Muhammad Jamil

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.

Slip Effects on Fractional Viscoelastic Fluids

Unsteady flow of an incompressible Maxwell fluid with fractional derivative induced by a sudden moved plate has been studied, where the no-slip assumption between the wall and the fluid is no longer valid. The solutions obtained for the velocity field and shear stress, written in terms of Wright generalized hypergeometric functions 𝑝Ψ𝑞, by using discrete Laplace transform of the sequential fractional derivatives, satisfy all imposed initial and boundary conditions. The no-slip contributions, that appeared in the general solutions, as expected, tend to zero when slip parameter is 𝜃→0. Furthermore, the solutions for ordinary Maxwell and Newtonian fluids, performing the same motion, are obtained as special cases of general solutions. The solutions for fractional and ordinary Maxwell fluid for no-slip condition also obtained as limiting cases, and they are equivalent to the previously known results. Finally, the influence of the material, slip, and the fractional parameters on the fluid motion as well as a comparison among fractional Maxwell, ordinary Maxwell, and Newtonian fluids is also discussed by graphical illustrations.

Solutions of the Force-Free Duffing-van der Pol Oscillator Equation

A new approximate method for solving the nonlinear Duffing-van der pol oscillator equation is proposed. The proposed scheme depends only on the two components of homotopy series, the Laplace transformation and, the Pade approximants. The proposed method introduces an alternative framework designed to overcome the difficulty of capturing the behavior of the solution and give a good approximation to the solution for a large time. The Runge-Kutta algorithm was used to solve the governing equation via numerical solution. Finally, to demonstrate the validity of the proposed method, the response of the oscillator, which was obtained from approximate solution, has been shown graphically and compared with that of numerical solution.

First Problem of Stokes for Generalized Burgers' Fluids

The velocity field and the adequate shear stress corresponding to the first problem of Stokes for generalized Burgers’ fluids are determined in simple forms by means of integral transforms. The solutions that have been obtained, presented as a sum of steady and transient solutions, satisfy all imposed initial and boundary conditions. They can be easily reduced to the similar solutions for Burgers, Oldroyd-B, Maxwell, and second-grade and Newtonian fluids. Furthermore, as a check of our calculi, for small values of the corresponding material parameters, their diagrams are almost identical to those corresponding to the known solutions for Newtonian and Oldroyd-B fluids. Finally, the influence of the rheological parameters on the fluid motions, as well as a comparison between models, is graphically illustrated. The non-Newtonian effects disappear in time, and the required time to reach steady-state is the lowest for Newtonian fluids.

Helices of fractionalized Maxwell fluid

Abstract In this paper the helical flows of fractionalized Maxwell fluid model, through a circular cylinder, is studied. The motion is produced by the cylinder that at the initial moment begins to rotate around its axis with an angular velocity Omegatp, and to slide along the same axis with linear velocity Utp. The solutions that have been obtained using Laplace and finite Hankel transforms and presented in series form in terms of the newly defined special function M(z), satisfy all imposed initial and boundary conditions. Moreover, the corresponding solutions for ordinary Maxwell and Newtonian fluid obtained as special cases of the present general solution. Finally, the influence of various pertinent parameters on fluid motion as well as the comparison among different fluids models is analyzed by graphical illustrations.

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 ¨ ∂

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.

Oscillating Flows of Fractionalized Second Grade Fluid

New exact solutions for the motion of a fractionalized (this word is suitable when fractional derivative is used in constitutive or governing equations) second grade fluid due to longitudinal and torsional oscillations of an infinite circular cylinder are determined by means of Laplace and finite Hankel transforms. These solutions are presented in series form in term of generalized 𝐺𝑎,𝑏,𝑐(⋅,𝑡) functions and satisfy all imposed initial and boundary conditions. In special cases, solutions for ordinary second grade and Newtonian fluids are obtained. Furthermore, other equivalent forms of solutions for ordinary second grade and Newtonian fluids are presented and written as sum of steady-state and transient solutions. The solutions for Newtonian fluid coincide with the well-known classical solutions. Finally, by means of graphical illustrations, the influence of pertinent parameters on fluid motion as well as comparison among different models is discussed.