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.
I am solving some 2nd order differential equations using the finite element method. Doing so I represent the second order derivative at a given point as:
∂2ψi/∂x2 = 1/(Δx)2 (ψi-1+ψi+1-2ψi)
And solve the differential equation by setting up a matrix of N entries and solving for the eigenvectors...
I am doing a lab which involves the determination of vapour pressure of a particular compound by Knudsen Effusion. Like other analytical methods it involves calibration but we are using the know vapour pressure of naphthalene because its in the accepted range. So we have an equation for the...
Homework Statement
Use the Heun method to compute the period of small oscillations about the equilibrium position of a nitrogen atom.
xi = 1.1
Um = 7.37
x0 = 1.2
alpha = 2.287
m = 2.325e-26
Homework Equations
[/B]
U(x) = Um((1-e^(-alpha(x-x0)))^2 - 1)
The Attempt at a Solution
I was told to...
How do you get deflections of a fully restrained beam? :)
I already solved for the propped reactions and end moments.. I'm not sure how to do that part. :/
ありがとうございます。 :)
Hello,
I have already studied first year mathematics but I am doing some basic revision incase I missed anything,
(I skipped two grades and am thus missing some small simple things I should know)
When simplifying the single term expression,
2x^2y^2 times 5xy^4
my initial thought was they...
Homework Statement
Find the voltage, V, across the 6A current source
Homework Equations
V=IR
Node Voltage Method
The Attempt at a Solution
Did I set this up correctly to find my voltage?
Description of the Method
We are given a trinomial of the form $ax^{2}+bx+c$, and asked to factor it into a product of two dissimilar binomials $(fx+u)(gx+v)$. The method that follows assumes $a,b,c$ have no common factor; if they do, you must factor out the greatest common factor before...
Prove both by method of contrapositive.
1. If a ≤ b + ε, where ε > 0, then b > a.
2. If 0 ≤ a - b < ε, where ε > 0, then a = b.
I'll start with problem 1.:
p: If a ≤ b + ε, where ε > 0
q: b > a
neg q: b < a
neg p: for some ε' > 0 1/2(a - b), a > b + ε'
define ε' = 1/2(a - b)
I...
The pole method of magnetostatics is presented in many E&M textbooks, particularly the older ones, to do computations in magnetostatics and even to try to explain permanent magnets. An equation that arises in the pole method is B=H+4*pi*M (c.g.s. units), where H consists of contributions...
Is there anybody on the forum who is up to speed with the ins and outs of these devices?
99% of research papers and all manufacturers comment I have read say these devices are working because the distant object emits radiation that the detector absorbs and therefore this absorbed energy *heats*...
If there are any experimentalists out there maybe you could help?
I need to bubble gas through a liquid at ~80 mL/min is there a specific instrument designed to do this? I'm assuming it's some sort of pump, what would it be called in a chemistry lab?
Thank you for any suggestions / help.
Frobenius method - recurrance relation question
If, using the Frobenius method, I get a 3 term recurrence relation of the form $a_{j+2} = a_j .f(k,j) + a_{j-2}. g(k,j)$ ( j even), how do I treat the $a_{j-2}$ term at first? I have found $a_1 = 0$, but how do I find a value for $a_{-2}$ so as...
Hermite's ODE is $y'' - 2xy' + 2\alpha y = 0$
Let $y = \sum_{\lambda = 0}^{\infty} {a}_{\lambda} x^{k+\lambda}, y' = \sum a_{\lambda} (k+\lambda)x^{k+\lambda-1}, y'' = \sum a_\lambda (k+\lambda)(k+\lambda-1)x^{k+\lambda-2}$
I get the indicial eqtn of k(k-1) = 0, therefore k = 0 or 1...
So, when you use the Frobenius method on a differential equation, you assume a solution Σa_k*x^(k+s). Sometimes you get more than one solution for s in the indicial equation. Is the sum of these two solutions you get from evaluating the rest of the problem with each s solution the...
Good Day
Let's say I have developed a new method to extract, more efficiently (yes, "more efficiently" is ill-defined; but bear with me) the differential equations that describe a specific phenomena (please just assume it).
So now I have a system of coupled second order differential...
In the chapter 9-5 "The Linear Variation Method" p. 363 from the book: Basic Principles and Techniques of Molecular Quantum Mechanics by Ralph Christoffersen, the first thing he does is to minimize the energy, E = c†Hc/c†Sc, by requiring its derivative with respect to the...
