Asmat Ara

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.

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.

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...

Approximate solution of couple stress fluid with expanding or contracting porous channel

Purpose – The purpose of this paper is to investigate the approximate solution of the couple stress fluid equations in a semi‐infinite rectangular channel with porous and uniformly expanding or contracting walls.Design/methodology/approach – Perturbation method is a traditional method depending on a small parameter which is difficult to be found for real‐life nonlinear problems. The governing partial differential equations are transformed using a transformation into an ordinary differential equation that is solved by homotopy analysis method (HAM) and shooting technique.Findings – To assess the accuracy of the solutions, the comparison of the obtained results reveals that both methods are tremendously effective. Analytical and numerical solutions comparison indicates an excellent agreement and this comparison is also presented. Graphs are portrayed for the effects of some values of parameters.Practical implications – Expansion or contraction problems occur naturally in the transport of biological fluids, ...

Fractional-order Riccati differential equation: Analytical approximation and numerical results

The aim of this article is to introduce the Laplace-Adomian-Padé method (LAPM) to the Riccati differential equation of fractional order. This method presents accurate and reliable results and has a great perfection in the Adomian decomposition method (ADM) truncated series solution which diverges promptly as the applicable domain increases. The approximate solutions are obtained in a broad range of the problem domain and are compared with the generalized Euler method (GEM). The comparison shows a precise agreement between the results, the applicable one of which needs fewer computations.

Multiple-Parameter Hamiltonian Approach for Higher Accurate Approximations of a Nonlinear Oscillator with Discontinuity

We applied a new approach to obtain natural frequency of the nonlinear oscillator with discontinuity. He’s Hamiltonian approach is modified for nonlinear oscillator with discontinuity for ¨ ∂

Orthogonal Flow Impinging on a Wall with Suction or Blowing

The goal of this work is the approximate solutions of a viscous incompressible fluid impinging orthogonally on a porous flat plate. The equation governing the flow of an incompressible fluid is investigated using the homotopy perturbation method (HPM) with the aid of Padé-approximants. The approximate solutions can be successfully applied to provide the value of the skin-friction. The reliability and efficiency of the approximate solutions were verified using numerical solutions in the literature.