# What is Euler method: Definition and 44 Discussions

In mathematics and computational science, the Euler method (also called forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who treated it in his book Institutionum calculi integralis (published 1768–1870).The Euler method is a first-order method, which means that the local error (error per step) is proportional to the square of the step size, and the global error (error at a given time) is proportional to the step size.
The Euler method often serves as the basis to construct more complex methods, e.g., predictor–corrector method.

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1. ### Use Euler method to determine the approximation of given problem

There is a mistake in my opinion on the text. In my working i have, ##y_1= 3 + 0.2 e^{\cos1} = 3+ 0.54357 = 3.54357## ##y_2 = 3.54357 + 0.2 e^{\cos 1.2} = 4.0871## ##y_3 = 4.0871 + 0.2 e^{\cos 1.4} = 4.6305## I also noted that we do not have an exact solution for this problem.
2. ### I Understanding Euler Method: Finding Initial Condition of y(0)=1

The Euler method is straightforward to me; i.e ##y_{n+1}=y_n+ hf(t_0, y_0)## where the smaller the steps i.e ##h## size the better the approximation. My question is 'how does one go about in determining the initial condition ##y(0)=1## in this problem? am assuming that this has to be a point...
3. ### Solve Euler Method in C++ for Beginners

Summary: Problem with Euler Method in C++ Hello, I have a very difficult problem for me (a beginner in programming) how to make the version of the euler method presented in c ++ with the void, float functions, so that the program will calculate from the data that I enter during the program...
4. ### I Derive local truncation error for the Improved Euler Method

I'm trying to find the local truncation error of the autonomous ODE: fx/ft = f(x). I know that the error is |x(t1) − x1|, but I can't successfully figure out the Taylor expansion to get to the answer, which I believe is O(h^3). Any help would be greatly appreciated!
5. ### Parametric distance of a line in a grid (Line Integral Convolution)

Hi, the above image is from the Line Integral Convolution paper by Cabral and Leedom. However, I am having a hard time implementing it, and I am quite certain I am misreading it. It is supposed to give me the distances of the lines like in the example below, but I am not sure how it can. First...
6. ### I Does the Euler method give a better insight into adiabatic processess?

I verified with others the equation below is an Euler method as well with ##a## can be any value such that it give the same ##\frac{dE}{dv}=-1.4\frac{p}{v}## but with ##a## other than one, it have no meaning in physics. For anyone that already understand Euler method can omit the part i have...
7. ### A Implicit Euler method with adaptive time step and step doubling

For Initial Value problems I want to implement an ODE solver for implicit Euler method with adaptive time step and use step doubling to estimate error. I have found some reading stuff about adaptive time step and error estimation using step doubling but those are mostly related to RK methods. I...
8. ### Coding a numerical approximation for a damped pendulum

Hi there. I have a question about the damped pendulum. I am working on an exercise where I have already numerically approximated the solution for a simple pendulum without dampening. Now, the excercise says that I can simply change the code of this simple situation to describe a pendulum with...
9. ### Determine stopping distance of a train - modified Euler method

Homework Statement 12.81[/B] A train traveling at 20m/s is brought to an emergency stop. During braking, the acceleration is a=-(7/4)+(t/16) m/s^2, where t is the time in seconds measured from when the brakes were applied. (a) Integrate the acceleration from t=0 to t=16s using Euler's method...
10. ### Determine stopping distance of a train traveling at 20m/s

This is problem 12.81 from Pytels Dynamics 2nd edition 1. Homework Statement A train traveling at 20m/s is brought to an emergency stop. During braking, the acceleration is a=-(7/4)+(t/16) m/s^2, where t is the time in seconds measured from when the brakes were applied. (a) Integrate the...
11. ### MHB Bound of Euler method- nonuniform partition

Hello! (Wave) Consider a nonuniform partition $a=t_0< t_1< \dots < t_{\nu}=b$ and assume that if $h_n=t^{n+1}-t^n, 0 \leq n \leq N-1$ is the changeable step, then $\min_{n} h_n > \lambda \max_{n} h_n, \lambda>0$ independent of $n$. Show a bound of the error of Euler method analogous to...
12. ### Exploring the Reliability of Euler Method for Orbital Gravity Equations

Hello everyone, I am curious as to if it is possible to use the Euler Method to solve the gravity differential equations? Would the approximations quickly diverge to inaccurate solutions, or would it stay relatively reliable? Thanks
13. ### Euler's Method and Planetary Motion

Homework Statement Hi there, I wish to use Newton's Laws in conjunction with Euler's Method to model the motion of a planet around a star.Homework Equations 2nd Law F = m*a Law of Universal Gravitation F = -G*M1*M2/r^2 The Attempt at a Solution [/B] First I combined the two laws above...
14. ### MHB Can the Euler method accurately approximate solutions for stiff systems?

