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

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

Lanchester’s Law* “Modern Combat” Equations: * Published in 1914. An attrition model to illustrate the benefits of force concentration 3

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

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

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

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 = 52,735. All resistance eliminated by Day 36. Video: http://sonicbomb.com/xv1.php?vid=iwo_jima&id=343&ttitle=The%20Battle%20for%20 Iwo%20Jima&s=45&w=560&h=420 7

Matt’s Results

Matt’s Results

Matt’s Results

Matt’s Results

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

Engel’s Results

Matt’s Results

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 %

Richardson’s Extrapolation Error Analysis Richardson’s Extrapolation

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

Questions