4WoqTVsowUsprbyIpDw/Dr4qMfnlCbfmgAC/1va744w+xXi2hF64108Fk1Lg9lSLADAReTDf2mmqcB66oz+FfBTdKD8KT6s+CndkeTalTRyEpwt484tl3xVC9Up0HUTzrcZXPgJT/0niU0VvXTYUFVmdXuvh81gx5p/hR/7tjXy1t1jNHHbxVnJkQow3GrAFg5uQ7267PDvm+lf3fXAQv+/B8Lp6Fz2E0cCW8VE7mPY6Y8N9lRLhXXjB6X96FJ1aSYDP3os4k7Fx4ATYbSNl8t8x17RW/U4jyeYmhz57uRvVmWxZ7XvBEhkIb0YDGCwPLfdll27hcGW0+Jl3swhPOhS+l0klgpzxwOBo7mzdmr0jzrHxhhRHdKazMsL5eJVAdPHp2+V2pODwBz5vhzUqLI2qAQ6jcpG4ru8qPYng5w38gId0TT0wOYQXp+cTBaOycwjIbJY75D40okIHKDZJTrvgyMbhfSVTo9lIKmY35dhrilxhzAwNPeOfAQbtCK9OAxrEJBxhfkydN22TxJN7wYJUInsAPB6Wc9eR+yXM0dk7CmKUUCMPhAAMF1ZYwVMjQc+WZV88UPy7VxRPed7rgzwrA9jc5Y+ZtJwaw0IY1oDHG8DDA3H4517MsnsQHA+XEmpUg20Hz9jkeo7FzkeO20TwUD/XZJ+9r7gcqdxCxk8LtzSWrywTBYJi35cUIDjtyBE+TJ1de8rahAZuQc/vlXA8w/CyefGWMmPOdNhqNnZPwzDI8vKT9skjf/9CIQjHIJi9e/ksjGjpvMg+aLqebtNKGJ9owOeRWcjE/9sxMCoNp857ZGzIP7lHCrdeAGfZghVgD8UtnUA60eBTRh7oGtQ+PUuf4GsD4svvloaIJTxgoq7UuLz8aO5s2ZinIPdMzw/MhRh0eU3UonSVN5ywovgwq9cKsqjx+tbGpLfnwLrRRDWhsMTzGsa9XpqdteAJDO/EN/BrwMxo7mzVmKYIDKX52N/Wgto/ts3fG8Bm0CqFkXfB2A6BCePHDddQ3eq30gyWSkNk1rpsUl+QGNGBj3+VJ465k8eQrVPbLwg7nLzXsKW80djZnzFKCMzR1eup/aETHTAQ8ckqJgeQq/6Ux1cz5pM0ZBE/ao+tNeKJpMFIwrDTPtVNHRL3af3cksw9d9qm0hjrq+DPJgdfl+B9lTP0PjWJR/ksjSii014AwBr7wluALvEHvlc8KLPdM2FWI/txTP0rHUc5xONtx7y0orHycEVXEidhEEmV3Rzf/oUV3F9trSKk815vtBwrFDzCwRD/Kz+S097aULq2BKXiaip3NLbMXGAxm41EzYYMszMDwLHSeGpiCp0nYOXtj1mzISyDsX/Cok8jzaHrHdxLv0ngbGhiLpzmwc/bG7CHCRxltr3X2RRI85vTyfe9b6q1LA2PwNBk7Z79nNgz4mXH0z/6oPY/K3D99N54lPF8NDMHTXNj5P30UXTEffIJNAAAAAElFTkSuQmCC", "text/latex": [ "$\\displaystyle - \\frac{D \\frac{d^{2}}{d x^{2}} X{\\left(x \\right)}}{X{\\left(x \\right)}} + \\Sigma + \\frac{\\frac{d}{d t} T{\\left(t \\right)}}{T{\\left(t \\right)}}$" ], "text/plain": [ " 2 \n", " d \n", " D⋅───(X(x)) d \n", " 2 ──(T(t))\n", " dx dt \n", "- ─────────── + Σ + ────────\n", " X(x) T(t) " ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Input block: start with a generic separated ansatz.\n", "x, t = sy.symbols(\"x t\", real=True)\n", "D, Sigma, L, lam = sy.symbols(\"D Sigma L lambda\", positive=True)\n", "T = sy.Function(\"T\")\n", "X = sy.Function(\"X\")\n", "\n", "separable_ansatz = T(t) * X(x)\n", "separation_residual = sy.simplify(\n", " sy.diff(separable_ansatz, t)\n", " - D*sy.diff(separable_ansatz, x, 2)\n", " + Sigma*separable_ansatz\n", ")\n", "separation_ratio = sy.simplify(separation_residual / separable_ansatz)\n", "time_solution = sy.dsolve(sy.Eq(sy.diff(T(t), t), -lam*T(t)))\n", "\n", "print(\"Input\")\n", "print(\"v(x,t) = T(t)*X(x)\")\n", "print(\"Output\")\n", "print(f\"residual / (T*X): {separation_ratio}\")\n", "print(f\"time equation solution: {time_solution}\")\n", "separation_ratio" ] }, { "cell_type": "markdown", "id": "9ffbe3c3", "metadata": {}, "source": [ "## Boundary Conditions Select the Allowed Modes\n", "\n", "For the zero-value boundary conditions $v(0,t)=v(L,t)=0$, a compatible spatial shape is\n", "\n", "$$\n", "X_n(x)=\\sin\\left(\\frac{n\\pi x}{L}\\right),\n", "\\qquad n=1,2,3,\\ldots\n", "$$\n", "\n", "Substitute the full mode\n", "\n", "$$\n", "v_n(x,t)=e^{-\\lambda_n t}\\sin\\left(\\frac{n\\pi x}{L}\\right).\n", "$$\n", "\n", "Now the PDE residual reduces to one algebraic expression for $\\lambda_n$." ] }, { "cell_type": "code", "execution_count": null, "id": "b2746b19", "metadata": { "execution": { "iopub.execute_input": "2026-07-07T21:25:21.182510Z", "iopub.status.busy": "2026-07-07T21:25:21.182408Z", "iopub.status.idle": "2026-07-07T21:25:21.520425Z", "shell.execute_reply": "2026-07-07T21:25:21.519673Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Input\n", "v_n(x,t) = exp(-lambda*t)*sin(n*pi*x/L)\n", "Output\n", "mode residual / mode: pi**2*D*n**2/L**2 + Sigma - lambda\n", "lambda_n: pi**2*D*n**2/L**2 + Sigma\n" ] }, { "data": { "image/png": "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", "text/latex": [ "$\\displaystyle \\frac{\\pi^{2} D n^{2}}{L^{2}} + \\Sigma$" ], "text/plain": [ " 2 2 \n", "π ⋅D⋅n \n", "─────── + Σ\n", " 2 \n", " L " ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Input block: boundary-compatible separated reaction-diffusion mode.