This work deals with an analysis of two singularly perturbed partial differential equations with cubic nonlinearities using the conformable derivative approach. We investigate a modified Korteweg-de Vries type equation and a generalized KdV-Burgers equation, both incorporating conformable fractional derivatives and singular perturbation parameters. The conformable derivatives enable a treatment of fractional-order effects in these two nonlinear problems. Using asymptotic expansion methods, we derive solutions up to first-order corrections and examine the differences between perturbed and unperturbed cases. Our analysis shows how the conformable order and perturbation parameters influence wave propagation characteristics, and solution behavior. The results demonstrate qualitative differences from classical integer-order models, particularly in soliton dynamics and pattern formation.

The aim of this work is to prove the existence and uniqueness of solutions for a coupled system that generalizes hybrid pantograph equations incorporating some fractional operators of Caputo and Riemann-Liouville types. This is achieved throughout Banach and Leray-Schauder fixed point theorems. In addition, we investigate the stability by using the Ulam-Hyer technique, and an illustrative example is provided to show the validity of our findings.

. In this paper, we study a generalized nonlinear shallow water wave system with damping effects and cross operators involving the Khalil derivative, using regular perturbation techniques to investigate its traveling wave solutions. Applying the tanh-method, the system is reduced to a system of perturbed ordinary differential equations. By explicit substitution and algebraic manipulations, we obtain a set of compatibility conditions that govern the coefficients of the solution. A detailed case-by-case study is presented depending on the vanishing or non-vanishing of the parameters. The results provide explicit formulas for wave profiles and derive families of solutions parameterized by wave speed c and shape parameter & micro;. The approach confirms that the conformable fractional derivative framework is consistent with classical shallow-water solitary waves.

This paper deals with a new class of nonlinear coupled fractional differential equations with Caputo derivative. The existence and uniqueness of the solution is treated using the Banach mapping principle, also some results of the stability in the sens of Ulam-Hyers are introduced.

In this paper, we focus on the study of a hybrid system of sequential type that incorporates both Caputo and Hadamard fractional derivatives. Our approach leverages the fixed point principle to derive novel results concerning the existence and uniqueness of solutions to this system. Additionally, we establish further results by employing Schaefer’s fixed point theorem, which allows us to extend the applicability of our findings. To illustrate the practical relevance and application of our theoretical results, we also provide a detailed example at the conclusion of the paper. At the end, an example is given.

In this work, we begin by introducing a new notion of coupled closed fractional boundary conditions to study a class of nonlinear sequential systems of Caputo fractional differential equations. The existence and uniqueness of solutions for the class of systems is proved by applying Banach contraction principle. The existence of at least one solution is then accomplished by applying Schauder fixed point theorem. The Ulam Hyers stability, with a limiting-case example, is also discussed. In a second part of our work, we use the tanh method to obtain a new travelling wave solution for the coupled system of Burgers using time and space Khalil derivatives. By bridging these two aspects, we aim to present an understanding of the system’s behaviour.

In this paper, we employ Riemann-Liouville fractional integrals to derive two primary integral findings concerning the alpha- pre-Gruss inequality and the (alpha, beta)- pre-Gruss inequality. Our results extend existing integral inequalities reported in the literature. Additionally, we discuss some applications of our results for continuous random variables (CRV, for short) with bounded probability density functions. Some new estimates are provided in this context, along with classical results obtained as special cases from our results.

The aim of this work is to study two classes of stochastic fractional differential equations via the application of the method of upper and lower solutions combined with the Arzela-Ascoli theorem. We begin by proving an auxiliary result for the integral representation of an Airy-type stochastic problem. The specific symmetry features of an Airy-type stochastic problem depend on the form of the stochastic differential equations (SDE) and the relevant coefficients, it is vital to note. To comprehend each issue’s unique symmetries and their effects, a thorough investigation is necessary. Then, we prove an existence result for extremal solutions for the same problem. Another class of stochastic equations of higher-order type is also studied. We also present some examples to show the validity of the obtained results. At the end, a conclusion follows.

We prove two results regarding the existence of positive solutions for a p-Laplacian fractional high-order differential problem with initial conditions. We do this by applying the cones technique, lower and upper solutions, and the Leray-Schauder alternative. A detailed example illustrates one of the results.

Identifying and analyzing fixed points plays a crucial role in chaos theory for grasping the system behavior and advancing the understanding of its fundamental mechanics. This study explores new chaotic nonlinear integro-differential systems with four variables, employing Caputo–Fabrizio and Atangana–Baleanu derivatives. We confirm the presence and reliability of solutions and offer a real-life example. Additionally, we implement the suggested multi-step techniques on different nonlinear chaotic systems to demonstrate their accuracy.

In this paper, we apply the U-expansion method to finding new traveling wave solutions for two important classes of nonlinear fractional differential equations. The efficiency of this method is illustrated by some examples.

In this paper, we study a new problem of random differential equations of Airy type by means of the stochastic mean square theory. A new perturbation problem is introduced and some existence and uniqueness results for "stochastic process" solutions" are established. At the end, an example is discussed in details.

The conformable fractional derivatives in the sense of Khalil is considered and the Homotopy analysis method and the (G'/G)-Expansion G '/ G )-Expansion method are applied to solve the Klein- Gordon equation; new approximate solutions and new travelling wave solutions are obtained. Some numerical simulations are graphically illustrated. At the end, a conclusion is given.

In this paper, we establish the existence and uniqueness of solutions for a system of coupled differential equations of arbitrary order subject to nonlinear boundary conditions within generalized Banach spaces. The fractional derivatives used are of Caputo-Fabrizio type. We provide the exact solution for an auxiliary boundary value problem and use this expression for the analysis of the nonlinear problem of interest. The analytical approach involves the application of the nonlinear alternative of Leray-Schauder type and the Banach contraction principle to a mapping whose fixed points are the solutions to the problem. Additionally, the paper concludes with an illustrative example to enhance understanding and highlight the practical implications of the obtained results.

.in this paper, we apply the(G ' G)-expansion and the Tanh methods to find-ing exact traveling wave solutions of the space and time fractional Zakharov-Kuznetsov(FZK) equation and the space and time fractional Zoomeron (FZZ) equation using theconformable fractional derivatives. For the FZK equation,we obtain six solutions withthe Tanh method and three solutions with the(G ' G)-expansion method. On the other hand,for the FZZ equation, we find one solution with the Tanh methodand three solutions withthe second method. From this study, we observe that the(G ' G)-expansion method andthe Tanh method are not equivalent, it is shown that the Tanh method is more effectivethan the(G ' G)-expansion method, for solving fractional differential equation in specialconditions

In this work, we suggest a new numerical scheme called the fractional higher order Taylor method (FHOTM) to solve fractional differential equations (FDEs). Using the generalized Taylor’s theorem is the fundamental concept of this approach. Then, the local truncation error generated by the suggested FHOTM is estimated by proving suitable theoretical results. At last, several numerical applications are given to demonstrate the applicability of the suggested approach in relation to their exact solutions.