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Differential Equations and Integration

Differential Equations and Integration. Module 6 Lecture 3. Topic: Numerical Integration of Differential Equations – Euler’s Method.

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Differential Equations and Integration

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  1. Differential Equations and Integration Module 6 Lecture 3

  2. Topic: Numerical Integration of Differential Equations – Euler’s Method

  3. Water flows out of the bottom of a tank of muddy water through a small tap at a rate which is proportional to the volume. However, with time, the tap slowly clogs up and so the rate of flow of water is inversely proportional to the time mud Water flow

  4. Water flows out of the bottom of a tank of muddy water through a small tap at a rate which is proportional to the volume. However, with time, the tap slowly clogs up and so the rate of flow of water is inversely proportional to the time mud Water flow

  5. litres hours Experiment shows that the constant of proportionality is 0.5 when the water is measured in litres and time is measured in hours The tap is turned on at midnight and by 10am the tank holds 24000 litres. How much water is left in the tank at 3pm?

  6. litres hours Experiment shows that the constant of proportionality is 0.5 when the water is measured in litres and time is measured in hours The tap is turned on at midnight and by 10am the tank holds 24000 litres. How much water is left in the tank at 3pm?

  7. litres hours Experiment shows that the constant of proportionality is 0.5 when the water is measured in litres and time is measured in hours The tap is turned on at midnight and by 10am the tank holds 24000 litres. How much water is left in the tank at 3pm?

  8. We can solve this problem … by the method of separation of variables. In fact, can you show that But let us imagine that the DE is too difficult to solve (as with 99% of the DE’s which arise in “real/interesting” problems) We will try to get an approximate/numerical solution to the DE/problem

  9. In Module 5, lecture 1, we had a set of data and we obtained an approximate/numerical solution to the derivative (numerical differentiation) Here, we have the opposite problem, we have the derivative ( dV/dt ) and we want to get an approximate/numerical solution for V (numerical integration) As we shall see, these two methods are quite similar

  10. Let us begin with the initial condition. We know what V is here. We immediately know dV/dt and we can use this information to move forward in time

  11. We can estimate the value at t = 15 as follows …

  12. We can estimate the value at t = 15 as follows …

  13. For a general problem, use Then, if we know the value of f at t, we can estimate the value at t+h through

  14. Our estimate of V(15) is very crude. We can increase the accuracy by reducing the “step size” h We can use a table …

  15. Another example Given with use Euler’s method with a step size 1 to find

  16. Class Exercise Given with use Euler’s method with a step size 0.5 to find Challenge Question: Find with

  17. Euler’s Method … what you need to know • Use Euler’s method to solve differential equations

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