\n", "n = sy.symbols(\"n\", integer=True, positive=True)\n", "\n", "v_mode = sy.exp(-lam*t) * sy.sin(n*sy.pi*x/L)\n", "pde_residual = sy.simplify(sy.diff(v_mode, t) - D*sy.diff(v_mode, x, 2) + Sigma*v_mode)\n", "mode_factor = sy.simplify(pde_residual / v_mode)\n", "lam_n = sy.solve(sy.Eq(mode_factor, 0), lam)[0]\n", "\n", "print(\"Input\")\n", "print(\"v_n(x,t) = exp(-lambda*t)*sin(n*pi*x/L)\")\n", "print(\"Output\")\n", "print(f\"mode residual / mode: {mode_factor}\")\n", "print(f\"lambda_n: {lam_n}\")\n", "lam_n" ] }, { "cell_type": "code", "execution_count": null, "id": "865940eb", "metadata": { "execution": { "iopub.execute_input": "2026-07-07T21:25:21.521566Z", "iopub.status.busy": "2026-07-07T21:25:21.521461Z", "iopub.status.idle": "2026-07-07T21:25:21.657240Z", "shell.execute_reply": "2026-07-07T21:25:21.656822Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Input\n", "D=0.08, Sigma=0.12, L=10.0\n", "Output\n", "mode 1: lambda = 0.1279 1/time, tau = 7.819 time\n", "mode 2: lambda = 0.1516 1/time, tau = 6.597 time\n", "mode 3: lambda = 0.1911 1/time, tau = 5.234 time\n" ] }, { "data": { "image/png": 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", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Input block: plot the first three separated modes.\n", "mode_values = {D: 0.08, Sigma: 0.12, L: 10.0}\n", "xgrid = np.linspace(0.0, mode_values[L], 300)\n", "times = [0.0, 5.0, 15.0]\n", "mode_numbers = [1, 2, 3]\n", "\n", "print(\"Input\")\n", "print(f\"D={mode_values[D]}, Sigma={mode_values[Sigma]}, L={mode_values[L]}\")\n", "print(\"Output\")\n", "for mode in mode_numbers:\n", " decay = float(lam_n.subs(mode_values).subs(n, mode))\n", " print(f\"mode {mode}: lambda = {decay:.4f} 1/time, tau = {1/decay:.3f} time\")\n", "\n", "fig, axes = plt.subplots(1, 3, figsize=(10.5, 3.2), sharey=True, constrained_layout=True)\n", "for ax, time_value in zip(axes, times):\n", " for mode, color in zip(mode_numbers, [COLORS[\"purple\"], COLORS[\"orange\"], COLORS[\"blue\"]]):\n", " decay = float(lam_n.subs(mode_values).subs(n, mode))\n", " shape = np.exp(-decay*time_value) * np.sin(mode*np.pi*xgrid/mode_values[L])\n", " ax.plot(xgrid, shape, lw=2.2, color=color, label=f\"n={mode}\")\n", " ax.set_title(f\"t = {time_value:g}\")\n", " ax.set_xlabel(\"x\")\n", " style_axes(ax)\n", "axes[0].set_ylabel(\"mode amplitude\")\n", "axes[0].legend(frameon=False, loc=\"upper right\")\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "aa22fddd", "metadata": {}, "source": [ "## What SymPy Is Really Buying Us\n", "\n", "Symbolic work is not a replacement for numerical work. It is a way to expose structure before we compute:\n", "\n", "- the pure reaction solution shows the equilibrium $s_0/\\Sigma$,\n", "- the advection-reaction solution shows the relaxation length $w/\\Sigma$,\n", "- the characteristic equation turns a differential problem into roots,\n", "- separation of variables turns a PDE into a time ODE and a spatial eigenvalue problem,\n", "- each separated mode carries a decay rate $\\lambda_n$ and time constant $1/\\lambda_n$.\n", "\n", "The pattern to carry forward is boring in the best possible way: define the variables, write the equation, solve, apply data, check by substitution, and only then evaluate or plot." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.13.13" }, "ne415_schedule": { "lesson": "L27", "notes": "Symbolic reductions and separation scaffold", "worksheet": "W27" } }, "nbformat": 4, "nbformat_minor": 5 }