1. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1 Part 6 - Chapter 24 Boundary Value Problems

2. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Review of Initial Value Problems The previous 2 chapters focused on solving initial value problems If you have –a first order differential equation you need a single initial value –System of n equation you need n initial values –An nth order differential equation you need n initial values … because you reformulate into a system of n equations 2

3. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Boundary Value Problems Instead of knowing the starting value of the function you know the value at some other point You need the same number of conditions –a first order differential equation you need a single boundary value –System of n equation you need n boundary values –An nth order differential equation you need n boundary values … because you reformulate into a system of n equations 3 They just aren’t the starting conditions

4. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 24.1 Temperature of a heated rod subject to convection 4

5. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Second Order Differential Equation Therefore you need two boundary conditions The two end temperatures are fixed – which are the conditions we need 5

6. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Approaches MATLAB offers two approaches –Symbolic Algebra –Numerical Solution – bvp4c 6

7. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 7 Initial Condition Boundary Condition at t=10

8. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. dsolve With dsolve there is no need to decompose the problem into a system of equations The boundary conditions can be –Initial conditions –Conditions at some other value of the independent variable –Values of the dependent variable –Values of derivatives of the dependent variable 8

9. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Numerical Approach Chapter 24 discusses a variety of numerical approaches –Shooting technique –Finite difference technique We will focus on the built-in bvp4c function which uses a finite difference approach 9

10. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. bvp4c This function requires 3 inputs –function handle to a function that evaluates the differential equations, just like we used in the odesolvers –Function handle to a function that evaluates the residual (the error) in the boundary condition approximations –A structure array representing the initial guess for the solution 10

11. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 11

12. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 12 This is our system of equations that represent the original second order equation

13. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 13 T is an array T(1) is the initial value of T T(2) is the initial value of dT/dx The trouble is… we don’t know T(2)

14. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 14

15. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 15 bvpinit returns a structure array of guesses for the initial values of x and y (In our case x, T and dT/dx)

16. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 16