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In the study of biological systems, nonlinear models are commonly employed, although exact solutions are often unattainable. Therefore, it is imperative to develop techniques that offer approximate solutions. This study utilizes the Elzaki residual power series method (ERPSM) to analyze the fractional nonlinear smoking model concerning the Caputo derivative. The outcomes of the proposed technique exhibit good agreement with the Laplace decomposition method, demonstrating that our technique is an excellent alternative to various series solution methods. Our approach utilizes the simple limit principle at zero, making it the easiest way to extract series solutions, while variational iteration, Adomian decomposition, and homotopy perturbation methods require integration. Moreover, our technique is also superior to the residual method by eliminating the need for derivatives, as fractional integration and differentiation are particularly challenging in fractional contexts. Significantly, our technique is simpler than other series solution techniques by not relying on Adomian's and He's polynomials, thereby offering a more efficient way of solving nonlinear problems. [ABSTRACT FROM AUTHOR]

This paper aims to establish the existence and uniqueness of mild solutions to abstract sequential fractional differential equations. The approach employed involves the utilization of resolvent operators and the fixed-point theorem. Additionally, we investigate a specific example concerning a partial differential equation incorporating the Caputo fractional derivative. [ABSTRACT FROM AUTHOR]

In this research work, we use the concepts of contraction mapping to establish the existence and uniqueness results and also study the averaging principle in L p space by using Jensen's, Grönwall–Bellman's, Hölder's, and Burkholder–Davis–Gundy's inequalities, and the interval translation technique for a class of fractional neutral stochastic differential equations. We establish the results within the framework of the Ψ -Caputo derivative. We generalize the two situations of p = 2 and the Caputo derivative with the findings that we obtain. To help with the understanding of the theoretical results, we provide two applied examples at the end. [ABSTRACT FROM AUTHOR]

This paper focuses on studying the uniqueness of the mild solution for an abstract fractional differential equation. We use Banach's fixed point theorem to prove this uniqueness. Additionally, we examine the stability properties of the equation using Ulam's stability. To analyze these properties, we consider the involvement of Hadamard fractional derivatives. Throughout this study, we put significant emphasis on the role and properties of resolvent operators. Furthermore, we investigate Ulam-type stability by providing examples of partial fractional differential equations that incorporate Hadamard derivatives. [ABSTRACT FROM AUTHOR]

In this paper, we consider a weakly system of fractional θ -evolution equations. Using the fixed-point theorem, a global-in-time existence of small data solutions to the Cauchy problem is proved for one single equation. Using these results, we prove the global existence for the system under some mixed symmetrical conditions that describe the interaction between the equations of the system. [ABSTRACT FROM AUTHOR]

In this paper we consider a fractional nonlinearity for the wave equation with friction and viscoelastic damping. Using Fixed point theorem a global in time existence of small data solutions to the Cauchy problem is investigated in this research. Our main interest is to show the influence of the fractional nonlinearity parameter to the admissible range of exponent ς comparing with power nonlinearity and also the generating of loss of decay. [ABSTRACT FROM AUTHOR]

We prove the global existence of small data solution in all spaces of all dimensions n ≥ 1 for weakly coupled systems of semilinear effectively damped wave, with different time-dependent coefficients in the dissipation terms. Moreover, we assume that the nonlinearity terms f (t , u) and g (t , v) satisfy some properties of parabolic equations. We study the problem in several classes of regularity. [ABSTRACT FROM AUTHOR]

We study the global existence of small data solutions to the Cauchy problem for semilinear damped wave equations with an effective dissipation term, where the data are supposed to belong to different classes of regularity. We apply these results to the Cauchy problem for weakly coupled systems of semilinear effectively damped waves with respect to the defined classes of regularity for different power nonlinearities. [ABSTRACT FROM AUTHOR]