chapter 4 updated

examples/chap_04.ipynb

@@ -16,13 +16,14 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 8,
+   "execution_count": 1,
    "metadata": {},
    "outputs": [],
    "source": [
     "import sympy as sp\n",
     "import numpy as np\n",
-    "from scipy.linalg import eig"
+    "from scipy.linalg import eig\n",
+    "from IPython.display import display, Math"
    ]
   },
   {
@@ -33,87 +34,47 @@
    ]
   },
   {
-   "cell_type": "code",
-   "execution_count": 2,
+   "cell_type": "markdown",
    "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "array([-3.5+14.72243186j, -3.5-14.72243186j, -8.  +0.j        ])"
-      ]
-     },
-     "execution_count": 2,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
    "source": [
-    "A = np.array([[-5, -10, 7], [7, -5, -10], [-10, 7, -5]])\n",
-    "B = np.array([1,1,1])\n",
-    "X0 = np.array([0, 0, 0])\n",
-    "\n",
-    "eigenvalues, eigenvectors = eig(A)\n",
-    "eigenvalues"
+    "$$\n",
+    "\\frac{d}{dt}\n",
+    "\\begin{pmatrix}\n",
+    "E_1 \\\\\n",
+    "E_2 \\\\\n",
+    "E_3\n",
+    "\\end{pmatrix} =\n",
+    "\\begin{pmatrix}\n",
+    "-5 & -10 & 7 \\\\\n",
+    " 7&  -5& -10 \\\\\n",
+    " -10& 7 & -5 \n",
+    "\\end{pmatrix} \n",
+    "\\begin{pmatrix}\n",
+    "E_1 \\\\\n",
+    "E_2 \\\\\n",
+    "E_3\n",
+    "\\end{pmatrix}\n",
+    "$$"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 3,
+   "execution_count": 2,
    "metadata": {},
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "x1(t) = exp(-8*t) + 4*sqrt(3)*exp(-7*t/2)*sin(17*sqrt(3)*t/2)\n",
-      "x2(t) = exp(-8*t) - 2*sqrt(3)*exp(-7*t/2)*sin(17*sqrt(3)*t/2) - 6*exp(-7*t/2)*cos(17*sqrt(3)*t/2)\n",
-      "x3(t) = exp(-8*t) - 2*sqrt(3)*exp(-7*t/2)*sin(17*sqrt(3)*t/2) + 6*exp(-7*t/2)*cos(17*sqrt(3)*t/2)\n"
-     ]
-    }
-   ],
+   "outputs": [],
    "source": [
-    "# Define the symbolic variables\n",
-    "t = sp.Symbol('t')\n",
-    "x1, x2, x3 = sp.Function('x1')(t), sp.Function('x2')(t), sp.Function('x3')(t)\n",
-    "\n",
-    "# Define the matrix A\n",
-    "A = sp.Matrix([[-5, -10, 7],\n",
-    "               [7, -5, -10],\n",
-    "               [-10, 7, -5]])\n",
-    "\n",
-    "# Define the vector X\n",
-    "X = sp.Matrix([x1, x2, x3])\n",
-    "\n",
-    "# Define the system of differential equations\n",
-    "dX_dt = A * X\n",
-    "\n",
-    "# Create the system of differential equations\n",
-    "system = [sp.Eq(sp.diff(x1, t), dX_dt[0]),\n",
-    "          sp.Eq(sp.diff(x2, t), dX_dt[1]),\n",
-    "          sp.Eq(sp.diff(x3, t), dX_dt[2])]\n",
-    "\n",
-    "# Solve the system of differential equations\n",
-    "solution = sp.dsolve(system)\n",
-    "\n",
-    "# Define the initial conditions\n",
-    "X0 = sp.Matrix([1, -5, 7])\n",
-    "\n",
-    "# Apply the initial conditions to find the constants\n",
-    "t0 = 0\n",
-    "constants = sp.solve([sol.rhs.subs(t, t0) - x0 for sol, x0 in zip(solution, X0)])\n",
-    "\n",
-    "# Substitute the constants back into the solution\n",
-    "final_solution = [sol.rhs.subs(constants) for sol in solution]\n",
+    "A = np.array([[-5, -10, 7], [7, -5, -10], [-10, 7, -5]])\n",
+    "B = np.array([0,0,0])\n",
+    "X0 = np.array([1, -5, 7])\n",
+    "t_range = np.linspace(0, 2, 1000)\n",
     "\n",
-    "# Print the final solution\n",
-    "for i, sol in enumerate(final_solution, 1):\n",
-    "    print(f\"x{i}(t) = {sol}\")"
+    "# eigenvalues, eigenvectors = eig(A)\n",
+    "# eigenvalues"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 4,
+   "execution_count": 3,
    "metadata": {},
    "outputs": [
     {
@@ -154,22 +115,24 @@
     }
    ],
    "source": [
-    "from IPython.display import display, Math\n",
+    "from spikes.solver import solve_system_of_equations\n",
+    "\n",
+    "final_solution, x_values = solve_system_of_equations(A, B, X0, t_range)\n",
     "\n",
     "for i, sol in enumerate(final_solution, 1):\n",
-    "    display(Math(f\"x_{i}(t) = {sp.latex(sol)}\"))"
+    "    display(Math(f'x_{i}(t) = {sp.latex(sol)}'))\n"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "- Evaluating over a given time interval:"
+    "- plot the evaluation of analytical solution over a given time interval:"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 5,
+   "execution_count": 4,
    "metadata": {},
    "outputs": [
     {
@@ -188,16 +151,6 @@
     "from sympy import lambdify\n",
     "import matplotlib.pyplot as plt\n",
     "\n",
-    "# Create a time range\n",
-    "t_range = np.linspace(0, 2, 1000)  # From 0 to 10 seconds, with 1000 points\n",
-    "\n",
-    "# Convert symbolic solutions to numerical functions\n",
-    "x_functions = []\n",
-    "x_values = []\n",
-    "for sol in final_solution:\n",
-    "    x_functions.append(lambdify(t, sol, 'numpy'))\n",
-    "    x_values.append(x_functions[-1](t_range))\n",
-    "\n",
     "plt.figure(figsize=(6, 4))\n",
     "for i in range(3):\n",
     "    plt.plot(t_range, x_values[i], label=f\"x{i+1}(t)\")\n",
@@ -218,7 +171,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 6,
+   "execution_count": 5,
    "metadata": {},
    "outputs": [
     {
@@ -249,7 +202,6 @@
     "t = np.linspace(0, 2, 1000)\n",
     "solution_numerical = odeint(system, X0, t, args=(A,))\n",
     "\n",
-    "\n",
     "plt.figure(figsize=(6, 4))\n",
     "for i in range(3):\n",
     "    plt.plot(t_range, x_values[i], label=f\"x{i+1}(t)\", alpha=0.5)\n",