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In this study, we present and examine a fractional integral operator with an I$$ I $$‐function in its kernel. This operator is used to solve several fractional differential equations (FDEs). FDE has a set of particular cases whose solutions represent different physical phenomena. Much mathematical physics, biology, engineering, and chemistry problems are identified and solved using FDE. We first solve the FDE and the integral operator for the incomplete I$$ I $$‐function (I I$$ I $$F) for the generalized composite fractional derivative (GCFD). This is followed by the discovery and investigation of several important exceptional cases. The significant finding of this study is a first‐order integer‐differential equation of the Volterra type that clearly describes the unsaturated nature of free‐electron lasers. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a numerical and analytical study of the HIV‐1 infection of CD4+ T‐cells conformable fractional mathematical model. This model is studied analytically by Kudryashov and modified Kudryashov methods and numerically by finite difference method. A comparison between the results of the analytical and numerical methods is investigated. We also give some figures that show how accurate the solutions will be obtained from analytical and numerical methods. [ABSTRACT FROM AUTHOR]

Bernoulli sub‐equation function method is applied to obtain exact solutions of Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) nonlinear partial differential equation. As a result of this, exact traveling‐wave and some new oscillating solutions to CDGSK are obtained. It may be observed that Bernoulli sub‐equation function method employed here is very effective and reliable to get explicit solutions for this nonlinear partial differential equation. Profiles of all constructed solutions are graphically illustrated entirely as well. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence results of Ψ‐Hilfer fractional impulsive integro‐differential equations involving almost sectorial operators. The mild solutions of the problems are proved by using Schauder fixed‐point theorem along with measure of noncompactness. We have discussed the two cases of operators associated semigroup. Also, we consider an abstract application via Hilfer fractional derivative system to verify the results. [ABSTRACT FROM AUTHOR]

In this article, we determine a set of appropriate conditions for approximate controllability of second‐order Sobolev‐type impulsive neutral differential evolution inclusions with infinite delay. We verify the main results by applying ideas about the cosine function, sine functions, and fixed point approach. Next, we continue the discussion to the nonlocal second‐order Sobolev‐type differential system. In the end, we provide a theoretical application to assist in the effectiveness of our discussion. [ABSTRACT FROM AUTHOR]

The pivotal aim of the present work is to find the numerical solution for fractional Benney–Lin equation by using two efficient methods, called q‐homotopy analysis transform method and fractional natural decomposition method. The considered equation exemplifies the long waves on the liquid films. Projected methods are distinct with solution procedure and they are modified with different transform algorithms. To illustrate the reliability and applicability of the considered solution procedures we consider eight special cases with different initial conditions. The fractional operator is considered in Caputo sense. The achieved results are drowned through two and three‐dimensional plots for different Brownian motions and classical order. The numerical simulations are presented to ensure the efficiency of considered techniques. The behavior of the obtained results for distinct fractional order is captured in the present framework. The outcomes of the present investigation show that, the considered schemes are efficient and powerful to solve nonlinear differential equations arise in science and technology. [ABSTRACT FROM AUTHOR]

In this paper, the q‐homotopy analysis transform method (q‐HATM) is applied to find the solution for the fractional Lakshmanan‐Porsezian‐Daniel (LPD) model. The LPD model is the generalization of the non‐linear Schrödinger (NLS) equation. The proposed method is graceful fusions of Laplace transform technique with q‐homotopy analysis scheme, and the derivative is considered in Caputo sense. In order to validate and illustrate the efficiency of the proposed method, we analysed the projected model in terms of fractional order. Moreover, the physical behaviour of the obtained solution has been captured for the three different cases in terms of 3D and contour plots for diverse values of the fractional order. The obtained results confirm that the future method is easy to implement, highly methodical, and very effective to analyse the behaviour of complex non‐linear fractional differential equations exist in the connected areas of science and engineering. [ABSTRACT FROM AUTHOR]

In this study, with the aid of Wolfram Mathematica 11, the modified exp [ABSTRACT FROM AUTHOR]