Hello! (Wave)
I want to solve the following linear programming problem:
$$\min (3y_1-y_2+2y_3) \\ 3y_1+2y_2-y_3 \leq 9 \\ 5y_2-y_3 \leq 1 \\ 4y_1-y_2 \geq 1 \\ y_1+y_2+y_3 \leq 3 \\ y_1, y_2, y_3 \geq 0$$
In this case we use the $M$-method.The canonical form of the problem is the...
I'm solving a problem numerically where I have some charge density above an infinite, grounded conducting plane and want to determine the electrical potential at a given point. My intuition says that this is not simply given by the potential of the charge density above the plane, since this will...
I would like to know how to implement internal hinges in a program I'm developing. A hinge is created by changing the stiffness matrix of the beam. The problem is when two interconnected beams have a hinge at the same location, so basically we have a hinged joint, in this scenario I will obtain...
If we have a linear programming problem that is of the form as the following:
$$\max (- x_1+ 2 x_2-3x_3) \\ x_1- \frac{1}{2} x_2+x_3+x_{4}=11 \\ 2x_2-x_3+x_5=0 \\ 2x_4+x_6=8 \\ x_i \geq 0, i \in \{ 1, \dots, 6 \}$$
we cannot use the simplex method since we cannot find a basic feasible...
Homework Statement
What is the volume of a solid formed by the area trapped between y= -x^2 and y= -2x rotated 360° around x-axis?
Homework Equations
V = ∫A(x)dx
The Attempt at a Solution
y=y
-x^2 = -2x
x^2 -2x = 0
x(x-2) = 0
This means that the two functions cross at x = 0 and x = 2
From x...
I've written a poly phase merge sort using 3 "stacks" in C++. The key part of the logic is to merge from containers A and B into container C. It's expected for A or B to go empty with lots of data remaining on B or A. To deal with this, when A or B go empty, the empty container is swapped with C...
Homework Statement
Solve
\begin{equation*}
36x^2y''+(5-9x^2)y=0
\end{equation*}
using the Frobenius method
Homework Equations
Assume a solution of the form
\begin{equation*}
y=\sum_{n=0}^{\infty}{a_nx^{n+s}}
\end{equation*}
then
\begin{equation*}...
Homework Statement
Determine the components of the forces acting on each member of the pin-connected frame shown.
(the frame shown is drawn as a free-body diagram in the image provided below (the top-most portion) and is correct)
Homework Equations
Equilibrium equations:
∑M=0
∑Fx=0
∑Fy=0
The...
Use the method of undetermined coefficients to find a general solution of the ODE:
$y''+3y'+2y=2x^{2}+4x+5$$r^{2}+3r+2=0$
$r=-2$ and $r=-1$
$y_{h}=c_{1}e^{-2x}+c_{2}e^{-x}$
I'm not sure how to get $y_{p}$ here
So here's what I've done so far. I have my final exam tomorrow and I have a few...
Homework Statement
I am trying to solve the following equation using the method of characteristics:
∂u/∂x + (xy)(∂u/∂y) + 2x2zLog[y](∂u/∂z) = 0
I'm really just trying to follow along the solution provided here:
http://www.ucl.ac.uk/~ucahhwi/LTCC/sectionA-firstorder.pdf
on page 9.
The...
I studied this from Griffith Chapter 2, with the algebraic (raising and lowering operator) method, we reached the ground state by setting a_Ψ0 = 0 , then we got what the ground state is, and then plugged it in the Schrodinger equation to know the energy, and it turned out to be 0.5 ħω.
My...
Homework Statement
I have an object of a known mass, and known dimension along one axis (length).
Mass: 10 kg
Length: 2 meters
I wish to calculate what the mass of an object that is similarly constituted and shaped will be if it has a greatly expanded length of: 1,000 meters.
I am seeking to...
Homework Statement
I've been asked to graphically verify that the system of equations F (that I've uploaded) has exactly 4 roots. And so I did, using the ContourPlot function in Mathematica and also calculated them using FindRoot. Now, I've to approximate the zeros of F using the fixed point...
Homework Statement
Find the extremizing (maximum) value of the function f(x) = sin x / x using Newton's 1D method.
Homework Equations
[/B]The Attempt at a Solution
I know the maximum point in this equation is (0, 1). When I differentiated the equation twice and used the formula above, I...
Homework Statement
A motorist drives along a straight road at a constant speed of 14.4m/s. Just as she passes a parked motorcycle police officer, the officer starts to accelerate at 1.8m/s^2 to overtake her. Assuming the officer maintains this acceleration, determine the time it takes the...
