By using the monophonic paths in the theory of directed graphs, this paper constructs a new topology, called the out mondirected topology, and characterizes the graphs that induce the indiscrete or discrete topology. We give and study some relations and properties such as the relationship between the isomorphic relation in directed graphs and the homeomorphic property in out mondirected topological spaces, compactness, ${\mathbb{D}}_{\pm }$-connectedness, connectedness, and ${\mathbb{D}}_{\pm }$-discrete properties. Finally, we apply our results of out mondirected topological spaces in the nervous system of the human body, such as in the messenger signal network, in diagrams of sensory neuron cells, and in models of two distinct nicotinic receptor types based on second messenger signal.

The Windkessel model is a lumped-parameter representation that simplifies the complex cardiovascular system into an equivalent hydraulic circuit comprising resistive and compliant components. It plays a crucial role in understanding the dynamics of blood flow and pressure within the circulatory system, especially in the arterial network. In this study, we employ the Caputo fractional operator to obtain approximate solutions for the Windkessel model within the framework of usual topological Banach spaces. An efficient hybrid analytical technique, termed the Aboodh residual power series method (ARPSM), has been developed by integrating the Aboodh transform with the residual power series method. This approach is used to investigate and derive approximate solutions of the modified Caputo fractional Windkessel model. The accuracy, reliability, and applicability of the proposed ARPSM are demonstrated through numerical and graphical analyses, confirming its effectiveness in solving fractional differential equations of this type.