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.
For this,
I tried solving the differential equation using an alternative method. My alternative method starts at
##tv^{''} + v^{'} = 0##
I substitute ##v(t) = e^{rt}## into the equation getting,
##tr^2e^{rt} + re^{rt} = 0##
##e^{rt}[tr^2 + r] = 0##
##e^{rt} = 0## or ##tr^2 + r = 0##
Note that...
Using the complete the square method I got;
-3+root8 or -3-root8
But using the quadratic formula (to check my answer) I got;
-3+root 10 or -3-root 10
I've checked both answers several times but can't get to the bottom of it :)
If we have a first order d.e. like: $$\frac {dy} {dx} - \frac y x = 1$$
I would use two subs, namely: ##y = uv## and ##\frac {dy} {dx} = u \frac {dv} {dx} + v \frac {du} {dx}##
So I get: ##\frac {dy} {dx} = u \frac {dv} {dx} + v \frac {du} {dx} - \frac {uv} {x} = 1##
I then factor like this...
Hello,
I'm looking for some sources and bibliography focused on nuclear fuel fabrication.
In this context, I am referring to the creation of fuel pellets, excluding the uranium enrichment procedures. I am aware of at least two methods, the mechanical and the ceramic one.
With fabrication I...
I am working on HPLC method Beta alanine and unable to get a good enough peak. I'm using an agilent system HPLC ,gradient mobile phase(a: 0.1% formic Acid in Ro water,0.1% Formic acid in ACN) 250 mm C18 4/6x 250 mm column Flow rate 1.0 ul Detector: 214 nm 10 ul injected temp:ambient
Gradient...
I recently made a Python library for modelling (very basic) finite difference problems. The Github readme goes into details of what it does and how it works, and I put together a Google Colab with some examples (diffusion, advection, water wave refraction) with interactive visuals.
I'd love to...
Hi guys!
I have problems with this mechanism...finding the aceleration with graphic method.
I already have drawn the speed diagram and the values of all of them.
But, i dont know if my aceleration diagram it is well drawn. I'm stuck especially when trying to plot and calculate the...
The set of equations is
I have first tried to solve only first two equations (removing the components of other 4 equations from them.)
This is the output, where first column is the time, 2nd - X_p, and 3rd - X_n
Problem:
I have done part a) in spherical polar coordinates.
For part b) I thought it would be just:
$$\sigma = -\epsilon_0 \frac{\partial V}{\partial r}$$
But I got confused by "You may want to use different coordinate systems .." So I assume partial derivative w.r.t to r is the spherical...
There is a question that puzzle me when I apply numerical method to principal value integral. Let me descibe it. We make use of the fact that the integral ##\int_0^\infty \frac{dk}{k^2-k_0^2}## vanishes, namely,
$$
\int_0^\infty \frac{dk}{k^2-k_0^2} = 0 .
$$
We use this formula to express a...
I am trying to apply finite difference scheme for Beam propagation method by following this paper.
I was wondering if anyone can share their code if they have implemented this method. I can share my code which is not working as expected and can get some insights if possible.
Hi,
I want to solve the problem by method of mirroring and by using the electric field by doing superposition and then adding them up to use in Lorentz law to get the force. I have attached a figure that represents the problem.
How do I know from the figure that $-p_l$ is from the...
I'm reading a book called Asymptotic Methods and Perturbation Theory, and I came across a derivation that I just couldn't follow. Maybe its simple and I am missing something. Equation 3.3.3b below. y(x) takes the form A(x)*(x-x0)^α and A(x) is expanded in a taylor series.
Hi,
This is my first post on PF. I am engaged in a project seeking to find a simple way to reduce ethanol content in wine for those wishing to minimize alcohol intake for health or personal reasons. The alcohol-reduced wine will be used for cooking. Seems heating would do the trick however, the...
The classical "power method" for solving one special eigenvalue of an operator works, in a finite-dimensional vector space, as follows: suppose an operator ##\hat{A}## can be written as an ##n\times n## matrix, and its unknown eigenvectors are (in Dirac bra-ket notation) ##\left|\psi_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...
Can anyone explain to me why grounded means zero electric potential. I confuse what's the relation between infinite ground conducting plane and its electric potential (the method of images).
I have a several question:
1. Why the conductor plane must be infinite, while in reality there's no...
Can anyone explain to me why grounded means zero electric potential. I confuse what's the relation between infinite ground conducting plane and its electric potential (the method of images).
I have a several question:
1. Why the conductor plane must be infinite, while in reality there's no...
I'm trying to code Newton Raphson's method for finding zeros. I realize that even if the solution is real, it's possible for guesses to be complex. For example:
$$y=\sqrt{x-6}-2$$
While 10 is a valid real root, for any guess less than 6, the result is complex.
I tried to run the code allowing...
For this problem,
The solution is,
However, why can the differential area not be:
I tried integrating and got,
Can someone please tell me what I have done wrong?
Thank you!
This is from an examination paper -A level. My interest is on part (ii). Ok my take;
i. ##KE_{initial} = \dfrac {1}{2} mu^2= \dfrac {1}{2}× 0.4 ×12^2=28.8## Joules.
ii. ##\dfrac {1}{2} mv^2=\dfrac {1}{2} mu^2-mgh##
##0=28.8-(0.4×10×h)## where h is the vertical perpendiculor distance...
Here is my code so far and solution.
When using fsolve, we obtain the result that the equation does get solved. We find the values to be 1.0000, 0.0000, and 2.0000. fsolve completed because the vector of function values is near zero as measured by the value of the function tolerance, and the...
