Use the strategy of Example 3 to find a value of h for Euler's method such that y(1) is approxim

Use the strategy of Example 3 to find a value of h for...

Key Concepts

Euler's Method

Euler's method is a numerical technique for approximating the solution of ordinary differential equations (ODEs) with a given initial value. It employs a stepwise process in which the tangent line (given by the derivative function) at the current point is used to estimate the next point in the solution curve. This method is simple but introduces errors that accumulate over successive steps, and understanding its mechanics is key in numerical analysis of differential equations.

Step Size and Error Tolerance

The step size (h) in Euler's method determines the intervals at which the approximation is computed and directly influences the accuracy of the numerical solution. A smaller step size usually results in a more accurate approximation by reducing local truncation error, but at the cost of increased computational effort. Selecting h to meet a specific error tolerance involves analyzing the propagation of errors and balancing accuracy with efficiency.

Initial Value Problem (IVP)

An initial value problem is a differential equation coupled with an initial condition that specifies the value of the unknown function at a particular point. IVPs are fundamental to the study of differential equations because they provide a unique solution that describes the behavior of a system over an interval. Understanding IVPs is critical when applying numerical methods like Euler's method to approximate the solution.

Exact Solution Verification

Comparing numerical approximations with an exact solution is a common practice for verifying the accuracy and reliability of a numerical method. The exact solution, if available, serves as a benchmark, and the differences between the exact and numerical solutions help assess the effectiveness of the chosen step size and the method's error characteristics.

Graphical Comparison of Numerical and Analytical Solutions

Plotting both the numerical (polygonal-line approximation from Euler's method) and analytical (exact solution) curves on the same graph provides a visual assessment of the accuracy of the numerical method. This comparison helps to illustrate where the numerical method closely follows the actual behavior of the system and where deviations occur, thus giving insights into the performance and limitations of the numerical approximation.

Transcript

00:01 Okay, so we're given this differential equation and this initial condition.

00:05 And this time we're asked to find using oilers method concepts, we want to find an h value, like a step size, so that we can approximate y of 1 within 1 ,100th of the correct value.

00:22 And just to be clear, our x not in this case is zero and our y not in this case is zero.

00:29 And now i won't go through all the steps of using oilers method because we've done that a lot in this section.

00:38 But i will show you that i tried an h going from x is zero to x is one.

00:45 So our final x is x equals 1.

00:49 So we're going from x equals 0 to x equal 1 using different step sizes.

00:54 So i used an h of 0 .1.

00:57 I used an h of 0 .05, and i looked again at an h of 0 .02.

01:05 So decreasing values of h to approximate the solution at y of 1.

01:14 For h is 0 .1, i got a y of 1 to be 0 .3487.

01:24 If i used a smaller h of 0 .05, i got the solution at y of 1 .1.

01:30 One to be 0 .3585.

01:38 And again, decreasing our step size again, i got a solution of 0 .36442.

01:54 Okay.

01:55 And so we're looking for a tolerance of 1 ,100, 0 .01.

02:03 And if you think about rounding these just to 100.

02:07 This would give me 0 .35.

02:11 This guy would give me 0 .36.

02:14 This guy would give me also 0 .36.

02:16 So i've converted by shrinking our step size from 0 .05 to 0 .02, i didn't gain any more accuracy to 2 decimal places, right? these are the same to two decimal places.

02:31 So what that tells me is i could have just stuck with this h value of, 0 .05 and gotten an accuracy of 0 .01.

02:41 So for the first part, find an h value that gives us accuracy within this amount, then that is our solution.

02:47 An h value of 0 .05 will give you that accuracy.

02:53 Okay.

02:54 So the next step of this problem says, can we find some values they're calling x not, such that when i find the solution at that x value, it gives me the value.

03:08 0 .2, it approximates solution 0 .2 within 500s.

03:15 And so in a similar, similar strategy, i used an h value, looking at all the steps along the way for an h value of 0 .05 and also for 0 .02...

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