{ "cells": [ { "cell_type": "markdown", "id": "6365d295-b7bd-42b3-90cf-d9e6aaa24a28", "metadata": {}, "source": [ "# Title: Introduction to Differential Equations and Numerical Methods\n" ] }, { "cell_type": "markdown", "id": "168105f3-2e4b-44b4-a07e-530f51a5714a", "metadata": {}, "source": [ "## Libraries\n" ] }, { "cell_type": "code", "execution_count": 1, "id": "610bf72c-1fb1-4cb4-a25b-0a2f308a375a", "metadata": {}, "outputs": [], "source": [ "import numpy as np ### Linear algebra package\n", "import matplotlib.pyplot as plt ### To plot \n", "import time ### Take the time to run cells\n", "%matplotlib inline" ] }, { "cell_type": "markdown", "id": "dff70b68-5ba3-4b89-9449-2dcbe6d9ca59", "metadata": {}, "source": [ "## Goals \n", "The goal is solve diferential equations, numerically and analitically, and test different methods to solve problems in physics, starting from ODE of first order \n", "\n", "$$\\frac{dy}{dt}=f(t,y), \\qquad a\\leq t\\leq b, \\qquad y(a)=\\alpha \\qquad \\qquad (1)$$\n" ] }, { "cell_type": "markdown", "id": "e4485c03-cda5-4bfa-aa25-dd607bdc4dda", "metadata": {}, "source": [ "- In order, to determine an **unique** solution, we must specify a set of initial conditions, aditionally the number of initial conditions must be equal to the **order** of the diferential equation.\n", "- Computationally it is imposible to get a continous solution y(t); instead, we must choose a range [a,b] and a subdivision of this range to get the approximate solution evaluated at this points, it is important to note that the magnitude of the division of the range is associated with the accuracy of the numerical solution\n", "- Once we find the solution in the set of points in the range, the continuos solution can be obtained thorugh an interpolation" ] }, { "cell_type": "markdown", "id": "7269835f-a2ed-4193-8055-4f5a82bcfe31", "metadata": {}, "source": [ "\n", "\n", "# 1. Introduction to Differential Equations\n", "\n", "Differential equations are mathematical equations that involve derivatives. They are used to model many phenomena in physics, engineering, and other fields. There are different types of differential equations, including ordinary differential equations (ODEs) and partial differential equations (PDEs).\n", "\n", "Examples of differential equations commonly encountered in physics include:\n", "\n", "- Simple Harmonic Oscillator\n", "- Pendulum Motion\n", "- Projectile Motion\n", "- Planetasing the formula:\n", "\n" ] }, { "cell_type": "markdown", "id": "dbf9ea66-4a36-4151-a1da-25b7d5f5aa68", "metadata": {}, "source": [ "\n", "## 2. Numerical Methods for Solving Differential Equations\n", "\n", "Numerical methods are used to approximate the solutions of differential equations when analytical solutions are not feasible. Some of the commonly used numerical methods include:\n", "\n", "**Euler's Method**\n", "\n", "Euler's method is a simple and intuitive method for solving first-order ODEs. It approximates the solution at discrete time steps using the formula:\n" ] }, { "cell_type": "code", "execution_count": 2, "id": "14380834-ab47-4777-ae91-d2b07b323f24", "metadata": {}, "outputs": [], "source": [ "def euler_method(y0, t0, dt, t_end, f):\n", " t_values = np.arange(t0, t_end, dt)\n", " y_values = [y0]\n", "\n", " for t in t_values[1:]:\n", " y_next = y_values[-1] + dt * f(t, y_values[-1])\n", " y_values.append(y_next)\n", "\n", " return t_values, y_values" ] }, { "cell_type": "markdown", "id": "a0d2c35e-8961-4414-baca-1964ba4447d1", "metadata": {}, "source": [ "## Euler's Method " ] }, { "cell_type": "markdown", "id": "48a67af3-9463-4f91-9f54-b6adb69293af", "metadata": {}, "source": [ "First, we define a set of points equally spaced in an interval [a,b], then we choose to divide the interval in $N$ with subdivisions of magnitude $h = (b-a)/N $\n", "$$t_i=a+ih, \\qquad i=0,1,2,..,N.