1. A Numerical Analysis of the Battle of Iwo Jima
Engel’s Lanchester Law Verification
Presented By:
LT Matthew J. Malinowski

2. Introduction What I solved numerically Lanchester’s Laws Equations
Spoiler: System of Coupled 1st Order Linear ODE’s
Lanchester’s Laws Equations
Background on Iwo Jima
Results
Validation to Engel’s Results 2

3. Lanchester’s Law* “Modern Combat” Equations:
* Published in An attrition model to illustrate the benefits of force concentration 3

4. Initial Conditions Battle of total annihilation* Mo = 0 No = 21,500
Reinforcements
* Not Numerically 4

5. Battle of Iwo Jima (Operation Detachment)
Capture of Island of Iwo Jima (IJ) from Japan in Pacific Campaign of WWII. 19 Feb – 26 Mar 1945.
Location & Strategic Importance
Early warning of bombers to main land (Jap).
Interception of B-29 bombers, provision of support to naval units (Jap).
Staging area for invasion of mainland (US).
Reduce B-29 bombing dist, allow P-51 Mustangs to escort and protect bombers (US). 5

6. Japan’s Strategy Commander – Lt Gen Kuribayashi
Strong supporting defensive positions, network of tunnels, bunkers & pillboxes, hidden artillery & mortars
Importance of Mt Suribachi
“No Japanese survivors”
Iwo Jima was Japanese home soil, part of Japan, only 650 miles from Tokyo
To the US, Iwo Jima's importance lay in its location, midway between Japan and American bomber bases in the Marianas.
550-foot volcanic cone at the islands southern tip, dominates both possible landing beaches. From here, Japanese gunners zeroed in on every inch of the landing beach. Blockhouses and pillboxes flanked the landing areas. Within, more heavy weapons stood ready to blast the attacking Marines. Machine guns criss-crossed the beaches with deadly interlocking fire. Rockets, anti-boat and anti-tank guns were also trained on the beaches. Every Marine, everywhere on the island was always in range of Japanese guns. 6

7. The Battle
19 Feb 1945, 0859 hrs.
1st Day Landing of 54,000 US Marines, 21,500 Japanese held fire. Heavy fire and the devastating blows from machine gunners.
Isolating Mt Suribachi from rest of the island, capturing of Mt Suribachi.
After securing landing area, airfields towards the north and remainder of IJ was captured.
Island secured by Day 28. M = 52, All resistance eliminated by Day 36.
Video: Iwo%20Jima&s=45&w=560&h=420 7

8. Matt’s Results

9. Matt’s Results

10. Matt’s Results

11. Matt’s Results

12. A Verification of Lanchester's Law* J. H. Engel (December 1953)
* A Verifications of Lanchester's Law, J.H. Engel, Massachusetts Institute of Technology, 1953 12

13. Engel’s Results

14. Matt’s Results

15. Error Analysis E(28) = 5,032 E(28) = 35,315 E(28) = 9.5% E(28) = 67 %
Exact Value
ODE45
RK4
E(28) = 𝑁𝑢𝑚𝑒𝑟𝑖𝑐𝑎𝑙 −𝐸𝑥𝑎𝑐𝑡
E(28) = 47,403 −52,735
E(28) = 5,032
E(28) = 9.5%
E(28) = 𝑁𝑢𝑚𝑒𝑟𝑖𝑐𝑎𝑙 −𝐸𝑥𝑎𝑐𝑡
E(28) = 17,420 −52,735
E(28) = 35,315
E(28) = 67 %

16. Richardson’s Extrapolation
Error Analysis
Richardson’s Extrapolation

17. Conclusion Validated the ode45 results
Solved System of Coupled 1st Order Linear ODE’s Numerically
ode45 & RK4
Analyzed the Error
Between the Methods
To Exact Values
Validated the ode45 results
Found that if we use RK4 the Japanese may have won Iwo Jima 17

18. Questions