Fractal Structures in Some Non-linear Partial Differential Equations

Fractal-type solutions of the non-linear partial differential equation are obtained by using the Riccati projection method. To obtain the solutions, necessary symbolic computation and the method of separation of variable are performed sequentially. The arbitrariness of the functions in the solution leads to several fractal structures. The ordinary and random fractal structures for different non-linear partial differential equations are depicted. It is significant to note that the structures are self-similar at different scales. Repeating patterns are observed when the frame is in magnification. These types of self-similar structures may be obtained in the formation of water waves and in the space of plasma.