$$" ] }, { "cell_type": "markdown", "id": "a5afcd03-e076-42e3-ba07-afab92235221", "metadata": {}, "source": [ "Now, we use a taylor expasion to deduce the Eulers method. Assuming that we have an **unique** solution y(t) and its derivate is continuos in the interval [a,b], then we can expand the solution in powers of $t_{i+1} -t_i$\n", "\n", "$$y(t_ {i+1})=y(t_i)+(t_{i+1}-t_i)y'(t_i)+\\frac{(t_{i+1}-t_i)^2}{2}y''(\\xi_i),$$\n", "\n", "and using the equation (1), we get: \n", "\n", "$$y(t_{i+1})=y(t_i)+hf(t_i,y(t_i))+\\frac{h^2}{2}y''(\\xi_i)$$\n", "\n", "asuming that $h$ is small, then we get the Euler's method \n", "$$y_0=\\alpha$$\n", "$$y_{i+1}=y_i +hf(t_i,y_i), \\qquad i =0,1,...,N-1$$" ] }, { "cell_type": "code", "execution_count": 3, "id": "8ce91b69-bbb9-4267-839a-f71a50599a02", "metadata": {}, "outputs": [], "source": [ "def euler(f,x0,xn,y0,n):\n", " h = (xn-x0)/n\n", " x0_l=[] #list of steps\n", " yn_l=[] #list of solutions\n", " for i in range(n):\n", " slope = f(x0, y0) #Initial value of the function \n", " yn = y0 + h * slope\n", " y0 = yn #update the solution to yn\n", " x0 = x0+h #update the step \n", " x0_l.append(x0)\n", " yn_l.append(y0)\n", " \n", " return(yn_l) #return the numerical solution to the step n " ] }, { "cell_type": "markdown", "id": "343024a9-bacc-4ec3-b1d3-48ef35ffe4d1", "metadata": {}, "source": [ "### Newton's Law of Cooling " ] }, { "cell_type": "markdown", "id": "f1a2f235-1a3c-42ae-bbcb-8c40f1cae59a", "metadata": {}, "source": [ "This law decribe how an object reach a given temperature $T_f$ of the surroundings(Thermalize) starting from an initial temperature $T_i$, the law that describes this procedure is: \n", "\n", "$$\\frac{dT(t)}{dt}=-k\\Delta T =-k(T(t)-T_f), \\qquad T_i=T(t_0)$$\n", "\n", "In other words, this law says that **the rate of change of the temperature is proportional to the difference between the initial and the equilibrium temperature** $T_f$." ] }, { "cell_type": "markdown", "id": "faced891-2c4b-4d89-b1b9-d4b12221df03", "metadata": {}, "source": [ "The analitical solution can be obtained using stardar integration techniques, getting: \n", "\n", "$$T(t)=T_f+(T(0)-T_f)e^{-kt}$$" ] }, { "cell_type": "code", "execution_count": 4, "id": "5f39ab6b-6b21-4b13-acd0-a3685053c9c9", "metadata": {}, "outputs": [], "source": [ "#EDO \n", "def newtoncooling(time, temp):\n", "\treturn -0.07 * (temp - 20)\n", "### Analitycal solution\n", "def newtoncooling_ana(t,TR=20,T0=100,k=0.07):\n", " return TR+(T0-TR)*np.exp(-k*t) #TR Temperatura de entorno\n" ] }, { "cell_type": "code", "execution_count": 5, "id": "dbe5d574-9b44-418a-a2d7-1b88900b8e6f", "metadata": {}, "outputs": [], "source": [ "a=0\n", "b=100 #segundos \n", "T0=100\n", "N=100\n", "t_T=np.linspace(a,b,N)\n", "Temp = euler(newtoncooling,a,b,T0,N)\n", "Temp_ana = newtoncooling_ana(t_T)" ] }, { "cell_type": "code", "execution_count": 6, "id": "0fb89fb2-d2fb-41e3-afa3-aed5746e1160", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig = plt.figure()\n", "plt.plot(t_T,Temp, '--',lw=3)\n", "plt.plot(t_T,Temp_ana, lw=3)\n", "plt.legend([\"Euler's Method\", \"Analitycal sol.\"])\n", "plt.xlabel(\"Time\")\n", "plt.ylabel(\"Temperature\")\n", "plt.show() " ] }, { "cell_type": "markdown", "id": "8ab065f0-9543-484e-93c3-31850c1c3c63", "metadata": {}, "source": [ "### Radioactive decay " ] }, { "cell_type": "markdown", "id": "8f4425dc-8012-48be-98e4-afdf9020cfcf", "metadata": {}, "source": [ "- Radioactive decay is a process by which the nucleus of an unstable atom undergoes a spontaneous transformation, resulting in the emission of radiation. This transformation occurs due to the instability of the atomic nucleus, which seeks to achieve a more stable state.