Section 7 1 7 2 Regular Perturbation Theory

Section 7. 1 & 7. 2 Regular Perturbation Theory Applied to Differential Equations Benjamin G. Deaver February 15, 2017

2 Agenda Review of Perturbation Method In Depth Example 7. 2 – Solving a Differential Equation Summary of Perturbation Method Questions & Answers Benjamin G. P. Deaver 10/30/2021

Review of Perturbation Method 3 Benjamin G. P. Deaver 10/30/2021

Example 2 – Solving a Differential Equation 4 Benjamin G. P. Deaver 10/30/2021

Step 1 – Introduce Small Parameter 5 Benjamin G. P. Deaver 10/30/2021

Step 2 – Expand y(x) into power series 6 Benjamin G. P. Deaver 10/30/2021

Step 2 – Evaluate Boundary Conditions 7 Benjamin G. P. Deaver 10/30/2021

Step 2 – Evaluate Boundary Conditions 8 Benjamin G. P. Deaver 10/30/2021

Step 2 – Evaluate the Zeroth Ordered Problem 9 Benjamin G. P. Deaver 10/30/2021

Step 2 – Evaluate the Zeroth Ordered Problem Part 2 10 Benjamin G. P. Deaver 10/30/2021

Step 2 – Determine the nth order problem 11 Benjamin G. P. Deaver 10/30/2021

Step 2 – nth order problem continued 12 Benjamin G. P. Deaver 10/30/2021

Step 2 – nth order problem continued 13 Benjamin G. P. Deaver 10/30/2021

Step 2 – Solving this Differential Equation 14 Benjamin G. P. Deaver 10/30/2021

Step 3 – Summing the Perturbation Series 15 Benjamin G. P. Deaver 10/30/2021

Step 3 – Summing the Perturbation Series 16 Benjamin G. P. Deaver 10/30/2021

Step 3 – Summing Up the Perturbation Series 17 Benjamin G. P. Deaver 10/30/2021

18 This is an example from the book of how few terms are needed to converge to the exact solution using the Example 7. 2 differential equation with a determined f(x) = -e-x Note that the 11 term Taylor Series approximation is only good around x=0. Typically, the perturbation method is only done for 2 terms because it is quick; if more accuracy is required, other methods are utilized. Benjamin G. P. Deaver 10/30/2021

Review of Perturbation Method 19 Benjamin G. P. Deaver 10/30/2021

20 Questions Benjamin G. P. Deaver 10/30/2021