機械系応用数学(Applied Mathematics for Mechanical Engineering)

The objective of this course is to brush up the mathematical skills required in advanced mechanical engineering through solving the example problems. Besides, solutions of partial differential equations are explained with the emphasis of the related physics. The methods using Green’s functions for nonhomogeneous differential equations are explained. Singular perturbation problems and renormalization technique are also explained as tools to obtain the approximate solutions. Furthermore, homogenization method, optimal design, structural optimization and sensitivity analysis are explained.

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Contents: 1.Delta Function and Distribution Theory (デルタ関数と超関数理論) 2.Review of Ordinary Differential Equations and Green’s Function　 (常微分方程式とグリーン関数の復習） 3.Review of Partial Differential Equations and Green’s Function　 (偏微分方程式とグリーン関数の復習） 4.Regular Perturbation Method （正則摂動法） 5.Domain Perturbation Method (領域摂動法) 6.Multiple-Time-Scale-Type Singular Perturbation Problems（多重時間スケール型特異摂動問題） 7.Boundary-Layer-Type Singular Perturbation Problems（境界層型特異摂動問題） 8.Fractal Structure and Renormalization Method（フラクタル構造と繰り込み法） 9.Renormalization Method for Singular Perturbation Problems-1（特異摂動問題に対する繰り込み法） 10.Homogenization Method Using the Asymptotic Expansions（漸近展開による均質化法） 11.Optimal Design（最適設計） 12.Structural Optimization（構造最適化） 13.Sensitivity Analysis（感度解析） 14. Final Exam　（期末テスト）