In this work, we first develop the modified time Caputo fractional Kawahara Equations (MTCFKEs) in the usual Hilbert spaces and extend them to analogous structures within the theory of Hilbert algebras. Next, we employ the residual power series method, combined with the Laplace transform, to introduce a new effective technique called the Laplace Residual Power Series Method (LRPSM). This method is applied to derive the coefficients of the series solution for MTCFKEs in the context of Hilbert algebras. In real Hilbert algebras, we obtain approximate solutions for MTCFKEs under both exact and approximate initial conditions. We present both graphical and numerical results of the approximate analytical solutions to demonstrate the capability, efficiency, and reliability of the LRPSM. Furthermore, we compare our results with solutions obtained using the homotopy analysis method and the natural transform decomposition method.

One of the important techniques for solving several partial differential equations is the residual power series method, which provides the approximate solutions of differential equations in power series form.In this work, we use Aboodh transform in the analogical structure of the residual power series method to obtain a new method called the Aboodh residual power series method (ARPSM). By using this technique, we calculate the coefficients of some power series solutions of time Caputo fractional Kawahara equations. To obtain analytical and numerical solutions for the TCFKEs, we use ARPSM, first with the approximate initial condition and then with the exact initial condition. We present ARPSM’s reliability, efficiency, and capability by graphically describing the numerical results for analytical solutions and by comparing our solutions with other solutions for the TCFKEs obtained using two alternative methods, namely, the residual power series method and the natural transform decomposition method.

In the theory of Banach algebras, we use the Schauder fixed-point theorem to obtain some results that concern the existence property for mild solutions of fractional Cauchy problems that involve the Lie bracket operator, the almost sectorial operator, and the $\phi -$Hilfer derivative operator. For any Banach algebra and in two types of non-compact associated semigroups and compact associated semigroups, we prove some properties of the existence of these mild solutions using the Hausdorff measure of a non-compact associated semigroup in the collection of bounded sets. That is, we obtain the existence property of mild solutions when the semigroup associated with an almost sectorial operator is compact as well as non-compact. Some examples are introduced as applications for our results in commutative real Banach algebra $\mathbb{R}$ and commutative Banach algebra of the collection of continuous functions in $\mathbb{R}$.

Using the monophonic paths in the theory of directed graphs, this paper constructs a new topology called the out mondirected topology, which characterizes the graphs that induce indiscrete or discrete topology. We give and study some of the 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 the second messenger signal.

In this work, we use the monophonic eccentric open neighborhood system and the upper approximation neighborhood system to construct a new class of topologies in the theory of undirected simple graphs. We study some fundamental topological properties of this class and characterize the graphs that induce the indiscrete or discrete topology. Next, we present the openness, the connectedness and the continuity properties with isomorphic maps of graphs. Some applications concerning the topological discrete and connectedness properties for some corresponding graphs of the COVID-19 form and its diffusion are introduced.

This study introduces a new class of coupled differential systems described by fractal–fractional Caputo derivatives with both constant and state-dependent delays. In contrast to traditional delay differential equations, the proposed framework integrates memory effects and geometric complexity while capturing adaptive feedback delays that vary with the system’s state. Such a formulation provides a closer representation of biological and physical processes in which delays are not fixed but evolve dynamically. Sufficient conditions for the existence and uniqueness of solutions are established using fixed-point theory, while the stability of the solution is investigated via the Hyers–Ulam (HU) stability approach. To demonstrate applicability, the approach is applied to two illustrative examples, including a predator–prey interaction model. The findings advance the theory of fractional-order systems with mixed delays and offer a rigorous foundation for developing realistic, application-driven dynamical models.