Is there any method to define weather I have to calculate for determine buckling of column lead to fail or not. Is there is any rule of thumb or formula that show it is required to check for buckling or not (like very short column)
If I have the following relations:
X = sqrt(1-V^2)*cos(U)
Y = sqrt(1-V^2)*sin(U)
Z = V
where (-pi < U < pi) and (-1 < V < 1) are independent random variables, both with uniform distributions.
How do I use the CDF method to find X_pdf(x)?
X_pdf(x) =
X_cdf'(x) =
( P( X < x ) )' =
( P(...
Hello, a dubt arose while doing some exercise.
If I have a charge q at a distance d from the above-mentioned plane, i can find the solution to the laplace equations (thanks to the uniqueness theorems) finding a collection of image charges that satisfies the boundary conditions.
These conditions...
Homework Statement
In the attachment.
Homework Equations
Chart of pressure loss due to friction for steel ducts : http://postimg.org/image/4iml9x761/
The Attempt at a Solution
I don't understand the deltaP/L column of the results table.
Where did the values come from? Shouldn't they all be...
I am trying to solve the differential equation
##\frac{d^{2}y}{dr^2}+(\frac{1}{r}+1)y=0##
with the boundary conditions
##y(r) \rightarrow r \frac{dy}{dr}(0)## as ##r \rightarrow 0## and ##y(r) \rightarrow \sin(kr+\delta)## as ##r \rightarrow \infty##.
I know that the shooting method is the...
regarding question number 10, we have h = f + λg where g is the constraint (the ellipsoid) and f is the function we need to maximize or minimize (the rectangular parallelpiped volume),
now my question : is it right that f is 8xyz ? i mean if we take f to be xyz not 8xyz and solved till we got...
Homework Statement
Hi! I've been interpolating a data set using Legendre polynomials and Newton's method and now I've been asked to, given a data set, approximate the function using the Least Squares Method with the following fitting function: ##g(r)=a+be^{cr}##, where ##a##, ##b## and ##c##...
Hello, I am new in the field of FORTRAN. I started to write a code using Newton Raphson method. Below is the code of main program portion only. During my calculation, I have found that my results are not converging, Any help or advice will be highly appreciated in this case.
Thank you for the...
I have a question where f(x) = 20-2x^2/(x-1)(x+2)^2 and have solved for constants A,B and C.
A = 2
B = -4
C = -4
I have worked this out myself. Now I am told to compute the indefinite integral and I am getting this answer but apparently it is wrong and I don't understand how?
My answer...
Question:
Why equations
x(1-x)\frac{d^2y}{dx^2}+[\gamma-(\alpha+\beta+1)x]\frac{dy}{dx}-\alpha \beta y(x)=0
should be solved by choosing
##y(x)=\sum^{\infty}_{m=0}a_mx^{m+k}##
and not
##y(x)=\sum^{\infty}_{m=0}a_mx^{m}##?
How to know when we need to choose one of the forms.
Also when I sum over...
Hello everyone!
I am building set of Fortran code to solve integral equation. I have read "Numerical recipe" and heard about "Nystrom method". But there's no sample problem, I found it difficult to understand. Can anyone explain "Nystrom method" for me with a simple problem?
Many thanks
The Transverse resonance method is used to determine the propagation constant of a wave in several waveguides, like the rectangular waveguide, or also dielectric waveguides.
It takes advantage of the fact that a standing wave is present along a certain direction (transverse with respect to the...
This is not part of my homework, but it can make my life much easier. I try to prepare the Excel file as instructed in the link, but I can not find information on how to get the correct value of Theta Dot.
http://www.esm.psu.edu/courses/emch12/IntDyn/course-docs/Euler-tutorial/
I'm sorry that...
G'Day Everybody, I am computing buckling of a rectangular(edges: a*b) specially orthotropic composite plate. The boundary conditions are clamped-clamped at the opposing loaded edges and the other two edges are free (also known as CFCF). After having the boundary conditions and the governing...
Hello All,
I'm working on a problem which uses the largest remainder method
https://en.wikipedia.org/wiki/Largest_remainder_method
I need to allocate a trade quantity among 2 or more strategies.
e.g.
Trade Qty = 99
Strategy A ratio = 0.61
Strategy B ratio = 0.09
Strategy C ratio = 0.23...
So I'm getting ready for an exam on tuesday, and I'm using each method for volumes of revolutions for every problem but I'm not getting the same answers. So, let's use this as an example:
y = 5x; the shaded region is from [1,2]
Using the disk method (about the x-axis) I find:
R(x) = 5x; r(x)...