In the present article, fractional-order heat and wave equations are solved by using the natural transform decomposition method. The series form solutions are obtained for fractional-order heat and wave equations, using the proposed method. Some numerical examples are presented to understand the procedure of natural transform decomposition method. The natural transform decomposition method procedure has shown that less volume of calculations and a high rate of convergence can be easily applied to other nonlinear problems. Therefore, the natural transform decomposition method is considered to be one of the best analytical techniques, in order to solve fractional-order linear and nonlinear Partial deferential equations, particularly fractional-order heat and wave equation.

In the present article, we related the analytical solution of the fractional-order dispersive partial differential equations, using the Laplace–Adomian decomposition method. The Caputo operator is used to define the derivative of fractional-order. Laplace–Adomian decomposition method solutions for both fractional and integer orders are obtained in series form, showing higher convergence of the proposed method. Illustrative examples are considered to confirm the validity of the present method. The fractional order solutions that are convergent to integer order solutions are also investigated.

In this research paper, a hybrid method called Laplace Adomian Decomposition Method (LADM) is used for the analytical solution of the system of time fractional Navier-Stokes equation. The solution of this system can be obtained with the help of Maple software, which provide LADM algorithm for the given problem. Moreover, the results of the proposed method are compared with the exact solution of the problems, which has confirmed, that as the terms of the series increases the approximate solutions are convergent to the exact solution of each problem. The accuracy of the method is examined with help of some examples. The LADM, results have shown that, the proposed method has higher rate of convergence as compare to ADM and HPM. ). Although it is considered an old topic, but for the last few decade, fractional calculus is launched as an important topic by the scientists and researchers [10,11].The Navier-Stokes equation is known as Newton second Law for fluid substance, has been derived in 1822 by Claude Louis Navier and Gabriel Stokes. Navier-Stokes equation is an important model to describe many physical phenomena in applied sciences. This model have the capacity of modelling weather, ocean current, water flow in pipes and air flow around a wing. A very special case was considered, which has established the relationship between pressure and external forces acting on the fluid to the responses of fluid flow [12]. The Navier-Stock equation is also used to derive the connection between viscous fluid with rigid bodies and considered a best tool in the field of thermohydraulics, meteorology, petroleum industry, plasma physics and technology [13].Several mathematicians have applied different techniques for the solution of Navier-Stock equation. Among these methods, Kumar et al. have implemented modified Laplace decomposition method for the analytical solution of fractional Navier-Stokes equation [14] coupled method is the combination of He-Laplace transform (HLT) and Fractional Complex Transform (FCT) is used to solve Navier-Stock equation [15]. Fractional Reduced Differential Transformation Method (FRDM) is also implemented for the numerical solution of time fractional Navier-Stock equation [16], see also [17]. Definitions and Preliminaries ConceptsIn this unit, among few definitions of fractional calculus, presented in the article due to Riemann Liouville, Grunwald Letnikov, Caputo, etc., first folks simple descriptions and introductions are reconsidered, which we want to comprehend our education.

In this work, the novel iterative transformation technique and homotopy perturbation transformation technique are used to calculate the fractional-order gas dynamics equation. In this technique, the novel iteration method and homotopy perturbation method are combined with the Elzaki transformation. The current methods are implemented with four examples to show the efficacy and validation of the techniques. The approximate solutions obtained by the given techniques show that the methods are accurate and easy to apply to other linear and nonlinear problems.

In the present article, fractional-order partial differential equations with proportional delay, including generalized Burger equations with proportional delay are solved by using Natural transform decomposition method. Natural transform decomposition method solutions for both fractional and integer orders are obtained in series form, showing higher convergence of the proposed method. Illustrative examples are considered to confirm the validity of the present method. Therefore, Natural transform decomposition method is considered to be one of the best analytical technique, to solve fractional-order linear and non-linear Partial deferential equations particularly fractional-order partial differential equations with proportional delay.