Applications: almost every field in science

• Financial engineering, see for example Patriarca et al, Physica 340, page 334 (2004).
• Neuroscience, see for example Lipinski, Physics Medical Biology 35, page 441 (1990) or Farnell and Gibson, Journal of Computational Physics 208, page 253 (2005)
• Tons of applications in physics
• and chemistry
• and biology, medicine
• Nobel prize in economy to Black and Scholes

$$\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2} V}{\partial S^{2}}+rS\frac{\partial V}{\partial S}-rV=0.$$ The Black and Scholes equation is a partial differential equation, which describes the price of the option over time. It is a diffusion equation with a random term.

The list of applications is endless

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