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After extensive examination, scholars have determined that many dynamic systems exhibit intricate connections not only with their current and past states but also with the delay function itself. As a result, their focus shifts towards fractional neutral stochastic differential equations, which find applications in diverse fields such as biology, physics, signal processing, economics, and others. The fundamental principles of existence and uniqueness of solutions to differential equations, which guarantee the presence of a solution and its uniqueness for a specified equation, are pivotal in both the mathematical and physical realms. A crucial approach for analyzing complex systems of differential equations is the utilization of the averaging principle, which simplifies problems by approximating existing ones. Applying contraction mapping principles, we present results concerning the concepts of existence and uniqueness for the solutions of fractional neutral stochastic differential equations. Additionally, we present Ulam-type stability and the averaging principle results within the framework of Lp space. This exploration involved the utilization of Jensen’s, Gröenwall-Bellman’s, Hölder’s, Burkholder-Davis-Gundy’s inequalities, and the interval translation technique. Our findings are established within the context of the conformable fractional derivative, and we provide several examples to aid in comprehending the theoretical outcomes. [ABSTRACT FROM AUTHOR]

One kind of stochastic delay differential equation in which the delay term is dependent on a proportion of the current time is the pantograph stochastic differential equation. Electric current collection, nonlinear dynamics, quantum mechanics, and electrodynamics are among the phenomena modeled using this equation. A key idea in physics and mathematics is the well-posedness of a differential equation, which guarantees that the solution to the problem exists and is a unique and meaningful solution that relies continuously on the initial condition and the value of the fractional derivative. Ulam-Hyers stability is a property of equations that states that if a function is approximately satisfying the equation, then there exists an exact solution that is close to the function. Inspired by these findings, in this research work, we established the Ulam-Hyers stability and well-posedness of solutions of pantograph fractional stochastic differential equations (PFSDEs) in the framework of conformable derivatives. In addition, we provided examples to analyze the theoretical results. [ABSTRACT FROM AUTHOR]

The mathematical efficiency of fractional-order differential equations in modeling real systems has been established. The first-order and second-order time derivatives are substituted in integer-order problems by a fractional derivative of order 0 < ω ≤ 1 , resulting in time-fractional heat and wave problems with variable coefficients. In this research, we analyze fractional-order wave and heat problems with variable coefficients within the framework of a Caputo derivative (CD) using the Elzaki residual power series method (ERPSM), which is a coupling of the residual power series method (RPSM) and the Elzaki transform (E-T). It relies on a novel form of fractional power series (FPS), which provides a convergent series as a solution. The accuracy and convergence rates have been proven by the relative, absolute, and recurrence error analyses, demonstrating the validity of the recommended approach. By employing the simple limit principle at zero, the ERPSM excels at calculating the coefficients of terms in a FPS, but other well-known approaches such as Adomian decomposition, variational iteration, and homotopy perturbation need integration, while the RPSM needs the derivative, both of which are challenging in fractional contexts. ERPSM is also more effective than various series solution methods due to the avoidance of Adomian's and He's polynomials to solve nonlinear problems. The results obtained using the ERPSM show excellent agreement with the natural transform decomposition method and homotopy analysis transform method, demonstrating that the ERPSM is an effective approach for obtaining the approximate and closed-form solutions of fractional models. We established that our approach for fractional models is accurate and straightforward and researcher can use this approach to solve various problems. [ABSTRACT FROM AUTHOR]

Some researchers have combined two powerful techniques to establish a new method for solving fractional-order differential equations. In this study, we used a new combined technique, known as the Elzaki residual power series method (ERPSM), to offer approximate and exact solutions for fractional multipantograph systems (FMPS) and pantograph differential equations (PDEs). In Caputo logic, the fractional-order derivative operator is measured. The Elzaki transform method and the residual power series method (RPSM) are combined in this novel technique. The suggested technique is based on a new version of Taylor's series that generates a convergent series as a solution. Establishing the coefficients for a series, like the RPSM, necessitates computing the fractional derivatives each time. As ERPSM just requires the concept of a zero limit, we simply need a few computations to get the coefficients. The novel technique solves nonlinear problems without the need for He's and Adomian polynomials, which is an advantage over the other combined methods based on homotopy perturbation and Adomian decomposition methods. The relative, recurrence, and absolute errors of the problems are analyzed to evaluate the efficiency and consistency of the presented method. Graphical significances are also identified for various values of fractional-order derivatives. As a result, the procedure is quick, precise, and easy to implement, and it yields outstanding results. [ABSTRACT FROM AUTHOR]