\n", "Radioactive decay is a random process, and the rate of decay for a particular radioactive substance is measured by its characteristic life time $(\\tau)$. The characteristic life-time is the time it takes forthe radioactive atoms in a sample to decay at $63 \\%$. Different radioactive isotopes have different half-lives, ranging from fractions of a second to billions of years. \n", "Radioactive decay plays a significant role in various fields, including nuclear physics, geology, archaeology, medicine (such as radioactive tracers in imaging), and power generation (in nuclear reactor)s The equation that describes its procces is given by:\n", "\n", "$$\\frac{dN_U}{dt} = -\\frac{N_U}{\\tau}.$$\n", "- The analitycal solotuion is given by:\n", "\n", "$$N_U= N_U(t=0)e^{-t/\\tau} $$.\n", "\n", "\n", "The differential equation that appears in this problem is almost identical to the Newton's law of cooling. Now, lets observe some of the problems that could appear, when we solve the differential equations numerically. For this reason **we must be careful with the choice of the method to solve the differential equation and the step size** " ] }, { "cell_type": "code", "execution_count": 7, "id": "0e816b85-6bcf-4c40-9eeb-63fa4195e61b", "metadata": {}, "outputs": [], "source": [ "tau = 1\n", "def Nuclei(t, N):\n", " return -N/tau \n", "\n", "def Nuclei_exc(t):\n", " return 100*np.exp(-t)\n", "\n", "a=0\n", "b=5\n", "#dt=0.1\n", "times1= np.arange(a,b,0.05)\n", "n1=int(5/0.05)\n", "times2= np.arange(a,b,0.2)\n", "n2=int(5/0.2) \n", "times3 = np.arange(a,b,0.5)\n", "n3=int(5/0.5)\n", "times4 = np.arange(a,b,0.7)\n", "n4=int(5/0.7) +1\n", "\n", "y1 = euler(Nuclei,a,b, 100, n1)\n", "y2 = euler(Nuclei,a,b, 100, n2)\n", "y3 = euler(Nuclei,a,b, 100, n3)\n", "y4 = euler(Nuclei,a,b, 100, n4)" ] }, { "cell_type": "code", "execution_count": 8, "id": "cb88012e-5c2c-4875-af28-190bbe5dc5d3", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.figure()\n", "plt.plot(times1,Nuclei_exc(times1),color='red',label= 'Exact')\n", "plt.scatter(times1,y1,marker ='o', color='black',label ='dt=0.05')\n", "plt.scatter(times2,y2,marker ='P',color='black',label ='dt=0.2')\n", "plt.scatter(times3,y3,marker = 's',color='black',label ='dt=0.5')\n", "plt.scatter(times4,y4,marker = 'x',color = 'black',label ='dt=0.7')\n", "plt.xlabel('Time/Ï„')\n", "plt.ylabel('Number of Nuclei ')\n", "plt.legend()\n" ] }, { "cell_type": "markdown", "id": "74f30577-7243-43d6-94ac-d5dfa4d457ac", "metadata": {}, "source": [ "## Euler-Cromers method\n", "\n", "The Euler-Cromer method is a modification of Euler's method that provides better stability and accuracy for certain problems, such as conservative systems. It updates the position first and then updates the velocity using the formula:" ] }, { "cell_type": "code", "execution_count": 9, "id": "6300f52c-acf1-4e5f-888d-bd915d43cd0c", "metadata": {}, "outputs": [], "source": [ "def euler_cromer_method(y0, v0, t0, dt, t_end, f):\n", " t_values = np.arange(t0, t_end, dt)\n", " y_values = [y0]\n", " v_values = [v0]\n", "\n", " for t in t_values[1:]:\n", " v_next = v_values[-1] + dt * f(t, y_values[-1])\n", " y_next = y_values[-1] + dt * v_next\n", " v_values.append(v_next)\n", " y_values.append(y_next)\n", "\n", " return t_values, y_values, v_values" ] }, { "cell_type": "markdown", "id": "928c7303-83bd-4379-b175-2f635a37e3f5", "metadata": {}, "source": [ "## Mid point method\n", "\n", "The Midpoint method, also known as the Modified Euler method, is a second-order method that provides better accuracy than Euler's method. It approximates the solution at the midpoint of each time step using the formula:" ] }, { "cell_type": "code", "execution_count": 10, "id": "466d6306-8844-43aa-93a2-584600191b67", "metadata": {}, "outputs": [], "source": [ "def midpoint_method(y0, t0, dt, t_end, f):\n", " t_values = np.arange(t0, t_end, dt)\n", " y_values = [y0]\n", "\n", " for t in t_values[1:]:\n", " k1 = f(t, y_values[-1])\n", " k2 = f(t + dt/2, y_values[-1] + (dt/2) * np.array(k1) )\n", " y_next = y_values[-1] + dt *np.array( k2)\n", " y_values.append(y_next)\n", "\n", " return t_values, y_values" ] }, { "cell_type": "markdown", "id": "6d00bb6b-a663-42e2-9e93-7df44973ec8e", "metadata": {}, "source": [ "### Projectile motion" ] }, { "cell_type": "code", "execution_count": 11, "id": "107c62ed-29a2-4006-a6e7-bc4be6fc7932", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Parameters\n", "g = 9.81 # Acceleration due to gravity\n", "v0 = 30.0 # Initial velocity\n", "theta = np.pi/4 # Launch angle (in radians)\n", "k = 0.1 # Air resistance coefficient\n", "m = 0.1 # Mass of the projectile\n", "t0 = 0.0 # Initial time\n", "t_end = 5.0 # End time\n", "dt = 0.01 # Time step\n", "\n", "# Define the differential equations for projectile motion\n", "def projectile_motion(t, y):\n", " vx, vy, x, y = y\n", " ax = -k * vx / m\n", " ay = -g - k * vy / m\n", " return ax, ay, vx, vy\n", "\n", "# Initial conditions\n", "vx0 = v0 * np.cos(theta)\n", "vy0 = v0 * np.sin(theta)\n", "y0 = [vx0, vy0, 0.0, 0.0]\n", "\n", "# Solve using Midpoint Method\n", "t_values, y_values = midpoint_method(y0, t0, dt, t_end, projectile_motion)\n", "y_valus = [y_values[i][3] for i in range(t_values.size) ]\n", "#Plot the result\n", "plt.plot( np.array(y_valus) )\n", "plt.xlabel('Time')\n", "plt.ylabel('Height')\n", "plt.title('Projectile Motion with Air Resistance')\n", "plt.show()\n", "\n", "#y_values[3]" ] }, { "cell_type": "markdown", "id": "dc77b802-f24c-4b00-bb0f-7232356246b4", "metadata": {}, "source": [ "## Verlet algorithm\n", "\n" ] }, { "cell_type": "markdown", "id": "997822e0-3f69-40ab-a3df-3731559b6105", "metadata": {}, "source": [ "One of the widely used drift-free higher-order algorithms is often credited to Verlet [L. Verlet, \"Computer experiments on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules,\" Physical Review 159, 98 (1967)]. In this paper, Verlet discusses the method and its application to molecular dynamics simulations.\n", "\n", "To derive the algorithm, we express the Taylor series expansion for $x_{n-1}$ and $x_{n+1}$ and we manipulate them in order to get the evolution of $x_{n+1}$ in function of $x_{n}$ and $x_{n-1}$, then we get:\n", "\n", "- $u_{n+1}(i)=2 u_n(i)-u_{n-1}(i)+\\frac{1}{m} F_n(\\Delta t)^2$\n", "\n", "In this case $F_n$ corresponds to the Force function acting over the particles at nth time step \n", "\n", "because the method requires the two past elements, usually the first term in the evolution is computed using other method(e.g. Euler)\n", "\n", "- $u_2(i)=u_1(i)+v_1 \\Delta t$\n", "\n", "The global error associated with the Verlet algorithm is third order for the position and second-order for the velocity. However, the velocity plays no part in the integration of the equations of motions. In the numerical analysis literature, the Verlet method is also knows as the ``explicit central difference method''\n", "\n", "\n", "In order to determine an unique solution the we must specify the initial conditions:\n", "\\begin{equation}\n", "\\left\\{\n", "\t \\begin{array}{ll}\n", "\t\t y'' = F(y) \\\\\n", "\t\t y(0) = \\alpha \\\\\n", " y'(0) = \\beta\n", "\t \\end{array}\n", "\t \\right.\n", "\\end{equation}\n" ] }, { "cell_type": "code", "execution_count": 14, "id": "e8ab0e35-0f05-41b1-a26b-f4627d0c1544", "metadata": {}, "outputs": [], "source": [ "def verlet(f,a,b,xf,niter):\n", " dx = xf/(niter-1) # corresponding dx taking into account the desired number of steps\n", " xs = np.arange(0,xf+dx,dx) # \n", " ys = np.zeros(np.size(xs))\n", " yinit = a-dx*b+0.5*dx*dx*f(a) # Initial step\n", " ys[0] = yinit\n", " ys[1] = a\n", " for i in range(1,niter-1):\n", " ys[i+1] = 2*ys[i]-ys[i-1]+f(ys[i])*dx**2\n", " return xs, ys" ] }, { "cell_type": "markdown", "id": "b75ceac5-0523-4599-bd48-39e0e467cf51", "metadata": {}, "source": [ "### Free fall problem \n", "\n", "An object initially at a height of 30 m is dropped\n", "\n", "\\begin{equation}\n", "\\left\\{\n", "\t \\begin{array}{ll}\n", "\t\t y'' = -g \\\\\n", "\t\t y(0) = 50 \\\\\n", " y'(0) = 0\n", "\t \\end{array}\n", "\t \\right.