Two-Point Boundary Value Problem

1. Two-Point Boundary Value Problem DE Weak Form 1 2 3 4 Linear System Discrete Form

2. spaces Definition: The space of all square integrable funcions defined in the domain Definition: The function and its first derivative are square integrable Remark: Both spaces are Hilbert spaces. R2 is also a Hilbert space R2 is also a Hilbert space with inner product

3. Triangle inequality Triangle inequality: Triangle inequality: Triangle inequality:

4. Cauchy-Schwarz inequality Cauchy-Schwarz inequality: Cauchy-Schwarz inequality: Cauchy-Schwarz inequality: (integral form) Example: verify CS-inequality

5. Cauchy-Schwarz inequality Is this true?: Cauchy-Schwarz inequality: (integral form)

6. Bilinear form Definition: Definition: The bilinear form is said to be symmetricif a(w, v) = a(v,w), ∀v,w ∈ V, A bilinear form on V is a function : V × V → R, which is linear in each argument separately Definition: the bilinear form a(・, ・) on V is bounded if there is a constant M such that. Example: Example: prove that a is bounded bilinear form on

7. Bilinear form Linear functional Definition: Definition: the bilinear form a(・, ・) on is coercive if there is a constant α > 0such that. A linear functional L : V → R is said to be bounded is the smallest constant c Example: prove that a is coercive on Remark: Example:

8. Lax-Milgram lemma Consider: where Hilbert space bilinear form on linear functional on Lax-Milgram lemma 2 Hilbert space bounded coercive bilinear form on bounded linear functional on Then there exists a unique vector u ∈ V such that (2) is satisfied

9. Lax-Milgram lemma DE Weak Form 1 2 1 2 Example: Show that there exist a unique solution for (2)

10. Lax-Milgram lemma Example: 1 2 Show that there exist a unique solution for (2) In order to show that there exist a unique solution for (2), we need to satisfy all the conditions of Lax-Milgram lemma solution: Poincare’s inequality (HW) proof Show that: Later we will do another proof for a symmetric a

12. Lax-Milgram lemma Example: 1 2 3 Show that there exist a unique solution for (3) solution: Thm: A finite dimensional subspace of a Hilbert space is Hilbert In order to show that there exist a unique solution for (3), we need to satisfy all the conditions of Lax-Milgram lemma

13. Stability Example: Definintion: A problem that satisfies the three conditions is said to be well posed 1)existence of solutions, 2)uniqueness of solutions, 3)stability 2 Poincare’s inequality Stability: continuous dependence of solutions with respect to perturbations of data Setting ϕ = u in (2) and using (coercive) and (Poincare), we find Solution bounded by the data of the problem Small change in the data produce small chang in the solution

14. Linear System of Equations Remark: A is symmetric and positive definite 3 4 Remark: Definition: Under what condition that (4) has solution An nxn matrix A is symmetric and positive definite if (4) has a unique solution iff that the matrix A is invertible ( non-singular ) Example: show that A is SPD

15. Linear System of Equations Remark: A is symmetric and positive definite Remark: Proof: 3 4 Under what condition that (4) has solution (4) has a unique solution iff that the matrix A is invertible ( non-singular ) coercively