I am marking this now...just need your insight...the second part has [2] marks. I am giving [1] mark for the correct method...i expect use of second derivative test or otherwise for Method Mark...then the correct answer ##(0,0)## would realize the Answer mark.
The student has not shown any...
I have solved many finite element problems using nodal based (rectangular element) for higher order. now i am trying to solve electromagnetic problem using vector element (Nedelec or Whitney). I know only triangular edge based element with first order only and not higher order. i am searching...
Hello guys, I have conducted an experiment and got some results.
I have 3 variables to vary, for example, five x1, five x2, and two x3
and 2 observation results, like y1, y2
I already make y1 y2 and x1 x2 x3 dimensionless
since plot is 2D, what I am doing now is just plot when x3=1, x2=1, plot...
Suppose an optical scalar wave traveling in Z direction. Using the diffraction theory of Fourier Optics, we can predict its new distribution after a distance Z. The core idea of Fourier Optics is to decompose a scalar wave into plane waves traveling in different directions. But this...
We are given f(x)=(1/2)(xT)Qx+qTx-B where xk+1=xk+αksk, the search direction is sk=-∇f(xk). Q is a 2x2 matrix and q is 2x1 matrix and B=6. My issue is I'm confused what -∇f(xk) is, is ∇f(xk)=Q(xk)-q? Just like how it is in Conjugate Gradient/Fletcher Reeve's method? Or is it Q(xk)+q?
Thank you
I am finding the potential everywhere in space due to a point charge a distance 'a' on the z-axis above an infinite xy-plane held at zero potential. This problem is fairly straight forward; place an image charge q' = -q at position -a on the z-axis. I have the solution in cartesian coordinates...
I have written FE codes.
I have seen FE classes from mechanical engineering
I have seen FE classes from the math departments
When offered from a math department, the focus is the theory (with a lot of attention paid to general differential equations)
They are good courses and they suggest, I...
My function is f(x1,x2)=(x2-x1 2)2+ex1^2
I understand the equation for finding a point is xk+1=xk+αkdk, where dk=(0,1) and x0=(0,1). For the first step I can choose α1=0.618 (GR), but how do we choose α2? In the solution manual I see they chose 0.382, was this just some arbitrary number? My...
Summary: Current using jω method with 3 components in the following series : Resistor, capacitor and Inductor.
I have the following information:
R=1 * 10^3 ohm
L = 0.5 H
C = 0.1 * 10^-6 F
v(t) = 230 * sqrt(2) * sin(2pi * 400 * t)
All the components are in parallel as shown in the image...
I know how to apply boundary condition like Dirichlet, Neumann and Robin but i have been struggling to apply divergence free condition for Maxwells or Stokes equations in nodal finite element method. to overcome this difficulties a special element was developed called as edge element but i don't...
The image attached above from a textbook, explains that we should refrain from using (a+b)/2 while applying Bisection method but I am unable to get the reason why it is asking to do so?
While the image above is from another textbook. This book uses (a+b)/2.
I am really confused by two different...
the simple rectangular isoparametric element (curved edges element) can be used to generate many complex shapes like..
square to circle
square to triangle
two square elements to annular.
and the results i calculated in python code (2D case) are very accurate, then i confused why to use complex...
Here's the image that a lot of pop science articles use:
Evidently, a lot of articles say to look for the Great Square of Pegasus, but it seems to me to not be a very noticeable asterism. I use the Cassiopeia's W deeper-side (i.e., right) arrowhead and count 3 lengths of the depth of the...
I have worked through the basics of the FEM method in 2D with rectangular elements and now am trying to learn triangular elements but need a simple enough numerical example. It seems triangular elements are far more complicated and also depend on the global coordinates thus each local stiffness...
Let's say that my_mock.method_calls gives me this array:
[call.thing.pop(3, 4), call.thing.bob.smash(6, 7), call.thing.jig.slurp(4)]
How can I extract the names 'thing', 'pop', 'bob', 'smash' etc. from this array?
for the wihelmy blade method we want to measure the interfacial tension exerted on a paracetamol compact in the shape of a rectangular parallepiped perimeter submerged 0.12 m in water we measure a force of 0.21 mN for a contact angle of 88.29°
answer whether they are true or false
Surface...
Hello there,
I would like to get some help for this problem if possible.
The problem here is that, after looking at the table I don't really understand how I'd be able to incorporate the moment at the pin A into the deformation. as if you look at the first page 2nd block, there's that simply...
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...
Write a program that uses the Monte Carlo method to approximate the double integral $\displaystyle\iint\limits_R e^{xy}dA$ where $R = [-1,1] \times [0, x^2]$. Show the program output for N = 10, 100, 1000, 10000, 100000 and 1000000 random points.
My correct answer:
My Java program...
(This is from W. Greiner Quantum Mechanics, p. 293 from the topic of Ritz Variational Method)
1) Are ##\frac{\delta}{\delta \psi^{*}}## derivatives in equations 11.35a and 11.35b? If this is so, we can differentiate under the integral sign to get ##\int d^3x (\hat{H}\psi)## in equation 11.35a...
(I do understand there are many ways to formulate the FE method, but I wish to understand this one.)
I understand that Hamilton's Principle is just that: a principle ( a law the recapitulates the equations of motion)
I also understand that the Principle of Virtual Work (PVW) is a reformulation...
So I was looking at this example problem in my textbook and I don’t understand how they got -168.1 degrees. The part I’m confused with is the first part of the 2nd picture, there’s a coma in the inverse tan, I don’t know what that means.