Hello! (Wave) We consider the initial value problem $$\left\{\begin{matrix} y'=\lambda y, & t \in [0,\infty), \lambda \in \mathbb{C}, Re(\lambda)<0 \\ y(0)=1 & \end{matrix}\right.$$ Since $y^n=(1+h \lambda)^n, n \in \mathbb{N}_0$ is the sequence of approximations that the Euler method...
15. ### Euler method for modeling simple harmonic oscillation

Hello! An assignment for my computational modeling course is to demonstrate the use of the Standard Euler method for modeling a simple harmonic oscillator; in this case, a mass attached to the end of a spring. I have the two coupled first-order differential equations satisfying hookes law...
16. ### Help with implicit euler method

Hey I don't understand this backward euler solution, in particular why the f(y,t+h) is equal to y(t+2)
17. ### 2nd Order Differential Equation with Improved Euler Method (Heun's)

Homework Statement I would like to solve a 2nd Order Differential Equation using the Improved Euler Method. The 2nd ODE is a Mass-Spring-Damper equation. I tried coming up with an solution for the Improved Euler Method, but not entirely sure. Can you help me and have a look if this is correct...
18. ### MHB Harmonic oscillator and symplectic Euler method

Given the equations for the harmonic oscillator $\frac{dy}{dz}=z, \frac{dz}{dt}= -y$if the system is approximated by the symplectic Euler method, then it gives$z_{n+1}= z_{n}-hy_{n}, \\ y_{n+1}= y_{n}+hz_{n+1}$which shows that the circle $y^2_{n} + z^2_{n} = 1$ is mapped into an ellipse...
19. ### Forward Euler Method for ODE system

Homework Statement Solve the following system for 0<t<5 u^\prime = u-e^{-2t} v, u(0) = 1 v^\prime = u+3v, v(0) = -2 using Forward Euler method and implement the numerical scheme into a MATLAB code. Homework Equations Forward Euler : \vec x^(\prime)_{n+1} = \vec F(t,\vec x)...
20. ### Fortran How to fix the Euler method error?

Hi everybody, I am programming a new code for a problem. The problem is numerically solving the Simple Harmonic Motion using the Euler method. This approach is just an approximate solution and not a exact solution, however when I run the code successfully and plot my data, it comes up as an...
21. ### Fortran Fortran program for oscillator using Euler method

I am trying to run a program with fortran. The program is about solving the Oscillator using Euler Method. I am trying to run this code and applying array arguments (as I want to extend it to 3 dimensions afterwards). When I try to compile, it comes up with an error "Unclassifiable statement at...
22. ### Euler Method with h=0.05 for y'=3+t-y, y(0)=1 at t=0.1, 0.2, 0.3, and 0.4

Homework Statement Use the Euler method with h=0.05 to find approximate values of y'=3+t-y, y(0)=1 at t=0.1, 0.2, 0.3, and 0.4. Homework Equations I don't even know what formula and how to use it. The Attempt at a Solution The answers are 1.1975, 1.38549, 1.56491, 1.73658.
23. ### Midpoint Euler method, second order system

Hello I need to plot this simple system: x'' = -x using midpoint Euler. u1 = -x , u2 = -x' u1' = u2 u2' = -x u1(n+1) = u1(n) + h*? u2(n+1) = u2(n) + h*f((1/2)*(u1(n) + u1(n+1)) We don't know u1(n+1). I tried approximating it with u1(n+1) = u1(n) + h*u2(n) u2(1+i) =...
24. ### MHB Backward Euler Method for 2x2 systems

Hi! I want to write a code in Matlab for the Backward Euler Method for 2x2 systems, using the fixed point iteration to find the yn+1. y1n+1=y1n+h*f(tn+1,y1n+1,y2n+1) (1) y2n+1=y2n+h*g(tn+1,y1n+1,y2n+1) (2) Could you tell how I use the fixed point iteration?? At (1) the fixed point iteration...
25. ### Backward euler method for heat equation with neumann b.c.

I am trying to solve the following pde numerically using backward f.d. for time and central di fference approximation for x, in MATLAB but i can't get correct results. \frac{\partial u}{\partial t}=\alpha\frac{\partial^{2}u}{\partial x^{2}},\qquad u(x,0)=f(x),\qquad u_{x}(0,t)=0,\qquad...
26. ### How can the Improved Euler Method be used in programming assignments?

The "improved" Euler method Homework Statement Using it on a programming assignment. The description in our course notes is a little confusing, so I need to know whether I'm doing it correctly. Homework Equations Go to p. 22 of this, if you're so inclined...
27. ### Comp Sci FORTRAN: second-order ODE with Euler Method

Homework Statement Dear all, please help. I have tried this question and came up with strange numbers, my fortran is definitely not correct. Please help! When the effect of the air resistance is taken into account, the equation of motion for a particle of mass m falling vertically in a...
28. ### Back Euler method for 2nd order d.e

Hi, How can one use back Euler method for 2nd order d.e? Is it possible this method to be expanded for a system of 4 odes? Thanks
29. ### Understanding the Limitations of the Euler Method in Computational Physics

