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The main objective of this study is to provide a new computational procedure for extracting approximate and exact solutions of the temporal-fractional Swift–Hohenberg (S–H) equations in the context of conformable derivatives using the conformable natural transform (CNT) and Daftardar–Jafari method (DJM). We refer to it as the “natural conformable Daftardar–Jafari method” (CNDJM). The three types of errors are assessed in order to gauge the efficiency and consistency of the proposed method. Furthermore, 2D and 3D graphics are used to compare the exact and approximate solutions. This method offers a considerable benefit over homotopy analysis and Adomian decomposition methods in terms of computational work because it does not require Adomian and He’s polynomials. The procedure is quick and easy to use.

There is considerable literature on solutions to the gas-dynamic equation (GDE) and Fokker–Planck equation (FPE), where the fractional derivative is expressed in terms of the Caputo fractional derivative. There is hardly any work on analytical and numerical GDE and FPE solutions involving conformable fractional derivative (CFD). For the reasons stated above, we are required to solve the GDE and FPE in the form of CFD. The main goal of this research is to offer a novel combined method by employing the conformable Shehu transform (CST) and the homotopy perturbation method (HPM) for extracting analytical and numerical solutions of the time-fractional conformable GDE and FPE. The proposed method is called the conformable Shehu homotopy perturbation method (CSHPM). To evaluate its efficiency and consistency, relative and absolute errors among the approximate and exact solutions to three nonlinear problems of GDE and FPE are considered numerically and graphically. Moreover, fifth-term approximate and exact solutions are also compared by 2D and 3D graphs. This method has the benefit of not requiring any minor or major physical parameter assumptions in the problem. As a result, it may be used to solve both weakly and strongly nonlinear problems, overcoming some of the inherent constraints of classic perturbation approaches. Second, while addressing nonlinear problems, the CSHPM does not require Adomian polynomials. Therefore, to solve nonlinear GDE and FPE, just a few computations are necessary. As a consequence, it outperforms homotopy analysis and Adomian decomposition approaches significantly. It does not require discretization or linearization, unlike traditional numerical methods. The convergence and error analysis of the series solutions are also presented.

In this paper, the existence results for the solutions of the multi-term ABC-fractional differential boundary value problem (BVP) [Formula: see text] of order [Formula: see text] with nonlocal boundary conditions have been derived by using Krasnoselskii’s fixed point theorem. The uniqueness of the solution is obtained with the help of Banach contraction principle. Examples are provided to confirm our obtained results.

In this research paper, we find the numerical solutions of fractional order scalers and coupled system of differential equations under initial conditions using shifted Legendre polynomials. By using the properties of shifted Legendre polynomials, we establish operational matrices of integration and differentiation in order to simplify our considered problems under initial conditions. In order to check the accuracy of the proposed model, some test problems are solved along with the graphical representations. For coupled system, we applied the algorithm to the Pharmacokinetic two-compartment model. As the proposed method is computer-oriented, we use therefore the MATLAB for required calculations. Numerical results are shown graphically.

In this paper, we use Krasnoselskii’s fixed point theorem to find existence results for the solution of the following nonlinear fractional differential equations (FDEs) for a coupled system involving AB-Caputo fractional derivative [Formula: see text] with boundary conditions [Formula: see text] We discuss uniqueness with the help of the Banach contraction principle. The criteria for Hyers–Ulam stability of given AB-Caputo fractional-coupled boundary value problem (BVP) is also discussed. Some examples are provided to validate our results. In Example 1, we find a unique and stable solution of AB-Caputo fractional-coupled BVP. In Example 2, the analysis of approximate and exact solutions with errors of nonlinear integral equations is elaborated with graphs.

This paper is devoted to investigating a class of impulsive fractional order differential equations (FODEs) with integral boundary condition. For the proposed paper, we use non-singular type derivative of fractional order which has been introduced by Atangana, Baleanu and Caputo (ABC). The aforesaid type problems have numerous applications in fluid mechanics and hydrodynamics to model various problems of flow phenomenons. We establish some sufficient conditions for the existence and uniqueness of solution to the proposed problem by using classical fixed point results due to Banach and Krasnoselskii. Further, on using tools of the nonlinear analysis, sufficient conditions are developed for Hyers–Ulam (HU) type stability results. A pertinent example is given to justify our results.