Numerical analysis w/euler's method

In summary: You can then use this method to solve for the concentration at t=1 day and plot the solution on a semilog graph.In summary, the amount of a uniform distributed radioactive contaminant contained in a closed reactor is measured by its concentration (c) in Becquerel/liter or Bq/L. The contaminant decreases at a decay rate proportional to its concentration, given by the equation decay rate(dc/dt)= -kc. In order to solve this equation, we can use Euler's Method with a step size of Delta t = 0.1 day and a constant value of k=0.2 day^-1. By plugging in the initial values of t=0 and c=10, we can solve for
  • #1
solowa4
3
0
The amount of a uniform distributed radioactive contaminant contained in a closed reactor is measured by its concentration (c) (Becquerel/liter or Bq/L). The contaminant decreases at a decay rate proportional to its concentration; that is

Decay rate(dc/dt)= -kc
Where (dc/dt) is the change in mass, (k) is a constant with units of (day^-1), and (-kc) is the decrease by decay.

a-use euler's method to solve this eq from t=0 to 1 day with k=0.2 day^-1. Employ a step size of Delta t = 0.1 day. The concentration at t=0 is 10 Bq/L.

b-plot the solution on a semilog graph (ie (ln c) versus (t) and determine th slope.

I am at a loss at where to begin or accomplish this.
 
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  • #2
Welcome to PF!

Well, the first step is understanding how Euler's Method works. What do you know about it?
 
  • #3
Not much as I just started the class :( That's why it is giving me difficult time
 
  • #5
I have already looked at this and other sources with no help to my problem.
Thanks for your help
 
  • #6
solowa4 said:
I have already looked at this and other sources with no help to my problem

Sure it does. It explains the methodology:

y(n+1) = y(n) + h*y'(n)
t(n+1) = t(n) + h

For the Wiki example (i.e. y' = y and h = 1)

t(0) = 0, y(0) = 1, y'(0) = y(0) = 1
t(1) = 0 + 1 = 1, y(1) = y(0) + h*y'(0) = 1 + 1*1 = 2, y'(1) = y(1) = 2
t(2) = 1 + 1 = 2, y(2) = y(1) + h*y'(1) = 2 + 1*2 = 4, y'(2) = y(2) = 4
t(3) = 3 + 1 = 3, y(3) = y(2) + h*y'(2) = 4 + 1*4 = 8, y'(3) = y(3) = 8

In your case, you know that t(0) = 0, y(0) = 10, h = 0.1, and y' = -ky with known value for k.
 

What is numerical analysis?

Numerical analysis is a branch of mathematics that deals with the development and application of algorithms for solving mathematical problems using computers or other numerical methods. It involves the use of mathematical models and numerical approximations to find solutions to problems that are difficult or impossible to solve analytically.

What is Euler's method?

Euler's method is a numerical method for solving ordinary differential equations. It involves approximating the solution of an initial value problem by using a sequence of small, linear steps. This method is based on the idea of using the tangent line to approximate the solution at each step, and it is named after the Swiss mathematician Leonhard Euler.

How does Euler's method work?

Euler's method works by approximating the solution of an initial value problem at discrete points along the domain. It starts with an initial value and uses the derivative at that point to calculate the slope of the tangent line. Then, it uses this slope to estimate the value of the solution at the next point. This process is repeated until the desired accuracy is achieved.

What are the advantages of using Euler's method?

Euler's method is a simple and easy-to-implement numerical method. It is also computationally efficient and can be used to approximate the solution of a wide range of differential equations. Additionally, it can handle nonlinear equations and does not require advanced mathematical knowledge to use.

What are the limitations of Euler's method?

Euler's method has some limitations, including the fact that it can produce inaccurate results if the step size is too large. It also assumes that the function is continuous and differentiable, which may not always be the case in real-world problems. Furthermore, it may not converge to the correct solution if the initial value is far from the actual solution.

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