4 1. INTRODUCTION

5. Heat (or diﬀusion) equation

ut − Δu = 0.

6. Schr¨odinger’s equation

iut + Δu = 0.

7. Kolmogorov’s equation

ut −

n

i,j=1

aijuxixj

+

n

i=1

biuxi

= 0.

8. Fokker–Planck equation

ut −

n

i,j=1

(aiju)xixj

−

n

i=1

(biu)xi

= 0.

9. Wave equation

utt − Δu = 0.

10. Klein–Gordon equation

utt − Δu +

m2u

= 0.

11. Telegraph equation

utt + 2dut − uxx = 0.

12. General wave equation

utt −

n

i,j=1

aijuxixj

+

n

i=1

biuxi

= 0.

13. Airy’s equation

ut + uxxx = 0.

14. Beam equation

utt + uxxxx = 0.