This is an extract from my third year notes on 'Computational Physics': The Euler method is inaccurate because it uses the gradient evaluated at the initial point to calculate the next point. This only gives a good estimate if the function is linear since the truncation error is quadratic in...
30. ### MHB Pharaoh's modified Euler method question from Yahoo Answers

Part 2 of Pharaoh's Taylor series and modified Euler question from Yahoo Answers There is a standard method of converting higher order ODEs into first order systems, in this case it is to introduce the state vector:$$Y(t)=\left[ \begin{array}{c} y(t) \\ y'(t) \end{array} \right]$$Then the ODE...
31. ### Convergence of implicit Euler method

Homework Statement The implicit Euler method is yn = yn-1 + hf(xn,yn). Find the local truncation error and hence show that the method is convergent. Homework Equations The Attempt at a Solution I found the error to be ln = (-h2/2)y''(xn-1) + O(h3). For convergence I am up to...
32. ### C/C++ Solve ODEs Backwards in Time with C++ Euler Method

Hi Everybody I am beginner in c++ and I need your help please. I implemented euler method for solving simple ODEs (y' = x -y, y(0)=1)and it is forward in time(from t=0 to t=1) and it worked well, my question is : I want to run this code backward in time(t=1 to t=0) what i have to change in my...
33. ### Solving Diff. Eq. with Euler Method: x=0, y=5

Homework Statement Using the Euler method solve the following differential equation. At x = 0, y = 5. y' + x/y = 0 Calculate the Numerical solution using step sizes of .5; .1; and .01 The Attempt at a Solution From my textbook I have coded Euler's method function [t,y]...
34. ### Error in Numerical Solution of ODE by Euler Method - Patrick

Hi, I recently need to do some numerical simulation by Euler method to solve a PDE. However, I noticed that there are some errors which are obtained with bigger numerical steps, when applying Euler scheme. Since my major is not mathematics, I do not know what this phenomenon is called. I...
35. ### Numerical Computation (Euler method)

Homework Statement Differential equations are important in physics.Consider Solving the differential equation dy/dx = Σ(from i to N) (ai) (x^i) Using Euler ,Midpoint ,4th order Runge -Kutta methods.For each of these methods what is the largest value of N that would lead to an exact solution ...
36. ### Advanced Engineering Mathematics: Euler Method

Do 10 steps. Solve the problem exactly. Compute the error (Show all details). The problems says do 10 steps, but 3-4 steps will suffice! Problem: y(prime) = (y-x)^2 y(0) = 0 h = 0.1 I don't understand how to get the exact solution and what to do from there! I know that, f(x,y) =...
37. ### MATLAB Exploring Euler Method: Solving y'=x^2-y^2 w/ MATLAB

I'm looking at the following example (http://www.cyclismo.org/tutorial/matlab/control.html): y'=x^2-y^2, y(0)=1, where they use the Euler method to approximate numerical solutions. This is the code: I don't understand the third and fourth line at the top. What is y = 0*x and how did they...
38. ### Finding slope fields using Euler method

hi guys, can someone give me a quick tutorial on how to solve and explain to me the concept of slope field of the following differential equation: sketch the slope field for dy/dt = 2t+1 showing the solution y=t^2+t-4, which satisfies the initial condition y(-2)= -2 Also how to use...
39. ### Temperature Change with Cream Addition

Matlab Help! Euler method Use Euler's method: cooling constant r = 0.2 , initial temperature of 86 C, and room temperature at 17 C. Choose your time step h and integrate to 5 minutes. You can add some cream which will drop the temperature instantly by 5 C. (a) Now, add the cream at the...
40. ### Euler methond and the improved Euler method

I am confused by both of these. I get what it does, and how it does it, but when using it, somehow I ultimately screw up. Could someone just post a quick description of what to do in conversational English, please? I am pretty sure that would solve this random mental block I have...
41. ### Fortran Euler Method in Fortran - HELP

Euler Method in Fortran - HELP! Using Euler method I want to calculate the equation below, deltat=0.3, 10 times. Problem: When n=1, Yn1(1) is calculated right. But, in the following steps it should assume that Yn(2)=Yn1(1) and the program is assuming Yn(2)=0, as well as Yn(3)=0,...and so on...
42. ### MATLAB Euler Method in MATLAB: Solving a Double Pendulum

Hi all Im working on a systems dynamics problem which involves modelling a double pendulum, a chaotic system I have a non linear system of 4 1st order differential equations which I need to solve using numerical methods in MATLAB The methods are ode45, ode23s and euler method Ive...
43. ### Is the Advanced Euler Method Accurate for Solving Differential Equations?

Homework Statement integrate dy/dx = (1+e^y)(1+x) from x=0 using the improved euler method taking step sizes of 0.125. compare the numerical solution of y(0.5) with the exact value The Attempt at a Solution Firstly to find the exact value I use separation of variables and the initial...
44. ### Improving Euler's Approximation for Differential Equations: Tips and Resources

Would you please tell me how to improve Euler's approximation to be better in solving differerential equations ? Can you give me some links to this? Thank you,