Dear Sir, there is a useful book for (A.P S. Selvadurai) called "Paritial differential equation in mechanics 1". it will be interesting.

The linear partial differential equations of first-order is used in some topics such as:

traffic flow

gas dynamics

waves in shallow water

thermal efficiency of heat exchangers

heat propagation between two superconducting cables

👇👇👇👇👇👇👇👇👇👇

Dear Dr. Zamani, discussion of some quasi-linear first order PDE can be found in

Ian Sneddon, Elements of Partial Differential Equations

Note that Burger's equation on mechanical transport of fluids is perhaps the most known application of a quasi-linear first order PDE.

Gianluca

Systems of first-order quasilinear PDEs especially often arise in fluid and gas mechanics; the most prominent example is provided by the Euler equations

https://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics)

and their various special cases and reductions.

It should be recalled that systems of first-order quasilinear PDEs are also known as dispersionless systems or hydrodynamic-type systems, see e.g. the article by G. El

Data HYDRODYNAMIC TYPE SYSTEMS AND THEIR INTEGRABILITY: Introduct...

and my article

Article New integrable (3+1)-dimensional systems and contact geometry

and references therein.