This work presents an analytical study of several partial differential equations commonly used to model physical phenomena such as heat diffusion, wave propagation, and fluid flow. Emphasis is placed on the use of trigonometric functions to derive exact or synthetic solutions. The heat equations are then is examined using Fourier series and a complex-variable approach. The linearized Saint-Venant equations are then analyzed to describe shallow water wave propagation. The Burgers, in both inviscid and viscous forms, is used to illustrate nonlinear effects, damping, and shock formation. Finally, the Korteweg-de Vrie equation is discussed through its soliton solution, highlighting the balance between nonlinearity and dispersion. These results underline the importance of analytical and trigonometric methods in the modeling of thermal and hydraulic phenomena.
Kader et al. (2025) studied this question.