\n", "\\end{equation}\n" ] }, { "cell_type": "code", "execution_count": 28, "id": "dc23f9b1-1338-4c38-9264-0cc56ec166b3", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "### Solution to the free fall problem \n", "def f(y):\n", " return -9.8\n", "\n", "pos0 = 4.5\n", "t, y = verlet(f,100.0,0.0,pos0,10)\n", "\n", "### Exact solution \n", "def sol_ex(t):\n", " return 100 -9.8/2.0*pow(t,2)\n", "t_ex = np.arange(0.0,pos0+0.1,0.1)\n", "y_ex = np.zeros(np.size(t_ex))\n", "for i in range(np.size(t_ex)):\n", " y_ex[i] = sol_ex(t_ex[i])\n", "plt.figure()\n", "plt.plot(t,y,'k',label = 'Numerical solution')\n", "plt.plot(t_ex,y_ex,c='red',label = 'Exact solution')\n", "plt.xlabel(\"time \")\n", "plt.ylabel(\" Height\")\n", "plt.legend(frameon= False)" ] }, { "cell_type": "markdown", "id": "94455a00-36d6-4c8b-b6f2-29dd8a613205", "metadata": {}, "source": [ "### Pendulum " ] }, { "cell_type": "markdown", "id": "0ad30285-6916-4bc1-b818-db658aebc3e5", "metadata": {}, "source": [ "**Excercise 1):** Derive step by step the equation of motion for the pendulum and solve it using the verlet method \n", "\n", "**Excercise 2):** Aproximate ODE for small angles and solve it analitically.\n", "\n", "**Excercise 3):** Compare the solutions (Linear and non-Linear) making a plot " ] }, { "cell_type": "markdown", "id": "c9f7ad4e-f050-444b-9b9c-67655485c7f6", "metadata": {}, "source": [ "## 3. Local and Global Error\n", "\n", "When using numerical methods to solve differential equations, we encounter two types of errors:\n", "\n", "- **Local Error**: The error that occurs in a single time step due to the approximation method used. It is the difference between the exact solution and the solution obtained in one step.\n", "- **Global Error**: The accumulated error over multiple time steps. It is the difference between the exact solution and the numerical solution at a specific time point.\n", "\n", "The accuracy of a numerical method is affected by both the step size (dt) and the method used.\n", "\n" ] }, { "cell_type": "markdown", "id": "9f355c21-66e7-4a26-92b2-d950c155f6af", "metadata": {}, "source": [ "## 4. Conclusion\n", "\n", "In this Jupyter Notebook, we explored different numerical methods for solving ordinary differential equations (ODEs) and applied them to various physics-based problems. We discussed Euler's Method, Euler-Cromer Method, and the Midpoint Method, along with their implementation in Python. Additionally, we introduced the concepts of local and global error in numerical methods. Understanding these methods is essential for solving dynamic systems in physics, engineering, and other scientific fields.\n" ] }, { "cell_type": "markdown", "id": "027deec5-bff9-4000-b78c-74d2d3600aac", "metadata": {}, "source": [ "## Summary: \n", "\n", "- The Euler method is a simple method to implement, but it has low accuracy.\n", "- There is an Error introduced by the discretization. The solution must preserve the Energy(Euler's method fails with this )\n", "- \n", "The Verlet algorithm is particularly suitable in situations where the expression for the second derivative is only a function of the variables, dependent or independent, without involving the first derivative. This is the case for numerous problems in Newtonian dynamics, which is why it is frequently used in astronomy and molecular mechanic.\n" ] }, { "cell_type": "code", "execution_count": null, "id": "932ab623-5f5c-454e-888a-c9f73db18c72", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "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": "3.9.13" } }, "nbformat": 4, "nbformat_minor": 5 }