How to resolve the algorithm Euler method step by step in the Mathematica / Wolfram Language programming language

Published on 22 June 2024 08:30 PM
#Wolfram

How to resolve the algorithm Euler method step by step in the Mathematica / Wolfram Language programming language

Table of Contents

Problem Statement

Euler's method numerically approximates solutions of first-order ordinary differential equations (ODEs) with a given initial value.   It is an explicit method for solving initial value problems (IVPs), as described in the wikipedia page. The ODE has to be provided in the following form: with an initial value To get a numeric solution, we replace the derivative on the   LHS   with a finite difference approximation: then solve for

y ( t + h )

{\displaystyle y(t+h)}

: which is the same as The iterative solution rule is then: where

h

{\displaystyle h}

is the step size, the most relevant parameter for accuracy of the solution.   A smaller step size increases accuracy but also the computation cost, so it has always has to be hand-picked according to the problem at hand.

Example: Newton's Cooling Law Newton's cooling law describes how an object of initial temperature

T (

t

0

)

T

0

{\displaystyle T(t_{0})=T_{0}}

cools down in an environment of temperature

T

R

{\displaystyle T_{R}}

: or

It says that the cooling rate

d T ( t )

d t

{\displaystyle {\frac {dT(t)}{dt}}}

of the object is proportional to the current temperature difference

Δ T

( T ( t ) −

T

R

)

{\displaystyle \Delta T=(T(t)-T_{R})}

to the surrounding environment. The analytical solution, which we will compare to the numerical approximation, is

Implement a routine of Euler's method and then to use it to solve the given example of Newton's cooling law with it for three different step sizes of: and to compare with the analytical solution.

A reference solution (Common Lisp) can be seen below.   We see that bigger step sizes lead to reduced approximation accuracy.

Let's start with the solution:

Step by Step solution about How to resolve the algorithm Euler method step by step in the Mathematica / Wolfram Language programming language

The provided Wolfram code solves a differential equation using the Explicit Euler method and calculates the temperature (T) at a specific value (val) of the independent variable (t). Here's a detailed explanation of the code:

  1. euler[step_, val_] :=: This is a function definition that takes two arguments: step (the step size) and val (the value of t at which we want to calculate the temperature). It returns the temperature value calculated using the Explicit Euler method.

  2. NDSolve[...]:: This is the main differential equation solver function provided by Wolfram. It takes several arguments:

    • {T'[t] == -0.07 (T[t] - 20), T[0] == 100}: This is the differential equation we want to solve. It represents the change in temperature (T) over time (t). The specific equation given here describes the cooling of an object with an initial temperature of 100 degrees and a cooling rate proportional to the temperature difference between the object and the ambient temperature (20 degrees).
    • T: The dependent variable, which is the temperature.
    • {t, 0, 100}: The independent variable (t) and its range. We are solving for t values from 0 to 100.
    • Method -> "ExplicitEuler": This specifies the method to use for solving the differential equation. In this case, we are using the Explicit Euler method.
    • StartingStepSize -> step: This sets the initial step size for the Explicit Euler method. The step size determines the accuracy and speed of the numerical solution.

  3. [[1, 1, 2]][val]: This part of the code extracts the temperature value at the specified val from the solution obtained by NDSolve.

    • [[1, 1, 2]]: This retrieves the second element ([[2]]) of the first element ([[1]]) of the first element ([[1]]) of the solution returned by NDSolve. In this case, the first element is a list of solutions for different step sizes, the second element is a list of time points, and the third element is a list of temperature values.
    • [val]: This evaluates the temperature value at the specified val of t.

In summary, this code defines a function (euler) that solves a differential equation using the Explicit Euler method and calculates the temperature at a given value of the independent variable. It can be used to model cooling processes or other scenarios described by differential equations.

Source code in the wolfram programming language

euler[step_, val_] := NDSolve[
    {T'[t] == -0.07 (T[t] - 20), T[0] == 100},
    T, {t, 0, 100},
    Method -> "ExplicitEuler",
    StartingStepSize -> step
][[1, 1, 2]][val]

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