Najeeb Alam Khan

An efficient approach for solving the Riccati equation with fractional orders

The present study introduces a novel and simple analytical method for the solution of fractional order Riccati differential equation. In this approach, the solution considered as a Taylor series expansion converges rapidly to the nonlinear problem. New homotopy perturbation method (NHPM) depends only on two components of the homotopy series. The method is illustrated by applications and the results obtained are compared with those of the exact solution. Moreover, comparing the methodology with some known techniques shows that the present approach is relatively easy and efficient.

Analytical Study of Navier-Stokes Equation with Fractional Orders Using He's Homotopy Perturbation and Variational Iteration Methods

In the present work, by introducing the fractional derivative in the sense of Caputo, the Hes homotopy perturbation method (HPM) and variational iteration method (VIM) are used to study the Navier-Stokes equation with fractional orders. The analytical solutions are calculated in the form of series with easily computable components. Two examples are given. The present methods perform well in terms of efficiency and simplicity. Λ good agreement of the result is observed.

Approximate Solutions to Time-Fractional Schrödinger Equation via Homotopy Analysis Method

We construct the approximate solutions of the time-fractional Schrodinger equations, with zero and nonzero trapping potential, by homotopy analysis method (HAM). The fractional derivatives, in the Caputo sense, are used. The method is capable of reducing the size of calculations and handles nonlinear-coupled equations in a direct manner. The results show that HAM is more promising, convenient, efficient and less computational than differential transform method (DTM), and easy to apply in spaces of higher dimensions as well.

New exact analytical solutions for Stokes' first problem of Maxwell fluid with fractional derivative approach

The unsteady flow of an incompressible Maxwell fluid with fractional derivative induced by a sudden moved plate has been studied using Fourier sine and Laplace transforms. The obtained solutions for the velocity field and shear stress, written in terms of generalized G functions, are presented as sum of the similar Newtonian solutions and the corresponding non-Newtonian contributions. The non-Newtonian contributions, as expected, tend to zero for @l->0. Furthermore, the solutions for ordinary Maxwell fluid, performing the same motion, are obtained as limiting cases of general solutions and verified by comparison with previously known results. Finally, the influence of the material and the fractional parameters on the fluid motion, as well as a comparison among fractional Maxwell, ordinary Maxwell and Newtonian fluids is also analyzed by graphical illustrations.

On Efficient Method for System of Fractional Differential Equations

The present study introduces a new version of homotopy perturbation method for the solution of system of fractional-order differential equations. In this approach, the solution is considered as a Taylor series expansion that converges rapidly to the nonlinear problem. The systems include fractional-order stiff system, the fractional-order Genesio system, and the fractional-order matrix Riccati-type differential equation. The new approximate analytical procedure depends only on two components. Comparing the methodology with some known techniques shows that the present method is relatively easy, less computational, and highly accurate.

Approximate Solution of Time-Fractional Chemical Engineering Equations: A Comparative Study

In this paper, we present the approximate solutions of the time fractional chemical engineering equations by means of the variational iteration method (VIM) and homotopy perturbation method (HPM). The fractional derivatives are described in the Caputo sense. The solutions of the chemical reactor, reaction, and concentration equations are calculated in the form of convergent series with easily computable components. We compared the HPM against the VIM; an additional comparison will be made against the conventional numerical method. The results show that HPM is more promising, convenient, and efficient than VIM.

Translational flows of an Oldroyd-B fluid with fractional derivatives

The objective of this paper is to study the unsteady flow of an Oldroyd-B fluid with fractional derivative model, between two infinite coaxial circular cylinders, using Laplace and finite Hankel transforms. The motion of the fluid is produced by the inner cylinder that, at time t=0^+, applies a time dependent longitudinal shear stress to the fluid. Velocity field and the adequate shear stress are presented in series form in terms of the generalized G and R functions. The solutions that have been obtained satisfy all imposed initial and boundary conditions. The corresponding solutions for ordinary Oldroyd-B, fractional Maxwell, ordinary Maxwell, fractional second grade, ordinary second grade and Newtonian fluids performing the same motion are obtained as limiting cases of general solutions. In particular, the existing solutions for ordinary Oldroyd-B and second grade fluids are compared with the present solutions. Finally, the influence of the pertinent parameters on the fluid motion as well as a comparison between models is underlined by graphical illustrations.

Numerical solutions of time‐fractional Burgers equations

Purpose – The purpose of this paper is to use the generalized differential transform method (GDTM) and homotopy perturbation method (HPM) for solving time‐fractional Burgers and coupled Burgers equations. The fractional derivatives are described in the Caputo sense.Design/methodology/approach – In these schemes, the solutions takes the form of a convergent series. In GDTM, the differential equation and related initial conditions are transformed into a recurrence relation that finally leads to the solution of a system of algebraic equations as coefficients of a power series solution. HPM requires a homotopy with an embedding parameter which is considered as a small parameter.Findings – The paper extends the application and numerical comparison of the GDTM and HPM to obtain analytic and approximate solutions to the time‐fractional Burgers and coupled Burgers equations.Research limitations/implications – Burgers and coupled Burgers equations with time‐fractional derivative used.Practical implications – The i...

Analytical methods for solving the time-fractional Swift-Hohenberg (S-H) equation

In this paper, the most effective methods, the homotopy perturbation method (HPM) and the differential transform method (DTM), are applied to obtain the approximate solutions of the nonlinear time-fractional Swift-Hohenberg (S-H) equation. The basic philosophy of these methods does not involve linearization, weak nonlinearity assumptions or perturbation theory. Numerical solutions for various combinations of the parameters a (eigenvalue parameter), L (length) and @a (fractional index) are obtained. The solutions of the S-H equation are useful for studies of shear thinning effects in non-Newtonian fluid flows. At the end, the solutions obtained are also presented graphically.

On the double diffusive convection flow of Eyring-Powell fluid due to cone through a porous medium with Soret and Dufour effects

This paper devotedly study the double diffusive Darcian convection flow of Eyring-Powell fluid from a cone embedded in a homogeneous porous medium with the effects of Soret and Dufour. Arising set of non-linear partial differential equations are transformed through a suitable self-similar transformation into a set of nonlinear ordinary differential equations. Further, the numerical and the analytical solutions of the governing equations are elucidated by using numerical method as well as non-perturbation scheme. Numerical values are presented through tables for the skin friction coefficients, Nusselt number and Sherwood number. The obtained results are validated by comparing the analytical results with previously published results obtained by bvp4c for the numerical values of physical quantities. The effect of various parameters on the velocity, temperature and concentration profiles is discussed and also shown graphically.