Saleh M. Hassan

Solitary waves for the MRLW equation

Abstract This work presents a computational comparison study of quadratic, cubic, quartic and quintic splines for solving the modified regularized long wave (MRLW) equation. Collocation schemes with quadratic and cubic splines are found to be unconditionally stable. The fourth-order Runge–Kutta method has been used to solve the collocation schemes when quartic and quintic B-splines are used. The three invariants of motion have been evaluated to determine the conservation properties of the suggested algorithms. Comparisons of results due to different schemes with the exact values shows the accuracy and efficiency of the proposed schemes. Results corresponding to higher order splines are more accurate than those corresponding to lower order splines.

Transverse vibration of a circular plate with arbitrary thickness variation

Abstract Rayleigh–Ritz method has been employed to obtain approximations to frequencies and mode shapes of circular plates with variable thickness. The boundary is either clamped, simply supported or completely free. The main distinguishing feature of the present investigations is that the thickness approximation is done by measuring thickness at a suitable set of sample points and then using interpolation to get the approximating polynomial. Thus, unlike other methods already available in literature where either linear or quadratic variation of thickness has been examined, here one can have a polynomial of arbitrary degree depending upon the number and locations of the sample points. The results have been tabulated in a large number of cases and three-dimensional mode shapes have been plotted for some selected cases. Comparison has been made with available results. A short tables are also given to depict the rate of convergence with the order of approximation.

Transverse vibrations of elliptical plate of linearly varying thickness with half of the boundary clamped and the rest simply supported

Abstract The Rayleigh–Ritz method has been applied to study the problem of transverse vibrations of elliptical plate with half of the boundary clamped and the rest simply supported. The thickness of the plate is varying linearly in the space coordinates. For the same thickness variation, the tabulated results in case of mixed boundary conditions lies between those reported for the two extreme cases when the entire boundary is either clamped or simply supported. Convergence of the results is indicated by increasing the order of approximation. Some results for a circular plate of linearly varying thickness with half of the boundary clamped and the rest simply supported have been obtained as special case. Comparisons have been made with the known results. Three-dimensional mode shapes have been drawn for some selected cases.

Solitary wave solutions of the improved KdV equation by VIM

The variational iteration method (VIM) is applied to solve numerically the improved Korteweg-de Vries equation (IKdV). A correction function is constructed with a general Lagrange multiplier that can be identified optimally via the variational theory. This technique provides a sequence of functions with easily computable components that converge rapidly to the exact solution of the IKdV equation. Propagation of single, interaction of two, and three solitary waves, and also birth of solitons have been discussed. Three invariants of motion have been evaluated to determine the conservation properties of the problem. This procedure is promising for solving other nonlinear equations.

Modified Legendre Wavelets Technique for Fractional Oscillation Equations

Physical Phenomena’s located around us are primarily nonlinear in nature and their solutions are of highest significance for scientists and engineers. In order to have a better representation of these physical models, fractional calculus is used. Fractional order oscillation equations are included among these nonlinear phenomena’s. To tackle with the nonlinearity arising, in these phenomena’s we recommend a new method. In the proposed method, Picard’s iteration is used to convert the nonlinear fractional order oscillation equation into a fractional order recurrence relation and then Legendre wavelets method is applied on the converted problem. In order to check the efficiency and accuracy of the suggested modification, we have considered three problems namely: fractional order force-free Duffing–van der Pol oscillator, forced Duffing–van der Pol oscillator and higher order fractional Duffing equations. The obtained results are compared with the results obtained via other techniques.

Numerical investigation of magnetohydrodynamic flow and heat transfer of copper–water nanofluid in a channel with non-parallel walls considering different shapes of nanoparticles:

This article presents the magnetohydrodynamic flow and heat transfer of water-based nanofluid in divergent and convergent channels. Equations governing the flow are transformed to a set of ordinary differential equations by employing suitable similarity transforms. Resulting system is solved using a strong numerical procedure called Runge–Kutta–Fehlberg method. Results are compared with existing solutions available in the literature and an excellent agreement is seen. Three shapes of nanoparticles, namely, platelet-, cylinder-, and brick-shaped particles, are considered to perform the analysis. Influence of emerging parameters such as channel opening, Reynolds number, magnetic parameter, Eckert number, and the nanoparticle volume fraction are heighted with the help of graphs coupled with comprehensive discussions. The magnetic field can be used as a controlling parameter to reduce the backflow regions for the divergent channel case. Temperature of the fluid can be controlled with the help of strong magnetic field. It is also observed that platelet-shaped particles have higher temperature values as compared to cylinder- and brick-shaped particles.

Some numerical experiments on high accuracy fast direct finite difference method for elliptic problems

A high accuracy finite difference scheme has been developed for solving some elliptic problems which appear in engineering and applied sciences. These include Laplace, Poisson, Helmholtz and related equations. The second- and fourth-order problems dealing with vibration of membranes and plates have also been examined. Numerically, the problem reduces to a block tridiagonal system which can be solved by suitably modifying the fast direct method developed by Hockney. Comparison has been made with results obtained from some alternative numerical methods or analytical methods whenever available

Unsteady mixed convection squeezing flow of nanofluid between parallel disks

In this article, mixed convection squeezing flow of a nanofluid between parallel disks is considered. The partial differential equations governing the flow problem are converted into coupled system of ordinary differential equation with the help of suitable similarity transforms. Homotopy analysis method is employed to solve the coupled system of ordinary differential equations. The influence of involved parameters, on velocity, temperature, and concentration profile, is presented graphically coupled with detailed discussion. The results for skin friction coefficient and Nusselt and Sherwood numbers are also a part of this study. Numerical solution is also obtained with the help of Runge–Kutta method of order 4. An excellent agreement is found between analytical and numerical solutions. From the results obtained, we observe that the skin friction coefficient decreases with increasing squeeze number for the case of injection and increases with increase in squeeze number for the case of injection at the walls. Furthermore, Nusselt number gets a rise with increment in squeeze number for the case of injection at the wall and a drop in Nusselt number for the case of suction at the wall is observed when there is suction at the wall. Sherwood number is seen to drop quite steeply with higher values of squeeze number for the injection case and a rise in Sherwood number for the suction is observed when there is suction at the wall.

Differential Transform Method with Complex Transforms to Some Nonlinear Fractional Problems in Mathematical Physics

This paper witnesses the coupling of an analytical series expansion method which is called reduced differential transform with fractional complex transform. The proposed technique is applied on three mathematical models, namely, fractional Kaup-Kupershmidt equation, generalized fractional Drinfeld-Sokolov equations, and system of coupled fractional Sine-Gordon equations subject to the appropriate initial conditions which arise frequently in mathematical physics. The derivatives are defined in Jumarie’s sense. The accuracy, efficiency, and convergence of the proposed technique are demonstrated through the numerical examples. It is observed that the presented coupling is an alternative approach to overcome the demerit of complex calculation of fractional differential equations. The proposed technique is independent of complexities arising in the calculation of Lagrange multipliers, Adomian’s polynomials, linearization, discretization, perturbation, and unrealistic assumptions and hence gives the solution in the form of convergent power series with elegantly computed components. All the examples show that the proposed combination is a powerful mathematical tool to solve other nonlinear equations also.

On Modified Algorithm for Fourth-Grade Fluid

This paper shows the analysis of the thin film flow of fourth-grade fluid on the outer side of a vertical cylinder. Solution of the governing nonlinear equation is obtained by Rational Homotopy Perturbation Method (RHPM); comparison with exact solution reflects the reliability of the method. Analysis shows that this method is reliable for even high nonlinearity. Graphs and tables strengthen the idea.