<p>This study investigates critical aspects of tsunami wave dynamics, highlighting their potential threat to coastal infrastructure. The model studied is a complex system governed by non-linear partial differential equations, forming the foundational structure for our exploration. The study delves into the impact of fractional orders on the shallow water wave equation associated with tsunami waves. The temporal variable is discretized using the finite difference method to solve the mathematical model, while the spatial variable undergoes discretization through radial basis functions (RBFs). The reliability of the computational code is confirmed by comparing it against a known analytical solution to an integer-order model. We explore the influence of time- and space-fractional orders, coast slope, and sea depth on tsunami wave velocity and run-up height. The results are presented through graphical representations, and tabular data support the findings. This comprehensive exploration contributes valuable insights into the intricate dynamics of tsunami waves, providing a deeper understanding of their behaviour under the influence of fractional orders and various environmental factors.</p>