12 boundary value problem
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This document summarizes a lecture on numerical differentiation and finite difference methods given by Dr. Eng. Mohammad Tawfik. The key points covered include: - Numerical differentiation techniques like forward, backward, and central differences to approximate derivatives from discrete data. - Applying these finite difference methods to transform differential equations into difference equations by substituting derivative approximations. - An example of using forward differences to derive the Euler method for numerically solving differential equations. - An example of using finite differences to solve a 1D heat transfer boundary value problem.
12 boundary value problem
- 1. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Boundary Value Problems
- 2. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Objectives • Applying finite difference as a numerical method for differentiation • Solve a Boundary Value Problem using finite difference method • Applying weighted residual methods for the solution of BVP’s
- 3. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Numerical Differentiation
- 4. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Numerical Differentiation • In most of the practical applications, the differentiation of the function does not present a great challenge! • In some cases, the differentiation becomes very expensive in terms of the computational requirements • If you are using the computer, you need to teach it how to differentiate!!! • If the function is given in the form of a table, what are you going to do?!
- 5. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Numerical Differentiation • In the previous lectures, we learned that the slope of the function may be approximated by a difference expression • Recall, the parachutist: • Note that this evaluates the derivative at the starting point in terms of what is going to happen in the future! 12 12 tt vv t v dt dv − − = ∆ ∆ ≈
- 6. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Forward Difference • When you evaluate the slope at some point in terms of the value of the function at higher values of time/space, then we call this forward difference t vv tt vv dt dv tt ∆ − = − − ≈ = 12 12 12 1
- 7. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik How about the second derivative? • The second derivative of a function is defined as the rate of change of slope. − ∆ ≈ === 121 1 2 2 tttttt dt dv dt dv tdt vd
- 8. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Substitute for the 1st derivative − ∆ ≈ === 121 1 2 2 tttttt dt dv dt dv tdt vd ∆ − − ∆ − ∆ ≈ = t vv t vv tdt vd tt 1223 2 2 1 1 2 123 2 2 2 1 t vvv dt vd tt ∆ +− ≈ =
- 9. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Backward Difference • When you evaluate the slope at some point in terms of the value of the function at lower values of time/space, then we call this backward difference • Note that this is the same formula for the forward difference! t vv tt vv dt dv tt ∆ − = − − ≈ = 21 21 21 2
- 10. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik How about the second derivative? • The second derivative of a function is defined as the rate of change of slope. − ∆ ≈ === 212 1 2 2 tttttt dt dv dt dv tdt vd
- 11. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Substitute for the 1st derivative − ∆ ≈ === 212 1 2 2 tttttt dt dv dt dv tdt vd ∆ − − ∆ − ∆ ≈ = t vv t vv tdt vd tt 2110 2 2 1 2 2 210 2 2 2 2 t vvv dt vd tt ∆ +− ≈ =
- 12. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Central Difference • If you know some information about the previous and the upcoming points, why don’t you make use of it? t vv tt vv dt dv tt ∆ − = − − ≈ = 2 02 02 02 1 2 012 2 2 2 1 t vvv dt vd tt ∆ +− ≈ =
- 13. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Applications Finite Difference
- 14. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Finite Difference • The finite difference is a direct application of the numerical differentiation. • Substitute each derivative in a DE by its FD equivalent to transform the differential equation into difference equation
- 15. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Example: The Parachutist • The DE is: • Applying the FD relation: • You get the Euler formula! m cvmg v − = m cv g t vv dt dv −= ∆ − ≈ 12 −∆+= m cv gtvv 1 12
- 16. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Example: Heat transfer problem dx q1=-kAdT/dx q2=q1+(dq1/dx)dx q3=hPdx(T-Ta) Tin Tout 0 L x Ta d2 T/dx2 - b2 T = -b2 Ta Area A Perim. P (b2 = hP/kA) We need 2 conditions: T(0) = Tin; T(L) = Tout Conditions at 2 boundaries: Boundary value problemBoundary value problem
- 17. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Finite difference methods Discretize into n sub-domains 0 x1 x2 xn=Lxn-1 (equidistant for simplicity) Approximate derivatives using neighboring points d2 T/dx2 |i ≈ (Ti-1-2Ti+Ti+1)/h2 h Apply equation at each internal point: (Ti-1-2Ti+Ti+1)/h2 - b2 Ti = -b2 Ta Solve n-1 equations in n-1 unknowns
- 18. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Homework #10 • Chapter 27, p. 779, numbers 27.1, 27.3 • Due week 26 May 2007