Amir Mahmood

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 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...

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.

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, ...

Some Exact Solutions for Helical Flows of Maxwell Fluid in an Annular Pipe due to Accelerated Shear Stresses

Some exact solutions corresponding to helical flows of Maxwell fluid between two infinite coaxial circular cylinders are determined by means of finite Hankel transform, presented in series form, satisfied all imposed initial and boundary conditions. The motion is produced by the inner cylinder that applies the torsional and longitudinal time-dependent accelerated shear stresses to the fluid. The corresponding solutions for Newtonian fluid and large-time solutions are also obtained as limiting cases and effect of material parameters on the large-time solutions are discussed. Finally, the influence of pertinent parameters as well as a comparison between Maxwell and Newtonian fluids on the velocity components and shear stresses profiles is also examined by graphical illustrations.

Analytical Study of Nonlinear Fractional-Order Integrodifferential Equation: Revisit Volterra's Population Model

e approximation was effectively used in this method to capture the essential behavior of solutions for the mathematical model of accumulated effect of toxins on a population living in a closed system. The behavior of the solutions and the effects of different values of fractional-order α are indicated graphically. The study outlines significant features of this method as well as sheds some light on advantages of the method over the other. The results show that this method is very efficient, convenient, and can be adapted to fit a larger class of problems.

Approximate Solution for the Electrohydrodynamic Flow in a Circular Cylindrical Conduit

This paper considers the nonlinear boundary value problem (BVP) for the electrohydrodynamic flow of a fluid in an ion drag configuration in a circular cylindrical conduit. The velocity field was solved using the new homotopy perturbation method (NHPM), considering the electrical field and strength of the nonlinearity. The approximate analytical procedure depends only on two components and polynomial initial condition. The analytical solution is obtained and the numerical results presented graphically. The effects of the Hartmann electric number ℋ𝑎 and the strength of nonlinearity 𝛼 are discussed and presented graphically. We also compare this method with numerical solution (N.S) and show that the present approach is less computational and is applicable for solving nonlinear boundary value problem (BVP).

Traveling Wave Solutions for MHD Aligned Flow of a Second Grade Fluid

In this work, an approach based on traveling wave phenomenon is implemented for finding exact solutions of MHD aligned flow of an incompressible second grade fluid. The partial differential equations (PDEs) are reduced to ordinary differential equations (ODEs) by using wave parameter. The methodology used in this work is independent of symmetry consideration and other restrictive assumption. Comparison is made with the results obtained previously.