Chapter 2 Theory of First Order Differential Equations

1. Chapter 2Theory of First Order Differential Equations Shurong Sun University of Jinan Semester 1, 2011-2012

2. 1Existence-uniqueness theorem (Picard’s Method of Successive Approximations) Consider the initial-value problem (1) Q1. How to know that the IVP (1) actually has a solution if we can’t exhibit it? Q2. How do we know that there is only one solution y(t) of (1)?

3. The short answer to the question, how do youknow the equation has a solution, is ‘Picarditeration’. Picard’s Existence and Uniqueness Theorem is one of the most important theorems in ODE! Why is Picard’s Theorem so important? Iteration means to do something over and over.

4. (i) Rewrite the differential equation as an integral equation: (ii) Pick a function satisfying the initial condition. Pick Define Then we iterate: define

5. define and having defined This gives a sequence of functions Picard iteration The solution is when Picard iteration works. Intuitively, is the solution since

6. and But then

7. Remark: Was that a foolproof proof? Almost, but the formula is called ‘taking the limit under the integral sign’ and it is not always correct. We need to add some further conditions on f to make it work. But basically this is the idea.

8. Theorem 1.1 (Existence and uniqueness for the Cauchy problem) Let and suppose that • is continuous, • satisfies a Lipschitz condition with respect to y. Then the Cauchy problem (1) has a unique solution on where

11. Proof of the Theorem We will prove the theorem in five steps: • Show that the initial value problem (1) is equivalent to the integral equation (2)

12. (ii) Set up a sequence of Picard iteration and show that they are well defined. (iii) Show that our sequence of Picard iteration converges uniformly. (iv) Show that the function to which our sequence converges is a solution to the IVP. (v) Show that this solution is unique.

13. (i) The initial-value problem (1) is equivalent to the integral equation (2) Specifically, if y(t) satisfies (1), then (2) Hence

14. Conversely, if y(t) is continuous and satisfies (2), then Moreover, Therefore, y(t) is a solution of (1) if, and only if, it is a continuous solution of (2).

15. (ii)Construction of the approximating sequence Let us start by guessing a solution of (2). The simplest possible guess is To check whether is a solution, we compute If then is indeed a solution of (2). If not, then we try as our next guess.

16. To check whether is a solution, we compute and so on. In this manner, we define a sequence of functions (3) where are called successive approximation, These functions or Picard iteration, after the French mathematician Picard who first discovered them.

17. Define the sequence of Picard iteration as To ensure that Picard iteration are well defined it is sufficient to ensure that each stays within [-b,b], that is (4) It’s convenient to assume a similar proof will hold for

18. We prove (4) by induction on n. Observe first that (4) is obviously true for n=0, since Next, we must show that (4) is true for n=j+1 if it is true for n=j. But this follows immediately, for then . for

19. (iii) We now show that our sequence of Picard iteration converges uniformly for t in the interval This is accomplished by writing in the form Clearly, the sequence converges if, and only if, the infinite series converges .

20. To prove the infinite series converges, it suffices to show that Observe that (5)

21. Setting n=1 in (5) gives This, in turn, implies that

22. Proceeding inductively, we see that Therefore, for Consequently, converges uniformly on the interval since

23. We denote the limit of the sequence by y(t), which is continuous on (iv) Prove that y(t) satisfies the initial-value problem (1) We will show that y(t) satisfies the integral equation To this end, recall that the Picard iteration are defined recursively through the equation (6) Taking limits of both sides of (6) gives

24. Note that converges uniformly to f(s,y(s)) on since and converges uniformly to y(t) on Then one can exchange integral and the limit. Thus and y(t) satisfies

25. (v) Show that the solution is unique. (i) Let z(t) be also a solution of (2). By induction we show that (7) We have and therefore which (7) holds for n = 0.

26. If (7) holds for n − 1 we have and this proves for all

27. Consequently, since This implies that converges uniformly to z(t) on And so

28. (v) (ii) It remains to prove uniqueness of the solution. Let y(t) and z(t) be two solutions of (2). By recurrence we show that (7) We have and therefore which (7) for k = 0.

29. If (7) for k − 1 holds we have and this proves for all

30. Consequently, since This implies that

31. Remark: (1) Geometrically, this means the graph of is constrained to lie in the shaded region shown below.

34. Remark: (2) Geometrically, this means the graph of is constrained to lie in the shaded region shown below.

35. (3) If is continuous in the rectangle A , then f satisfies a Lipschitz condition, since where Note that f satisfies a Lipschitz condition is continuous. For example: f(t,y)=|y|.

36. Corollary : Let f and be continuous in the rectangle Compute Then the initial problem has a unique solution y(t) on the interval

37. Example 1.1 Show that the solution y(t) of the initial-value problem exists for and in this interval, Solution: Let Computing we see that y(t) exists for and in this interval,

38. Remark (4)The initial value problem (IVP) where has a unique solution valid on [a, b].

39. Proof (Method I) The unique solution is to the IVP is

40. Method II Successive approximation • Show that the initial value problem (1) is equivalent to the integral equation (2) Specifically, y(t) is a solution of (1) on [a,b] if, and only if, it is a continuous solution of (2) on [a,b].

41. (ii)Construction of the approximating sequence Then are well defined on [a,b] for all (iii) Show that the sequence of Picard iteration converges uniformly on the interval converges if, and only if, the infinite series converges .

42. Take Then (iv) Show that the limit of the sequence is a continuous solution to the integral equation (2) on [a,b]. (v) Show that this solution is unique. Take Then

43. Theorem 1.2 (Peano 1890) Let and suppose that is continuous with Set Then the Cauchy problem (1) has a solution on

44. Estimates of error and approximate calculation The method of Picard iteration not only give the solution but also found functions which approximate it well, i.e. Let y(t) be the exact solution of the IVP and the Picard scheme for solving this is

45. If we make in approximating the solution y(t) by then the error is called the nth approximate solution to the IVP.

46. Example1.2 Let (i) Determine the interval of existence that the Existence Theorem predicts for the solution y(x) of the initial-value problem (ii) Determine an approximate solution of the IVP above such that the error is less than 0.05.

47. Solution: the existence and uniqueness theorem holds. From the Existence Theorem, has a unique solution on [-h,h]. Here Thus, the interval of solution is

48. Since we can take Take then .

49. The successive approximates are: is the approximate solution desired,

50. Example 1.3The initial value problem is equivalent to solving the integral equation Let We obtain